Integrand size = 22, antiderivative size = 364 \[ \int \frac {\left (a+b x^2\right )^{7/2}}{x (c+d x)} \, dx=-\frac {b c \left (b^2 c^4+3 a b c^2 d^2+3 a^2 d^4\right ) \sqrt {a+b x^2}}{d^6}+\frac {b \left (8 b^2 c^4+26 a b c^2 d^2+29 a^2 d^4\right ) x \sqrt {a+b x^2}}{16 d^5}+\frac {b^2 \left (6 b c^2+19 a d^2\right ) x^3 \sqrt {a+b x^2}}{24 d^3}+\frac {b^3 x^5 \sqrt {a+b x^2}}{6 d}-\frac {b c \left (b c^2+2 a d^2\right ) \left (a+b x^2\right )^{3/2}}{3 d^4}-\frac {b c \left (a+b x^2\right )^{5/2}}{5 d^2}+\frac {\sqrt {b} \left (16 b^3 c^6+56 a b^2 c^4 d^2+70 a^2 b c^2 d^4+35 a^3 d^6\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 d^7}+\frac {\left (b c^2+a d^2\right )^{7/2} \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{c d^7}-\frac {a^{7/2} \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{c} \] Output:
-b*c*(3*a^2*d^4+3*a*b*c^2*d^2+b^2*c^4)*(b*x^2+a)^(1/2)/d^6+1/16*b*(29*a^2* d^4+26*a*b*c^2*d^2+8*b^2*c^4)*x*(b*x^2+a)^(1/2)/d^5+1/24*b^2*(19*a*d^2+6*b *c^2)*x^3*(b*x^2+a)^(1/2)/d^3+1/6*b^3*x^5*(b*x^2+a)^(1/2)/d-1/3*b*c*(2*a*d ^2+b*c^2)*(b*x^2+a)^(3/2)/d^4-1/5*b*c*(b*x^2+a)^(5/2)/d^2+1/16*b^(1/2)*(35 *a^3*d^6+70*a^2*b*c^2*d^4+56*a*b^2*c^4*d^2+16*b^3*c^6)*arctanh(b^(1/2)*x/( b*x^2+a)^(1/2))/d^7+(a*d^2+b*c^2)^(7/2)*arctanh((-b*c*x+a*d)/(a*d^2+b*c^2) ^(1/2)/(b*x^2+a)^(1/2))/c/d^7-a^(7/2)*arctanh((b*x^2+a)^(1/2)/a^(1/2))/c
Time = 1.75 (sec) , antiderivative size = 327, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x^2\right )^{7/2}}{x (c+d x)} \, dx=\frac {b \sqrt {a+b x^2} \left (29 a^2 d^4 (-32 c+15 d x)+2 a b d^2 \left (-400 c^3+195 c^2 d x-128 c d^2 x^2+95 d^3 x^3\right )-4 b^2 \left (60 c^5-30 c^4 d x+20 c^3 d^2 x^2-15 c^2 d^3 x^3+12 c d^4 x^4-10 d^5 x^5\right )\right )}{240 d^6}+\frac {2 \left (-b c^2-a d^2\right )^{7/2} \arctan \left (\frac {\sqrt {-b c^2-a d^2} x}{\sqrt {a} (c+d x)-c \sqrt {a+b x^2}}\right )}{c d^7}+\frac {\sqrt {b} \left (16 b^3 c^6+56 a b^2 c^4 d^2+70 a^2 b c^2 d^4+35 a^3 d^6\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{-\sqrt {a}+\sqrt {a+b x^2}}\right )}{8 d^7}-\frac {a^{7/2} \log (x)}{c}+\frac {a^{7/2} \log \left (-\sqrt {a}+\sqrt {a+b x^2}\right )}{c} \] Input:
Integrate[(a + b*x^2)^(7/2)/(x*(c + d*x)),x]
Output:
(b*Sqrt[a + b*x^2]*(29*a^2*d^4*(-32*c + 15*d*x) + 2*a*b*d^2*(-400*c^3 + 19 5*c^2*d*x - 128*c*d^2*x^2 + 95*d^3*x^3) - 4*b^2*(60*c^5 - 30*c^4*d*x + 20* c^3*d^2*x^2 - 15*c^2*d^3*x^3 + 12*c*d^4*x^4 - 10*d^5*x^5)))/(240*d^6) + (2 *(-(b*c^2) - a*d^2)^(7/2)*ArcTan[(Sqrt[-(b*c^2) - a*d^2]*x)/(Sqrt[a]*(c + d*x) - c*Sqrt[a + b*x^2])])/(c*d^7) + (Sqrt[b]*(16*b^3*c^6 + 56*a*b^2*c^4* d^2 + 70*a^2*b*c^2*d^4 + 35*a^3*d^6)*ArcTanh[(Sqrt[b]*x)/(-Sqrt[a] + Sqrt[ a + b*x^2])])/(8*d^7) - (a^(7/2)*Log[x])/c + (a^(7/2)*Log[-Sqrt[a] + Sqrt[ a + b*x^2]])/c
Time = 1.25 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.10, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.818, Rules used = {606, 243, 60, 60, 60, 73, 221, 682, 27, 682, 27, 682, 27, 719, 224, 219, 488, 219}
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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{7/2}}{x (c+d x)} \, dx\) |
| \(\Big \downarrow \) 606 |
| \(\displaystyle \frac {a \int \frac {\left (b x^2+a\right )^{5/2}}{x}dx}{c}-\frac {\int \frac {(a d-b c x) \left (b x^2+a\right )^{5/2}}{c+d x}dx}{c}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {a \int \frac {\left (b x^2+a\right )^{5/2}}{x^2}dx^2}{2 c}-\frac {\int \frac {(a d-b c x) \left (b x^2+a\right )^{5/2}}{c+d x}dx}{c}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {a \left (a \int \frac {\left (b x^2+a\right )^{3/2}}{x^2}dx^2+\frac {2}{5} \left (a+b x^2\right )^{5/2}\right )}{2 c}-\frac {\int \frac {(a d-b c x) \left (b x^2+a\right )^{5/2}}{c+d x}dx}{c}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {a \left (a \left (a \int \frac {\sqrt {b x^2+a}}{x^2}dx^2+\frac {2}{3} \left (a+b x^2\right )^{3/2}\right )+\frac {2}{5} \left (a+b x^2\right )^{5/2}\right )}{2 c}-\frac {\int \frac {(a d-b c x) \left (b x^2+a\right )^{5/2}}{c+d x}dx}{c}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {a \left (a \left (a \left (a \int \frac {1}{x^2 \sqrt {b x^2+a}}dx^2+2 \sqrt {a+b x^2}\right )+\frac {2}{3} \left (a+b x^2\right )^{3/2}\right )+\frac {2}{5} \left (a+b x^2\right )^{5/2}\right )}{2 c}-\frac {\int \frac {(a d-b c x) \left (b x^2+a\right )^{5/2}}{c+d x}dx}{c}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {a \left (a \left (a \left (\frac {2 a \int \frac {1}{\frac {x^4}{b}-\frac {a}{b}}d\sqrt {b x^2+a}}{b}+2 \sqrt {a+b x^2}\right )+\frac {2}{3} \left (a+b x^2\right )^{3/2}\right )+\frac {2}{5} \left (a+b x^2\right )^{5/2}\right )}{2 c}-\frac {\int \frac {(a d-b c x) \left (b x^2+a\right )^{5/2}}{c+d x}dx}{c}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {a \left (a \left (a \left (2 \sqrt {a+b x^2}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )\right )+\frac {2}{3} \left (a+b x^2\right )^{3/2}\right )+\frac {2}{5} \left (a+b x^2\right )^{5/2}\right )}{2 c}-\frac {\int \frac {(a d-b c x) \left (b x^2+a\right )^{5/2}}{c+d x}dx}{c}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {a \left (a \left (a \left (2 \sqrt {a+b x^2}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )\right )+\frac {2}{3} \left (a+b x^2\right )^{3/2}\right )+\frac {2}{5} \left (a+b x^2\right )^{5/2}\right )}{2 c}-\frac {\frac {\int \frac {b \left (a d \left (b c^2+6 a d^2\right )-b c \left (6 b c^2+11 a d^2\right ) x\right ) \left (b x^2+a\right )^{3/2}}{c+d x}dx}{6 b d^2}+\frac {\left (a+b x^2\right )^{5/2} \left (6 \left (a d^2+b c^2\right )-5 b c d x\right )}{30 d^2}}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \left (a \left (a \left (2 \sqrt {a+b x^2}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )\right )+\frac {2}{3} \left (a+b x^2\right )^{3/2}\right )+\frac {2}{5} \left (a+b x^2\right )^{5/2}\right )}{2 c}-\frac {\frac {\int \frac {\left (a d \left (b c^2+6 a d^2\right )-b c \left (6 b c^2+11 a d^2\right ) x\right ) \left (b x^2+a\right )^{3/2}}{c+d x}dx}{6 d^2}+\frac {\left (a+b x^2\right )^{5/2} \left (6 \left (a d^2+b c^2\right )-5 b c d x\right )}{30 d^2}}{c}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {a \left (a \left (a \left (2 \sqrt {a+b x^2}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )\right )+\frac {2}{3} \left (a+b x^2\right )^{3/2}\right )+\frac {2}{5} \left (a+b x^2\right )^{5/2}\right )}{2 c}-\frac {\frac {\frac {\int \frac {3 b \left (a d \left (2 b^2 c^4+5 a b d^2 c^2+8 a^2 d^4\right )-b c \left (8 b^2 c^4+22 a b d^2 c^2+19 a^2 d^4\right ) x\right ) \sqrt {b x^2+a}}{c+d x}dx}{4 b d^2}+\frac {\left (a+b x^2\right )^{3/2} \left (8 \left (a d^2+b c^2\right )^2-b c d x \left (11 a d^2+6 b c^2\right )\right )}{4 d^2}}{6 d^2}+\frac {\left (a+b x^2\right )^{5/2} \left (6 \left (a d^2+b c^2\right )-5 b c d x\right )}{30 d^2}}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \left (a \left (a \left (2 \sqrt {a+b x^2}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )\right )+\frac {2}{3} \left (a+b x^2\right )^{3/2}\right )+\frac {2}{5} \left (a+b x^2\right )^{5/2}\right )}{2 c}-\frac {\frac {\frac {3 \int \frac {\left (a d \left (2 b^2 c^4+5 a b d^2 c^2+8 a^2 d^4\right )-b c \left (8 b^2 c^4+22 a b d^2 c^2+19 a^2 d^4\right ) x\right ) \sqrt {b x^2+a}}{c+d x}dx}{4 d^2}+\frac {\left (a+b x^2\right )^{3/2} \left (8 \left (a d^2+b c^2\right )^2-b c d x \left (11 a d^2+6 b c^2\right )\right )}{4 d^2}}{6 d^2}+\frac {\left (a+b x^2\right )^{5/2} \left (6 \left (a d^2+b c^2\right )-5 b c d x\right )}{30 d^2}}{c}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {a \left (a \left (a \left (2 \sqrt {a+b x^2}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )\right )+\frac {2}{3} \left (a+b x^2\right )^{3/2}\right )+\frac {2}{5} \left (a+b x^2\right )^{5/2}\right )}{2 c}-\frac {\frac {\frac {3 \left (\frac {\int \frac {b \left (a d \left (8 b^3 c^6+26 a b^2 d^2 c^4+29 a^2 b d^4 c^2+16 a^3 d^6\right )-b c \left (16 b^3 c^6+56 a b^2 d^2 c^4+70 a^2 b d^4 c^2+35 a^3 d^6\right ) x\right )}{(c+d x) \sqrt {b x^2+a}}dx}{2 b d^2}+\frac {\sqrt {a+b x^2} \left (16 \left (a d^2+b c^2\right )^3-b c d x \left (19 a^2 d^4+22 a b c^2 d^2+8 b^2 c^4\right )\right )}{2 d^2}\right )}{4 d^2}+\frac {\left (a+b x^2\right )^{3/2} \left (8 \left (a d^2+b c^2\right )^2-b c d x \left (11 a d^2+6 b c^2\right )\right )}{4 d^2}}{6 d^2}+\frac {\left (a+b x^2\right )^{5/2} \left (6 \left (a d^2+b c^2\right )-5 b c d x\right )}{30 d^2}}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \left (a \left (a \left (2 \sqrt {a+b x^2}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )\right )+\frac {2}{3} \left (a+b x^2\right )^{3/2}\right )+\frac {2}{5} \left (a+b x^2\right )^{5/2}\right )}{2 c}-\frac {\frac {\frac {3 \left (\frac {\int \frac {a d \left (8 b^3 c^6+26 a b^2 d^2 c^4+29 a^2 b d^4 c^2+16 a^3 d^6\right )-b c \left (16 b^3 c^6+56 a b^2 d^2 c^4+70 a^2 b d^4 c^2+35 a^3 d^6\right ) x}{(c+d x) \sqrt {b x^2+a}}dx}{2 d^2}+\frac {\sqrt {a+b x^2} \left (16 \left (a d^2+b c^2\right )^3-b c d x \left (19 a^2 d^4+22 a b c^2 d^2+8 b^2 c^4\right )\right )}{2 d^2}\right )}{4 d^2}+\frac {\left (a+b x^2\right )^{3/2} \left (8 \left (a d^2+b c^2\right )^2-b c d x \left (11 a d^2+6 b c^2\right )\right )}{4 d^2}}{6 d^2}+\frac {\left (a+b x^2\right )^{5/2} \left (6 \left (a d^2+b c^2\right )-5 b c d x\right )}{30 d^2}}{c}\) |
| \(\Big \downarrow \) 719 |
| \(\displaystyle \frac {a \left (a \left (a \left (2 \sqrt {a+b x^2}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )\right )+\frac {2}{3} \left (a+b x^2\right )^{3/2}\right )+\frac {2}{5} \left (a+b x^2\right )^{5/2}\right )}{2 c}-\frac {\frac {\frac {3 \left (\frac {\frac {16 \left (a d^2+b c^2\right )^4 \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}-\frac {b c \left (35 a^3 d^6+70 a^2 b c^2 d^4+56 a b^2 c^4 d^2+16 b^3 c^6\right ) \int \frac {1}{\sqrt {b x^2+a}}dx}{d}}{2 d^2}+\frac {\sqrt {a+b x^2} \left (16 \left (a d^2+b c^2\right )^3-b c d x \left (19 a^2 d^4+22 a b c^2 d^2+8 b^2 c^4\right )\right )}{2 d^2}\right )}{4 d^2}+\frac {\left (a+b x^2\right )^{3/2} \left (8 \left (a d^2+b c^2\right )^2-b c d x \left (11 a d^2+6 b c^2\right )\right )}{4 d^2}}{6 d^2}+\frac {\left (a+b x^2\right )^{5/2} \left (6 \left (a d^2+b c^2\right )-5 b c d x\right )}{30 d^2}}{c}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {a \left (a \left (a \left (2 \sqrt {a+b x^2}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )\right )+\frac {2}{3} \left (a+b x^2\right )^{3/2}\right )+\frac {2}{5} \left (a+b x^2\right )^{5/2}\right )}{2 c}-\frac {\frac {\frac {3 \left (\frac {\frac {16 \left (a d^2+b c^2\right )^4 \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}-\frac {b c \left (35 a^3 d^6+70 a^2 b c^2 d^4+56 a b^2 c^4 d^2+16 b^3 c^6\right ) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}}{2 d^2}+\frac {\sqrt {a+b x^2} \left (16 \left (a d^2+b c^2\right )^3-b c d x \left (19 a^2 d^4+22 a b c^2 d^2+8 b^2 c^4\right )\right )}{2 d^2}\right )}{4 d^2}+\frac {\left (a+b x^2\right )^{3/2} \left (8 \left (a d^2+b c^2\right )^2-b c d x \left (11 a d^2+6 b c^2\right )\right )}{4 d^2}}{6 d^2}+\frac {\left (a+b x^2\right )^{5/2} \left (6 \left (a d^2+b c^2\right )-5 b c d x\right )}{30 d^2}}{c}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {a \left (a \left (a \left (2 \sqrt {a+b x^2}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )\right )+\frac {2}{3} \left (a+b x^2\right )^{3/2}\right )+\frac {2}{5} \left (a+b x^2\right )^{5/2}\right )}{2 c}-\frac {\frac {\frac {3 \left (\frac {\frac {16 \left (a d^2+b c^2\right )^4 \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}-\frac {\sqrt {b} c \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (35 a^3 d^6+70 a^2 b c^2 d^4+56 a b^2 c^4 d^2+16 b^3 c^6\right )}{d}}{2 d^2}+\frac {\sqrt {a+b x^2} \left (16 \left (a d^2+b c^2\right )^3-b c d x \left (19 a^2 d^4+22 a b c^2 d^2+8 b^2 c^4\right )\right )}{2 d^2}\right )}{4 d^2}+\frac {\left (a+b x^2\right )^{3/2} \left (8 \left (a d^2+b c^2\right )^2-b c d x \left (11 a d^2+6 b c^2\right )\right )}{4 d^2}}{6 d^2}+\frac {\left (a+b x^2\right )^{5/2} \left (6 \left (a d^2+b c^2\right )-5 b c d x\right )}{30 d^2}}{c}\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \frac {a \left (a \left (a \left (2 \sqrt {a+b x^2}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )\right )+\frac {2}{3} \left (a+b x^2\right )^{3/2}\right )+\frac {2}{5} \left (a+b x^2\right )^{5/2}\right )}{2 c}-\frac {\frac {\frac {3 \left (\frac {-\frac {16 \left (a d^2+b c^2\right )^4 \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}}{d}-\frac {\sqrt {b} c \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (35 a^3 d^6+70 a^2 b c^2 d^4+56 a b^2 c^4 d^2+16 b^3 c^6\right )}{d}}{2 d^2}+\frac {\sqrt {a+b x^2} \left (16 \left (a d^2+b c^2\right )^3-b c d x \left (19 a^2 d^4+22 a b c^2 d^2+8 b^2 c^4\right )\right )}{2 d^2}\right )}{4 d^2}+\frac {\left (a+b x^2\right )^{3/2} \left (8 \left (a d^2+b c^2\right )^2-b c d x \left (11 a d^2+6 b c^2\right )\right )}{4 d^2}}{6 d^2}+\frac {\left (a+b x^2\right )^{5/2} \left (6 \left (a d^2+b c^2\right )-5 b c d x\right )}{30 d^2}}{c}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {a \left (a \left (a \left (2 \sqrt {a+b x^2}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )\right )+\frac {2}{3} \left (a+b x^2\right )^{3/2}\right )+\frac {2}{5} \left (a+b x^2\right )^{5/2}\right )}{2 c}-\frac {\frac {\frac {3 \left (\frac {\sqrt {a+b x^2} \left (16 \left (a d^2+b c^2\right )^3-b c d x \left (19 a^2 d^4+22 a b c^2 d^2+8 b^2 c^4\right )\right )}{2 d^2}+\frac {-\frac {\sqrt {b} c \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (35 a^3 d^6+70 a^2 b c^2 d^4+56 a b^2 c^4 d^2+16 b^3 c^6\right )}{d}-\frac {16 \left (a d^2+b c^2\right )^{7/2} \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{d}}{2 d^2}\right )}{4 d^2}+\frac {\left (a+b x^2\right )^{3/2} \left (8 \left (a d^2+b c^2\right )^2-b c d x \left (11 a d^2+6 b c^2\right )\right )}{4 d^2}}{6 d^2}+\frac {\left (a+b x^2\right )^{5/2} \left (6 \left (a d^2+b c^2\right )-5 b c d x\right )}{30 d^2}}{c}\) |
Input:
Int[(a + b*x^2)^(7/2)/(x*(c + d*x)),x]
Output:
-((((6*(b*c^2 + a*d^2) - 5*b*c*d*x)*(a + b*x^2)^(5/2))/(30*d^2) + (((8*(b* c^2 + a*d^2)^2 - b*c*d*(6*b*c^2 + 11*a*d^2)*x)*(a + b*x^2)^(3/2))/(4*d^2) + (3*(((16*(b*c^2 + a*d^2)^3 - b*c*d*(8*b^2*c^4 + 22*a*b*c^2*d^2 + 19*a^2* d^4)*x)*Sqrt[a + b*x^2])/(2*d^2) + (-((Sqrt[b]*c*(16*b^3*c^6 + 56*a*b^2*c^ 4*d^2 + 70*a^2*b*c^2*d^4 + 35*a^3*d^6)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2] ])/d) - (16*(b*c^2 + a*d^2)^(7/2)*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 + a*d^ 2]*Sqrt[a + b*x^2])])/d)/(2*d^2)))/(4*d^2))/(6*d^2))/c) + (a*((2*(a + b*x^ 2)^(5/2))/5 + a*((2*(a + b*x^2)^(3/2))/3 + a*(2*Sqrt[a + b*x^2] - 2*Sqrt[a ]*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]]))))/(2*c)
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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
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[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, x]
Int[(((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_))/(x_), x_Symbol] : > Simp[a/c Int[(c + d*x)^(n + 1)*((a + b*x^2)^(p - 1)/x), x], x] - Simp[1 /c Int[(c + d*x)^n*(a*d - b*c*x)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[p, 0] && ILtQ[n, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))) Int[(d + e*x) ^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x ], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || ! RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1354\) vs. \(2(322)=644\).
Time = 0.38 (sec) , antiderivative size = 1355, normalized size of antiderivative = 3.72
Input:
int((b*x^2+a)^(7/2)/x/(d*x+c),x,method=_RETURNVERBOSE)
Output:
1/c*(1/7*(b*x^2+a)^(7/2)+a*(1/5*(b*x^2+a)^(5/2)+a*(1/3*(b*x^2+a)^(3/2)+a*( (b*x^2+a)^(1/2)-a^(1/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)))))-1/c*(1/7 *(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(7/2)-b*c/d*(1/12*(2*b*(x +c/d)-2*b*c/d)/b*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(5/2)+5/2 4*(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^2)/b*(1/8*(2*b*(x+c/d)-2*b*c/d)/b*(b* (x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(3/2)+3/16*(4*b*(a*d^2+b*c^2) /d^2-4*b^2*c^2/d^2)/b*(1/4*(2*b*(x+c/d)-2*b*c/d)/b*(b*(x+c/d)^2-2*b*c/d*(x +c/d)+(a*d^2+b*c^2)/d^2)^(1/2)+1/8*(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^2)/b ^(3/2)*ln((-b*c/d+b*(x+c/d))/b^(1/2)+(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b *c^2)/d^2)^(1/2)))))+(a*d^2+b*c^2)/d^2*(1/5*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+( a*d^2+b*c^2)/d^2)^(5/2)-b*c/d*(1/8*(2*b*(x+c/d)-2*b*c/d)/b*(b*(x+c/d)^2-2* b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(3/2)+3/16*(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c ^2/d^2)/b*(1/4*(2*b*(x+c/d)-2*b*c/d)/b*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2 +b*c^2)/d^2)^(1/2)+1/8*(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^2)/b^(3/2)*ln((- b*c/d+b*(x+c/d))/b^(1/2)+(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^( 1/2))))+(a*d^2+b*c^2)/d^2*(1/3*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/ d^2)^(3/2)-b*c/d*(1/4*(2*b*(x+c/d)-2*b*c/d)/b*(b*(x+c/d)^2-2*b*c/d*(x+c/d) +(a*d^2+b*c^2)/d^2)^(1/2)+1/8*(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^2)/b^(3/2 )*ln((-b*c/d+b*(x+c/d))/b^(1/2)+(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2) /d^2)^(1/2)))+(a*d^2+b*c^2)/d^2*((b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*...
Timed out. \[ \int \frac {\left (a+b x^2\right )^{7/2}}{x (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((b*x^2+a)^(7/2)/x/(d*x+c),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\left (a+b x^2\right )^{7/2}}{x (c+d x)} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {7}{2}}}{x \left (c + d x\right )}\, dx \] Input:
integrate((b*x**2+a)**(7/2)/x/(d*x+c),x)
Output:
Integral((a + b*x**2)**(7/2)/(x*(c + d*x)), x)
Time = 0.13 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.13 \[ \int \frac {\left (a+b x^2\right )^{7/2}}{x (c+d x)} \, dx=\frac {{\left (\frac {120 \, \sqrt {b x^{2} + a} b^{3} c^{5} x}{d^{6}} + \frac {60 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2} c^{3} x}{d^{4}} + \frac {330 \, \sqrt {b x^{2} + a} a b^{2} c^{3} x}{d^{4}} + \frac {40 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b c x}{d^{2}} + \frac {110 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a b c x}{d^{2}} + \frac {285 \, \sqrt {b x^{2} + a} a^{2} b c x}{d^{2}} + \frac {240 \, b^{\frac {7}{2}} c^{7} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{d^{8}} + \frac {840 \, a b^{\frac {5}{2}} c^{5} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{d^{6}} + \frac {1050 \, a^{2} b^{\frac {3}{2}} c^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{d^{4}} + \frac {525 \, a^{3} \sqrt {b} c \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{d^{2}} - \frac {240 \, {\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {7}{2}} \operatorname {arsinh}\left (\frac {2 \, b c x}{\sqrt {a b} {\left | 2 \, d x + 2 \, c \right |}} - \frac {2 \, a d}{\sqrt {a b} {\left | 2 \, d x + 2 \, c \right |}}\right )}{d} - \frac {240 \, a^{\frac {7}{2}} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{d} - \frac {240 \, \sqrt {b x^{2} + a} b^{3} c^{6}}{d^{7}} - \frac {80 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2} c^{4}}{d^{5}} - \frac {720 \, \sqrt {b x^{2} + a} a b^{2} c^{4}}{d^{5}} - \frac {48 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b c^{2}}{d^{3}} - \frac {160 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a b c^{2}}{d^{3}} - \frac {720 \, \sqrt {b x^{2} + a} a^{2} b c^{2}}{d^{3}}\right )} d}{240 \, c} \] Input:
integrate((b*x^2+a)^(7/2)/x/(d*x+c),x, algorithm="maxima")
Output:
1/240*(120*sqrt(b*x^2 + a)*b^3*c^5*x/d^6 + 60*(b*x^2 + a)^(3/2)*b^2*c^3*x/ d^4 + 330*sqrt(b*x^2 + a)*a*b^2*c^3*x/d^4 + 40*(b*x^2 + a)^(5/2)*b*c*x/d^2 + 110*(b*x^2 + a)^(3/2)*a*b*c*x/d^2 + 285*sqrt(b*x^2 + a)*a^2*b*c*x/d^2 + 240*b^(7/2)*c^7*arcsinh(b*x/sqrt(a*b))/d^8 + 840*a*b^(5/2)*c^5*arcsinh(b* x/sqrt(a*b))/d^6 + 1050*a^2*b^(3/2)*c^3*arcsinh(b*x/sqrt(a*b))/d^4 + 525*a ^3*sqrt(b)*c*arcsinh(b*x/sqrt(a*b))/d^2 - 240*(a + b*c^2/d^2)^(7/2)*arcsin h(2*b*c*x/(sqrt(a*b)*abs(2*d*x + 2*c)) - 2*a*d/(sqrt(a*b)*abs(2*d*x + 2*c) ))/d - 240*a^(7/2)*arcsinh(a/(sqrt(a*b)*abs(x)))/d - 240*sqrt(b*x^2 + a)*b ^3*c^6/d^7 - 80*(b*x^2 + a)^(3/2)*b^2*c^4/d^5 - 720*sqrt(b*x^2 + a)*a*b^2* c^4/d^5 - 48*(b*x^2 + a)^(5/2)*b*c^2/d^3 - 160*(b*x^2 + a)^(3/2)*a*b*c^2/d ^3 - 720*sqrt(b*x^2 + a)*a^2*b*c^2/d^3)*d/c
Exception generated. \[ \int \frac {\left (a+b x^2\right )^{7/2}}{x (c+d x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((b*x^2+a)^(7/2)/x/(d*x+c),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E rror: Bad Argument Value
Timed out. \[ \int \frac {\left (a+b x^2\right )^{7/2}}{x (c+d x)} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{7/2}}{x\,\left (c+d\,x\right )} \,d x \] Input:
int((a + b*x^2)^(7/2)/(x*(c + d*x)),x)
Output:
int((a + b*x^2)^(7/2)/(x*(c + d*x)), x)
Time = 0.21 (sec) , antiderivative size = 844, normalized size of antiderivative = 2.32 \[ \int \frac {\left (a+b x^2\right )^{7/2}}{x (c+d x)} \, dx =\text {Too large to display} \] Input:
int((b*x^2+a)^(7/2)/x/(d*x+c),x)
Output:
(480*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**3*d**6 + 1440*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x** 2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**2*b*c**2*d**4 + 1440*sqrt(a*d** 2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a *b**2*c**4*d**2 + 480*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a *d**2 + b*c**2) - a*d + b*c*x)*b**3*c**6 - 480*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**3*d**6 - 1440*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**2*b*c**2*d* *4 - 1440*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b**2*c**4*d**2 - 480*sqrt(a *d**2 + b*c**2)*log(c + d*x)*b**3*c**6 - 1856*sqrt(a + b*x**2)*a**2*b*c**2 *d**5 + 870*sqrt(a + b*x**2)*a**2*b*c*d**6*x - 1600*sqrt(a + b*x**2)*a*b** 2*c**4*d**3 + 780*sqrt(a + b*x**2)*a*b**2*c**3*d**4*x - 512*sqrt(a + b*x** 2)*a*b**2*c**2*d**5*x**2 + 380*sqrt(a + b*x**2)*a*b**2*c*d**6*x**3 - 480*s qrt(a + b*x**2)*b**3*c**6*d + 240*sqrt(a + b*x**2)*b**3*c**5*d**2*x - 160* sqrt(a + b*x**2)*b**3*c**4*d**3*x**2 + 120*sqrt(a + b*x**2)*b**3*c**3*d**4 *x**3 - 96*sqrt(a + b*x**2)*b**3*c**2*d**5*x**4 + 80*sqrt(a + b*x**2)*b**3 *c*d**6*x**5 + 240*sqrt(a)*log(sqrt(a + b*x**2) - sqrt(a))*a**3*d**7 - 240 *sqrt(a)*log(sqrt(a + b*x**2) + sqrt(a))*a**3*d**7 - 525*sqrt(b)*log(sqrt( a + b*x**2) - sqrt(b)*x)*a**3*c*d**6 - 1050*sqrt(b)*log(sqrt(a + b*x**2) - sqrt(b)*x)*a**2*b*c**3*d**4 - 840*sqrt(b)*log(sqrt(a + b*x**2) - sqrt(b)* x)*a*b**2*c**5*d**2 - 240*sqrt(b)*log(sqrt(a + b*x**2) - sqrt(b)*x)*b**...