\(\int \frac {(A+B x^2) \sqrt [3]{d+e x^2}}{a+c x^4} \, dx\) [25]

Optimal result

Integrand size = 28, antiderivative size = 466 \[ \int \frac {\left (A+B x^2\right ) \sqrt [3]{d+e x^2}}{a+c x^4} \, dx=\frac {\left (\sqrt {c} (B d+A e)-\frac {A c d-a B e}{\sqrt {-a}}\right ) x \left (1+\frac {e x^2}{d}\right )^{2/3} \operatorname {AppellF1}\left (\frac {1}{2},\frac {2}{3},1,\frac {3}{2},-\frac {e x^2}{d},-\frac {\sqrt {c} x^2}{\sqrt {-a}}\right )}{2 \sqrt {-a} c \left (d+e x^2\right )^{2/3}}-\frac {\left (\sqrt {c} (B d+A e)+\frac {A c d-a B e}{\sqrt {-a}}\right ) x \left (1+\frac {e x^2}{d}\right )^{2/3} \operatorname {AppellF1}\left (\frac {1}{2},\frac {2}{3},1,\frac {3}{2},-\frac {e x^2}{d},\frac {\sqrt {c} x^2}{\sqrt {-a}}\right )}{2 \sqrt {-a} c \left (d+e x^2\right )^{2/3}}-\frac {3^{3/4} \sqrt {2-\sqrt {3}} B \left (\sqrt [3]{d}-\sqrt [3]{d+e x^2}\right ) \sqrt {\frac {d^{2/3}+\sqrt [3]{d} \sqrt [3]{d+e x^2}+\left (d+e x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{d}-\sqrt [3]{d+e x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{d}-\sqrt [3]{d+e x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{d}-\sqrt [3]{d+e x^2}}\right ),-7+4 \sqrt {3}\right )}{c x \sqrt {-\frac {\sqrt [3]{d} \left (\sqrt [3]{d}-\sqrt [3]{d+e x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{d}-\sqrt [3]{d+e x^2}\right )^2}}} \] Output:

1/2*(c^(1/2)*(A*e+B*d)-(A*c*d-B*a*e)/(-a)^(1/2))*x*(1+e*x^2/d)^(2/3)*Appel 
lF1(1/2,1,2/3,3/2,-c^(1/2)*x^2/(-a)^(1/2),-e*x^2/d)/(-a)^(1/2)/c/(e*x^2+d) 
^(2/3)-1/2*(c^(1/2)*(A*e+B*d)+(A*c*d-B*a*e)/(-a)^(1/2))*x*(1+e*x^2/d)^(2/3 
)*AppellF1(1/2,1,2/3,3/2,c^(1/2)*x^2/(-a)^(1/2),-e*x^2/d)/(-a)^(1/2)/c/(e* 
x^2+d)^(2/3)-3^(3/4)*(1/2*6^(1/2)-1/2*2^(1/2))*B*(d^(1/3)-(e*x^2+d)^(1/3)) 
*((d^(2/3)+d^(1/3)*(e*x^2+d)^(1/3)+(e*x^2+d)^(2/3))/((1-3^(1/2))*d^(1/3)-( 
e*x^2+d)^(1/3))^2)^(1/2)*EllipticF(((1+3^(1/2))*d^(1/3)-(e*x^2+d)^(1/3))/( 
(1-3^(1/2))*d^(1/3)-(e*x^2+d)^(1/3)),2*I-I*3^(1/2))/c/x/(-d^(1/3)*(d^(1/3) 
-(e*x^2+d)^(1/3))/((1-3^(1/2))*d^(1/3)-(e*x^2+d)^(1/3))^2)^(1/2)
 

Mathematica [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt [3]{d+e x^2}}{a+c x^4} \, dx=\int \frac {\left (A+B x^2\right ) \sqrt [3]{d+e x^2}}{a+c x^4} \, dx \] Input:

Integrate[((A + B*x^2)*(d + e*x^2)^(1/3))/(a + c*x^4),x]
 

Output:

Integrate[((A + B*x^2)*(d + e*x^2)^(1/3))/(a + c*x^4), x]
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.37, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2257, 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.

\(\displaystyle \int \frac {\left (A+B x^2\right ) \sqrt [3]{d+e x^2}}{a+c x^4} \, dx\)

\(\Big \downarrow \) 2257

\(\displaystyle \int \left (\frac {\sqrt [3]{d+e x^2} \left (\sqrt {-a} B-A \sqrt {c}\right )}{2 \sqrt {-a} \sqrt {c} \left (\sqrt {-a}+\sqrt {c} x^2\right )}-\frac {\sqrt [3]{d+e x^2} \left (\sqrt {-a} B+A \sqrt {c}\right )}{2 \sqrt {-a} \sqrt {c} \left (\sqrt {-a}-\sqrt {c} x^2\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {x \sqrt [3]{d+e x^2} \left (A-\frac {\sqrt {-a} B}{\sqrt {c}}\right ) \operatorname {AppellF1}\left (\frac {1}{2},1,-\frac {1}{3},\frac {3}{2},-\frac {\sqrt {c} x^2}{\sqrt {-a}},-\frac {e x^2}{d}\right )}{2 a \sqrt [3]{\frac {e x^2}{d}+1}}+\frac {x \sqrt [3]{d+e x^2} \left (\frac {\sqrt {-a} B}{\sqrt {c}}+A\right ) \operatorname {AppellF1}\left (\frac {1}{2},1,-\frac {1}{3},\frac {3}{2},\frac {\sqrt {c} x^2}{\sqrt {-a}},-\frac {e x^2}{d}\right )}{2 a \sqrt [3]{\frac {e x^2}{d}+1}}\)

Input:

Int[((A + B*x^2)*(d + e*x^2)^(1/3))/(a + c*x^4),x]
 

Output:

((A - (Sqrt[-a]*B)/Sqrt[c])*x*(d + e*x^2)^(1/3)*AppellF1[1/2, 1, -1/3, 3/2 
, -((Sqrt[c]*x^2)/Sqrt[-a]), -((e*x^2)/d)])/(2*a*(1 + (e*x^2)/d)^(1/3)) + 
((A + (Sqrt[-a]*B)/Sqrt[c])*x*(d + e*x^2)^(1/3)*AppellF1[1/2, 1, -1/3, 3/2 
, (Sqrt[c]*x^2)/Sqrt[-a], -((e*x^2)/d)])/(2*a*(1 + (e*x^2)/d)^(1/3))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2257

Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol 
] :> Int[ExpandIntegrand[Px*(d + e*x^2)^q*(a + c*x^4)^p, x], x] /; FreeQ[{a 
, c, d, e, q}, x] && PolyQ[Px, x] && IntegerQ[p]
 

Maple [F]

\[\int \frac {\left (B \,x^{2}+A \right ) \left (e \,x^{2}+d \right )^{\frac {1}{3}}}{c \,x^{4}+a}d x\]

Input:

int((B*x^2+A)*(e*x^2+d)^(1/3)/(c*x^4+a),x)
 

Output:

int((B*x^2+A)*(e*x^2+d)^(1/3)/(c*x^4+a),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (A+B x^2\right ) \sqrt [3]{d+e x^2}}{a+c x^4} \, dx=\text {Timed out} \] Input:

integrate((B*x^2+A)*(e*x^2+d)^(1/3)/(c*x^4+a),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt [3]{d+e x^2}}{a+c x^4} \, dx=\int \frac {\left (A + B x^{2}\right ) \sqrt [3]{d + e x^{2}}}{a + c x^{4}}\, dx \] Input:

integrate((B*x**2+A)*(e*x**2+d)**(1/3)/(c*x**4+a),x)
 

Output:

Integral((A + B*x**2)*(d + e*x**2)**(1/3)/(a + c*x**4), x)
 

Maxima [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt [3]{d+e x^2}}{a+c x^4} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{\frac {1}{3}}}{c x^{4} + a} \,d x } \] Input:

integrate((B*x^2+A)*(e*x^2+d)^(1/3)/(c*x^4+a),x, algorithm="maxima")
 

Output:

integrate((B*x^2 + A)*(e*x^2 + d)^(1/3)/(c*x^4 + a), x)
 

Giac [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt [3]{d+e x^2}}{a+c x^4} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{\frac {1}{3}}}{c x^{4} + a} \,d x } \] Input:

integrate((B*x^2+A)*(e*x^2+d)^(1/3)/(c*x^4+a),x, algorithm="giac")
 

Output:

integrate((B*x^2 + A)*(e*x^2 + d)^(1/3)/(c*x^4 + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (A+B x^2\right ) \sqrt [3]{d+e x^2}}{a+c x^4} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (e\,x^2+d\right )}^{1/3}}{c\,x^4+a} \,d x \] Input:

int(((A + B*x^2)*(d + e*x^2)^(1/3))/(a + c*x^4),x)
 

Output:

int(((A + B*x^2)*(d + e*x^2)^(1/3))/(a + c*x^4), x)
 

Reduce [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt [3]{d+e x^2}}{a+c x^4} \, dx=\left (\int \frac {\left (e \,x^{2}+d \right )^{\frac {1}{3}}}{c \,x^{4}+a}d x \right ) a +\left (\int \frac {\left (e \,x^{2}+d \right )^{\frac {1}{3}} x^{2}}{c \,x^{4}+a}d x \right ) b \] Input:

int((B*x^2+A)*(e*x^2+d)^(1/3)/(c*x^4+a),x)
 

Output:

int((d + e*x**2)**(1/3)/(a + c*x**4),x)*a + int(((d + e*x**2)**(1/3)*x**2) 
/(a + c*x**4),x)*b