Integrand size = 23, antiderivative size = 306 \[ \int \frac {(c+d x)^{5/2}}{x \left (a-b x^2\right )^3} \, dx=\frac {\sqrt {c+d x} \left (a d^2+b c (c+2 d x)\right )}{4 a b \left (a-b x^2\right )^2}-\frac {\sqrt {c+d x} \left (a d^2-b c (8 c+11 d x)\right )}{16 a^2 b \left (a-b x^2\right )}-\frac {2 c^{5/2} \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3}+\frac {\sqrt {\sqrt {b} c-\sqrt {a} d} \left (32 b c^2-14 \sqrt {a} \sqrt {b} c d-3 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{32 a^3 b^{5/4}}+\frac {\sqrt {\sqrt {b} c+\sqrt {a} d} \left (32 b c^2+14 \sqrt {a} \sqrt {b} c d-3 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{32 a^3 b^{5/4}} \] Output:
1/4*(d*x+c)^(1/2)*(a*d^2+b*c*(2*d*x+c))/a/b/(-b*x^2+a)^2-1/16*(d*x+c)^(1/2 )*(a*d^2-b*c*(11*d*x+8*c))/a^2/b/(-b*x^2+a)-2*c^(5/2)*arctanh((d*x+c)^(1/2 )/c^(1/2))/a^3+1/32*(b^(1/2)*c-a^(1/2)*d)^(1/2)*(32*b*c^2-14*a^(1/2)*b^(1/ 2)*c*d-3*a*d^2)*arctanh(b^(1/4)*(d*x+c)^(1/2)/(b^(1/2)*c-a^(1/2)*d)^(1/2)) /a^3/b^(5/4)+1/32*(b^(1/2)*c+a^(1/2)*d)^(1/2)*(32*b*c^2+14*a^(1/2)*b^(1/2) *c*d-3*a*d^2)*arctanh(b^(1/4)*(d*x+c)^(1/2)/(b^(1/2)*c+a^(1/2)*d)^(1/2))/a ^3/b^(5/4)
Time = 2.19 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.02 \[ \int \frac {(c+d x)^{5/2}}{x \left (a-b x^2\right )^3} \, dx=-\frac {\frac {2 a \sqrt {c+d x} \left (-3 a^2 d^2+b^2 c x^2 (8 c+11 d x)-a b \left (12 c^2+19 c d x+d^2 x^2\right )\right )}{b \left (a-b x^2\right )^2}+\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \left (32 b c^2+14 \sqrt {a} \sqrt {b} c d-3 a d^2\right ) \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{b^{3/2}}+\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \left (32 b c^2-14 \sqrt {a} \sqrt {b} c d-3 a d^2\right ) \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{b^{3/2}}+64 c^{5/2} \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{32 a^3} \] Input:
Integrate[(c + d*x)^(5/2)/(x*(a - b*x^2)^3),x]
Output:
-1/32*((2*a*Sqrt[c + d*x]*(-3*a^2*d^2 + b^2*c*x^2*(8*c + 11*d*x) - a*b*(12 *c^2 + 19*c*d*x + d^2*x^2)))/(b*(a - b*x^2)^2) + (Sqrt[-(b*c) - Sqrt[a]*Sq rt[b]*d]*(32*b*c^2 + 14*Sqrt[a]*Sqrt[b]*c*d - 3*a*d^2)*ArcTan[(Sqrt[-(b*c) - Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqrt[b]*c + Sqrt[a]*d)])/b^(3/2) + ( Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d]*(32*b*c^2 - 14*Sqrt[a]*Sqrt[b]*c*d - 3*a* d^2)*ArcTan[(Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqrt[b]*c - Sqrt[a]*d)])/b^(3/2) + 64*c^(5/2)*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/a^3
Leaf count is larger than twice the leaf count of optimal. \(796\) vs. \(2(306)=612\).
Time = 2.88 (sec) , antiderivative size = 796, normalized size of antiderivative = 2.60, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.913, Rules used = {561, 25, 27, 1650, 1598, 27, 25, 1492, 27, 1480, 221, 1652, 25, 25, 27, 1484, 1492, 27, 1406, 221, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(c+d x)^{5/2}}{x \left (a-b x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 561 |
| \(\displaystyle \frac {2 \int \frac {(c+d x)^3}{x \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^3}d\sqrt {c+d x}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2 \int -\frac {(c+d x)^3}{x \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^3}d\sqrt {c+d x}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 \int -\frac {(c+d x)^3}{d x \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^3}d\sqrt {c+d x}\) |
| \(\Big \downarrow \) 1650 |
| \(\displaystyle -2 \left (\frac {c^2 \int -\frac {c+d x}{d x \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{a}-\frac {\int \frac {(c+d x) \left (c \left (a-\frac {b c^2}{d^2}\right )+\left (\frac {b c^2}{d^2}+a\right ) (c+d x)\right )}{\left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^3}d\sqrt {c+d x}}{a}\right )\) |
| \(\Big \downarrow \) 1598 |
| \(\displaystyle -2 \left (\frac {c^2 \int -\frac {c+d x}{d x \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{a}-\frac {\frac {d^2 \int -\frac {2 a \left (\left (a-\frac {b c^2}{d^2}\right ) d^2-10 b c (c+d x)\right )}{d^2 \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{16 a b}+\frac {\sqrt {c+d x} \left (d^2 \left (a-\frac {b c^2}{d^2}\right )+2 b c (c+d x)\right )}{8 b \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}}{a}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 \left (\frac {c^2 \int -\frac {c+d x}{d x \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{a}-\frac {\frac {\sqrt {c+d x} \left (d^2 \left (a-\frac {b c^2}{d^2}\right )+2 b c (c+d x)\right )}{8 b \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}-\frac {\int -\frac {b c^2+10 b (c+d x) c-a d^2}{\left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{8 b}}{a}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \left (\frac {c^2 \int -\frac {c+d x}{d x \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{a}-\frac {\frac {\int \frac {b c^2+10 b (c+d x) c-a d^2}{\left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{8 b}+\frac {\sqrt {c+d x} \left (d^2 \left (a-\frac {b c^2}{d^2}\right )+2 b c (c+d x)\right )}{8 b \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}}{a}\right )\) |
| \(\Big \downarrow \) 1492 |
| \(\displaystyle -2 \left (\frac {c^2 \int -\frac {c+d x}{d x \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{a}-\frac {\frac {\frac {d^4 \int \frac {2 b \left (b c^2-a d^2\right ) \left (11 b c^2+11 b (c+d x) c-3 a d^2\right )}{d^4 \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )}d\sqrt {c+d x}}{8 a b \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (\left (b c^2-a d^2\right ) \left (a d^2+11 b c^2\right )-11 b c (c+d x) \left (b c^2-a d^2\right )\right )}{4 a \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}}{8 b}+\frac {\sqrt {c+d x} \left (d^2 \left (a-\frac {b c^2}{d^2}\right )+2 b c (c+d x)\right )}{8 b \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}}{a}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 \left (\frac {c^2 \int -\frac {c+d x}{d x \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{a}-\frac {\frac {\frac {\int \frac {11 b c^2+11 b (c+d x) c-3 a d^2}{-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}d\sqrt {c+d x}}{4 a}-\frac {\sqrt {c+d x} \left (\left (b c^2-a d^2\right ) \left (a d^2+11 b c^2\right )-11 b c (c+d x) \left (b c^2-a d^2\right )\right )}{4 a \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}}{8 b}+\frac {\sqrt {c+d x} \left (d^2 \left (a-\frac {b c^2}{d^2}\right )+2 b c (c+d x)\right )}{8 b \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}}{a}\right )\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle -2 \left (\frac {c^2 \int -\frac {c+d x}{d x \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{a}-\frac {\frac {\frac {\frac {\sqrt {b} \left (11 \sqrt {a} \sqrt {b} c d-3 a d^2+22 b c^2\right ) \int \frac {1}{\frac {\sqrt {b} \left (\sqrt {b} c+\sqrt {a} d\right )}{d^2}-\frac {b (c+d x)}{d^2}}d\sqrt {c+d x}}{2 \sqrt {a} d}-\frac {\sqrt {b} \left (-11 \sqrt {a} \sqrt {b} c d-3 a d^2+22 b c^2\right ) \int \frac {1}{\frac {\sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right )}{d^2}-\frac {b (c+d x)}{d^2}}d\sqrt {c+d x}}{2 \sqrt {a} d}}{4 a}-\frac {\sqrt {c+d x} \left (\left (b c^2-a d^2\right ) \left (a d^2+11 b c^2\right )-11 b c (c+d x) \left (b c^2-a d^2\right )\right )}{4 a \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}}{8 b}+\frac {\sqrt {c+d x} \left (d^2 \left (a-\frac {b c^2}{d^2}\right )+2 b c (c+d x)\right )}{8 b \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}}{a}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -2 \left (\frac {c^2 \int -\frac {c+d x}{d x \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{a}-\frac {\frac {\frac {\frac {d \left (11 \sqrt {a} \sqrt {b} c d-3 a d^2+22 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {a} d+\sqrt {b} c}}-\frac {d \left (-11 \sqrt {a} \sqrt {b} c d-3 a d^2+22 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {b} c-\sqrt {a} d}}}{4 a}-\frac {\sqrt {c+d x} \left (\left (b c^2-a d^2\right ) \left (a d^2+11 b c^2\right )-11 b c (c+d x) \left (b c^2-a d^2\right )\right )}{4 a \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}}{8 b}+\frac {\sqrt {c+d x} \left (d^2 \left (a-\frac {b c^2}{d^2}\right )+2 b c (c+d x)\right )}{8 b \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}}{a}\right )\) |
| \(\Big \downarrow \) 1652 |
| \(\displaystyle -2 \left (\frac {c^2 \left (\frac {\int -\frac {\left (a-\frac {b c^2}{d^2}\right ) d^2+b c (c+d x)}{d^2 \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{a}+\frac {c \int -\frac {1}{d x \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )}d\sqrt {c+d x}}{a}\right )}{a}-\frac {\frac {\frac {\frac {d \left (11 \sqrt {a} \sqrt {b} c d-3 a d^2+22 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {a} d+\sqrt {b} c}}-\frac {d \left (-11 \sqrt {a} \sqrt {b} c d-3 a d^2+22 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {b} c-\sqrt {a} d}}}{4 a}-\frac {\sqrt {c+d x} \left (\left (b c^2-a d^2\right ) \left (a d^2+11 b c^2\right )-11 b c (c+d x) \left (b c^2-a d^2\right )\right )}{4 a \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}}{8 b}+\frac {\sqrt {c+d x} \left (d^2 \left (a-\frac {b c^2}{d^2}\right )+2 b c (c+d x)\right )}{8 b \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}}{a}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \left (\frac {c^2 \left (\frac {c \int -\frac {1}{d x \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )}d\sqrt {c+d x}}{a}-\frac {\int -\frac {b c^2-b (c+d x) c-a d^2}{d^2 \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{a}\right )}{a}-\frac {\frac {\frac {\frac {d \left (11 \sqrt {a} \sqrt {b} c d-3 a d^2+22 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {a} d+\sqrt {b} c}}-\frac {d \left (-11 \sqrt {a} \sqrt {b} c d-3 a d^2+22 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {b} c-\sqrt {a} d}}}{4 a}-\frac {\sqrt {c+d x} \left (\left (b c^2-a d^2\right ) \left (a d^2+11 b c^2\right )-11 b c (c+d x) \left (b c^2-a d^2\right )\right )}{4 a \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}}{8 b}+\frac {\sqrt {c+d x} \left (d^2 \left (a-\frac {b c^2}{d^2}\right )+2 b c (c+d x)\right )}{8 b \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}}{a}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \left (\frac {c^2 \left (\frac {\int \frac {b c^2-b (c+d x) c-a d^2}{d^2 \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{a}+\frac {c \int -\frac {1}{d x \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )}d\sqrt {c+d x}}{a}\right )}{a}-\frac {\frac {\frac {\frac {d \left (11 \sqrt {a} \sqrt {b} c d-3 a d^2+22 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {a} d+\sqrt {b} c}}-\frac {d \left (-11 \sqrt {a} \sqrt {b} c d-3 a d^2+22 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {b} c-\sqrt {a} d}}}{4 a}-\frac {\sqrt {c+d x} \left (\left (b c^2-a d^2\right ) \left (a d^2+11 b c^2\right )-11 b c (c+d x) \left (b c^2-a d^2\right )\right )}{4 a \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}}{8 b}+\frac {\sqrt {c+d x} \left (d^2 \left (a-\frac {b c^2}{d^2}\right )+2 b c (c+d x)\right )}{8 b \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}}{a}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 \left (\frac {c^2 \left (\frac {\int \frac {b c^2-b (c+d x) c-a d^2}{\left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{a d^2}+\frac {c \int -\frac {1}{d x \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )}d\sqrt {c+d x}}{a}\right )}{a}-\frac {\frac {\frac {\frac {d \left (11 \sqrt {a} \sqrt {b} c d-3 a d^2+22 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {a} d+\sqrt {b} c}}-\frac {d \left (-11 \sqrt {a} \sqrt {b} c d-3 a d^2+22 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {b} c-\sqrt {a} d}}}{4 a}-\frac {\sqrt {c+d x} \left (\left (b c^2-a d^2\right ) \left (a d^2+11 b c^2\right )-11 b c (c+d x) \left (b c^2-a d^2\right )\right )}{4 a \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}}{8 b}+\frac {\sqrt {c+d x} \left (d^2 \left (a-\frac {b c^2}{d^2}\right )+2 b c (c+d x)\right )}{8 b \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}}{a}\right )\) |
| \(\Big \downarrow \) 1484 |
| \(\displaystyle -2 \left (\frac {c^2 \left (\frac {\int \frac {b c^2-b (c+d x) c-a d^2}{\left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{a d^2}+\frac {c \int \left (-\frac {b d x}{a \left (-b c^2+2 b (c+d x) c+a d^2-b (c+d x)^2\right )}-\frac {1}{a d x}\right )d\sqrt {c+d x}}{a}\right )}{a}-\frac {\frac {\frac {\frac {d \left (11 \sqrt {a} \sqrt {b} c d-3 a d^2+22 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {a} d+\sqrt {b} c}}-\frac {d \left (-11 \sqrt {a} \sqrt {b} c d-3 a d^2+22 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {b} c-\sqrt {a} d}}}{4 a}-\frac {\sqrt {c+d x} \left (\left (b c^2-a d^2\right ) \left (a d^2+11 b c^2\right )-11 b c (c+d x) \left (b c^2-a d^2\right )\right )}{4 a \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}}{8 b}+\frac {\sqrt {c+d x} \left (d^2 \left (a-\frac {b c^2}{d^2}\right )+2 b c (c+d x)\right )}{8 b \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}}{a}\right )\) |
| \(\Big \downarrow \) 1492 |
| \(\displaystyle -2 \left (\frac {c^2 \left (\frac {c \int \left (-\frac {b d x}{a \left (-b c^2+2 b (c+d x) c+a d^2-b (c+d x)^2\right )}-\frac {1}{a d x}\right )d\sqrt {c+d x}}{a}+\frac {\frac {d^4 \int -\frac {6 a b \left (b c^2-a d^2\right )}{d^2 \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )}d\sqrt {c+d x}}{8 a b \left (b c^2-a d^2\right )}-\frac {d^2 \sqrt {c+d x}}{4 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}}{a d^2}\right )}{a}-\frac {\frac {\frac {\frac {d \left (11 \sqrt {a} \sqrt {b} c d-3 a d^2+22 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {a} d+\sqrt {b} c}}-\frac {d \left (-11 \sqrt {a} \sqrt {b} c d-3 a d^2+22 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {b} c-\sqrt {a} d}}}{4 a}-\frac {\sqrt {c+d x} \left (\left (b c^2-a d^2\right ) \left (a d^2+11 b c^2\right )-11 b c (c+d x) \left (b c^2-a d^2\right )\right )}{4 a \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}}{8 b}+\frac {\sqrt {c+d x} \left (d^2 \left (a-\frac {b c^2}{d^2}\right )+2 b c (c+d x)\right )}{8 b \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}}{a}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 \left (\frac {c^2 \left (\frac {-\frac {3}{4} d^2 \int \frac {1}{-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}d\sqrt {c+d x}-\frac {d^2 \sqrt {c+d x}}{4 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}}{a d^2}+\frac {c \int \left (-\frac {b d x}{a \left (-b c^2+2 b (c+d x) c+a d^2-b (c+d x)^2\right )}-\frac {1}{a d x}\right )d\sqrt {c+d x}}{a}\right )}{a}-\frac {\frac {\frac {\frac {d \left (11 \sqrt {a} \sqrt {b} c d-3 a d^2+22 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {a} d+\sqrt {b} c}}-\frac {d \left (-11 \sqrt {a} \sqrt {b} c d-3 a d^2+22 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {b} c-\sqrt {a} d}}}{4 a}-\frac {\sqrt {c+d x} \left (\left (b c^2-a d^2\right ) \left (a d^2+11 b c^2\right )-11 b c (c+d x) \left (b c^2-a d^2\right )\right )}{4 a \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}}{8 b}+\frac {\sqrt {c+d x} \left (d^2 \left (a-\frac {b c^2}{d^2}\right )+2 b c (c+d x)\right )}{8 b \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}}{a}\right )\) |
| \(\Big \downarrow \) 1406 |
| \(\displaystyle -2 \left (\frac {c^2 \left (\frac {-\frac {3}{4} d^2 \left (\frac {\sqrt {b} \int \frac {1}{\frac {\sqrt {b} \left (\sqrt {b} c+\sqrt {a} d\right )}{d^2}-\frac {b (c+d x)}{d^2}}d\sqrt {c+d x}}{2 \sqrt {a} d}-\frac {\sqrt {b} \int \frac {1}{\frac {\sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right )}{d^2}-\frac {b (c+d x)}{d^2}}d\sqrt {c+d x}}{2 \sqrt {a} d}\right )-\frac {d^2 \sqrt {c+d x}}{4 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}}{a d^2}+\frac {c \int \left (-\frac {b d x}{a \left (-b c^2+2 b (c+d x) c+a d^2-b (c+d x)^2\right )}-\frac {1}{a d x}\right )d\sqrt {c+d x}}{a}\right )}{a}-\frac {\frac {\frac {\frac {d \left (11 \sqrt {a} \sqrt {b} c d-3 a d^2+22 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {a} d+\sqrt {b} c}}-\frac {d \left (-11 \sqrt {a} \sqrt {b} c d-3 a d^2+22 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {b} c-\sqrt {a} d}}}{4 a}-\frac {\sqrt {c+d x} \left (\left (b c^2-a d^2\right ) \left (a d^2+11 b c^2\right )-11 b c (c+d x) \left (b c^2-a d^2\right )\right )}{4 a \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}}{8 b}+\frac {\sqrt {c+d x} \left (d^2 \left (a-\frac {b c^2}{d^2}\right )+2 b c (c+d x)\right )}{8 b \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}}{a}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -2 \left (\frac {c^2 \left (\frac {c \int \left (-\frac {b d x}{a \left (-b c^2+2 b (c+d x) c+a d^2-b (c+d x)^2\right )}-\frac {1}{a d x}\right )d\sqrt {c+d x}}{a}+\frac {-\frac {3}{4} d^2 \left (\frac {d \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {a} d+\sqrt {b} c}}-\frac {d \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {b} c-\sqrt {a} d}}\right )-\frac {d^2 \sqrt {c+d x}}{4 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}}{a d^2}\right )}{a}-\frac {\frac {\frac {\frac {d \left (11 \sqrt {a} \sqrt {b} c d-3 a d^2+22 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {a} d+\sqrt {b} c}}-\frac {d \left (-11 \sqrt {a} \sqrt {b} c d-3 a d^2+22 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {b} c-\sqrt {a} d}}}{4 a}-\frac {\sqrt {c+d x} \left (\left (b c^2-a d^2\right ) \left (a d^2+11 b c^2\right )-11 b c (c+d x) \left (b c^2-a d^2\right )\right )}{4 a \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}}{8 b}+\frac {\sqrt {c+d x} \left (d^2 \left (a-\frac {b c^2}{d^2}\right )+2 b c (c+d x)\right )}{8 b \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}}{a}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \left (\frac {c^2 \left (\frac {c \left (\frac {\text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a \sqrt {c}}-\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 a \sqrt {\sqrt {b} c-\sqrt {a} d}}-\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{2 a \sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{a}+\frac {-\frac {3}{4} \left (\frac {d \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {b} c+\sqrt {a} d}}-\frac {d \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {b} c-\sqrt {a} d}}\right ) d^2-\frac {\sqrt {c+d x} d^2}{4 \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )}}{a d^2}\right )}{a}-\frac {\frac {\sqrt {c+d x} \left (\left (a-\frac {b c^2}{d^2}\right ) d^2+2 b c (c+d x)\right )}{8 b \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}+\frac {\frac {\frac {d \left (22 b c^2+11 \sqrt {a} \sqrt {b} d c-3 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {b} c+\sqrt {a} d}}-\frac {d \left (22 b c^2-11 \sqrt {a} \sqrt {b} d c-3 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {b} c-\sqrt {a} d}}}{4 a}-\frac {\sqrt {c+d x} \left (\left (b c^2-a d^2\right ) \left (11 b c^2+a d^2\right )-11 b c \left (b c^2-a d^2\right ) (c+d x)\right )}{4 a \left (b c^2-a d^2\right ) \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )}}{8 b}}{a}\right )\) |
Input:
Int[(c + d*x)^(5/2)/(x*(a - b*x^2)^3),x]
Output:
-2*((c^2*((c*(ArcTanh[Sqrt[c + d*x]/Sqrt[c]]/(a*Sqrt[c]) - (b^(1/4)*ArcTan h[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c - Sqrt[a]*d]])/(2*a*Sqrt[Sqrt[b]* c - Sqrt[a]*d]) - (b^(1/4)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c + Sqrt[a]*d]])/(2*a*Sqrt[Sqrt[b]*c + Sqrt[a]*d])))/a + (-1/4*(d^2*Sqrt[c + d*x])/(a - (b*c^2)/d^2 + (2*b*c*(c + d*x))/d^2 - (b*(c + d*x)^2)/d^2) - ( 3*d^2*(-1/2*(d*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c - Sqrt[a]*d] ])/(Sqrt[a]*b^(1/4)*Sqrt[Sqrt[b]*c - Sqrt[a]*d]) + (d*ArcTanh[(b^(1/4)*Sqr t[c + d*x])/Sqrt[Sqrt[b]*c + Sqrt[a]*d]])/(2*Sqrt[a]*b^(1/4)*Sqrt[Sqrt[b]* c + Sqrt[a]*d])))/4)/(a*d^2)))/a - ((Sqrt[c + d*x]*((a - (b*c^2)/d^2)*d^2 + 2*b*c*(c + d*x)))/(8*b*(a - (b*c^2)/d^2 + (2*b*c*(c + d*x))/d^2 - (b*(c + d*x)^2)/d^2)^2) + (-1/4*(Sqrt[c + d*x]*((b*c^2 - a*d^2)*(11*b*c^2 + a*d^ 2) - 11*b*c*(b*c^2 - a*d^2)*(c + d*x)))/(a*(b*c^2 - a*d^2)*(a - (b*c^2)/d^ 2 + (2*b*c*(c + d*x))/d^2 - (b*(c + d*x)^2)/d^2)) + (-1/2*(d*(22*b*c^2 - 1 1*Sqrt[a]*Sqrt[b]*c*d - 3*a*d^2)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt [b]*c - Sqrt[a]*d]])/(Sqrt[a]*b^(1/4)*Sqrt[Sqrt[b]*c - Sqrt[a]*d]) + (d*(2 2*b*c^2 + 11*Sqrt[a]*Sqrt[b]*c*d - 3*a*d^2)*ArcTanh[(b^(1/4)*Sqrt[c + d*x] )/Sqrt[Sqrt[b]*c + Sqrt[a]*d]])/(2*Sqrt[a]*b^(1/4)*Sqrt[Sqrt[b]*c + Sqrt[a ]*d]))/(4*a))/(8*b))/a)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{k = Denominator[n]}, Simp[k/d Subst[Int[x^(k*(n + 1) - 1)*(-c /d + x^k/d)^m*Simp[(b*c^2 + a*d^2)/d^2 - 2*b*c*(x^k/d^2) + b*(x^(2*k)/d^2), x]^p, x], x, (c + d*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, m, p}, x] && Frac tionQ[n] && IntegerQ[p] && IntegerQ[m]
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^ 2 - 4*a*c, 2]}, Simp[c/q Int[1/(b/2 - q/2 + c*x^2), x], x] - Simp[c/q I nt[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c , 0] && PosQ[b^2 - 4*a*c]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symb ol] :> Int[ExpandIntegrand[(d + e*x^2)^q/(a + b*x^2 + c*x^4), x], x] /; Fre eQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[q]
Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb ol] :> Simp[x*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && IntegerQ[2*p]
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*( x_)^4)^(p_.), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(a + b*x^2 + c*x^4)^(p + 1) *((b*d - 2*a*e - (b*e - 2*c*d)*x^2)/(2*(p + 1)*(b^2 - 4*a*c))), x] - Simp[f ^2/(2*(p + 1)*(b^2 - 4*a*c)) Int[(f*x)^(m - 2)*(a + b*x^2 + c*x^4)^(p + 1 )*Simp[(m - 1)*(b*d - 2*a*e) - (4*p + 4 + m + 1)*(b*e - 2*c*d)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && GtQ[m, 1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
Int[(((f_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[-f^4/(c*d^2 - b*d*e + a*e^2) Int[(f*x)^ (m - 4)*(a*d + (b*d - a*e)*x^2)*(a + b*x^2 + c*x^4)^p, x], x] + Simp[d^2*(f ^4/(c*d^2 - b*d*e + a*e^2)) Int[(f*x)^(m - 4)*((a + b*x^2 + c*x^4)^(p + 1 )/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && GtQ[m, 2]
Int[(((f_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[f^2/(c*d^2 - b*d*e + a*e^2) Int[(f*x)^( m - 2)*(a*e + c*d*x^2)*(a + b*x^2 + c*x^4)^p, x], x] - Simp[d*e*(f^2/(c*d^2 - b*d*e + a*e^2)) Int[(f*x)^(m - 2)*((a + b*x^2 + c*x^4)^(p + 1)/(d + e* x^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ [p, -1] && GtQ[m, 0]
Time = 0.78 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.19
| method | result | size |
| pseudoelliptic | \(-\frac {3 \left (\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (\frac {\left (11 a \,d^{2} c +32 b \,c^{3}\right ) \sqrt {a b \,d^{2}}}{3}+a \,d^{2} \left (a \,d^{2}-\frac {46 b \,c^{2}}{3}\right )\right ) b \left (-b \,x^{2}+a \right )^{2} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )+\left (\left (\frac {\left (-11 a \,d^{2} c -32 b \,c^{3}\right ) \sqrt {a b \,d^{2}}}{3}+a \,d^{2} \left (a \,d^{2}-\frac {46 b \,c^{2}}{3}\right )\right ) b \left (-b \,x^{2}+a \right )^{2} \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )-2 \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (-\frac {32 b \,c^{\frac {5}{2}} \left (-b \,x^{2}+a \right )^{2} \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{3}+\left (-\frac {8 x^{2} c \left (\frac {11 d x}{8}+c \right ) b^{2}}{3}+4 \left (\frac {1}{12} d^{2} x^{2}+\frac {19}{12} c d x +c^{2}\right ) a b +a^{2} d^{2}\right ) a \sqrt {d x +c}\right ) \sqrt {a b \,d^{2}}\right ) \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\right )}{32 \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (-b \,x^{2}+a \right )^{2} a^{3} b}\) | \(365\) |
| derivativedivides | \(-2 d^{6} \left (\frac {c^{\frac {5}{2}} \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{a^{3} d^{6}}-\frac {\frac {-\frac {11 a b c \,d^{2} \left (d x +c \right )^{\frac {7}{2}}}{32}+\frac {\left (a \,d^{2}+25 b \,c^{2}\right ) a \,d^{2} \left (d x +c \right )^{\frac {5}{2}}}{32}+\frac {17 a c \,d^{2} \left (a \,d^{2}-b \,c^{2}\right ) \left (d x +c \right )^{\frac {3}{2}}}{32}+\frac {3 a \,d^{2} \left (a^{2} d^{4}-2 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) \sqrt {d x +c}}{32 b}}{\left (-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}\right )^{2}}-\frac {\left (3 a^{2} d^{4}-46 b \,c^{2} d^{2} a -11 \sqrt {a b \,d^{2}}\, a c \,d^{2}-32 \sqrt {a b \,d^{2}}\, b \,c^{3}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{64 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\left (-3 a^{2} d^{4}+46 b \,c^{2} d^{2} a -11 \sqrt {a b \,d^{2}}\, a c \,d^{2}-32 \sqrt {a b \,d^{2}}\, b \,c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{64 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}}{a^{3} d^{6}}\right )\) | \(382\) |
| default | \(2 d^{6} \left (-\frac {c^{\frac {5}{2}} \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{a^{3} d^{6}}+\frac {\frac {-\frac {11 a b c \,d^{2} \left (d x +c \right )^{\frac {7}{2}}}{32}+\frac {\left (a \,d^{2}+25 b \,c^{2}\right ) a \,d^{2} \left (d x +c \right )^{\frac {5}{2}}}{32}+\frac {17 a c \,d^{2} \left (a \,d^{2}-b \,c^{2}\right ) \left (d x +c \right )^{\frac {3}{2}}}{32}+\frac {3 a \,d^{2} \left (a^{2} d^{4}-2 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) \sqrt {d x +c}}{32 b}}{\left (-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}\right )^{2}}-\frac {\left (3 a^{2} d^{4}-46 b \,c^{2} d^{2} a -11 \sqrt {a b \,d^{2}}\, a c \,d^{2}-32 \sqrt {a b \,d^{2}}\, b \,c^{3}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{64 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\left (-3 a^{2} d^{4}+46 b \,c^{2} d^{2} a -11 \sqrt {a b \,d^{2}}\, a c \,d^{2}-32 \sqrt {a b \,d^{2}}\, b \,c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{64 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}}{a^{3} d^{6}}\right )\) | \(382\) |
Input:
int((d*x+c)^(5/2)/x/(-b*x^2+a)^3,x,method=_RETURNVERBOSE)
Output:
-3/32/((-b*c+(a*b*d^2)^(1/2))*b)^(1/2)/(a*b*d^2)^(1/2)/((b*c+(a*b*d^2)^(1/ 2))*b)^(1/2)*(((b*c+(a*b*d^2)^(1/2))*b)^(1/2)*(1/3*(11*a*c*d^2+32*b*c^3)*( a*b*d^2)^(1/2)+a*d^2*(a*d^2-46/3*b*c^2))*b*(-b*x^2+a)^2*arctan(b*(d*x+c)^( 1/2)/((-b*c+(a*b*d^2)^(1/2))*b)^(1/2))+((1/3*(-11*a*c*d^2-32*b*c^3)*(a*b*d ^2)^(1/2)+a*d^2*(a*d^2-46/3*b*c^2))*b*(-b*x^2+a)^2*arctanh(b*(d*x+c)^(1/2) /((b*c+(a*b*d^2)^(1/2))*b)^(1/2))-2*((b*c+(a*b*d^2)^(1/2))*b)^(1/2)*(-32/3 *b*c^(5/2)*(-b*x^2+a)^2*arctanh((d*x+c)^(1/2)/c^(1/2))+(-8/3*x^2*c*(11/8*d *x+c)*b^2+4*(1/12*d^2*x^2+19/12*c*d*x+c^2)*a*b+a^2*d^2)*a*(d*x+c)^(1/2))*( a*b*d^2)^(1/2))*((-b*c+(a*b*d^2)^(1/2))*b)^(1/2))/(-b*x^2+a)^2/a^3/b
Leaf count of result is larger than twice the leaf count of optimal. 1926 vs. \(2 (246) = 492\).
Time = 2.18 (sec) , antiderivative size = 3861, normalized size of antiderivative = 12.62 \[ \int \frac {(c+d x)^{5/2}}{x \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)^(5/2)/x/(-b*x^2+a)^3,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {(c+d x)^{5/2}}{x \left (a-b x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate((d*x+c)**(5/2)/x/(-b*x**2+a)**3,x)
Output:
Timed out
\[ \int \frac {(c+d x)^{5/2}}{x \left (a-b x^2\right )^3} \, dx=\int { -\frac {{\left (d x + c\right )}^{\frac {5}{2}}}{{\left (b x^{2} - a\right )}^{3} x} \,d x } \] Input:
integrate((d*x+c)^(5/2)/x/(-b*x^2+a)^3,x, algorithm="maxima")
Output:
-integrate((d*x + c)^(5/2)/((b*x^2 - a)^3*x), x)
Leaf count of result is larger than twice the leaf count of optimal. 658 vs. \(2 (246) = 492\).
Time = 0.35 (sec) , antiderivative size = 658, normalized size of antiderivative = 2.15 \[ \int \frac {(c+d x)^{5/2}}{x \left (a-b x^2\right )^3} \, dx=\frac {2 \, c^{3} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{a^{3} \sqrt {-c}} - \frac {{\left (46 \, \sqrt {a b} a^{2} b^{3} c^{3} d^{2} - 3 \, \sqrt {a b} a^{3} b^{2} c d^{4} - {\left (32 \, \sqrt {a b} b c^{3} + 11 \, \sqrt {a b} a c d^{2}\right )} a^{2} b^{2} d^{2} + {\left (32 \, a b^{3} c^{4} - 35 \, a^{2} b^{2} c^{2} d^{2} + 3 \, a^{3} b d^{4}\right )} {\left | a \right |} {\left | b \right |} {\left | d \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {a^{3} b^{2} c + \sqrt {a^{6} b^{4} c^{2} - {\left (a^{3} b^{2} c^{2} - a^{4} b d^{2}\right )} a^{3} b^{2}}}{a^{3} b^{2}}}}\right )}{32 \, {\left (a^{4} b^{3} c - \sqrt {a b} a^{4} b^{2} d\right )} \sqrt {-b^{2} c - \sqrt {a b} b d} {\left | a \right |} {\left | d \right |}} + \frac {{\left (46 \, a^{2} b^{3} c^{3} d^{2} - 3 \, a^{3} b^{2} c d^{4} - {\left (32 \, b c^{3} + 11 \, a c d^{2}\right )} a^{2} b^{2} d^{2} - {\left (32 \, \sqrt {a b} b^{2} c^{4} - 35 \, \sqrt {a b} a b c^{2} d^{2} + 3 \, \sqrt {a b} a^{2} d^{4}\right )} {\left | a \right |} {\left | b \right |} {\left | d \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {a^{3} b^{2} c - \sqrt {a^{6} b^{4} c^{2} - {\left (a^{3} b^{2} c^{2} - a^{4} b d^{2}\right )} a^{3} b^{2}}}{a^{3} b^{2}}}}\right )}{32 \, {\left (a^{4} b^{2} d + \sqrt {a b} a^{3} b^{2} c\right )} \sqrt {-b^{2} c + \sqrt {a b} b d} {\left | a \right |} {\left | d \right |}} - \frac {11 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{2} c d^{2} - 25 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{2} c^{2} d^{2} + 17 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{2} c^{3} d^{2} - 3 \, \sqrt {d x + c} b^{2} c^{4} d^{2} - {\left (d x + c\right )}^{\frac {5}{2}} a b d^{4} - 17 \, {\left (d x + c\right )}^{\frac {3}{2}} a b c d^{4} + 6 \, \sqrt {d x + c} a b c^{2} d^{4} - 3 \, \sqrt {d x + c} a^{2} d^{6}}{16 \, {\left ({\left (d x + c\right )}^{2} b - 2 \, {\left (d x + c\right )} b c + b c^{2} - a d^{2}\right )}^{2} a^{2} b} \] Input:
integrate((d*x+c)^(5/2)/x/(-b*x^2+a)^3,x, algorithm="giac")
Output:
2*c^3*arctan(sqrt(d*x + c)/sqrt(-c))/(a^3*sqrt(-c)) - 1/32*(46*sqrt(a*b)*a ^2*b^3*c^3*d^2 - 3*sqrt(a*b)*a^3*b^2*c*d^4 - (32*sqrt(a*b)*b*c^3 + 11*sqrt (a*b)*a*c*d^2)*a^2*b^2*d^2 + (32*a*b^3*c^4 - 35*a^2*b^2*c^2*d^2 + 3*a^3*b* d^4)*abs(a)*abs(b)*abs(d))*arctan(sqrt(d*x + c)/sqrt(-(a^3*b^2*c + sqrt(a^ 6*b^4*c^2 - (a^3*b^2*c^2 - a^4*b*d^2)*a^3*b^2))/(a^3*b^2)))/((a^4*b^3*c - sqrt(a*b)*a^4*b^2*d)*sqrt(-b^2*c - sqrt(a*b)*b*d)*abs(a)*abs(d)) + 1/32*(4 6*a^2*b^3*c^3*d^2 - 3*a^3*b^2*c*d^4 - (32*b*c^3 + 11*a*c*d^2)*a^2*b^2*d^2 - (32*sqrt(a*b)*b^2*c^4 - 35*sqrt(a*b)*a*b*c^2*d^2 + 3*sqrt(a*b)*a^2*d^4)* abs(a)*abs(b)*abs(d))*arctan(sqrt(d*x + c)/sqrt(-(a^3*b^2*c - sqrt(a^6*b^4 *c^2 - (a^3*b^2*c^2 - a^4*b*d^2)*a^3*b^2))/(a^3*b^2)))/((a^4*b^2*d + sqrt( a*b)*a^3*b^2*c)*sqrt(-b^2*c + sqrt(a*b)*b*d)*abs(a)*abs(d)) - 1/16*(11*(d* x + c)^(7/2)*b^2*c*d^2 - 25*(d*x + c)^(5/2)*b^2*c^2*d^2 + 17*(d*x + c)^(3/ 2)*b^2*c^3*d^2 - 3*sqrt(d*x + c)*b^2*c^4*d^2 - (d*x + c)^(5/2)*a*b*d^4 - 1 7*(d*x + c)^(3/2)*a*b*c*d^4 + 6*sqrt(d*x + c)*a*b*c^2*d^4 - 3*sqrt(d*x + c )*a^2*d^6)/(((d*x + c)^2*b - 2*(d*x + c)*b*c + b*c^2 - a*d^2)^2*a^2*b)
Time = 10.56 (sec) , antiderivative size = 8343, normalized size of antiderivative = 27.26 \[ \int \frac {(c+d x)^{5/2}}{x \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:
int((c + d*x)^(5/2)/(x*(a - b*x^2)^3),x)
Output:
(atan((d^20*(c^5)^(1/2)*(c + d*x)^(1/2)*81i)/(16384*((81*c^3*d^20)/16384 - (3573*b*c^5*d^18)/(8192*a) + (183025*b^2*c^7*d^16)/(16384*a^2) - (219245* b^3*c^9*d^14)/(2048*a^3) + (674625*b^4*c^11*d^12)/(1024*a^4) - (2277*b^5*c ^13*d^10)/(2*a^5) + (576*b^6*c^15*d^8)/a^6)) - (c^2*d^18*(c^5)^(1/2)*(c + d*x)^(1/2)*3573i)/(8192*((81*a*c^3*d^20)/(16384*b) - (3573*c^5*d^18)/8192 + (183025*b*c^7*d^16)/(16384*a) - (219245*b^2*c^9*d^14)/(2048*a^2) + (6746 25*b^3*c^11*d^12)/(1024*a^3) - (2277*b^4*c^13*d^10)/(2*a^4) + (576*b^5*c^1 5*d^8)/a^5)) + (b*c^4*d^16*(c^5)^(1/2)*(c + d*x)^(1/2)*183025i)/(16384*((1 83025*b*c^7*d^16)/16384 - (3573*a*c^5*d^18)/8192 - (219245*b^2*c^9*d^14)/( 2048*a) + (81*a^2*c^3*d^20)/(16384*b) + (674625*b^3*c^11*d^12)/(1024*a^2) - (2277*b^4*c^13*d^10)/(2*a^3) + (576*b^5*c^15*d^8)/a^4)) - (b^2*c^6*d^14* (c^5)^(1/2)*(c + d*x)^(1/2)*219245i)/(2048*((674625*b^3*c^11*d^12)/(1024*a ) - (219245*b^2*c^9*d^14)/2048 - (3573*a^2*c^5*d^18)/8192 + (81*a^3*c^3*d^ 20)/(16384*b) - (2277*b^4*c^13*d^10)/(2*a^2) + (576*b^5*c^15*d^8)/a^3 + (1 83025*a*b*c^7*d^16)/16384)) + (b^3*c^8*d^12*(c^5)^(1/2)*(c + d*x)^(1/2)*67 4625i)/(1024*((674625*b^3*c^11*d^12)/1024 - (3573*a^3*c^5*d^18)/8192 - (21 9245*a*b^2*c^9*d^14)/2048 + (183025*a^2*b*c^7*d^16)/16384 - (2277*b^4*c^13 *d^10)/(2*a) + (81*a^4*c^3*d^20)/(16384*b) + (576*b^5*c^15*d^8)/a^2)) - (b ^4*c^10*d^10*(c^5)^(1/2)*(c + d*x)^(1/2)*2277i)/(2*((674625*a*b^3*c^11*d^1 2)/1024 - (2277*b^4*c^13*d^10)/2 - (3573*a^4*c^5*d^18)/8192 + (183025*a...
Time = 5.61 (sec) , antiderivative size = 1575, normalized size of antiderivative = 5.15 \[ \int \frac {(c+d x)^{5/2}}{x \left (a-b x^2\right )^3} \, dx =\text {Too large to display} \] Input:
int((d*x+c)^(5/2)/x/(-b*x^2+a)^3,x)
Output:
( - 28*sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt( b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a**2*b*c*d + 56*sqrt(a)*sqrt(sqrt(b)*sq rt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b* c)))*a*b**2*c*d*x**2 - 28*sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt (c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*b**3*c*d*x**4 - 6*sq rt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(s qrt(b)*sqrt(a)*d - b*c)))*a**3*d**2 + 64*sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a**2* b*c**2 + 12*sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/( sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a**2*b*d**2*x**2 - 128*sqrt(b)*sqr t(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sq rt(a)*d - b*c)))*a*b**2*c**2*x**2 - 6*sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c )*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a*b**2*d **2*x**4 + 64*sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b) /(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*b**3*c**2*x**4 - 14*sqrt(a)*sqrt (sqrt(b)*sqrt(a)*d + b*c)*log( - sqrt(sqrt(b)*sqrt(a)*d + b*c) + sqrt(b)*s qrt(c + d*x))*a**2*b*c*d + 28*sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d + b*c)*log( - sqrt(sqrt(b)*sqrt(a)*d + b*c) + sqrt(b)*sqrt(c + d*x))*a*b**2*c*d*x**2 - 14*sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d + b*c)*log( - sqrt(sqrt(b)*sqrt(a)*d + b *c) + sqrt(b)*sqrt(c + d*x))*b**3*c*d*x**4 + 14*sqrt(a)*sqrt(sqrt(b)*sq...