Integrand size = 25, antiderivative size = 518 \[ \int \frac {x^3 \sqrt {a-b x^2}}{(c+d x)^{5/2}} \, dx=-\frac {2 \sqrt {c+d x} \left (2 c \left (32 b c^2-7 a d^2\right ) \left (b c^2-a d^2\right )-3 d \left (b c^2-a d^2\right ) \left (16 b c^2-a d^2\right ) x\right ) \sqrt {a-b x^2}}{15 d^4 \left (b c^2-a d^2\right )^2}-\frac {2 c^3 \left (a-b x^2\right )^{3/2}}{3 d^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}+\frac {6 c^2 \left (a-b x^2\right )^{3/2}}{d^2 \left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {4 \sqrt {a} \left (64 b^2 c^4-62 a b c^2 d^2+3 a^2 d^4\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{15 \sqrt {b} d^5 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {8 \sqrt {a} c \left (32 b c^2-7 a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{15 \sqrt {b} d^5 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
-2/15*(d*x+c)^(1/2)*(2*c*(-7*a*d^2+32*b*c^2)*(-a*d^2+b*c^2)-3*d*(-a*d^2+b* c^2)*(-a*d^2+16*b*c^2)*x)*(-b*x^2+a)^(1/2)/d^4/(-a*d^2+b*c^2)^2-2/3*c^3*(- b*x^2+a)^(3/2)/d^2/(-a*d^2+b*c^2)/(d*x+c)^(3/2)+6*c^2*(-b*x^2+a)^(3/2)/d^2 /(-a*d^2+b*c^2)/(d*x+c)^(1/2)+4/15*a^(1/2)*(3*a^2*d^4-62*a*b*c^2*d^2+64*b^ 2*c^4)*(d*x+c)^(1/2)*(1-b*x^2/a)^(1/2)*EllipticE(1/2*(1-b^(1/2)*x/a^(1/2)) ^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1/2)*d))^(1/2))/b^(1/2)/d^ 5/(-a*d^2+b*c^2)/(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)/(-b*x^2+a)^ (1/2)-8/15*a^(1/2)*c*(-7*a*d^2+32*b*c^2)*(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/ 2)*d))^(1/2)*(1-b*x^2/a)^(1/2)*EllipticF(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2 ^(1/2),2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1/2)*d))^(1/2))/b^(1/2)/d^5/(d*x+c )^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 24.17 (sec) , antiderivative size = 638, normalized size of antiderivative = 1.23 \[ \int \frac {x^3 \sqrt {a-b x^2}}{(c+d x)^{5/2}} \, dx=\frac {\sqrt {a-b x^2} \left (\frac {2 (c+d x) \left (-14 c+3 d x+\frac {5 c^3}{(c+d x)^2}+\frac {5 c^2 \left (11 b c^2-9 a d^2\right )}{\left (-b c^2+a d^2\right ) (c+d x)}\right )}{d^4}-\frac {4 \left (d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (64 b^2 c^4-62 a b c^2 d^2+3 a^2 d^4\right ) \left (a-b x^2\right )+i \sqrt {b} \left (64 b^{5/2} c^5-64 \sqrt {a} b^2 c^4 d-62 a b^{3/2} c^3 d^2+62 a^{3/2} b c^2 d^3+3 a^2 \sqrt {b} c d^4-3 a^{5/2} d^5\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )+i \sqrt {a} \sqrt {b} d \left (64 b^2 c^4-16 \sqrt {a} b^{3/2} c^3 d-62 a b c^2 d^2+11 a^{3/2} \sqrt {b} c d^3+3 a^2 d^4\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )\right )}{b d^6 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (b c^2-a d^2\right ) \left (-a+b x^2\right )}\right )}{15 \sqrt {c+d x}} \] Input:
Integrate[(x^3*Sqrt[a - b*x^2])/(c + d*x)^(5/2),x]
Output:
(Sqrt[a - b*x^2]*((2*(c + d*x)*(-14*c + 3*d*x + (5*c^3)/(c + d*x)^2 + (5*c ^2*(11*b*c^2 - 9*a*d^2))/((-(b*c^2) + a*d^2)*(c + d*x))))/d^4 - (4*(d^2*Sq rt[-c + (Sqrt[a]*d)/Sqrt[b]]*(64*b^2*c^4 - 62*a*b*c^2*d^2 + 3*a^2*d^4)*(a - b*x^2) + I*Sqrt[b]*(64*b^(5/2)*c^5 - 64*Sqrt[a]*b^2*c^4*d - 62*a*b^(3/2) *c^3*d^2 + 62*a^(3/2)*b*c^2*d^3 + 3*a^2*Sqrt[b]*c*d^4 - 3*a^(5/2)*d^5)*Sqr t[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/ (c + d*x))]*(c + d*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt [b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)] + I* Sqrt[a]*Sqrt[b]*d*(64*b^2*c^4 - 16*Sqrt[a]*b^(3/2)*c^3*d - 62*a*b*c^2*d^2 + 11*a^(3/2)*Sqrt[b]*c*d^3 + 3*a^2*d^4)*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*Ell ipticF[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)]))/(b*d^6*Sqrt[-c + (Sqrt[a]*d)/Sqrt [b]]*(b*c^2 - a*d^2)*(-a + b*x^2))))/(15*Sqrt[c + d*x])
Time = 1.03 (sec) , antiderivative size = 532, normalized size of antiderivative = 1.03, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {603, 27, 2182, 27, 682, 27, 600, 509, 508, 327, 512, 511, 321}
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.
| \(\displaystyle \int \frac {x^3 \sqrt {a-b x^2}}{(c+d x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 603 |
| \(\displaystyle \frac {2 \int -\frac {3 \sqrt {a-b x^2} \left (\frac {a c^2}{d}-\left (a-\frac {2 b c^2}{d^2}\right ) x c-\left (\frac {b c^2}{d}-a d\right ) x^2\right )}{2 (c+d x)^{3/2}}dx}{3 \left (b c^2-a d^2\right )}-\frac {2 c^3 \left (a-b x^2\right )^{3/2}}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {\sqrt {a-b x^2} \left (\frac {a c^2}{d}-\left (a-\frac {2 b c^2}{d^2}\right ) x c-\left (\frac {b c^2}{d}-a d\right ) x^2\right )}{(c+d x)^{3/2}}dx}{b c^2-a d^2}-\frac {2 c^3 \left (a-b x^2\right )^{3/2}}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 2182 |
| \(\displaystyle -\frac {\frac {2 \int -\frac {\left (2 a c \left (\frac {b c^2}{d}-a d\right )-\left (-\frac {16 b^2 c^4}{d^2}+17 a b c^2-a^2 d^2\right ) x\right ) \sqrt {a-b x^2}}{2 \sqrt {c+d x}}dx}{b c^2-a d^2}-\frac {6 c^2 \left (a-b x^2\right )^{3/2}}{d^2 \sqrt {c+d x}}}{b c^2-a d^2}-\frac {2 c^3 \left (a-b x^2\right )^{3/2}}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\int \frac {\left (2 a c \left (\frac {b c^2}{d}-a d\right )-\left (-\frac {16 b^2 c^4}{d^2}+17 a b c^2-a^2 d^2\right ) x\right ) \sqrt {a-b x^2}}{\sqrt {c+d x}}dx}{b c^2-a d^2}-\frac {6 c^2 \left (a-b x^2\right )^{3/2}}{d^2 \sqrt {c+d x}}}{b c^2-a d^2}-\frac {2 c^3 \left (a-b x^2\right )^{3/2}}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle -\frac {-\frac {\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (2 c \left (-7 a^2 d^2+39 a b c^2-\frac {32 b^2 c^4}{d^2}\right )-3 d x \left (-a^2 d^2+17 a b c^2-\frac {16 b^2 c^4}{d^2}\right )\right )}{15 d^2}-\frac {4 \int \frac {b \left (a c d \left (16 b c^2-11 a d^2\right ) \left (b c^2-a d^2\right )+\left (64 b^2 c^4-62 a b d^2 c^2+3 a^2 d^4\right ) x \left (b c^2-a d^2\right )\right )}{2 d^2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{15 b d^2}}{b c^2-a d^2}-\frac {6 c^2 \left (a-b x^2\right )^{3/2}}{d^2 \sqrt {c+d x}}}{b c^2-a d^2}-\frac {2 c^3 \left (a-b x^2\right )^{3/2}}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (2 c \left (-7 a^2 d^2+39 a b c^2-\frac {32 b^2 c^4}{d^2}\right )-3 d x \left (-a^2 d^2+17 a b c^2-\frac {16 b^2 c^4}{d^2}\right )\right )}{15 d^2}-\frac {2 \int \frac {a c d \left (16 b c^2-11 a d^2\right ) \left (b c^2-a d^2\right )+\left (64 b^2 c^4-62 a b d^2 c^2+3 a^2 d^4\right ) x \left (b c^2-a d^2\right )}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{15 d^4}}{b c^2-a d^2}-\frac {6 c^2 \left (a-b x^2\right )^{3/2}}{d^2 \sqrt {c+d x}}}{b c^2-a d^2}-\frac {2 c^3 \left (a-b x^2\right )^{3/2}}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle -\frac {-\frac {\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (2 c \left (-7 a^2 d^2+39 a b c^2-\frac {32 b^2 c^4}{d^2}\right )-3 d x \left (-a^2 d^2+17 a b c^2-\frac {16 b^2 c^4}{d^2}\right )\right )}{15 d^2}-\frac {2 \left (\frac {\left (b c^2-a d^2\right ) \left (3 a^2 d^4-62 a b c^2 d^2+64 b^2 c^4\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}-\frac {2 c \left (32 b c^2-7 a d^2\right ) \left (b c^2-a d^2\right )^2 \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{15 d^4}}{b c^2-a d^2}-\frac {6 c^2 \left (a-b x^2\right )^{3/2}}{d^2 \sqrt {c+d x}}}{b c^2-a d^2}-\frac {2 c^3 \left (a-b x^2\right )^{3/2}}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle -\frac {-\frac {\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (2 c \left (-7 a^2 d^2+39 a b c^2-\frac {32 b^2 c^4}{d^2}\right )-3 d x \left (-a^2 d^2+17 a b c^2-\frac {16 b^2 c^4}{d^2}\right )\right )}{15 d^2}-\frac {2 \left (\frac {\sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (3 a^2 d^4-62 a b c^2 d^2+64 b^2 c^4\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {2 c \left (32 b c^2-7 a d^2\right ) \left (b c^2-a d^2\right )^2 \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{15 d^4}}{b c^2-a d^2}-\frac {6 c^2 \left (a-b x^2\right )^{3/2}}{d^2 \sqrt {c+d x}}}{b c^2-a d^2}-\frac {2 c^3 \left (a-b x^2\right )^{3/2}}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle -\frac {-\frac {\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (2 c \left (-7 a^2 d^2+39 a b c^2-\frac {32 b^2 c^4}{d^2}\right )-3 d x \left (-a^2 d^2+17 a b c^2-\frac {16 b^2 c^4}{d^2}\right )\right )}{15 d^2}-\frac {2 \left (-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a^2 d^4-62 a b c^2 d^2+64 b^2 c^4\right ) \left (b c^2-a d^2\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {2 c \left (32 b c^2-7 a d^2\right ) \left (b c^2-a d^2\right )^2 \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{15 d^4}}{b c^2-a d^2}-\frac {6 c^2 \left (a-b x^2\right )^{3/2}}{d^2 \sqrt {c+d x}}}{b c^2-a d^2}-\frac {2 c^3 \left (a-b x^2\right )^{3/2}}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle -\frac {-\frac {\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (2 c \left (-7 a^2 d^2+39 a b c^2-\frac {32 b^2 c^4}{d^2}\right )-3 d x \left (-a^2 d^2+17 a b c^2-\frac {16 b^2 c^4}{d^2}\right )\right )}{15 d^2}-\frac {2 \left (-\frac {2 c \left (32 b c^2-7 a d^2\right ) \left (b c^2-a d^2\right )^2 \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a^2 d^4-62 a b c^2 d^2+64 b^2 c^4\right ) \left (b c^2-a d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^4}}{b c^2-a d^2}-\frac {6 c^2 \left (a-b x^2\right )^{3/2}}{d^2 \sqrt {c+d x}}}{b c^2-a d^2}-\frac {2 c^3 \left (a-b x^2\right )^{3/2}}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle -\frac {-\frac {\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (2 c \left (-7 a^2 d^2+39 a b c^2-\frac {32 b^2 c^4}{d^2}\right )-3 d x \left (-a^2 d^2+17 a b c^2-\frac {16 b^2 c^4}{d^2}\right )\right )}{15 d^2}-\frac {2 \left (-\frac {2 c \sqrt {1-\frac {b x^2}{a}} \left (32 b c^2-7 a d^2\right ) \left (b c^2-a d^2\right )^2 \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a^2 d^4-62 a b c^2 d^2+64 b^2 c^4\right ) \left (b c^2-a d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^4}}{b c^2-a d^2}-\frac {6 c^2 \left (a-b x^2\right )^{3/2}}{d^2 \sqrt {c+d x}}}{b c^2-a d^2}-\frac {2 c^3 \left (a-b x^2\right )^{3/2}}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle -\frac {-\frac {\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (2 c \left (-7 a^2 d^2+39 a b c^2-\frac {32 b^2 c^4}{d^2}\right )-3 d x \left (-a^2 d^2+17 a b c^2-\frac {16 b^2 c^4}{d^2}\right )\right )}{15 d^2}-\frac {2 \left (\frac {4 \sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \left (32 b c^2-7 a d^2\right ) \left (b c^2-a d^2\right )^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (b c^2-a d^2\right ) \left (3 a^2 d^4-62 a b c^2 d^2+64 b^2 c^4\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^4}}{b c^2-a d^2}-\frac {6 c^2 \left (a-b x^2\right )^{3/2}}{d^2 \sqrt {c+d x}}}{b c^2-a d^2}-\frac {2 c^3 \left (a-b x^2\right )^{3/2}}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle -\frac {-\frac {\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (2 c \left (-7 a^2 d^2+39 a b c^2-\frac {32 b^2 c^4}{d^2}\right )-3 d x \left (-a^2 d^2+17 a b c^2-\frac {16 b^2 c^4}{d^2}\right )\right )}{15 d^2}-\frac {2 \left (\frac {4 \sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \left (32 b c^2-7 a d^2\right ) \left (b c^2-a d^2\right )^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (b c^2-a d^2\right ) \left (3 a^2 d^4-62 a b c^2 d^2+64 b^2 c^4\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^4}}{b c^2-a d^2}-\frac {6 c^2 \left (a-b x^2\right )^{3/2}}{d^2 \sqrt {c+d x}}}{b c^2-a d^2}-\frac {2 c^3 \left (a-b x^2\right )^{3/2}}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
Input:
Int[(x^3*Sqrt[a - b*x^2])/(c + d*x)^(5/2),x]
Output:
(-2*c^3*(a - b*x^2)^(3/2))/(3*d^2*(b*c^2 - a*d^2)*(c + d*x)^(3/2)) - ((-6* c^2*(a - b*x^2)^(3/2))/(d^2*Sqrt[c + d*x]) - ((2*Sqrt[c + d*x]*(2*c*(39*a* b*c^2 - (32*b^2*c^4)/d^2 - 7*a^2*d^2) - 3*d*(17*a*b*c^2 - (16*b^2*c^4)/d^2 - a^2*d^2)*x)*Sqrt[a - b*x^2])/(15*d^2) - (2*((-2*Sqrt[a]*(b*c^2 - a*d^2) *(64*b^2*c^4 - 62*a*b*c^2*d^2 + 3*a^2*d^4)*Sqrt[c + d*x]*Sqrt[1 - (b*x^2)/ a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b ]*c)/Sqrt[a] + d)])/(Sqrt[b]*d*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[ a]*d)]*Sqrt[a - b*x^2]) + (4*Sqrt[a]*c*(32*b*c^2 - 7*a*d^2)*(b*c^2 - a*d^2 )^2*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 - (b*x^2)/a]* EllipticF[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c )/Sqrt[a] + d)])/(Sqrt[b]*d*Sqrt[c + d*x]*Sqrt[a - b*x^2])))/(15*d^4))/(b* c^2 - a*d^2))/(b*c^2 - a*d^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c *q))])) Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[Sq rt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[Sqrt[c + d*x]/Sqrt[1 + b*(x^2/a)], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt [c + d*x])) Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] , x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim p[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[1/(Sqrt[c + d*x]*Sqrt[1 + b*(x^ 2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] ), x_Symbol] :> Simp[B/d Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp [(B*c - A*d)/d Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, b, c, d, A, B}, x] && NegQ[b/a]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol ] :> With[{Qx = PolynomialQuotient[x^m, c + d*x, x], R = PolynomialRemainde r[x^m, c + d*x, x]}, Simp[d*R*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/((n + 1)*(b*c^2 + a*d^2))), x] + Simp[1/((n + 1)*(b*c^2 + a*d^2)) Int[(c + d*x) ^(n + 1)*(a + b*x^2)^p*ExpandToSum[(n + 1)*(b*c^2 + a*d^2)*Qx + b*c*R*(n + 1) - b*d*R*(n + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, p}, x] && IGt Q[m, 1] && LtQ[n, -1] && NeQ[b*c^2 + a*d^2, 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[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[e*R*(d + e*x)^(m + 1)*((a + b*x^2)^(p + 1)/((m + 1)*(b* d^2 + a*e^2))), x] + Simp[1/((m + 1)*(b*d^2 + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[(m + 1)*(b*d^2 + a*e^2)*Qx + b*d*R*(m + 1) - b *e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && LtQ[m, -1]
Time = 6.88 (sec) , antiderivative size = 844, normalized size of antiderivative = 1.63
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (\frac {2 c^{3} \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{3 d^{6} \left (x +\frac {c}{d}\right )^{2}}-\frac {2 \left (-b d \,x^{2}+a d \right ) c^{2} \left (9 a \,d^{2}-11 b \,c^{2}\right )}{3 d^{5} \left (a \,d^{2}-b \,c^{2}\right ) \sqrt {\left (x +\frac {c}{d}\right ) \left (-b d \,x^{2}+a d \right )}}+\frac {2 x \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{5 d^{3}}-\frac {28 c \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{15 d^{4}}+\frac {2 \left (-\frac {2 c \left (a \,d^{2}-2 b \,c^{2}\right )}{d^{5}}-\frac {b \,c^{3}}{3 d^{5}}-\frac {b \,c^{3} \left (9 a \,d^{2}-11 b \,c^{2}\right )}{3 d^{5} \left (a \,d^{2}-b \,c^{2}\right )}+\frac {8 a c}{15 d^{3}}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {2 \left (\frac {a \,d^{2}-3 b \,c^{2}}{d^{4}}-\frac {b \,c^{2} \left (9 a \,d^{2}-11 b \,c^{2}\right )}{3 d^{4} \left (a \,d^{2}-b \,c^{2}\right )}-\frac {3 a}{5 d^{2}}-\frac {28 b \,c^{2}}{15 d^{4}}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \left (\left (-\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{b}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\) | \(844\) |
| risch | \(\text {Expression too large to display}\) | \(1745\) |
| default | \(\text {Expression too large to display}\) | \(3114\) |
Input:
int(x^3*(-b*x^2+a)^(1/2)/(d*x+c)^(5/2),x,method=_RETURNVERBOSE)
Output:
((d*x+c)*(-b*x^2+a))^(1/2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)*(2/3/d^6*c^3*(-b *d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)/(x+c/d)^2-2/3*(-b*d*x^2+a*d)/d^5/(a*d^2-b* c^2)*c^2*(9*a*d^2-11*b*c^2)/((x+c/d)*(-b*d*x^2+a*d))^(1/2)+2/5/d^3*x*(-b*d *x^3-b*c*x^2+a*d*x+a*c)^(1/2)-28/15*c/d^4*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/ 2)+2*(-2*c*(a*d^2-2*b*c^2)/d^5-1/3*b*c^3/d^5-1/3*b*c^3/d^5*(9*a*d^2-11*b*c ^2)/(a*d^2-b*c^2)+8/15/d^3*a*c)*(c/d-1/b*(a*b)^(1/2))*((x+c/d)/(c/d-1/b*(a *b)^(1/2)))^(1/2)*((x-1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)*((x+1 /b*(a*b)^(1/2))/(-c/d+1/b*(a*b)^(1/2)))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c) ^(1/2)*EllipticF(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),((-c/d+1/b*(a*b)^(1 /2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2))+2*(1/d^4*(a*d^2-3*b*c^2)-1/3*b/d^4*c^2 *(9*a*d^2-11*b*c^2)/(a*d^2-b*c^2)-3/5/d^2*a-28/15*b/d^4*c^2)*(c/d-1/b*(a*b )^(1/2))*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*((x-1/b*(a*b)^(1/2))/(-c/d- 1/b*(a*b)^(1/2)))^(1/2)*((x+1/b*(a*b)^(1/2))/(-c/d+1/b*(a*b)^(1/2)))^(1/2) /(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*((-c/d-1/b*(a*b)^(1/2))*EllipticE(((x+ c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^ (1/2)))^(1/2))+1/b*(a*b)^(1/2)*EllipticF(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^( 1/2),((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2))))
Time = 0.17 (sec) , antiderivative size = 545, normalized size of antiderivative = 1.05 \[ \int \frac {x^3 \sqrt {a-b x^2}}{(c+d x)^{5/2}} \, dx=-\frac {2 \, {\left (4 \, {\left (32 \, b^{2} c^{7} - 55 \, a b c^{5} d^{2} + 18 \, a^{2} c^{3} d^{4} + {\left (32 \, b^{2} c^{5} d^{2} - 55 \, a b c^{3} d^{4} + 18 \, a^{2} c d^{6}\right )} x^{2} + 2 \, {\left (32 \, b^{2} c^{6} d - 55 \, a b c^{4} d^{3} + 18 \, a^{2} c^{2} d^{5}\right )} x\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) + 6 \, {\left (64 \, b^{2} c^{6} d - 62 \, a b c^{4} d^{3} + 3 \, a^{2} c^{2} d^{5} + {\left (64 \, b^{2} c^{4} d^{3} - 62 \, a b c^{2} d^{5} + 3 \, a^{2} d^{7}\right )} x^{2} + 2 \, {\left (64 \, b^{2} c^{5} d^{2} - 62 \, a b c^{3} d^{4} + 3 \, a^{2} c d^{6}\right )} x\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) + 3 \, {\left (64 \, b^{2} c^{5} d^{2} - 54 \, a b c^{3} d^{4} - 3 \, {\left (b^{2} c^{2} d^{5} - a b d^{7}\right )} x^{3} + 8 \, {\left (b^{2} c^{3} d^{4} - a b c d^{6}\right )} x^{2} + 10 \, {\left (8 \, b^{2} c^{4} d^{3} - 7 \, a b c^{2} d^{5}\right )} x\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}\right )}}{45 \, {\left (b^{2} c^{4} d^{6} - a b c^{2} d^{8} + {\left (b^{2} c^{2} d^{8} - a b d^{10}\right )} x^{2} + 2 \, {\left (b^{2} c^{3} d^{7} - a b c d^{9}\right )} x\right )}} \] Input:
integrate(x^3*(-b*x^2+a)^(1/2)/(d*x+c)^(5/2),x, algorithm="fricas")
Output:
-2/45*(4*(32*b^2*c^7 - 55*a*b*c^5*d^2 + 18*a^2*c^3*d^4 + (32*b^2*c^5*d^2 - 55*a*b*c^3*d^4 + 18*a^2*c*d^6)*x^2 + 2*(32*b^2*c^6*d - 55*a*b*c^4*d^3 + 1 8*a^2*c^2*d^5)*x)*sqrt(-b*d)*weierstrassPInverse(4/3*(b*c^2 + 3*a*d^2)/(b* d^2), -8/27*(b*c^3 - 9*a*c*d^2)/(b*d^3), 1/3*(3*d*x + c)/d) + 6*(64*b^2*c^ 6*d - 62*a*b*c^4*d^3 + 3*a^2*c^2*d^5 + (64*b^2*c^4*d^3 - 62*a*b*c^2*d^5 + 3*a^2*d^7)*x^2 + 2*(64*b^2*c^5*d^2 - 62*a*b*c^3*d^4 + 3*a^2*c*d^6)*x)*sqrt (-b*d)*weierstrassZeta(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b*c^3 - 9*a*c *d^2)/(b*d^3), weierstrassPInverse(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b *c^3 - 9*a*c*d^2)/(b*d^3), 1/3*(3*d*x + c)/d)) + 3*(64*b^2*c^5*d^2 - 54*a* b*c^3*d^4 - 3*(b^2*c^2*d^5 - a*b*d^7)*x^3 + 8*(b^2*c^3*d^4 - a*b*c*d^6)*x^ 2 + 10*(8*b^2*c^4*d^3 - 7*a*b*c^2*d^5)*x)*sqrt(-b*x^2 + a)*sqrt(d*x + c))/ (b^2*c^4*d^6 - a*b*c^2*d^8 + (b^2*c^2*d^8 - a*b*d^10)*x^2 + 2*(b^2*c^3*d^7 - a*b*c*d^9)*x)
\[ \int \frac {x^3 \sqrt {a-b x^2}}{(c+d x)^{5/2}} \, dx=\int \frac {x^{3} \sqrt {a - b x^{2}}}{\left (c + d x\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(x**3*(-b*x**2+a)**(1/2)/(d*x+c)**(5/2),x)
Output:
Integral(x**3*sqrt(a - b*x**2)/(c + d*x)**(5/2), x)
\[ \int \frac {x^3 \sqrt {a-b x^2}}{(c+d x)^{5/2}} \, dx=\int { \frac {\sqrt {-b x^{2} + a} x^{3}}{{\left (d x + c\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(x^3*(-b*x^2+a)^(1/2)/(d*x+c)^(5/2),x, algorithm="maxima")
Output:
integrate(sqrt(-b*x^2 + a)*x^3/(d*x + c)^(5/2), x)
\[ \int \frac {x^3 \sqrt {a-b x^2}}{(c+d x)^{5/2}} \, dx=\int { \frac {\sqrt {-b x^{2} + a} x^{3}}{{\left (d x + c\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(x^3*(-b*x^2+a)^(1/2)/(d*x+c)^(5/2),x, algorithm="giac")
Output:
integrate(sqrt(-b*x^2 + a)*x^3/(d*x + c)^(5/2), x)
Timed out. \[ \int \frac {x^3 \sqrt {a-b x^2}}{(c+d x)^{5/2}} \, dx=\int \frac {x^3\,\sqrt {a-b\,x^2}}{{\left (c+d\,x\right )}^{5/2}} \,d x \] Input:
int((x^3*(a - b*x^2)^(1/2))/(c + d*x)^(5/2),x)
Output:
int((x^3*(a - b*x^2)^(1/2))/(c + d*x)^(5/2), x)
\[ \int \frac {x^3 \sqrt {a-b x^2}}{(c+d x)^{5/2}} \, dx=\text {too large to display} \] Input:
int(x^3*(-b*x^2+a)^(1/2)/(d*x+c)^(5/2),x)
Output:
(6*sqrt(c + d*x)*sqrt(a - b*x**2)*a**2*d**3 - 80*sqrt(c + d*x)*sqrt(a - b* x**2)*a*b*c**2*d - 12*sqrt(c + d*x)*sqrt(a - b*x**2)*a*b*c*d**2*x + 96*sqr t(c + d*x)*sqrt(a - b*x**2)*b**2*c**3*x - 16*sqrt(c + d*x)*sqrt(a - b*x**2 )*b**2*c**2*d*x**2 + 6*sqrt(c + d*x)*sqrt(a - b*x**2)*b**2*c*d**2*x**3 - 3 *int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**2)/(a*c**3 + 3*a*c**2*d*x + 3*a*c* d**2*x**2 + a*d**3*x**3 - b*c**3*x**2 - 3*b*c**2*d*x**3 - 3*b*c*d**2*x**4 - b*d**3*x**5),x)*a**2*b*c**2*d**4 - 6*int((sqrt(c + d*x)*sqrt(a - b*x**2) *x**2)/(a*c**3 + 3*a*c**2*d*x + 3*a*c*d**2*x**2 + a*d**3*x**3 - b*c**3*x** 2 - 3*b*c**2*d*x**3 - 3*b*c*d**2*x**4 - b*d**3*x**5),x)*a**2*b*c*d**5*x - 3*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**2)/(a*c**3 + 3*a*c**2*d*x + 3*a*c *d**2*x**2 + a*d**3*x**3 - b*c**3*x**2 - 3*b*c**2*d*x**3 - 3*b*c*d**2*x**4 - b*d**3*x**5),x)*a**2*b*d**6*x**2 + 6*int((sqrt(c + d*x)*sqrt(a - b*x**2 )*x**2)/(a*c**3 + 3*a*c**2*d*x + 3*a*c*d**2*x**2 + a*d**3*x**3 - b*c**3*x* *2 - 3*b*c**2*d*x**3 - 3*b*c*d**2*x**4 - b*d**3*x**5),x)*a*b**2*c**4*d**2 + 12*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**2)/(a*c**3 + 3*a*c**2*d*x + 3* a*c*d**2*x**2 + a*d**3*x**3 - b*c**3*x**2 - 3*b*c**2*d*x**3 - 3*b*c*d**2*x **4 - b*d**3*x**5),x)*a*b**2*c**3*d**3*x + 6*int((sqrt(c + d*x)*sqrt(a - b *x**2)*x**2)/(a*c**3 + 3*a*c**2*d*x + 3*a*c*d**2*x**2 + a*d**3*x**3 - b*c* *3*x**2 - 3*b*c**2*d*x**3 - 3*b*c*d**2*x**4 - b*d**3*x**5),x)*a*b**2*c**2* d**4*x**2 + 192*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**2)/(a*c**3 + 3*a...