Integrand size = 25, antiderivative size = 553 \[ \int \frac {x^4}{(c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=-\frac {a (a d-b c x)}{b^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2} \sqrt {a-b x^2}}-\frac {\left (2 b^2 c^4+3 a b c^2 d^2+3 a^2 d^4\right ) \sqrt {a-b x^2}}{3 b^2 d \left (b c^2-a d^2\right )^2 (c+d x)^{3/2}}+\frac {c \left (4 b^2 c^4-27 a b c^2 d^2-9 a^2 d^4\right ) \sqrt {a-b x^2}}{3 b d \left (b c^2-a d^2\right )^3 \sqrt {c+d x}}-\frac {\sqrt {a} c \left (4 b^2 c^4-27 a b c^2 d^2-9 a^2 d^4\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 \sqrt {b} d^2 \left (b c^2-a d^2\right )^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {\sqrt {a} \left (4 b^2 c^4-15 a b c^2 d^2+3 a^2 d^4\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 b^{3/2} d^2 \left (b c^2-a d^2\right )^2 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
-a*(-b*c*x+a*d)/b^2/(-a*d^2+b*c^2)/(d*x+c)^(3/2)/(-b*x^2+a)^(1/2)-1/3*(3*a ^2*d^4+3*a*b*c^2*d^2+2*b^2*c^4)*(-b*x^2+a)^(1/2)/b^2/d/(-a*d^2+b*c^2)^2/(d *x+c)^(3/2)+1/3*c*(-9*a^2*d^4-27*a*b*c^2*d^2+4*b^2*c^4)*(-b*x^2+a)^(1/2)/b /d/(-a*d^2+b*c^2)^3/(d*x+c)^(1/2)-1/3*a^(1/2)*c*(-9*a^2*d^4-27*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^(1/ 2)/d^2/(-a*d^2+b*c^2)^3/(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)*(3*a^2*d^4-15*a*b*c^2*d^2+4*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^(3/2)/d^2/(-a*d^2+b*c^2)^2/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 24.04 (sec) , antiderivative size = 622, normalized size of antiderivative = 1.12 \[ \int \frac {x^4}{(c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\frac {\sqrt {a-b x^2} \left (-4 b^2 c^5+27 a b c^3 d^2+9 a^2 c d^4+4 b c^3 \left (b c^2-6 a d^2\right )-\frac {2 b c^4 \left (b c^2-a d^2\right )}{c+d x}+\frac {3 a d (c+d x) \left (a^2 d^3-b^2 c^3 x+3 a b c d (c-d x)\right )}{-a+b x^2}-\frac {i b c \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (4 b^2 c^4-27 a b c^2 d^2-9 a^2 d^4\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d^2 \left (-a+b x^2\right )}+\frac {i \sqrt {a} \left (4 b^{5/2} c^5+8 \sqrt {a} b^2 c^4 d-27 a b^{3/2} c^3 d^2+27 a^{3/2} b c^2 d^3-9 a^2 \sqrt {b} c d^4-3 a^{5/2} d^5\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}\right )}{3 b d \left (b c^2-a d^2\right )^3 \sqrt {c+d x}} \] Input:
Integrate[x^4/((c + d*x)^(5/2)*(a - b*x^2)^(3/2)),x]
Output:
(Sqrt[a - b*x^2]*(-4*b^2*c^5 + 27*a*b*c^3*d^2 + 9*a^2*c*d^4 + 4*b*c^3*(b*c ^2 - 6*a*d^2) - (2*b*c^4*(b*c^2 - a*d^2))/(c + d*x) + (3*a*d*(c + d*x)*(a^ 2*d^3 - b^2*c^3*x + 3*a*b*c*d*(c - d*x)))/(-a + b*x^2) - (I*b*c*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(4*b^2*c^4 - 27*a*b*c^2*d^2 - 9*a^2*d^4)*Sqrt[(d*(Sqr t[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)])/(d^2*(-a + b *x^2)) + (I*Sqrt[a]*(4*b^(5/2)*c^5 + 8*Sqrt[a]*b^2*c^4*d - 27*a*b^(3/2)*c^ 3*d^2 + 27*a^(3/2)*b*c^2*d^3 - 9*a^2*Sqrt[b]*c*d^4 - 3*a^(5/2)*d^5)*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)*EllipticF[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)])/(d*Sqr t[-c + (Sqrt[a]*d)/Sqrt[b]]*(-a + b*x^2))))/(3*b*d*(b*c^2 - a*d^2)^3*Sqrt[ c + d*x])
Time = 1.06 (sec) , antiderivative size = 575, normalized size of antiderivative = 1.04, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.560, Rules used = {602, 27, 2182, 27, 27, 688, 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.
| \(\displaystyle \int \frac {x^4}{\left (a-b x^2\right )^{3/2} (c+d x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 602 |
| \(\displaystyle \frac {\int -\frac {\frac {\left (2 b c^2+3 a d^2\right ) a^2}{b^2}-\frac {3 c d x a^2}{b}+2 \left (c^2-\frac {a d^2}{b}\right ) x^2 a}{2 (c+d x)^{5/2} \sqrt {a-b x^2}}dx}{a \left (b c^2-a d^2\right )}-\frac {a (a d-b c x)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {\frac {\left (2 b c^2+3 a d^2\right ) a^2}{b^2}-\frac {3 c d x a^2}{b}+2 \left (c^2-\frac {a d^2}{b}\right ) x^2 a}{(c+d x)^{5/2} \sqrt {a-b x^2}}dx}{2 a \left (b c^2-a d^2\right )}-\frac {a (a d-b c x)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 2182 |
| \(\displaystyle -\frac {\frac {2 \int \frac {a \left (12 a c \left (b c^2+a d^2\right )+b \left (\frac {4 b c^4}{d}-15 a d c^2+\frac {3 a^2 d^3}{b}\right ) x\right )}{2 b (c+d x)^{3/2} \sqrt {a-b x^2}}dx}{3 \left (b c^2-a d^2\right )}+\frac {2 a d \sqrt {a-b x^2} \left (\frac {3 a^2 d^2}{b^2}+\frac {3 a c^2}{b}+\frac {2 c^4}{d^2}\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}-\frac {a (a d-b c x)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {a \int \frac {12 a c d \left (b c^2+a d^2\right )+\left (4 b^2 c^4-15 a b d^2 c^2+3 a^2 d^4\right ) x}{d (c+d x)^{3/2} \sqrt {a-b x^2}}dx}{3 b \left (b c^2-a d^2\right )}+\frac {2 a d \sqrt {a-b x^2} \left (\frac {3 a^2 d^2}{b^2}+\frac {3 a c^2}{b}+\frac {2 c^4}{d^2}\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}-\frac {a (a d-b c x)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {a \int \frac {12 a c d \left (b c^2+a d^2\right )+\left (4 b^2 c^4-15 a b d^2 c^2+3 a^2 d^4\right ) x}{(c+d x)^{3/2} \sqrt {a-b x^2}}dx}{3 b d \left (b c^2-a d^2\right )}+\frac {2 a d \sqrt {a-b x^2} \left (\frac {3 a^2 d^2}{b^2}+\frac {3 a c^2}{b}+\frac {2 c^4}{d^2}\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}-\frac {a (a d-b c x)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle -\frac {\frac {a \left (\frac {2 \int \frac {a d \left (8 b^2 c^4+27 a b d^2 c^2-3 a^2 d^4\right )-b c \left (4 b^2 c^4-27 a b d^2 c^2-9 a^2 d^4\right ) x}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{b c^2-a d^2}-\frac {2 c \sqrt {a-b x^2} \left (-9 a^2 d^4-27 a b c^2 d^2+4 b^2 c^4\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{3 b d \left (b c^2-a d^2\right )}+\frac {2 a d \sqrt {a-b x^2} \left (\frac {3 a^2 d^2}{b^2}+\frac {3 a c^2}{b}+\frac {2 c^4}{d^2}\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}-\frac {a (a d-b c x)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {a \left (\frac {\int \frac {a d \left (8 b^2 c^4+27 a b d^2 c^2-3 a^2 d^4\right )-b c \left (4 b^2 c^4-27 a b d^2 c^2-9 a^2 d^4\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{b c^2-a d^2}-\frac {2 c \sqrt {a-b x^2} \left (-9 a^2 d^4-27 a b c^2 d^2+4 b^2 c^4\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{3 b d \left (b c^2-a d^2\right )}+\frac {2 a d \sqrt {a-b x^2} \left (\frac {3 a^2 d^2}{b^2}+\frac {3 a c^2}{b}+\frac {2 c^4}{d^2}\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}-\frac {a (a d-b c x)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle -\frac {\frac {a \left (\frac {\frac {\left (b c^2-a d^2\right ) \left (3 a^2 d^4-15 a b c^2 d^2+4 b^2 c^4\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {b c \left (-9 a^2 d^4-27 a b c^2 d^2+4 b^2 c^4\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}}{b c^2-a d^2}-\frac {2 c \sqrt {a-b x^2} \left (-9 a^2 d^4-27 a b c^2 d^2+4 b^2 c^4\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{3 b d \left (b c^2-a d^2\right )}+\frac {2 a d \sqrt {a-b x^2} \left (\frac {3 a^2 d^2}{b^2}+\frac {3 a c^2}{b}+\frac {2 c^4}{d^2}\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}-\frac {a (a d-b c x)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle -\frac {\frac {a \left (\frac {\frac {\left (b c^2-a d^2\right ) \left (3 a^2 d^4-15 a b c^2 d^2+4 b^2 c^4\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {b c \sqrt {1-\frac {b x^2}{a}} \left (-9 a^2 d^4-27 a b c^2 d^2+4 b^2 c^4\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}}{b c^2-a d^2}-\frac {2 c \sqrt {a-b x^2} \left (-9 a^2 d^4-27 a b c^2 d^2+4 b^2 c^4\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{3 b d \left (b c^2-a d^2\right )}+\frac {2 a d \sqrt {a-b x^2} \left (\frac {3 a^2 d^2}{b^2}+\frac {3 a c^2}{b}+\frac {2 c^4}{d^2}\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}-\frac {a (a d-b c x)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle -\frac {\frac {a \left (\frac {\frac {\left (b c^2-a d^2\right ) \left (3 a^2 d^4-15 a b c^2 d^2+4 b^2 c^4\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}+\frac {2 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (-9 a^2 d^4-27 a b c^2 d^2+4 b^2 c^4\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{b c^2-a d^2}-\frac {2 c \sqrt {a-b x^2} \left (-9 a^2 d^4-27 a b c^2 d^2+4 b^2 c^4\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{3 b d \left (b c^2-a d^2\right )}+\frac {2 a d \sqrt {a-b x^2} \left (\frac {3 a^2 d^2}{b^2}+\frac {3 a c^2}{b}+\frac {2 c^4}{d^2}\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}-\frac {a (a d-b c x)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle -\frac {\frac {a \left (\frac {\frac {\left (b c^2-a d^2\right ) \left (3 a^2 d^4-15 a b c^2 d^2+4 b^2 c^4\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}+\frac {2 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (-9 a^2 d^4-27 a b c^2 d^2+4 b^2 c^4\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{b c^2-a d^2}-\frac {2 c \sqrt {a-b x^2} \left (-9 a^2 d^4-27 a b c^2 d^2+4 b^2 c^4\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{3 b d \left (b c^2-a d^2\right )}+\frac {2 a d \sqrt {a-b x^2} \left (\frac {3 a^2 d^2}{b^2}+\frac {3 a c^2}{b}+\frac {2 c^4}{d^2}\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}-\frac {a (a d-b c x)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle -\frac {\frac {a \left (\frac {\frac {\sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (3 a^2 d^4-15 a b c^2 d^2+4 b^2 c^4\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (-9 a^2 d^4-27 a b c^2 d^2+4 b^2 c^4\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{b c^2-a d^2}-\frac {2 c \sqrt {a-b x^2} \left (-9 a^2 d^4-27 a b c^2 d^2+4 b^2 c^4\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{3 b d \left (b c^2-a d^2\right )}+\frac {2 a d \sqrt {a-b x^2} \left (\frac {3 a^2 d^2}{b^2}+\frac {3 a c^2}{b}+\frac {2 c^4}{d^2}\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}-\frac {a (a d-b c x)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle -\frac {\frac {a \left (\frac {\frac {2 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (-9 a^2 d^4-27 a b c^2 d^2+4 b^2 c^4\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (3 a^2 d^4-15 a b c^2 d^2+4 b^2 c^4\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}}{b c^2-a d^2}-\frac {2 c \sqrt {a-b x^2} \left (-9 a^2 d^4-27 a b c^2 d^2+4 b^2 c^4\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{3 b d \left (b c^2-a d^2\right )}+\frac {2 a d \sqrt {a-b x^2} \left (\frac {3 a^2 d^2}{b^2}+\frac {3 a c^2}{b}+\frac {2 c^4}{d^2}\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}-\frac {a (a d-b c x)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle -\frac {\frac {a \left (\frac {\frac {2 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (-9 a^2 d^4-27 a b c^2 d^2+4 b^2 c^4\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (3 a^2 d^4-15 a b c^2 d^2+4 b^2 c^4\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}}{b c^2-a d^2}-\frac {2 c \sqrt {a-b x^2} \left (-9 a^2 d^4-27 a b c^2 d^2+4 b^2 c^4\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{3 b d \left (b c^2-a d^2\right )}+\frac {2 a d \sqrt {a-b x^2} \left (\frac {3 a^2 d^2}{b^2}+\frac {3 a c^2}{b}+\frac {2 c^4}{d^2}\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}-\frac {a (a d-b c x)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}\) |
Input:
Int[x^4/((c + d*x)^(5/2)*(a - b*x^2)^(3/2)),x]
Output:
-((a*(a*d - b*c*x))/(b^2*(b*c^2 - a*d^2)*(c + d*x)^(3/2)*Sqrt[a - b*x^2])) - ((2*a*d*((3*a*c^2)/b + (2*c^4)/d^2 + (3*a^2*d^2)/b^2)*Sqrt[a - b*x^2])/ (3*(b*c^2 - a*d^2)*(c + d*x)^(3/2)) + (a*((-2*c*(4*b^2*c^4 - 27*a*b*c^2*d^ 2 - 9*a^2*d^4)*Sqrt[a - b*x^2])/((b*c^2 - a*d^2)*Sqrt[c + d*x]) + ((2*Sqrt [a]*Sqrt[b]*c*(4*b^2*c^4 - 27*a*b*c^2*d^2 - 9*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)])/(d*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + S qrt[a]*d)]*Sqrt[a - b*x^2]) - (2*Sqrt[a]*(b*c^2 - a*d^2)*(4*b^2*c^4 - 15*a *b*c^2*d^2 + 3*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)/Sqrt[a]]/Sqrt[2] ], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(Sqrt[b]*d*Sqrt[c + d*x]*Sqrt[a - b*x ^2]))/(b*c^2 - a*d^2)))/(3*b*d*(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[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/( (m + 1)*(c*d^2 + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] && LtQ[m, -1] && (IntegerQ[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]
Leaf count of result is larger than twice the leaf count of optimal. \(997\) vs. \(2(479)=958\).
Time = 8.33 (sec) , antiderivative size = 998, normalized size of antiderivative = 1.80
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (-\frac {2 c^{4} \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{3 d^{3} \left (a \,d^{2}-b \,c^{2}\right )^{2} \left (x +\frac {c}{d}\right )^{2}}+\frac {4 \left (-b d \,x^{2}+a d \right ) c^{3} \left (6 a \,d^{2}-b \,c^{2}\right )}{3 d^{2} \left (a \,d^{2}-b \,c^{2}\right )^{3} \sqrt {\left (x +\frac {c}{d}\right ) \left (-b d \,x^{2}+a d \right )}}-\frac {2 \left (-b d x -b c \right ) \left (-\frac {c \left (3 a \,d^{2}+b \,c^{2}\right ) a x}{2 \left (a \,d^{2}-b \,c^{2}\right )^{3} b}+\frac {a^{2} d \left (a \,d^{2}+3 b \,c^{2}\right )}{2 \left (a \,d^{2}-b \,c^{2}\right )^{3} b^{2}}\right )}{\sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d x -b c \right )}}+\frac {2 \left (-\frac {1}{b \,d^{2}}+\frac {c^{4} b}{3 d^{2} \left (a \,d^{2}-b \,c^{2}\right )^{2}}+\frac {2 c^{4} b \left (6 a \,d^{2}-b \,c^{2}\right )}{3 d^{2} \left (a \,d^{2}-b \,c^{2}\right )^{3}}+\frac {\left (a \,d^{2}+b \,c^{2}\right ) a}{\left (a \,d^{2}-b \,c^{2}\right )^{2} b}-\frac {a^{2} d^{2} \left (a \,d^{2}+3 b \,c^{2}\right )}{2 b \left (a \,d^{2}-b \,c^{2}\right )^{3}}+\frac {c^{2} \left (3 a \,d^{2}+b \,c^{2}\right ) a}{\left (a \,d^{2}-b \,c^{2}\right )^{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 {2 c^{3} b \left (6 a \,d^{2}-b \,c^{2}\right )}{3 d \left (a \,d^{2}-b \,c^{2}\right )^{3}}+\frac {a c d \left (3 a \,d^{2}+b \,c^{2}\right )}{2 \left (a \,d^{2}-b \,c^{2}\right )^{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}}}\, \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}}\) | \(998\) |
| default | \(\text {Expression too large to display}\) | \(3132\) |
Input:
int(x^4/(d*x+c)^(5/2)/(-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)*(-2/3/d^3/(a*d^2 -b*c^2)^2*c^4*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)/(x+c/d)^2+4/3*(-b*d*x^2+a *d)/d^2/(a*d^2-b*c^2)^3*c^3*(6*a*d^2-b*c^2)/((x+c/d)*(-b*d*x^2+a*d))^(1/2) -2*(-b*d*x-b*c)*(-1/2*c*(3*a*d^2+b*c^2)*a/(a*d^2-b*c^2)^3/b*x+1/2*a^2*d*(a *d^2+3*b*c^2)/(a*d^2-b*c^2)^3/b^2)/((x^2-a/b)*(-b*d*x-b*c))^(1/2)+2*(-1/b/ d^2+1/3*c^4/d^2*b/(a*d^2-b*c^2)^2+2/3*c^4/d^2*b*(6*a*d^2-b*c^2)/(a*d^2-b*c ^2)^3+1/(a*d^2-b*c^2)^2/b*(a*d^2+b*c^2)*a-1/2*a^2*d^2*(a*d^2+3*b*c^2)/b/(a *d^2-b*c^2)^3+c^2*(3*a*d^2+b*c^2)*a/(a*d^2-b*c^2)^3)*(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*(2/3*c^3/d*b*(6 *a*d^2-b*c^2)/(a*d^2-b*c^2)^3+1/2*a*c*d*(3*a*d^2+b*c^2)/(a*d^2-b*c^2)^3)*( 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))*El lipticE(((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))))
Leaf count of result is larger than twice the leaf count of optimal. 1113 vs. \(2 (478) = 956\).
Time = 0.16 (sec) , antiderivative size = 1113, normalized size of antiderivative = 2.01 \[ \int \frac {x^4}{(c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
integrate(x^4/(d*x+c)^(5/2)/(-b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
1/9*((4*a*b^3*c^8 - 3*a^2*b^2*c^6*d^2 + 72*a^3*b*c^4*d^4 - 9*a^4*c^2*d^6 - (4*b^4*c^6*d^2 - 3*a*b^3*c^4*d^4 + 72*a^2*b^2*c^2*d^6 - 9*a^3*b*d^8)*x^4 - 2*(4*b^4*c^7*d - 3*a*b^3*c^5*d^3 + 72*a^2*b^2*c^3*d^5 - 9*a^3*b*c*d^7)*x ^3 - (4*b^4*c^8 - 7*a*b^3*c^6*d^2 + 75*a^2*b^2*c^4*d^4 - 81*a^3*b*c^2*d^6 + 9*a^4*d^8)*x^2 + 2*(4*a*b^3*c^7*d - 3*a^2*b^2*c^5*d^3 + 72*a^3*b*c^3*d^5 - 9*a^4*c*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) + 3*(4*a*b^3* c^7*d - 27*a^2*b^2*c^5*d^3 - 9*a^3*b*c^3*d^5 - (4*b^4*c^5*d^3 - 27*a*b^3*c ^3*d^5 - 9*a^2*b^2*c*d^7)*x^4 - 2*(4*b^4*c^6*d^2 - 27*a*b^3*c^4*d^4 - 9*a^ 2*b^2*c^2*d^6)*x^3 - (4*b^4*c^7*d - 31*a*b^3*c^5*d^3 + 18*a^2*b^2*c^3*d^5 + 9*a^3*b*c*d^7)*x^2 + 2*(4*a*b^3*c^6*d^2 - 27*a^2*b^2*c^4*d^4 - 9*a^3*b*c ^2*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*(2*a*b^3 *c^6*d^2 - 31*a^2*b^2*c^4*d^4 - 3*a^3*b*c^2*d^6 - (4*b^4*c^5*d^3 - 27*a*b^ 3*c^3*d^5 - 9*a^2*b^2*c*d^7)*x^3 - (2*b^4*c^6*d^2 - 28*a*b^3*c^4*d^4 - 9*a ^2*b^2*c^2*d^6 + 3*a^3*b*d^8)*x^2 + (7*a*b^3*c^5*d^3 - 33*a^2*b^2*c^3*d^5 - 6*a^3*b*c*d^7)*x)*sqrt(-b*x^2 + a)*sqrt(d*x + c))/(a*b^5*c^8*d^3 - 3*a^2 *b^4*c^6*d^5 + 3*a^3*b^3*c^4*d^7 - a^4*b^2*c^2*d^9 - (b^6*c^6*d^5 - 3*a*b^ 5*c^4*d^7 + 3*a^2*b^4*c^2*d^9 - a^3*b^3*d^11)*x^4 - 2*(b^6*c^7*d^4 - 3*...
\[ \int \frac {x^4}{(c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {x^{4}}{\left (a - b x^{2}\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(x**4/(d*x+c)**(5/2)/(-b*x**2+a)**(3/2),x)
Output:
Integral(x**4/((a - b*x**2)**(3/2)*(c + d*x)**(5/2)), x)
\[ \int \frac {x^4}{(c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {x^{4}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(x^4/(d*x+c)^(5/2)/(-b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
integrate(x^4/((-b*x^2 + a)^(3/2)*(d*x + c)^(5/2)), x)
\[ \int \frac {x^4}{(c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {x^{4}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(x^4/(d*x+c)^(5/2)/(-b*x^2+a)^(3/2),x, algorithm="giac")
Output:
integrate(x^4/((-b*x^2 + a)^(3/2)*(d*x + c)^(5/2)), x)
Timed out. \[ \int \frac {x^4}{(c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {x^4}{{\left (a-b\,x^2\right )}^{3/2}\,{\left (c+d\,x\right )}^{5/2}} \,d x \] Input:
int(x^4/((a - b*x^2)^(3/2)*(c + d*x)^(5/2)),x)
Output:
int(x^4/((a - b*x^2)^(3/2)*(c + d*x)^(5/2)), x)
\[ \int \frac {x^4}{(c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\text {too large to display} \] Input:
int(x^4/(d*x+c)^(5/2)/(-b*x^2+a)^(3/2),x)
Output:
( - 20*sqrt(c + d*x)*sqrt(a - b*x**2)*a*c - 30*sqrt(c + d*x)*sqrt(a - b*x* *2)*a*d*x + 12*sqrt(c + d*x)*sqrt(a - b*x**2)*b*c*x**2 + 45*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**3)/(a**2*c**3 + 3*a**2*c**2*d*x + 3*a**2*c*d**2*x **2 + a**2*d**3*x**3 - 2*a*b*c**3*x**2 - 6*a*b*c**2*d*x**3 - 6*a*b*c*d**2* x**4 - 2*a*b*d**3*x**5 + b**2*c**3*x**4 + 3*b**2*c**2*d*x**5 + 3*b**2*c*d* *2*x**6 + b**2*d**3*x**7),x)*a**2*b*c**2*d**2 + 90*int((sqrt(c + d*x)*sqrt (a - b*x**2)*x**3)/(a**2*c**3 + 3*a**2*c**2*d*x + 3*a**2*c*d**2*x**2 + a** 2*d**3*x**3 - 2*a*b*c**3*x**2 - 6*a*b*c**2*d*x**3 - 6*a*b*c*d**2*x**4 - 2* a*b*d**3*x**5 + b**2*c**3*x**4 + 3*b**2*c**2*d*x**5 + 3*b**2*c*d**2*x**6 + b**2*d**3*x**7),x)*a**2*b*c*d**3*x + 45*int((sqrt(c + d*x)*sqrt(a - b*x** 2)*x**3)/(a**2*c**3 + 3*a**2*c**2*d*x + 3*a**2*c*d**2*x**2 + a**2*d**3*x** 3 - 2*a*b*c**3*x**2 - 6*a*b*c**2*d*x**3 - 6*a*b*c*d**2*x**4 - 2*a*b*d**3*x **5 + b**2*c**3*x**4 + 3*b**2*c**2*d*x**5 + 3*b**2*c*d**2*x**6 + b**2*d**3 *x**7),x)*a**2*b*d**4*x**2 + 12*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**3)/ (a**2*c**3 + 3*a**2*c**2*d*x + 3*a**2*c*d**2*x**2 + a**2*d**3*x**3 - 2*a*b *c**3*x**2 - 6*a*b*c**2*d*x**3 - 6*a*b*c*d**2*x**4 - 2*a*b*d**3*x**5 + b** 2*c**3*x**4 + 3*b**2*c**2*d*x**5 + 3*b**2*c*d**2*x**6 + b**2*d**3*x**7),x) *a*b**2*c**4 + 24*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**3)/(a**2*c**3 + 3 *a**2*c**2*d*x + 3*a**2*c*d**2*x**2 + a**2*d**3*x**3 - 2*a*b*c**3*x**2 - 6 *a*b*c**2*d*x**3 - 6*a*b*c*d**2*x**4 - 2*a*b*d**3*x**5 + b**2*c**3*x**4...