\(\int \frac {x^2 (2 b+a x^6)}{(-b+a x^6)^{3/4} (-b-2 c x^4+a x^6)} \, dx\) [1336]

Optimal result

Integrand size = 43, antiderivative size = 96 \[ \int \frac {x^2 \left (2 b+a x^6\right )}{\left (-b+a x^6\right )^{3/4} \left (-b-2 c x^4+a x^6\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x \left (-b+a x^6\right )^{3/4}}{b-a x^6}\right )}{2^{3/4} c^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x \left (-b+a x^6\right )^{3/4}}{b-a x^6}\right )}{2^{3/4} c^{3/4}} \]

[Out]

-1/2*arctan(2^(1/4)*c^(1/4)*x*(a*x^6-b)^(3/4)/(-a*x^6+b))*2^(1/4)/c^(3/4)+1/2*arctanh(2^(1/4)*c^(1/4)*x*(a*x^6
-b)^(3/4)/(-a*x^6+b))*2^(1/4)/c^(3/4)

Rubi [F]

\[ \int \frac {x^2 \left (2 b+a x^6\right )}{\left (-b+a x^6\right )^{3/4} \left (-b-2 c x^4+a x^6\right )} \, dx=\int \frac {x^2 \left (2 b+a x^6\right )}{\left (-b+a x^6\right )^{3/4} \left (-b-2 c x^4+a x^6\right )} \, dx \]

[In]

Int[(x^2*(2*b + a*x^6))/((-b + a*x^6)^(3/4)*(-b - 2*c*x^4 + a*x^6)),x]

[Out]

(Sqrt[(a*x^6)/(Sqrt[b] + Sqrt[-b + a*x^6])^2]*(Sqrt[b] + Sqrt[-b + a*x^6])*EllipticF[2*ArcTan[(-b + a*x^6)^(1/
4)/b^(1/4)], 1/2])/(3*a*b^(1/4)*x^3) + (2*c*x*(1 - (a*x^6)/b)^(3/4)*Hypergeometric2F1[1/6, 3/4, 7/6, (a*x^6)/b
])/(a*(-b + a*x^6)^(3/4)) - (2*b*c*Defer[Int][1/((b + 2*c*x^4 - a*x^6)*(-b + a*x^6)^(3/4)), x])/a - (4*c^2*Def
er[Int][x^4/((b + 2*c*x^4 - a*x^6)*(-b + a*x^6)^(3/4)), x])/a + 3*b*Defer[Int][x^2/((-b + a*x^6)^(3/4)*(-b - 2
*c*x^4 + a*x^6)), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2 c}{a \left (-b+a x^6\right )^{3/4}}+\frac {x^2}{\left (-b+a x^6\right )^{3/4}}+\frac {2 b c+3 a b x^2+4 c^2 x^4}{a \left (-b+a x^6\right )^{3/4} \left (-b-2 c x^4+a x^6\right )}\right ) \, dx \\ & = \frac {\int \frac {2 b c+3 a b x^2+4 c^2 x^4}{\left (-b+a x^6\right )^{3/4} \left (-b-2 c x^4+a x^6\right )} \, dx}{a}+\frac {(2 c) \int \frac {1}{\left (-b+a x^6\right )^{3/4}} \, dx}{a}+\int \frac {x^2}{\left (-b+a x^6\right )^{3/4}} \, dx \\ & = \frac {1}{3} \text {Subst}\left (\int \frac {1}{\left (-b+a x^2\right )^{3/4}} \, dx,x,x^3\right )+\frac {\int \left (-\frac {2 b c}{\left (b+2 c x^4-a x^6\right ) \left (-b+a x^6\right )^{3/4}}-\frac {4 c^2 x^4}{\left (b+2 c x^4-a x^6\right ) \left (-b+a x^6\right )^{3/4}}+\frac {3 a b x^2}{\left (-b+a x^6\right )^{3/4} \left (-b-2 c x^4+a x^6\right )}\right ) \, dx}{a}+\frac {\left (2 c \left (1-\frac {a x^6}{b}\right )^{3/4}\right ) \int \frac {1}{\left (1-\frac {a x^6}{b}\right )^{3/4}} \, dx}{a \left (-b+a x^6\right )^{3/4}} \\ & = \frac {2 c x \left (1-\frac {a x^6}{b}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {3}{4},\frac {7}{6},\frac {a x^6}{b}\right )}{a \left (-b+a x^6\right )^{3/4}}+(3 b) \int \frac {x^2}{\left (-b+a x^6\right )^{3/4} \left (-b-2 c x^4+a x^6\right )} \, dx-\frac {(2 b c) \int \frac {1}{\left (b+2 c x^4-a x^6\right ) \left (-b+a x^6\right )^{3/4}} \, dx}{a}-\frac {\left (4 c^2\right ) \int \frac {x^4}{\left (b+2 c x^4-a x^6\right ) \left (-b+a x^6\right )^{3/4}} \, dx}{a}+\frac {\left (2 \sqrt {\frac {a x^6}{b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^6}\right )}{3 a x^3} \\ & = \frac {\sqrt {\frac {a x^6}{\left (\sqrt {b}+\sqrt {-b+a x^6}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x^6}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{-b+a x^6}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{3 a \sqrt [4]{b} x^3}+\frac {2 c x \left (1-\frac {a x^6}{b}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {3}{4},\frac {7}{6},\frac {a x^6}{b}\right )}{a \left (-b+a x^6\right )^{3/4}}+(3 b) \int \frac {x^2}{\left (-b+a x^6\right )^{3/4} \left (-b-2 c x^4+a x^6\right )} \, dx-\frac {(2 b c) \int \frac {1}{\left (b+2 c x^4-a x^6\right ) \left (-b+a x^6\right )^{3/4}} \, dx}{a}-\frac {\left (4 c^2\right ) \int \frac {x^4}{\left (b+2 c x^4-a x^6\right ) \left (-b+a x^6\right )^{3/4}} \, dx}{a} \\ \end{align*}

Mathematica [A] (verified)

Time = 8.20 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.70 \[ \int \frac {x^2 \left (2 b+a x^6\right )}{\left (-b+a x^6\right )^{3/4} \left (-b-2 c x^4+a x^6\right )} \, dx=-\frac {-\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b+a x^6}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b+a x^6}}\right )}{2^{3/4} c^{3/4}} \]

[In]

Integrate[(x^2*(2*b + a*x^6))/((-b + a*x^6)^(3/4)*(-b - 2*c*x^4 + a*x^6)),x]

[Out]

-((-ArcTan[(2^(1/4)*c^(1/4)*x)/(-b + a*x^6)^(1/4)] + ArcTanh[(2^(1/4)*c^(1/4)*x)/(-b + a*x^6)^(1/4)])/(2^(3/4)
*c^(3/4)))

Maple [A] (verified)

Time = 3.43 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.88

method	result	size
pseudoelliptic	\(-\frac {2^{\frac {1}{4}} \left (\ln \left (\frac {-2^{\frac {1}{4}} c^{\frac {1}{4}} x -\left (a \,x^{6}-b \right )^{\frac {1}{4}}}{2^{\frac {1}{4}} c^{\frac {1}{4}} x -\left (a \,x^{6}-b \right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\left (a \,x^{6}-b \right )^{\frac {1}{4}} 2^{\frac {3}{4}}}{2 x \,c^{\frac {1}{4}}}\right )\right )}{4 c^{\frac {3}{4}}}\)	\(84\)

[In]

int(x^2*(a*x^6+2*b)/(a*x^6-b)^(3/4)/(a*x^6-2*c*x^4-b),x,method=_RETURNVERBOSE)

[Out]

-1/4*2^(1/4)/c^(3/4)*(ln((-2^(1/4)*c^(1/4)*x-(a*x^6-b)^(1/4))/(2^(1/4)*c^(1/4)*x-(a*x^6-b)^(1/4)))+2*arctan(1/
2*(a*x^6-b)^(1/4)/x*2^(3/4)/c^(1/4)))

Fricas [F(-1)]

Timed out. \[ \int \frac {x^2 \left (2 b+a x^6\right )}{\left (-b+a x^6\right )^{3/4} \left (-b-2 c x^4+a x^6\right )} \, dx=\text {Timed out} \]

[In]

integrate(x^2*(a*x^6+2*b)/(a*x^6-b)^(3/4)/(a*x^6-2*c*x^4-b),x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {x^2 \left (2 b+a x^6\right )}{\left (-b+a x^6\right )^{3/4} \left (-b-2 c x^4+a x^6\right )} \, dx=\text {Timed out} \]

[In]

integrate(x**2*(a*x**6+2*b)/(a*x**6-b)**(3/4)/(a*x**6-2*c*x**4-b),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {x^2 \left (2 b+a x^6\right )}{\left (-b+a x^6\right )^{3/4} \left (-b-2 c x^4+a x^6\right )} \, dx=\int { \frac {{\left (a x^{6} + 2 \, b\right )} x^{2}}{{\left (a x^{6} - 2 \, c x^{4} - b\right )} {\left (a x^{6} - b\right )}^{\frac {3}{4}}} \,d x } \]

[In]

integrate(x^2*(a*x^6+2*b)/(a*x^6-b)^(3/4)/(a*x^6-2*c*x^4-b),x, algorithm="maxima")

[Out]

integrate((a*x^6 + 2*b)*x^2/((a*x^6 - 2*c*x^4 - b)*(a*x^6 - b)^(3/4)), x)

Giac [F]

\[ \int \frac {x^2 \left (2 b+a x^6\right )}{\left (-b+a x^6\right )^{3/4} \left (-b-2 c x^4+a x^6\right )} \, dx=\int { \frac {{\left (a x^{6} + 2 \, b\right )} x^{2}}{{\left (a x^{6} - 2 \, c x^{4} - b\right )} {\left (a x^{6} - b\right )}^{\frac {3}{4}}} \,d x } \]

[In]

integrate(x^2*(a*x^6+2*b)/(a*x^6-b)^(3/4)/(a*x^6-2*c*x^4-b),x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \left (2 b+a x^6\right )}{\left (-b+a x^6\right )^{3/4} \left (-b-2 c x^4+a x^6\right )} \, dx=\int -\frac {x^2\,\left (a\,x^6+2\,b\right )}{{\left (a\,x^6-b\right )}^{3/4}\,\left (-a\,x^6+2\,c\,x^4+b\right )} \,d x \]

[In]

int(-(x^2*(2*b + a*x^6))/((a*x^6 - b)^(3/4)*(b - a*x^6 + 2*c*x^4)),x)

[Out]

int(-(x^2*(2*b + a*x^6))/((a*x^6 - b)^(3/4)*(b - a*x^6 + 2*c*x^4)), x)