Integrand size = 34, antiderivative size = 349 \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\frac {x \left ((A c+a C) d-a B e+(B c d-(A c+a C) e) x^2\right )}{2 a \left (c d^2-a e^2\right ) \sqrt {a-c x^4}}-\frac {(B c d-(A c+a C) e) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{a} c^{3/4} \left (c d^2-a e^2\right ) \sqrt {a-c x^4}}+\frac {\left (\sqrt {a} B \sqrt {c}+A c+a C\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{2 a^{3/4} c^{3/4} \left (\sqrt {c} d+\sqrt {a} e\right ) \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} \left (C d^2-B d e+A e^2\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} d \left (c d^2-a e^2\right ) \sqrt {a-c x^4}} \] Output:
1/2*x*((A*c+C*a)*d-B*a*e+(B*c*d-(A*c+C*a)*e)*x^2)/a/(-a*e^2+c*d^2)/(-c*x^4 +a)^(1/2)-1/2*(B*c*d-(A*c+C*a)*e)*(1-c*x^4/a)^(1/2)*EllipticE(c^(1/4)*x/a^ (1/4),I)/a^(1/4)/c^(3/4)/(-a*e^2+c*d^2)/(-c*x^4+a)^(1/2)+1/2*(a^(1/2)*B*c^ (1/2)+A*c+a*C)*(1-c*x^4/a)^(1/2)*EllipticF(c^(1/4)*x/a^(1/4),I)/a^(3/4)/c^ (3/4)/(c^(1/2)*d+a^(1/2)*e)/(-c*x^4+a)^(1/2)-a^(1/4)*(A*e^2-B*d*e+C*d^2)*( 1-c*x^4/a)^(1/2)*EllipticPi(c^(1/4)*x/a^(1/4),-a^(1/2)*e/c^(1/2)/d,I)/c^(1 /4)/d/(-a*e^2+c*d^2)/(-c*x^4+a)^(1/2)
Result contains complex when optimal does not.
Time = 11.96 (sec) , antiderivative size = 340, normalized size of antiderivative = 0.97 \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=-\frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} \sqrt {c} d x \left (-B c d x^2+A c \left (-d+e x^2\right )+a \left (-C d+B e+C e x^2\right )\right )+i \sqrt {a} d (-B c d+A c e+a C e) \sqrt {1-\frac {c x^4}{a}} E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )-i \left (\sqrt {a} B \sqrt {c}+A c+a C\right ) d \left (-\sqrt {c} d+\sqrt {a} e\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )-2 i a \sqrt {c} \left (C d^2+e (-B d+A e)\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )}{2 a^{3/2} \left (-\frac {\sqrt {c}}{\sqrt {a}}\right )^{3/2} \left (-c d^3+a d e^2\right ) \sqrt {a-c x^4}} \] Input:
Integrate[(A + B*x^2 + C*x^4)/((d + e*x^2)*(a - c*x^4)^(3/2)),x]
Output:
-1/2*(Sqrt[-(Sqrt[c]/Sqrt[a])]*Sqrt[c]*d*x*(-(B*c*d*x^2) + A*c*(-d + e*x^2 ) + a*(-(C*d) + B*e + C*e*x^2)) + I*Sqrt[a]*d*(-(B*c*d) + A*c*e + a*C*e)*S qrt[1 - (c*x^4)/a]*EllipticE[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] - I*(Sqrt[a]*B*Sqrt[c] + A*c + a*C)*d*(-(Sqrt[c]*d) + Sqrt[a]*e)*Sqrt[1 - (c *x^4)/a]*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] - (2*I)*a*Sq rt[c]*(C*d^2 + e*(-(B*d) + A*e))*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a] *e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1])/(a^(3/2)*(-( Sqrt[c]/Sqrt[a]))^(3/2)*(-(c*d^3) + a*d*e^2)*Sqrt[a - c*x^4])
Time = 0.67 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2259, 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.
| \(\displaystyle \int \frac {A+B x^2+C x^4}{\left (a-c x^4\right )^{3/2} \left (d+e x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2259 |
| \(\displaystyle \int \left (\frac {-A e^2+B d e-C d^2}{\sqrt {a-c x^4} \left (d+e x^2\right ) \left (c d^2-a e^2\right )}+\frac {x^2 (-a C e-A c e+B c d)-a B e+a C d+A c d}{\left (a-c x^4\right )^{3/2} \left (c d^2-a e^2\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {1-\frac {c x^4}{a}} \left (\sqrt {a} B \sqrt {c}+a C+A c\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{2 a^{3/4} c^{3/4} \sqrt {a-c x^4} \left (\sqrt {a} e+\sqrt {c} d\right )}-\frac {\sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right ) (-a C e-A c e+B c d)}{2 \sqrt [4]{a} c^{3/4} \sqrt {a-c x^4} \left (c d^2-a e^2\right )}-\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (A e^2-B d e+C d^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} d \sqrt {a-c x^4} \left (c d^2-a e^2\right )}+\frac {x \left (x^2 (B c d-e (a C+A c))-a B e+a C d+A c d\right )}{2 a \sqrt {a-c x^4} \left (c d^2-a e^2\right )}\) |
Input:
Int[(A + B*x^2 + C*x^4)/((d + e*x^2)*(a - c*x^4)^(3/2)),x]
Output:
(x*(A*c*d + a*C*d - a*B*e + (B*c*d - (A*c + a*C)*e)*x^2))/(2*a*(c*d^2 - a* e^2)*Sqrt[a - c*x^4]) - ((B*c*d - A*c*e - a*C*e)*Sqrt[1 - (c*x^4)/a]*Ellip ticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(2*a^(1/4)*c^(3/4)*(c*d^2 - a*e^2)* Sqrt[a - c*x^4]) + ((Sqrt[a]*B*Sqrt[c] + A*c + a*C)*Sqrt[1 - (c*x^4)/a]*El lipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(2*a^(3/4)*c^(3/4)*(Sqrt[c]*d + Sqrt[a]*e)*Sqrt[a - c*x^4]) - (a^(1/4)*(C*d^2 - B*d*e + A*e^2)*Sqrt[1 - (c *x^4)/a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), ArcSin[(c^(1/4)*x)/a^(1/4) ], -1])/(c^(1/4)*d*(c*d^2 - a*e^2)*Sqrt[a - c*x^4])
Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[1/Sqrt[a + c*x^4], Px*(d + e*x^2)^q*(a + c*x^4)^(p + 1/2), x], x] /; FreeQ[{a, c, d, e}, 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. 755 vs. \(2 (293 ) = 586\).
Time = 1.00 (sec) , antiderivative size = 756, normalized size of antiderivative = 2.17
| method | result | size |
| default | \(\frac {B e \left (\frac {x}{2 a \sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {\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}}\right )+C e \left (\frac {x^{3}}{2 a \sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {\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}\, \sqrt {c}}\right )-C d \left (\frac {x}{2 a \sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {\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}}\right )}{e^{2}}+\frac {\left (A \,e^{2}-B d e +C \,d^{2}\right ) \left (\frac {2 c \left (\frac {e \,x^{3}}{4 a \left (a \,e^{2}-c \,d^{2}\right )}-\frac {d x}{4 a \left (a \,e^{2}-c \,d^{2}\right )}\right )}{\sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}-\frac {c 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 )}{2 a \left (a \,e^{2}-c \,d^{2}\right ) \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {\sqrt {c}\, e \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 \sqrt {a}\, \left (a \,e^{2}-c \,d^{2}\right ) \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {\sqrt {c}\, e \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{2 \sqrt {a}\, \left (a \,e^{2}-c \,d^{2}\right ) \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {e^{2} \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}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{\left (a \,e^{2}-c \,d^{2}\right ) d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )}{e^{2}}\) | \(756\) |
| elliptic | \(\text {Expression too large to display}\) | \(1190\) |
Input:
int((C*x^4+B*x^2+A)/(e*x^2+d)/(-c*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/e^2*(B*e*(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*e*(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)*(EllipticF(x*(c^(1/2)/a^( 1/2))^(1/2),I)-EllipticE(x*(c^(1/2)/a^(1/2))^(1/2),I)))-C*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)*EllipticF(x*(c^(1/2)/a^(1 /2))^(1/2),I)))+(A*e^2-B*d*e+C*d^2)/e^2*(2*c*(1/4/a*e/(a*e^2-c*d^2)*x^3-1/ 4/a*d/(a*e^2-c*d^2)*x)/(-(x^4-a/c)*c)^(1/2)-1/2*c/a*d/(a*e^2-c*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)+1/2*c^(1/2)/a^ (1/2)*e/(a*e^2-c*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)-1/2*c^(1/2)/a^(1/2)*e/(a*e^2-c*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)*EllipticE(x*(c^(1/2)/a^(1/2))^(1/2),I)+1/(a*e^2-c*d^2)*e^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)*EllipticPi(x*(c^(1/2)/a^(1/2))^(1/2),-a^(1/2)*e/c^(1/2)/d ,(-c^(1/2)/a^(1/2))^(1/2)/(c^(1/2)/a^(1/2))^(1/2)))
Timed out. \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((C*x^4+B*x^2+A)/(e*x^2+d)/(-c*x^4+a)^(3/2),x, algorithm="fricas" )
Output:
Timed out
Timed out. \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((C*x**4+B*x**2+A)/(e*x**2+d)/(-c*x**4+a)**(3/2),x)
Output:
Timed out
\[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}} \,d x } \] Input:
integrate((C*x^4+B*x^2+A)/(e*x^2+d)/(-c*x^4+a)^(3/2),x, algorithm="maxima" )
Output:
integrate((C*x^4 + B*x^2 + A)/((-c*x^4 + a)^(3/2)*(e*x^2 + d)), x)
\[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}} \,d x } \] Input:
integrate((C*x^4+B*x^2+A)/(e*x^2+d)/(-c*x^4+a)^(3/2),x, algorithm="giac")
Output:
integrate((C*x^4 + B*x^2 + A)/((-c*x^4 + a)^(3/2)*(e*x^2 + d)), x)
Timed out. \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\int \frac {C\,x^4+B\,x^2+A}{{\left (a-c\,x^4\right )}^{3/2}\,\left (e\,x^2+d\right )} \,d x \] Input:
int((A + B*x^2 + C*x^4)/((a - c*x^4)^(3/2)*(d + e*x^2)),x)
Output:
int((A + B*x^2 + C*x^4)/((a - c*x^4)^(3/2)*(d + e*x^2)), x)
\[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right ) \left (a-c x^4\right )^{3/2}} \, dx=\left (\int \frac {\sqrt {-c \,x^{4}+a}}{c^{2} e \,x^{10}+c^{2} d \,x^{8}-2 a c e \,x^{6}-2 a c d \,x^{4}+a^{2} e \,x^{2}+a^{2} d}d x \right ) a +\left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{4}}{c^{2} e \,x^{10}+c^{2} d \,x^{8}-2 a c e \,x^{6}-2 a c d \,x^{4}+a^{2} e \,x^{2}+a^{2} d}d x \right ) c +\left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{c^{2} e \,x^{10}+c^{2} d \,x^{8}-2 a c e \,x^{6}-2 a c d \,x^{4}+a^{2} e \,x^{2}+a^{2} d}d x \right ) b \] Input:
int((C*x^4+B*x^2+A)/(e*x^2+d)/(-c*x^4+a)^(3/2),x)
Output:
int(sqrt(a - c*x**4)/(a**2*d + a**2*e*x**2 - 2*a*c*d*x**4 - 2*a*c*e*x**6 + c**2*d*x**8 + c**2*e*x**10),x)*a + int((sqrt(a - c*x**4)*x**4)/(a**2*d + a**2*e*x**2 - 2*a*c*d*x**4 - 2*a*c*e*x**6 + c**2*d*x**8 + c**2*e*x**10),x) *c + int((sqrt(a - c*x**4)*x**2)/(a**2*d + a**2*e*x**2 - 2*a*c*d*x**4 - 2* a*c*e*x**6 + c**2*d*x**8 + c**2*e*x**10),x)*b