Integrand size = 32, antiderivative size = 405 \[ \int \frac {x^8 \left (A+B x^2\right )}{\left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\frac {a x \left (A-\frac {a B d}{c^2}+\left (B-\frac {A d}{c}\right ) x^2\right )}{2 \left (c^3-a d^2\right ) \sqrt {a-c x^4}}+\frac {B x \sqrt {a-c x^4}}{3 c^2 d}+\frac {a^{3/4} (B c-A d) \left (2 c^3-3 a d^2\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 c^{7/4} d^2 \left (c^3-a d^2\right ) \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} \left (6 B c^6-6 A c^5 d+2 a B c^3 d^2+3 a A c^2 d^3-5 a^2 B d^4+3 \sqrt {a} \sqrt {c} d (B c-A d) \left (2 c^3-3 a d^2\right )\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{6 c^{9/4} d^3 \left (c^3-a d^2\right ) \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} c^{11/4} (B c-A d) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{d^3 \left (c^3-a d^2\right ) \sqrt {a-c x^4}} \] Output:
1/2*a*x*(A-a*B*d/c^2+(B-A*d/c)*x^2)/(-a*d^2+c^3)/(-c*x^4+a)^(1/2)+1/3*B*x* (-c*x^4+a)^(1/2)/c^2/d+1/2*a^(3/4)*(-A*d+B*c)*(-3*a*d^2+2*c^3)*(1-c*x^4/a) ^(1/2)*EllipticE(c^(1/4)*x/a^(1/4),I)/c^(7/4)/d^2/(-a*d^2+c^3)/(-c*x^4+a)^ (1/2)-1/6*a^(1/4)*(6*B*c^6-6*A*c^5*d+2*a*B*c^3*d^2+3*a*A*c^2*d^3-5*a^2*B*d ^4+3*a^(1/2)*c^(1/2)*d*(-A*d+B*c)*(-3*a*d^2+2*c^3))*(1-c*x^4/a)^(1/2)*Elli pticF(c^(1/4)*x/a^(1/4),I)/c^(9/4)/d^3/(-a*d^2+c^3)/(-c*x^4+a)^(1/2)+a^(1/ 4)*c^(11/4)*(-A*d+B*c)*(1-c*x^4/a)^(1/2)*EllipticPi(c^(1/4)*x/a^(1/4),-a^( 1/2)*d/c^(3/2),I)/d^3/(-a*d^2+c^3)/(-c*x^4+a)^(1/2)
Result contains complex when optimal does not.
Time = 11.77 (sec) , antiderivative size = 577, normalized size of antiderivative = 1.42 \[ \int \frac {x^8 \left (A+B x^2\right )}{\left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\frac {-2 a B \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^3 d^2 x-3 a A \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^2 d^3 x+5 a^2 B \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} d^4 x-3 a B \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^2 d^3 x^3+3 a A \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c d^4 x^3+2 B \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^4 d^2 x^5-2 a B \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c d^4 x^5-3 i \sqrt {a} \sqrt {c} d (B c-A d) \left (-2 c^3+3 a d^2\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+i \left (-c^{3/2}+\sqrt {a} d\right ) \left (-3 A \sqrt {c} d \left (2 c^3+4 \sqrt {a} c^{3/2} d+3 a d^2\right )+B \left (6 c^{9/2}+12 \sqrt {a} c^3 d+14 a c^{3/2} d^2+5 a^{3/2} d^3\right )\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+6 i B c^6 \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )-6 i A c^5 d \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )}{6 \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^2 d^3 \left (-c^3+a d^2\right ) \sqrt {a-c x^4}} \] Input:
Integrate[(x^8*(A + B*x^2))/((c + d*x^2)*(a - c*x^4)^(3/2)),x]
Output:
(-2*a*B*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^3*d^2*x - 3*a*A*Sqrt[-(Sqrt[c]/Sqrt[a]) ]*c^2*d^3*x + 5*a^2*B*Sqrt[-(Sqrt[c]/Sqrt[a])]*d^4*x - 3*a*B*Sqrt[-(Sqrt[c ]/Sqrt[a])]*c^2*d^3*x^3 + 3*a*A*Sqrt[-(Sqrt[c]/Sqrt[a])]*c*d^4*x^3 + 2*B*S qrt[-(Sqrt[c]/Sqrt[a])]*c^4*d^2*x^5 - 2*a*B*Sqrt[-(Sqrt[c]/Sqrt[a])]*c*d^4 *x^5 - (3*I)*Sqrt[a]*Sqrt[c]*d*(B*c - A*d)*(-2*c^3 + 3*a*d^2)*Sqrt[1 - (c* x^4)/a]*EllipticE[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + I*(-c^(3/2) + Sqrt[a]*d)*(-3*A*Sqrt[c]*d*(2*c^3 + 4*Sqrt[a]*c^(3/2)*d + 3*a*d^2) + B* (6*c^(9/2) + 12*Sqrt[a]*c^3*d + 14*a*c^(3/2)*d^2 + 5*a^(3/2)*d^3))*Sqrt[1 - (c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + (6*I)* B*c^6*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*d)/c^(3/2)), I*ArcSinh[Sqr t[-(Sqrt[c]/Sqrt[a])]*x], -1] - (6*I)*A*c^5*d*Sqrt[1 - (c*x^4)/a]*Elliptic Pi[-((Sqrt[a]*d)/c^(3/2)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1])/(6* Sqrt[-(Sqrt[c]/Sqrt[a])]*c^2*d^3*(-c^3 + a*d^2)*Sqrt[a - c*x^4])
Time = 0.99 (sec) , antiderivative size = 597, normalized size of antiderivative = 1.47, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2249, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^8 \left (A+B x^2\right )}{\left (a-c x^4\right )^{3/2} \left (c+d x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2249 |
| \(\displaystyle \int \left (\frac {a^2 \left (-a B d+c x^2 (B c-A d)+A c^2\right )}{c^2 \left (c^3-a d^2\right ) \left (a-c x^4\right )^{3/2}}-\frac {c^4 (B c-A d)}{d^3 \left (a d^2-c^3\right ) \sqrt {a-c x^4} \left (c+d x^2\right )}+\frac {A c^2 d-B \left (a d^2+c^3\right )}{c^2 d^3 \sqrt {a-c x^4}}+\frac {x^2 (B c-A d)}{c d^2 \sqrt {a-c x^4}}-\frac {B x^4}{c d \sqrt {a-c x^4}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^{3/4} \sqrt {1-\frac {c x^4}{a}} (B c-A d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{7/4} d^2 \sqrt {a-c x^4}}+\frac {a^{3/4} \sqrt {1-\frac {c x^4}{a}} (B c-A d) E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{7/4} d^2 \sqrt {a-c x^4}}+\frac {a^{5/4} \sqrt {1-\frac {c x^4}{a}} \left (\sqrt {a} B+A \sqrt {c}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{2 c^{9/4} \left (\sqrt {a} d+c^{3/2}\right ) \sqrt {a-c x^4}}-\frac {a^{7/4} \sqrt {1-\frac {c x^4}{a}} (B c-A d) E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 c^{7/4} \left (c^3-a d^2\right ) \sqrt {a-c x^4}}-\frac {a^{5/4} B \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{3 c^{9/4} d \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} c^{11/4} \sqrt {1-\frac {c x^4}{a}} (B c-A d) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{d^3 \left (c^3-a d^2\right ) \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (A c^2 d-B \left (a d^2+c^3\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{9/4} d^3 \sqrt {a-c x^4}}+\frac {a x \left (-a B d+c x^2 (B c-A d)+A c^2\right )}{2 c^2 \left (c^3-a d^2\right ) \sqrt {a-c x^4}}+\frac {B x \sqrt {a-c x^4}}{3 c^2 d}\) |
Input:
Int[(x^8*(A + B*x^2))/((c + d*x^2)*(a - c*x^4)^(3/2)),x]
Output:
(a*x*(A*c^2 - a*B*d + c*(B*c - A*d)*x^2))/(2*c^2*(c^3 - a*d^2)*Sqrt[a - c* x^4]) + (B*x*Sqrt[a - c*x^4])/(3*c^2*d) + (a^(3/4)*(B*c - A*d)*Sqrt[1 - (c *x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(7/4)*d^2*Sqrt[a - c*x^4]) - (a^(7/4)*(B*c - A*d)*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1 /4)*x)/a^(1/4)], -1])/(2*c^(7/4)*(c^3 - a*d^2)*Sqrt[a - c*x^4]) - (a^(5/4) *B*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(3*c^(9 /4)*d*Sqrt[a - c*x^4]) + (a^(5/4)*(Sqrt[a]*B + A*Sqrt[c])*Sqrt[1 - (c*x^4) /a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(2*c^(9/4)*(c^(3/2) + Sqrt [a]*d)*Sqrt[a - c*x^4]) - (a^(3/4)*(B*c - A*d)*Sqrt[1 - (c*x^4)/a]*Ellipti cF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(7/4)*d^2*Sqrt[a - c*x^4]) + (a^(1 /4)*(A*c^2*d - B*(c^3 + a*d^2))*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1 /4)*x)/a^(1/4)], -1])/(c^(9/4)*d^3*Sqrt[a - c*x^4]) + (a^(1/4)*c^(11/4)*(B *c - A*d)*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*d)/c^(3/2)), ArcSin[(c ^(1/4)*x)/a^(1/4)], -1])/(d^3*(c^3 - a*d^2)*Sqrt[a - c*x^4])
Int[(Px_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_) ^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[1/Sqrt[a + c*x^4], Px*(f*x)^m*(d + e*x^2)^q*(a + c*x^4)^(p + 1/2), x], x] /; FreeQ[{a, c, d, e, f, m}, x] & & PolyQ[Px, x] && IntegerQ[p + 1/2] && IntegerQ[q]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 703 vs. \(2 (349 ) = 698\).
Time = 12.37 (sec) , antiderivative size = 704, normalized size of antiderivative = 1.74
| method | result | size |
| risch | \(\frac {B x \sqrt {-c \,x^{4}+a}}{3 c^{2} d}-\frac {\frac {-\frac {3 \sqrt {c}\, d \left (A d -B c \right ) \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {3 B \,c^{3} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {3 A \,c^{2} d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {4 B a \,d^{2} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}}{d^{2}}+\frac {3 a^{2} d \left (\frac {2 c \left (-\frac {\left (A d -B c \right ) x^{3}}{4 a}+\frac {\left (A \,c^{2}-B a d \right ) x}{4 a c}\right )}{\sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {\left (A \,c^{2}-B a d \right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{2 a \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {\left (A d -B c \right ) \sqrt {c}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {a}\, \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )}{a \,d^{2}-c^{3}}-\frac {3 c^{5} \left (A d -B c \right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{d^{2} \left (a \,d^{2}-c^{3}\right ) \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}}{3 c^{2} d}\) | \(704\) |
| default | \(\text {Expression too large to display}\) | \(1099\) |
| elliptic | \(\text {Expression too large to display}\) | \(1327\) |
Input:
int(x^8*(B*x^2+A)/(d*x^2+c)/(-c*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/3*B*x*(-c*x^4+a)^(1/2)/c^2/d-1/3/c^2/d*(1/d^2*(-3*c^(1/2)*d*(A*d-B*c)*a^ (1/2)/(c^(1/2)/a^(1/2))^(1/2)*(1-c^(1/2)*x^2/a^(1/2))^(1/2)*(1+c^(1/2)*x^2 /a^(1/2))^(1/2)/(-c*x^4+a)^(1/2)*(EllipticF(x*(c^(1/2)/a^(1/2))^(1/2),I)-E llipticE(x*(c^(1/2)/a^(1/2))^(1/2),I))+3*B*c^3/(c^(1/2)/a^(1/2))^(1/2)*(1- c^(1/2)*x^2/a^(1/2))^(1/2)*(1+c^(1/2)*x^2/a^(1/2))^(1/2)/(-c*x^4+a)^(1/2)* EllipticF(x*(c^(1/2)/a^(1/2))^(1/2),I)-3*A*c^2*d/(c^(1/2)/a^(1/2))^(1/2)*( 1-c^(1/2)*x^2/a^(1/2))^(1/2)*(1+c^(1/2)*x^2/a^(1/2))^(1/2)/(-c*x^4+a)^(1/2 )*EllipticF(x*(c^(1/2)/a^(1/2))^(1/2),I)+4*B*a*d^2/(c^(1/2)/a^(1/2))^(1/2) *(1-c^(1/2)*x^2/a^(1/2))^(1/2)*(1+c^(1/2)*x^2/a^(1/2))^(1/2)/(-c*x^4+a)^(1 /2)*EllipticF(x*(c^(1/2)/a^(1/2))^(1/2),I))+3*a^2*d/(a*d^2-c^3)*(2*c*(-1/4 *(A*d-B*c)/a*x^3+1/4*(A*c^2-B*a*d)/a/c*x)/(-(x^4-a/c)*c)^(1/2)+1/2*(A*c^2- B*a*d)/a/(c^(1/2)/a^(1/2))^(1/2)*(1-c^(1/2)*x^2/a^(1/2))^(1/2)*(1+c^(1/2)* x^2/a^(1/2))^(1/2)/(-c*x^4+a)^(1/2)*EllipticF(x*(c^(1/2)/a^(1/2))^(1/2),I) -1/2*(A*d-B*c)*c^(1/2)/a^(1/2)/(c^(1/2)/a^(1/2))^(1/2)*(1-c^(1/2)*x^2/a^(1 /2))^(1/2)*(1+c^(1/2)*x^2/a^(1/2))^(1/2)/(-c*x^4+a)^(1/2)*(EllipticF(x*(c^ (1/2)/a^(1/2))^(1/2),I)-EllipticE(x*(c^(1/2)/a^(1/2))^(1/2),I)))-3*c^5*(A* d-B*c)/d^2/(a*d^2-c^3)/(c^(1/2)/a^(1/2))^(1/2)*(1-c^(1/2)*x^2/a^(1/2))^(1/ 2)*(1+c^(1/2)*x^2/a^(1/2))^(1/2)/(-c*x^4+a)^(1/2)*EllipticPi(x*(c^(1/2)/a^ (1/2))^(1/2),-a^(1/2)*d/c^(3/2),(-c^(1/2)/a^(1/2))^(1/2)/(c^(1/2)/a^(1/2)) ^(1/2)))
Timed out. \[ \int \frac {x^8 \left (A+B x^2\right )}{\left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(x^8*(B*x^2+A)/(d*x^2+c)/(-c*x^4+a)^(3/2),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {x^8 \left (A+B x^2\right )}{\left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(x**8*(B*x**2+A)/(d*x**2+c)/(-c*x**4+a)**(3/2),x)
Output:
Timed out
\[ \int \frac {x^8 \left (A+B x^2\right )}{\left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} x^{8}}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}} \,d x } \] Input:
integrate(x^8*(B*x^2+A)/(d*x^2+c)/(-c*x^4+a)^(3/2),x, algorithm="maxima")
Output:
integrate((B*x^2 + A)*x^8/((-c*x^4 + a)^(3/2)*(d*x^2 + c)), x)
\[ \int \frac {x^8 \left (A+B x^2\right )}{\left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} x^{8}}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}} \,d x } \] Input:
integrate(x^8*(B*x^2+A)/(d*x^2+c)/(-c*x^4+a)^(3/2),x, algorithm="giac")
Output:
integrate((B*x^2 + A)*x^8/((-c*x^4 + a)^(3/2)*(d*x^2 + c)), x)
Timed out. \[ \int \frac {x^8 \left (A+B x^2\right )}{\left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\int \frac {x^8\,\left (B\,x^2+A\right )}{{\left (a-c\,x^4\right )}^{3/2}\,\left (d\,x^2+c\right )} \,d x \] Input:
int((x^8*(A + B*x^2))/((a - c*x^4)^(3/2)*(c + d*x^2)),x)
Output:
int((x^8*(A + B*x^2))/((a - c*x^4)^(3/2)*(c + d*x^2)), x)
\[ \int \frac {x^8 \left (A+B x^2\right )}{\left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
int(x^8*(B*x^2+A)/(d*x^2+c)/(-c*x^4+a)^(3/2),x)
Output:
(9*sqrt(a - c*x**4)*a**2*d**2*x - 4*sqrt(a - c*x**4)*a*b*c*d*x - 3*sqrt(a - c*x**4)*a*c**2*d*x**3 + 3*sqrt(a - c*x**4)*b*c**3*x**3 - sqrt(a - c*x**4 )*b*c**2*d*x**5 - 9*int(sqrt(a - c*x**4)/(a**2*c + a**2*d*x**2 - 2*a*c**2* x**4 - 2*a*c*d*x**6 + c**3*x**8 + c**2*d*x**10),x)*a**4*c*d**2 + 4*int(sqr t(a - c*x**4)/(a**2*c + a**2*d*x**2 - 2*a*c**2*x**4 - 2*a*c*d*x**6 + c**3* x**8 + c**2*d*x**10),x)*a**3*b*c**2*d + 9*int(sqrt(a - c*x**4)/(a**2*c + a **2*d*x**2 - 2*a*c**2*x**4 - 2*a*c*d*x**6 + c**3*x**8 + c**2*d*x**10),x)*a **3*c**2*d**2*x**4 - 4*int(sqrt(a - c*x**4)/(a**2*c + a**2*d*x**2 - 2*a*c* *2*x**4 - 2*a*c*d*x**6 + c**3*x**8 + c**2*d*x**10),x)*a**2*b*c**3*d*x**4 - 9*int((sqrt(a - c*x**4)*x**6)/(a**2*c + a**2*d*x**2 - 2*a*c**2*x**4 - 2*a *c*d*x**6 + c**3*x**8 + c**2*d*x**10),x)*a**3*c*d**3 + 9*int((sqrt(a - c*x **4)*x**6)/(a**2*c + a**2*d*x**2 - 2*a*c**2*x**4 - 2*a*c*d*x**6 + c**3*x** 8 + c**2*d*x**10),x)*a**2*b*c**2*d**2 - 3*int((sqrt(a - c*x**4)*x**6)/(a** 2*c + a**2*d*x**2 - 2*a*c**2*x**4 - 2*a*c*d*x**6 + c**3*x**8 + c**2*d*x**1 0),x)*a**2*c**4*d + 9*int((sqrt(a - c*x**4)*x**6)/(a**2*c + a**2*d*x**2 - 2*a*c**2*x**4 - 2*a*c*d*x**6 + c**3*x**8 + c**2*d*x**10),x)*a**2*c**2*d**3 *x**4 + 3*int((sqrt(a - c*x**4)*x**6)/(a**2*c + a**2*d*x**2 - 2*a*c**2*x** 4 - 2*a*c*d*x**6 + c**3*x**8 + c**2*d*x**10),x)*a*b*c**5 - 9*int((sqrt(a - c*x**4)*x**6)/(a**2*c + a**2*d*x**2 - 2*a*c**2*x**4 - 2*a*c*d*x**6 + c**3 *x**8 + c**2*d*x**10),x)*a*b*c**3*d**2*x**4 + 3*int((sqrt(a - c*x**4)*x...