\(\int \frac {x^2 (-2 b+a x^6)}{(b+a x^6)^{3/4} (b+c x^4+a x^6)} \, dx\) [1843]

Optimal result

Integrand size = 38, antiderivative size = 125 \[ \int \frac {x^2 \left (-2 b+a x^6\right )}{\left (b+a x^6\right )^{3/4} \left (b+c x^4+a x^6\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{b+a x^6}}{-\sqrt {c} x^2+\sqrt {b+a x^6}}\right )}{\sqrt {2} c^{3/4}}+\frac {\text {arctanh}\left (\frac {\frac {\sqrt [4]{c} x^2}{\sqrt {2}}+\frac {\sqrt {b+a x^6}}{\sqrt {2} \sqrt [4]{c}}}{x \sqrt [4]{b+a x^6}}\right )}{\sqrt {2} c^{3/4}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 8.27 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.86 \[ \int \frac {x^2 \left (-2 b+a x^6\right )}{\left (b+a x^6\right )^{3/4} \left (b+c x^4+a x^6\right )} \, dx=\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{b+a x^6}}{\sqrt {c} x^2-\sqrt {b+a x^6}}\right )+\text {arctanh}\left (\frac {\sqrt {c} x^2+\sqrt {b+a x^6}}{\sqrt {2} \sqrt [4]{c} x \sqrt [4]{b+a x^6}}\right )}{\sqrt {2} c^{3/4}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.14 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.14

method	result	size
pseudoelliptic	\(\frac {\sqrt {2}\, \left (\ln \left (\frac {\left (a \,x^{6}+b \right )^{\frac {1}{4}} x \,c^{\frac {1}{4}} \sqrt {2}+\sqrt {c}\, x^{2}+\sqrt {a \,x^{6}+b}}{\sqrt {a \,x^{6}+b}-\left (a \,x^{6}+b \right )^{\frac {1}{4}} x \,c^{\frac {1}{4}} \sqrt {2}+\sqrt {c}\, x^{2}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \left (a \,x^{6}+b \right )^{\frac {1}{4}}+c^{\frac {1}{4}} x}{c^{\frac {1}{4}} x}\right )+2 \arctan \left (\frac {\sqrt {2}\, \left (a \,x^{6}+b \right )^{\frac {1}{4}}-c^{\frac {1}{4}} x}{c^{\frac {1}{4}} x}\right )\right )}{4 c^{\frac {3}{4}}}\)	\(142\)

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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