Integrand size = 32, antiderivative size = 367 \[ \int \frac {x^6 \left (A+B x^2\right )}{\left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\frac {x \left (a (B c-A d)+c^2 \left (A-\frac {a B d}{c^2}\right ) x^2\right )}{2 c \left (c^3-a d^2\right ) \sqrt {a-c x^4}}-\frac {a^{3/4} \left (2 B c^3+A c^2 d-3 a B d^2\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 c^{7/4} d \left (c^3-a d^2\right ) \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \left (2 B c^3+4 \sqrt {a} B c^{3/2} d-2 A c^2 d+3 a B d^2-\sqrt {a} A \sqrt {c} d^2\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{2 c^{7/4} d^2 \left (c^{3/2}+\sqrt {a} d\right ) \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} c^{7/4} (B c-A d) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{d^2 \left (c^3-a d^2\right ) \sqrt {a-c x^4}} \] Output:
1/2*x*(a*(-A*d+B*c)+c^2*(A-a*B*d/c^2)*x^2)/c/(-a*d^2+c^3)/(-c*x^4+a)^(1/2) -1/2*a^(3/4)*(A*c^2*d-3*B*a*d^2+2*B*c^3)*(1-c*x^4/a)^(1/2)*EllipticE(c^(1/ 4)*x/a^(1/4),I)/c^(7/4)/d/(-a*d^2+c^3)/(-c*x^4+a)^(1/2)+1/2*a^(1/4)*(2*B*c ^3+4*a^(1/2)*B*c^(3/2)*d-2*A*c^2*d+3*B*a*d^2-a^(1/2)*A*c^(1/2)*d^2)*(1-c*x ^4/a)^(1/2)*EllipticF(c^(1/4)*x/a^(1/4),I)/c^(7/4)/d^2/(c^(3/2)+a^(1/2)*d) /(-c*x^4+a)^(1/2)-a^(1/4)*c^(7/4)*(-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^2/(-a*d^2+c^3)/(-c*x^4+a)^(1/2)
Result contains complex when optimal does not.
Time = 11.62 (sec) , antiderivative size = 358, normalized size of antiderivative = 0.98 \[ \int \frac {x^6 \left (A+B x^2\right )}{\left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\frac {i \sqrt {a} d \left (-2 B c^3-A c^2 d+3 a B d^2\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )-i \left (-c^{3/2}+\sqrt {a} d\right ) \left (-A \sqrt {c} d \left (2 c^{3/2}+\sqrt {a} d\right )+B \left (2 c^3+4 \sqrt {a} c^{3/2} d+3 a d^2\right )\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+\sqrt {c} \left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} d^2 x \left (-A c^2 x^2+a \left (-B c+A d+B d x^2\right )\right )-2 i c^3 (B c-A d) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )\right )}{2 \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c^{3/2} d^2 \left (-c^3+a d^2\right ) \sqrt {a-c x^4}} \] Input:
Integrate[(x^6*(A + B*x^2))/((c + d*x^2)*(a - c*x^4)^(3/2)),x]
Output:
(I*Sqrt[a]*d*(-2*B*c^3 - A*c^2*d + 3*a*B*d^2)*Sqrt[1 - (c*x^4)/a]*Elliptic E[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] - I*(-c^(3/2) + Sqrt[a]*d)*(- (A*Sqrt[c]*d*(2*c^(3/2) + Sqrt[a]*d)) + B*(2*c^3 + 4*Sqrt[a]*c^(3/2)*d + 3 *a*d^2))*Sqrt[1 - (c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]* x], -1] + Sqrt[c]*(Sqrt[-(Sqrt[c]/Sqrt[a])]*d^2*x*(-(A*c^2*x^2) + a*(-(B*c ) + A*d + B*d*x^2)) - (2*I)*c^3*(B*c - A*d)*Sqrt[1 - (c*x^4)/a]*EllipticPi [-((Sqrt[a]*d)/c^(3/2)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1]))/(2*S qrt[-(Sqrt[c]/Sqrt[a])]*c^(3/2)*d^2*(-c^3 + a*d^2)*Sqrt[a - c*x^4])
Time = 0.84 (sec) , antiderivative size = 493, normalized size of antiderivative = 1.34, 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.
| \(\displaystyle \int \frac {x^6 \left (A+B x^2\right )}{\left (a-c x^4\right )^{3/2} \left (c+d x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2249 |
| \(\displaystyle \int \left (\frac {c^3 (B c-A d)}{d^2 \left (a d^2-c^3\right ) \sqrt {a-c x^4} \left (c+d x^2\right )}+\frac {a \left (x^2 \left (A c^2-a B d\right )+a (B c-A d)\right )}{c \left (c^3-a d^2\right ) \left (a-c x^4\right )^{3/2}}+\frac {B c-A d}{c d^2 \sqrt {a-c x^4}}-\frac {B x^2}{c d \sqrt {a-c x^4}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^{3/4} \sqrt {1-\frac {c x^4}{a}} \left (\sqrt {a} B+A \sqrt {c}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{2 c^{7/4} \left (\sqrt {a} d+c^{3/2}\right ) \sqrt {a-c x^4}}-\frac {a^{3/4} \sqrt {1-\frac {c x^4}{a}} \left (A c^2-a B d\right ) E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 c^{7/4} \left (c^3-a d^2\right ) \sqrt {a-c x^4}}+\frac {a^{3/4} B \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{7/4} d \sqrt {a-c x^4}}-\frac {a^{3/4} B \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{7/4} d \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} (B c-A d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{5/4} d^2 \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} c^{7/4} \sqrt {1-\frac {c x^4}{a}} (B c-A d) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{d^2 \left (c^3-a d^2\right ) \sqrt {a-c x^4}}+\frac {x \left (x^2 \left (A c^2-a B d\right )+a (B c-A d)\right )}{2 c \left (c^3-a d^2\right ) \sqrt {a-c x^4}}\) |
Input:
Int[(x^6*(A + B*x^2))/((c + d*x^2)*(a - c*x^4)^(3/2)),x]
Output:
(x*(a*(B*c - A*d) + (A*c^2 - a*B*d)*x^2))/(2*c*(c^3 - a*d^2)*Sqrt[a - c*x^ 4]) - (a^(3/4)*B*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)] , -1])/(c^(7/4)*d*Sqrt[a - c*x^4]) - (a^(3/4)*(A*c^2 - a*B*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^(3/4)*B*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^( 1/4)*x)/a^(1/4)], -1])/(c^(7/4)*d*Sqrt[a - c*x^4]) + (a^(3/4)*(Sqrt[a]*B + A*Sqrt[c])*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1] )/(2*c^(7/4)*(c^(3/2) + Sqrt[a]*d)*Sqrt[a - c*x^4]) + (a^(1/4)*(B*c - A*d) *Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(5/4)* d^2*Sqrt[a - c*x^4]) - (a^(1/4)*c^(7/4)*(B*c - A*d)*Sqrt[1 - (c*x^4)/a]*El lipticPi[-((Sqrt[a]*d)/c^(3/2)), ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(d^2*(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. 974 vs. \(2 (307 ) = 614\).
Time = 2.36 (sec) , antiderivative size = 975, normalized size of antiderivative = 2.66
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(975\) |
| elliptic | \(\text {Expression too large to display}\) | \(1087\) |
Input:
int(x^6*(B*x^2+A)/(d*x^2+c)/(-c*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/d^4*(d^2*(A*d-B*c)*(1/2/c*x/(-(x^4-a/c)*c)^(1/2)-1/2/c/(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))-c*d*(A*d-B*c)*(1/2/a*x^3 /(-(x^4-a/c)*c)^(1/2)+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)/c^(1/2)*(Elli pticF(x*(c^(1/2)/a^(1/2))^(1/2),I)-EllipticE(x*(c^(1/2)/a^(1/2))^(1/2),I)) )+A*c^2*d*(1/2/a*x/(-(x^4-a/c)*c)^(1/2)+1/2/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)*E llipticF(x*(c^(1/2)/a^(1/2))^(1/2),I))+B*d^3*(1/2/c*x^3/(-(x^4-a/c)*c)^(1/ 2)+3/2/c^(3/2)*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)))-B*c^3*(1/2/a*x/(- (x^4-a/c)*c)^(1/2)+1/2/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)))-c^3*(A*d-B*c)/d^4*(2*c*(1/4/a*d/(a*d^2-c^3)*x^3-1/4*c/a /(a*d^2-c^3)*x)/(-(x^4-a/c)*c)^(1/2)-1/2*c^2/a/(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)*EllipticF(x*(c^(1/2)/a^(1/2))^(1/2),I)+1/2*c^(1/2)/a^(1/2)*d/( 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)*EllipticF(x*(c^(1/2)/a^(1/2))^(1/...
Timed out. \[ \int \frac {x^6 \left (A+B x^2\right )}{\left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(x^6*(B*x^2+A)/(d*x^2+c)/(-c*x^4+a)^(3/2),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {x^6 \left (A+B x^2\right )}{\left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(x**6*(B*x**2+A)/(d*x**2+c)/(-c*x**4+a)**(3/2),x)
Output:
Timed out
\[ \int \frac {x^6 \left (A+B x^2\right )}{\left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} x^{6}}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}} \,d x } \] Input:
integrate(x^6*(B*x^2+A)/(d*x^2+c)/(-c*x^4+a)^(3/2),x, algorithm="maxima")
Output:
integrate((B*x^2 + A)*x^6/((-c*x^4 + a)^(3/2)*(d*x^2 + c)), x)
\[ \int \frac {x^6 \left (A+B x^2\right )}{\left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} x^{6}}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}} \,d x } \] Input:
integrate(x^6*(B*x^2+A)/(d*x^2+c)/(-c*x^4+a)^(3/2),x, algorithm="giac")
Output:
integrate((B*x^2 + A)*x^6/((-c*x^4 + a)^(3/2)*(d*x^2 + c)), x)
Timed out. \[ \int \frac {x^6 \left (A+B x^2\right )}{\left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\int \frac {x^6\,\left (B\,x^2+A\right )}{{\left (a-c\,x^4\right )}^{3/2}\,\left (d\,x^2+c\right )} \,d x \] Input:
int((x^6*(A + B*x^2))/((a - c*x^4)^(3/2)*(c + d*x^2)),x)
Output:
int((x^6*(A + B*x^2))/((a - c*x^4)^(3/2)*(c + d*x^2)), x)
\[ \int \frac {x^6 \left (A+B x^2\right )}{\left (c+d x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
int(x^6*(B*x^2+A)/(d*x^2+c)/(-c*x^4+a)^(3/2),x)
Output:
(3*sqrt(a - c*x**4)*a*b*d*x - sqrt(a - c*x**4)*b*c**2*x**3 - 3*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*b*c*d + 3*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 **2*d*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**2*b*c*d**2 + 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**3*d - 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**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**2*d** 2*x**4 - 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*c**4*d*x**4 + 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)*b*c**5*x**4 - 3*int((sqrt(a - c*x**4)*x**2)/(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*d**2 + 3*int((sqrt(a - c*x**4)*x**2)/(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 + 3*int((sqrt(a - c*x**4)*x**2)/(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*d**2*x**4 - 3*int((sq...