Integrand size = 25, antiderivative size = 496 \[ \int \frac {x^5}{(c+d x)^{3/2} \left (a-b x^2\right )^{3/2}} \, dx=\frac {a^2 (c-d x)}{b^2 \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {2 c \left (b^2 c^4+a^2 d^4\right ) \sqrt {a-b x^2}}{b^2 d^2 \left (b c^2-a d^2\right )^2 \sqrt {c+d x}}+\frac {2 \sqrt {c+d x} \sqrt {a-b x^2}}{3 b^2 d^2}-\frac {4 \sqrt {a} c \left (4 b^2 c^4-5 a b c^2 d^2+4 a^2 d^4\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 b^{3/2} d^3 \left (b c^2-a d^2\right )^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {\sqrt {a} \left (16 b^2 c^4-8 a b c^2 d^2-5 a^2 d^4\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 b^{5/2} d^3 \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
a^2*(-d*x+c)/b^2/(-a*d^2+b*c^2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)+2*c*(a^2*d^ 4+b^2*c^4)*(-b*x^2+a)^(1/2)/b^2/d^2/(-a*d^2+b*c^2)^2/(d*x+c)^(1/2)+2/3*(d* x+c)^(1/2)*(-b*x^2+a)^(1/2)/b^2/d^2-4/3*a^(1/2)*c*(4*a^2*d^4-5*a*b*c^2*d^2 +4*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^(3/ 2)/d^3/(-a*d^2+b*c^2)^2/(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)/(-b* x^2+a)^(1/2)+1/3*a^(1/2)*(-5*a^2*d^4-8*a*b*c^2*d^2+16*b^2*c^4)*(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^(5/2)/d^3/(-a*d^2+b*c^2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 24.44 (sec) , antiderivative size = 677, normalized size of antiderivative = 1.36 \[ \int \frac {x^5}{(c+d x)^{3/2} \left (a-b x^2\right )^{3/2}} \, dx=\frac {\sqrt {a-b x^2} \left (3 (c+d x) \left (\frac {2}{3 b^2 d^2}+\frac {2 c^5}{\left (b c^2 d-a d^3\right )^2 (c+d x)}-\frac {a^2 \left (a d^2+b c (c-2 d x)\right )}{b^2 \left (b c^2-a d^2\right )^2 \left (-a+b x^2\right )}\right )+\frac {4 c d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (4 b^2 c^4-5 a b c^2 d^2+4 a^2 d^4\right ) \left (a-b x^2\right )+4 i \sqrt {b} c \left (4 b^{5/2} c^5-4 \sqrt {a} b^2 c^4 d-5 a b^{3/2} c^3 d^2+5 a^{3/2} b c^2 d^3+4 a^2 \sqrt {b} c d^4-4 a^{5/2} d^5\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )+i \sqrt {a} d \left (16 b^{5/2} c^5-4 \sqrt {a} b^2 c^4 d-20 a b^{3/2} c^3 d^2-13 a^{3/2} b c^2 d^3+16 a^2 \sqrt {b} c d^4+5 a^{5/2} d^5\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{b^2 d^4 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (b c^2-a d^2\right )^2 \left (-a+b x^2\right )}\right )}{3 \sqrt {c+d x}} \] Input:
Integrate[x^5/((c + d*x)^(3/2)*(a - b*x^2)^(3/2)),x]
Output:
(Sqrt[a - b*x^2]*(3*(c + d*x)*(2/(3*b^2*d^2) + (2*c^5)/((b*c^2*d - a*d^3)^ 2*(c + d*x)) - (a^2*(a*d^2 + b*c*(c - 2*d*x)))/(b^2*(b*c^2 - a*d^2)^2*(-a + b*x^2))) + (4*c*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(4*b^2*c^4 - 5*a*b*c^ 2*d^2 + 4*a^2*d^4)*(a - b*x^2) + (4*I)*Sqrt[b]*c*(4*b^(5/2)*c^5 - 4*Sqrt[a ]*b^2*c^4*d - 5*a*b^(3/2)*c^3*d^2 + 5*a^(3/2)*b*c^2*d^3 + 4*a^2*Sqrt[b]*c* d^4 - 4*a^(5/2)*d^5)*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sq rt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticE[I*ArcSinh[Sq rt[-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]*d*(16*b^(5/2)*c^5 - 4*Sqrt[a]*b^2*c^4*d - 20*a*b^(3/2)*c^3*d^2 - 13*a^(3/2)*b*c^2*d^3 + 16*a^2*Sqrt[b]*c*d^4 + 5*a^( 5/2)*d^5)*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sq rt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-c + (Sq rt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqr t[a]*d)])/(b^2*d^4*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(b*c^2 - a*d^2)^2*(-a + b*x^2))))/(3*Sqrt[c + d*x])
Time = 1.28 (sec) , antiderivative size = 526, normalized size of antiderivative = 1.06, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {602, 27, 2182, 27, 2185, 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.
| \(\displaystyle \int \frac {x^5}{\left (a-b x^2\right )^{3/2} (c+d x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 602 |
| \(\displaystyle \frac {\int \frac {\frac {3 c d a^3}{b^2}-\frac {\left (2 b c^2-a d^2\right ) x a^2}{b^2}-2 \left (c^2-\frac {a d^2}{b}\right ) x^3 a}{2 (c+d x)^{3/2} \sqrt {a-b x^2}}dx}{a \left (b c^2-a d^2\right )}+\frac {a^2 (c-d x)}{b^2 \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\frac {3 c d a^3}{b^2}-\frac {\left (2 b c^2-a d^2\right ) x a^2}{b^2}-2 \left (c^2-\frac {a d^2}{b}\right ) x^3 a}{(c+d x)^{3/2} \sqrt {a-b x^2}}dx}{2 a \left (b c^2-a d^2\right )}+\frac {a^2 (c-d x)}{b^2 \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 2182 |
| \(\displaystyle \frac {\frac {2 \int \frac {\left (\frac {2 c^4}{d}+\frac {3 a d c^2}{b}-\frac {a^2 d^3}{b^2}\right ) a^2-\frac {2 \left (b c^2-a d^2\right )^2 x^2 a}{b d}-4 c \left (-\frac {b c^4}{d^2}+a c^2-\frac {a^2 d^2}{b}\right ) x a}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{b c^2-a d^2}+\frac {4 a c d \sqrt {a-b x^2} \left (\frac {a^2 d}{b^2}+\frac {c^4}{d^3}\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}+\frac {a^2 (c-d x)}{b^2 \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\left (\frac {2 c^4}{d}+\frac {3 a d c^2}{b}-\frac {a^2 d^3}{b^2}\right ) a^2-\frac {2 \left (b c^2-a d^2\right )^2 x^2 a}{b d}-4 c \left (-\frac {b c^4}{d^2}+a c^2-\frac {a^2 d^2}{b}\right ) x a}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{b c^2-a d^2}+\frac {4 a c d \sqrt {a-b x^2} \left (\frac {a^2 d}{b^2}+\frac {c^4}{d^3}\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}+\frac {a^2 (c-d x)}{b^2 \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {\frac {\frac {4 a \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )^2}{3 b^2 d^2}-\frac {2 \int -\frac {a \left (a d \left (4 b c^4+13 a d^2 c^2-\frac {5 a^2 d^4}{b}\right )+4 c \left (4 b^2 c^4-5 a b d^2 c^2+4 a^2 d^4\right ) x\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b d^2}}{b c^2-a d^2}+\frac {4 a c d \sqrt {a-b x^2} \left (\frac {a^2 d}{b^2}+\frac {c^4}{d^3}\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}+\frac {a^2 (c-d x)}{b^2 \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {a \int \frac {a d \left (4 b c^4+13 a d^2 c^2-\frac {5 a^2 d^4}{b}\right )+4 c \left (4 b^2 c^4-5 a b d^2 c^2+4 a^2 d^4\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b d^2}+\frac {4 a \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )^2}{3 b^2 d^2}}{b c^2-a d^2}+\frac {4 a c d \sqrt {a-b x^2} \left (\frac {a^2 d}{b^2}+\frac {c^4}{d^3}\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}+\frac {a^2 (c-d x)}{b^2 \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {\frac {\frac {a \left (\frac {4 c \left (4 a^2 d^4-5 a b c^2 d^2+4 b^2 c^4\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}-\frac {\left (b c^2-a d^2\right ) \left (-5 a^2 d^4-8 a b c^2 d^2+16 b^2 c^4\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{b d}\right )}{3 b d^2}+\frac {4 a \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )^2}{3 b^2 d^2}}{b c^2-a d^2}+\frac {4 a c d \sqrt {a-b x^2} \left (\frac {a^2 d}{b^2}+\frac {c^4}{d^3}\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}+\frac {a^2 (c-d x)}{b^2 \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {\frac {\frac {a \left (\frac {4 c \sqrt {1-\frac {b x^2}{a}} \left (4 a^2 d^4-5 a b c^2 d^2+4 b^2 c^4\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {\left (b c^2-a d^2\right ) \left (-5 a^2 d^4-8 a b c^2 d^2+16 b^2 c^4\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{b d}\right )}{3 b d^2}+\frac {4 a \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )^2}{3 b^2 d^2}}{b c^2-a d^2}+\frac {4 a c d \sqrt {a-b x^2} \left (\frac {a^2 d}{b^2}+\frac {c^4}{d^3}\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}+\frac {a^2 (c-d x)}{b^2 \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {\frac {\frac {a \left (-\frac {\left (b c^2-a d^2\right ) \left (-5 a^2 d^4-8 a b c^2 d^2+16 b^2 c^4\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{b d}-\frac {8 \sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (4 a^2 d^4-5 a b c^2 d^2+4 b^2 c^4\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{3 b d^2}+\frac {4 a \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )^2}{3 b^2 d^2}}{b c^2-a d^2}+\frac {4 a c d \sqrt {a-b x^2} \left (\frac {a^2 d}{b^2}+\frac {c^4}{d^3}\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}+\frac {a^2 (c-d x)}{b^2 \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\frac {\frac {a \left (-\frac {\left (b c^2-a d^2\right ) \left (-5 a^2 d^4-8 a b c^2 d^2+16 b^2 c^4\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{b d}-\frac {8 \sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (4 a^2 d^4-5 a b c^2 d^2+4 b^2 c^4\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{3 b d^2}+\frac {4 a \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )^2}{3 b^2 d^2}}{b c^2-a d^2}+\frac {4 a c d \sqrt {a-b x^2} \left (\frac {a^2 d}{b^2}+\frac {c^4}{d^3}\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}+\frac {a^2 (c-d x)}{b^2 \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {\frac {\frac {a \left (-\frac {\sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (-5 a^2 d^4-8 a b c^2 d^2+16 b^2 c^4\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{b d \sqrt {a-b x^2}}-\frac {8 \sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (4 a^2 d^4-5 a b c^2 d^2+4 b^2 c^4\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{3 b d^2}+\frac {4 a \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )^2}{3 b^2 d^2}}{b c^2-a d^2}+\frac {4 a c d \sqrt {a-b x^2} \left (\frac {a^2 d}{b^2}+\frac {c^4}{d^3}\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}+\frac {a^2 (c-d x)}{b^2 \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {\frac {\frac {a \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (-5 a^2 d^4-8 a b c^2 d^2+16 b^2 c^4\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{b^{3/2} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {8 \sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (4 a^2 d^4-5 a b c^2 d^2+4 b^2 c^4\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{3 b d^2}+\frac {4 a \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )^2}{3 b^2 d^2}}{b c^2-a d^2}+\frac {4 a c d \sqrt {a-b x^2} \left (\frac {a^2 d}{b^2}+\frac {c^4}{d^3}\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}+\frac {a^2 (c-d x)}{b^2 \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\frac {\frac {a \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (-5 a^2 d^4-8 a b c^2 d^2+16 b^2 c^4\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{b^{3/2} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {8 \sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (4 a^2 d^4-5 a b c^2 d^2+4 b^2 c^4\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{3 b d^2}+\frac {4 a \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )^2}{3 b^2 d^2}}{b c^2-a d^2}+\frac {4 a c d \sqrt {a-b x^2} \left (\frac {a^2 d}{b^2}+\frac {c^4}{d^3}\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}+\frac {a^2 (c-d x)}{b^2 \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
Input:
Int[x^5/((c + d*x)^(3/2)*(a - b*x^2)^(3/2)),x]
Output:
(a^2*(c - d*x))/(b^2*(b*c^2 - a*d^2)*Sqrt[c + d*x]*Sqrt[a - b*x^2]) + ((4* a*c*d*(c^4/d^3 + (a^2*d)/b^2)*Sqrt[a - b*x^2])/((b*c^2 - a*d^2)*Sqrt[c + d *x]) + ((4*a*(b*c^2 - a*d^2)^2*Sqrt[c + d*x]*Sqrt[a - b*x^2])/(3*b^2*d^2) + (a*((-8*Sqrt[a]*c*(4*b^2*c^4 - 5*a*b*c^2*d^2 + 4*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]) + (2*Sqrt[a]*(b*c^2 - a*d^2)*(16* b^2*c^4 - 8*a*b*c^2*d^2 - 5*a^2*d^4)*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)/Sqr t[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(b^(3/2)*d*Sqrt[c + d*x] *Sqrt[a - b*x^2])))/(3*b*d^2))/(b*c^2 - a*d^2))/(2*a*(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_Symbo l] :> With[{Qx = PolynomialQuotient[x^m, a + b*x^2, x], e = Coeff[Polynomia lRemainder[x^m, a + b*x^2, x], x, 0], f = Coeff[PolynomialRemainder[x^m, a + b*x^2, x], x, 1]}, Simp[(-(c + d*x)^(n + 1))*(a + b*x^2)^(p + 1)*((a*(d*e - c*f) + (b*c*e + a*d*f)*x)/(2*a*(p + 1)*(b*c^2 + a*d^2))), x] + Simp[1/(2 *a*(p + 1)*(b*c^2 + a*d^2)) Int[(c + d*x)^n*(a + b*x^2)^(p + 1)*ExpandToS um[2*a*(p + 1)*(b*c^2 + a*d^2)*Qx + e*(b*c^2*(2*p + 3) + a*d^2*(n + 2*p + 3 )) - a*c*d*f*n + d*(b*c*e + a*d*f)*(n + 2*p + 4)*x, x], x], x]] /; FreeQ[{a , b, c, d, n}, x] && IGtQ[m, 1] && LtQ[p, -1] && NeQ[b*c^2 + a*d^2, 0]
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]
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) ^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si mp[1/(b*e^q*(m + q + 2*p + 1)) Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x )^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p )*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d , e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && !(EqQ[d, 0] && True) && !(IGtQ[m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Leaf count of result is larger than twice the leaf count of optimal. \(923\) vs. \(2(424)=848\).
Time = 10.15 (sec) , antiderivative size = 924, normalized size of antiderivative = 1.86
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (\frac {2 b d \left (-\frac {\left (a^{2} d^{4}+b^{2} c^{4}\right ) c \,x^{2}}{b^{2} d^{3} \left (a^{2} d^{4}-2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}+\frac {a^{2} x}{2 b^{3} \left (a \,d^{2}-b \,c^{2}\right )}+\frac {a c \left (a^{2} d^{4}+b \,c^{2} d^{2} a +2 b^{2} c^{4}\right )}{2 d^{3} b^{3} \left (a^{2} d^{4}-2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}\right )}{\sqrt {-\left (x^{3}+\frac {c \,x^{2}}{d}-\frac {a x}{b}-\frac {a c}{b d}\right ) b d}}+\frac {2 \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{3 d^{2} b^{2}}+\frac {2 \left (-\frac {a \,d^{2}+b \,c^{2}}{b^{2} d^{3}}+\frac {3 a^{3} d^{6}-a^{2} b \,c^{2} d^{4}+2 b^{3} c^{6}}{2 d^{3} b^{2} \left (a^{2} d^{4}-2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}-\frac {d \,a^{2}}{b^{2} \left (a \,d^{2}-b \,c^{2}\right )}-\frac {a}{3 b^{2} d}\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 {5 c}{3 d^{2} b}+\frac {c \left (a^{2} d^{4}+b^{2} c^{4}\right )}{d^{2} \left (a^{2} d^{4}-2 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) b}\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}}\) | \(924\) |
| risch | \(\text {Expression too large to display}\) | \(1728\) |
| default | \(\text {Expression too large to display}\) | \(1818\) |
Input:
int(x^5/(d*x+c)^(3/2)/(-b*x^2+a)^(3/2),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*b*d*(-(a^2*d^ 4+b^2*c^4)/b^2*c/d^3/(a^2*d^4-2*a*b*c^2*d^2+b^2*c^4)*x^2+1/2*a^2/b^3/(a*d^ 2-b*c^2)*x+1/2*a*c*(a^2*d^4+a*b*c^2*d^2+2*b^2*c^4)/d^3/b^3/(a^2*d^4-2*a*b* c^2*d^2+b^2*c^4))/(-(x^3+c/d*x^2-a*x/b-a/b*c/d)*b*d)^(1/2)+2/3/d^2/b^2*(-b *d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)+2*(-(a*d^2+b*c^2)/b^2/d^3+1/2*(3*a^3*d^6-a ^2*b*c^2*d^4+2*b^3*c^6)/d^3/b^2/(a^2*d^4-2*a*b*c^2*d^2+b^2*c^4)-1/b^2*d*a^ 2/(a*d^2-b*c^2)-1/3*a/b^2/d)*(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*(5/3*c/d^2/b+c*(a^2*d^4+b^2*c^4)/d^2/(a ^2*d^4-2*a*b*c^2*d^2+b^2*c^4)/b)*(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 = 0.10 (sec) , antiderivative size = 843, normalized size of antiderivative = 1.70 \[ \int \frac {x^5}{(c+d x)^{3/2} \left (a-b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
integrate(x^5/(d*x+c)^(3/2)/(-b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
1/9*((16*a*b^3*c^7 - 32*a^2*b^2*c^5*d^2 - 23*a^3*b*c^3*d^4 + 15*a^4*c*d^6 - (16*b^4*c^6*d - 32*a*b^3*c^4*d^3 - 23*a^2*b^2*c^2*d^5 + 15*a^3*b*d^7)*x^ 3 - (16*b^4*c^7 - 32*a*b^3*c^5*d^2 - 23*a^2*b^2*c^3*d^4 + 15*a^3*b*c*d^6)* x^2 + (16*a*b^3*c^6*d - 32*a^2*b^2*c^4*d^3 - 23*a^3*b*c^2*d^5 + 15*a^4*d^7 )*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) + 12*(4*a*b^3*c^6*d - 5*a^2 *b^2*c^4*d^3 + 4*a^3*b*c^2*d^5 - (4*b^4*c^5*d^2 - 5*a*b^3*c^3*d^4 + 4*a^2* b^2*c*d^6)*x^3 - (4*b^4*c^6*d - 5*a*b^3*c^4*d^3 + 4*a^2*b^2*c^2*d^5)*x^2 + (4*a*b^3*c^5*d^2 - 5*a^2*b^2*c^3*d^4 + 4*a^3*b*c*d^6)*x)*sqrt(-b*d)*weier strassZeta(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*(8*a*b^3*c^5*d^2 - a^2*b^2*c^3*d^4 + 5*a^3*b*c*d^6 - 2*(b^4*c^4*d^3 - 2*a*b^3*c^2*d^5 + a^2*b^2*d^7)*x^3 - 4* (2*b^4*c^5*d^2 - a*b^3*c^3*d^4 + 2*a^2*b^2*c*d^6)*x^2 + (2*a*b^3*c^4*d^3 - 7*a^2*b^2*c^2*d^5 + 5*a^3*b*d^7)*x)*sqrt(-b*x^2 + a)*sqrt(d*x + c))/(a*b^ 5*c^5*d^4 - 2*a^2*b^4*c^3*d^6 + a^3*b^3*c*d^8 - (b^6*c^4*d^5 - 2*a*b^5*c^2 *d^7 + a^2*b^4*d^9)*x^3 - (b^6*c^5*d^4 - 2*a*b^5*c^3*d^6 + a^2*b^4*c*d^8)* x^2 + (a*b^5*c^4*d^5 - 2*a^2*b^4*c^2*d^7 + a^3*b^3*d^9)*x)
\[ \int \frac {x^5}{(c+d x)^{3/2} \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {x^{5}}{\left (a - b x^{2}\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(x**5/(d*x+c)**(3/2)/(-b*x**2+a)**(3/2),x)
Output:
Integral(x**5/((a - b*x**2)**(3/2)*(c + d*x)**(3/2)), x)
\[ \int \frac {x^5}{(c+d x)^{3/2} \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {x^{5}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^5/(d*x+c)^(3/2)/(-b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
integrate(x^5/((-b*x^2 + a)^(3/2)*(d*x + c)^(3/2)), x)
\[ \int \frac {x^5}{(c+d x)^{3/2} \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {x^{5}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^5/(d*x+c)^(3/2)/(-b*x^2+a)^(3/2),x, algorithm="giac")
Output:
integrate(x^5/((-b*x^2 + a)^(3/2)*(d*x + c)^(3/2)), x)
Timed out. \[ \int \frac {x^5}{(c+d x)^{3/2} \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {x^5}{{\left (a-b\,x^2\right )}^{3/2}\,{\left (c+d\,x\right )}^{3/2}} \,d x \] Input:
int(x^5/((a - b*x^2)^(3/2)*(c + d*x)^(3/2)),x)
Output:
int(x^5/((a - b*x^2)^(3/2)*(c + d*x)^(3/2)), x)
\[ \int \frac {x^5}{(c+d x)^{3/2} \left (a-b x^2\right )^{3/2}} \, dx=\text {too large to display} \] Input:
int(x^5/(d*x+c)^(3/2)/(-b*x^2+a)^(3/2),x)
Output:
( - 20*sqrt(c + d*x)*sqrt(a - b*x**2)*a**2*c*d**2 - 10*sqrt(c + d*x)*sqrt( a - b*x**2)*a**2*d**3*x - 96*sqrt(c + d*x)*sqrt(a - b*x**2)*a*b*c**3 - 48* sqrt(c + d*x)*sqrt(a - b*x**2)*a*b*c**2*d*x + 32*sqrt(c + d*x)*sqrt(a - b* x**2)*b**2*c**3*x**2 - 8*sqrt(c + d*x)*sqrt(a - b*x**2)*b**2*c**2*d*x**3 + 5*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**3)/(a**2*c**2 + 2*a**2*c*d*x + a **2*d**2*x**2 - 2*a*b*c**2*x**2 - 4*a*b*c*d*x**3 - 2*a*b*d**2*x**4 + b**2* c**2*x**4 + 2*b**2*c*d*x**5 + b**2*d**2*x**6),x)*a**3*b*c*d**4 + 5*int((sq rt(c + d*x)*sqrt(a - b*x**2)*x**3)/(a**2*c**2 + 2*a**2*c*d*x + a**2*d**2*x **2 - 2*a*b*c**2*x**2 - 4*a*b*c*d*x**3 - 2*a*b*d**2*x**4 + b**2*c**2*x**4 + 2*b**2*c*d*x**5 + b**2*d**2*x**6),x)*a**3*b*d**5*x + 44*int((sqrt(c + d* x)*sqrt(a - b*x**2)*x**3)/(a**2*c**2 + 2*a**2*c*d*x + a**2*d**2*x**2 - 2*a *b*c**2*x**2 - 4*a*b*c*d*x**3 - 2*a*b*d**2*x**4 + b**2*c**2*x**4 + 2*b**2* c*d*x**5 + b**2*d**2*x**6),x)*a**2*b**2*c**3*d**2 + 44*int((sqrt(c + d*x)* sqrt(a - b*x**2)*x**3)/(a**2*c**2 + 2*a**2*c*d*x + a**2*d**2*x**2 - 2*a*b* c**2*x**2 - 4*a*b*c*d*x**3 - 2*a*b*d**2*x**4 + b**2*c**2*x**4 + 2*b**2*c*d *x**5 + b**2*d**2*x**6),x)*a**2*b**2*c**2*d**3*x - 5*int((sqrt(c + d*x)*sq rt(a - b*x**2)*x**3)/(a**2*c**2 + 2*a**2*c*d*x + a**2*d**2*x**2 - 2*a*b*c* *2*x**2 - 4*a*b*c*d*x**3 - 2*a*b*d**2*x**4 + b**2*c**2*x**4 + 2*b**2*c*d*x **5 + b**2*d**2*x**6),x)*a**2*b**2*c*d**4*x**2 - 5*int((sqrt(c + d*x)*sqrt (a - b*x**2)*x**3)/(a**2*c**2 + 2*a**2*c*d*x + a**2*d**2*x**2 - 2*a*b*c...