\(\int \frac {x^2 (-1+3 x^4)}{(1+x^4)^2 (a-x+a x^4) \sqrt {x+x^5}} \, dx\) [819]

Optimal result

Integrand size = 39, antiderivative size = 62 \[ \int \frac {x^2 \left (-1+3 x^4\right )}{\left (1+x^4\right )^2 \left (a-x+a x^4\right ) \sqrt {x+x^5}} \, dx=\frac {2 \left (3 a+x+3 a x^4\right ) \sqrt {x+x^5}}{3 \left (1+x^4\right )^2}-2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {x+x^5}}{\sqrt {a} \left (1+x^4\right )}\right ) \]

[Out]

2/3*(3*a*x^4+3*a+x)*(x^5+x)^(1/2)/(x^4+1)^2-2*a^(3/2)*arctanh((x^5+x)^(1/2)/a^(1/2)/(x^4+1))

Rubi [F]

\[ \int \frac {x^2 \left (-1+3 x^4\right )}{\left (1+x^4\right )^2 \left (a-x+a x^4\right ) \sqrt {x+x^5}} \, dx=\int \frac {x^2 \left (-1+3 x^4\right )}{\left (1+x^4\right )^2 \left (a-x+a x^4\right ) \sqrt {x+x^5}} \, dx \]

[In]

Int[(x^2*(-1 + 3*x^4))/((1 + x^4)^2*(a - x + a*x^4)*Sqrt[x + x^5]),x]

[Out]

(7*x^3)/(8*a*Sqrt[x + x^5]) + x^3/(2*a*(1 + x^4)*Sqrt[x + x^5]) - (7*x^3*Sqrt[1 + x^4]*Hypergeometric2F1[1/2,
5/8, 13/8, -x^4])/(40*a*Sqrt[x + x^5]) - (8*Sqrt[x]*Sqrt[1 + x^4]*Defer[Subst][Defer[Int][x^4/((1 + x^8)^(5/2)
*(a - x^2 + a*x^8)), x], x, Sqrt[x]])/Sqrt[x + x^5] + (6*Sqrt[x]*Sqrt[1 + x^4]*Defer[Subst][Defer[Int][x^6/((1
+ x^8)^(5/2)*(a - x^2 + a*x^8)), x], x, Sqrt[x]])/(a*Sqrt[x + x^5])

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt {1+x^4}\right ) \int \frac {x^{3/2} \left (-1+3 x^4\right )}{\left (1+x^4\right )^{5/2} \left (a-x+a x^4\right )} \, dx}{\sqrt {x+x^5}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {1+x^4}\right ) \text {Subst}\left (\int \frac {x^4 \left (-1+3 x^8\right )}{\left (1+x^8\right )^{5/2} \left (a-x^2+a x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {1+x^4}\right ) \text {Subst}\left (\int \left (\frac {3 x^4}{a \left (1+x^8\right )^{5/2}}+\frac {x^4 \left (-4 a+3 x^2\right )}{a \left (1+x^8\right )^{5/2} \left (a-x^2+a x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {1+x^4}\right ) \text {Subst}\left (\int \frac {x^4 \left (-4 a+3 x^2\right )}{\left (1+x^8\right )^{5/2} \left (a-x^2+a x^8\right )} \, dx,x,\sqrt {x}\right )}{a \sqrt {x+x^5}}+\frac {\left (6 \sqrt {x} \sqrt {1+x^4}\right ) \text {Subst}\left (\int \frac {x^4}{\left (1+x^8\right )^{5/2}} \, dx,x,\sqrt {x}\right )}{a \sqrt {x+x^5}} \\ & = \frac {x^3}{2 a \left (1+x^4\right ) \sqrt {x+x^5}}+\frac {\left (2 \sqrt {x} \sqrt {1+x^4}\right ) \text {Subst}\left (\int \left (-\frac {4 a x^4}{\left (1+x^8\right )^{5/2} \left (a-x^2+a x^8\right )}+\frac {3 x^6}{\left (1+x^8\right )^{5/2} \left (a-x^2+a x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {x+x^5}}+\frac {\left (7 \sqrt {x} \sqrt {1+x^4}\right ) \text {Subst}\left (\int \frac {x^4}{\left (1+x^8\right )^{3/2}} \, dx,x,\sqrt {x}\right )}{2 a \sqrt {x+x^5}} \\ & = \frac {7 x^3}{8 a \sqrt {x+x^5}}+\frac {x^3}{2 a \left (1+x^4\right ) \sqrt {x+x^5}}-\frac {\left (8 \sqrt {x} \sqrt {1+x^4}\right ) \text {Subst}\left (\int \frac {x^4}{\left (1+x^8\right )^{5/2} \left (a-x^2+a x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}}-\frac {\left (7 \sqrt {x} \sqrt {1+x^4}\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {1+x^8}} \, dx,x,\sqrt {x}\right )}{8 a \sqrt {x+x^5}}+\frac {\left (6 \sqrt {x} \sqrt {1+x^4}\right ) \text {Subst}\left (\int \frac {x^6}{\left (1+x^8\right )^{5/2} \left (a-x^2+a x^8\right )} \, dx,x,\sqrt {x}\right )}{a \sqrt {x+x^5}} \\ & = \frac {7 x^3}{8 a \sqrt {x+x^5}}+\frac {x^3}{2 a \left (1+x^4\right ) \sqrt {x+x^5}}-\frac {7 x^3 \sqrt {1+x^4} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{8},\frac {13}{8},-x^4\right )}{40 a \sqrt {x+x^5}}-\frac {\left (8 \sqrt {x} \sqrt {1+x^4}\right ) \text {Subst}\left (\int \frac {x^4}{\left (1+x^8\right )^{5/2} \left (a-x^2+a x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}}+\frac {\left (6 \sqrt {x} \sqrt {1+x^4}\right ) \text {Subst}\left (\int \frac {x^6}{\left (1+x^8\right )^{5/2} \left (a-x^2+a x^8\right )} \, dx,x,\sqrt {x}\right )}{a \sqrt {x+x^5}} \\ \end{align*}

Mathematica [A] (verified)

Time = 22.20 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.98 \[ \int \frac {x^2 \left (-1+3 x^4\right )}{\left (1+x^4\right )^2 \left (a-x+a x^4\right ) \sqrt {x+x^5}} \, dx=\frac {2 \sqrt {x+x^5} \left (x+3 a \left (1+x^4\right )\right )}{3 \left (1+x^4\right )^2}-2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {x+x^5}}{\sqrt {a} \left (1+x^4\right )}\right ) \]

[In]

Integrate[(x^2*(-1 + 3*x^4))/((1 + x^4)^2*(a - x + a*x^4)*Sqrt[x + x^5]),x]

[Out]

(2*Sqrt[x + x^5]*(x + 3*a*(1 + x^4)))/(3*(1 + x^4)^2) - 2*a^(3/2)*ArcTanh[Sqrt[x + x^5]/(Sqrt[a]*(1 + x^4))]

Maple [A] (verified)

Time = 1.36 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.13

method	result	size
pseudoelliptic	\(\frac {-6 a^{\frac {3}{2}} \sqrt {x \left (x^{4}+1\right )}\, \left (x^{4}+1\right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (x^{4}+1\right )}\, \sqrt {a}}{x}\right )+6 x \left (a \,x^{4}+a +\frac {1}{3} x \right )}{\sqrt {x \left (x^{4}+1\right )}\, \left (3 x^{4}+3\right )}\)	\(70\)

[In]

int(x^2*(3*x^4-1)/(x^4+1)^2/(a*x^4+a-x)/(x^5+x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 231, normalized size of antiderivative = 3.73 \[ \int \frac {x^2 \left (-1+3 x^4\right )}{\left (1+x^4\right )^2 \left (a-x+a x^4\right ) \sqrt {x+x^5}} \, dx=\left [\frac {3 \, {\left (a x^{8} + 2 \, a x^{4} + a\right )} \sqrt {a} \log \left (\frac {a^{2} x^{8} + 2 \, a^{2} x^{4} + 6 \, a x^{5} - 4 \, {\left (a x^{4} + a + x\right )} \sqrt {x^{5} + x} \sqrt {a} + a^{2} + 6 \, a x + x^{2}}{a^{2} x^{8} + 2 \, a^{2} x^{4} - 2 \, a x^{5} + a^{2} - 2 \, a x + x^{2}}\right ) + 4 \, {\left (3 \, a x^{4} + 3 \, a + x\right )} \sqrt {x^{5} + x}}{6 \, {\left (x^{8} + 2 \, x^{4} + 1\right )}}, \frac {3 \, {\left (a x^{8} + 2 \, a x^{4} + a\right )} \sqrt {-a} \arctan \left (\frac {{\left (a x^{4} + a + x\right )} \sqrt {x^{5} + x} \sqrt {-a}}{2 \, {\left (a x^{5} + a x\right )}}\right ) + 2 \, {\left (3 \, a x^{4} + 3 \, a + x\right )} \sqrt {x^{5} + x}}{3 \, {\left (x^{8} + 2 \, x^{4} + 1\right )}}\right ] \]

[In]

integrate(x^2*(3*x^4-1)/(x^4+1)^2/(a*x^4+a-x)/(x^5+x)^(1/2),x, algorithm="fricas")

[Out]

[1/6*(3*(a*x^8 + 2*a*x^4 + a)*sqrt(a)*log((a^2*x^8 + 2*a^2*x^4 + 6*a*x^5 - 4*(a*x^4 + a + x)*sqrt(x^5 + x)*sqr
t(a) + a^2 + 6*a*x + x^2)/(a^2*x^8 + 2*a^2*x^4 - 2*a*x^5 + a^2 - 2*a*x + x^2)) + 4*(3*a*x^4 + 3*a + x)*sqrt(x^
5 + x))/(x^8 + 2*x^4 + 1), 1/3*(3*(a*x^8 + 2*a*x^4 + a)*sqrt(-a)*arctan(1/2*(a*x^4 + a + x)*sqrt(x^5 + x)*sqrt
(-a)/(a*x^5 + a*x)) + 2*(3*a*x^4 + 3*a + x)*sqrt(x^5 + x))/(x^8 + 2*x^4 + 1)]

Sympy [F(-1)]

Timed out. \[ \int \frac {x^2 \left (-1+3 x^4\right )}{\left (1+x^4\right )^2 \left (a-x+a x^4\right ) \sqrt {x+x^5}} \, dx=\text {Timed out} \]

[In]

integrate(x**2*(3*x**4-1)/(x**4+1)**2/(a*x**4+a-x)/(x**5+x)**(1/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {x^2 \left (-1+3 x^4\right )}{\left (1+x^4\right )^2 \left (a-x+a x^4\right ) \sqrt {x+x^5}} \, dx=\int { \frac {{\left (3 \, x^{4} - 1\right )} x^{2}}{{\left (a x^{4} + a - x\right )} \sqrt {x^{5} + x} {\left (x^{4} + 1\right )}^{2}} \,d x } \]

[In]

integrate(x^2*(3*x^4-1)/(x^4+1)^2/(a*x^4+a-x)/(x^5+x)^(1/2),x, algorithm="maxima")

[Out]

integrate((3*x^4 - 1)*x^2/((a*x^4 + a - x)*sqrt(x^5 + x)*(x^4 + 1)^2), x)

Giac [F]

\[ \int \frac {x^2 \left (-1+3 x^4\right )}{\left (1+x^4\right )^2 \left (a-x+a x^4\right ) \sqrt {x+x^5}} \, dx=\int { \frac {{\left (3 \, x^{4} - 1\right )} x^{2}}{{\left (a x^{4} + a - x\right )} \sqrt {x^{5} + x} {\left (x^{4} + 1\right )}^{2}} \,d x } \]

[In]

integrate(x^2*(3*x^4-1)/(x^4+1)^2/(a*x^4+a-x)/(x^5+x)^(1/2),x, algorithm="giac")

[Out]

integrate((3*x^4 - 1)*x^2/((a*x^4 + a - x)*sqrt(x^5 + x)*(x^4 + 1)^2), x)

Mupad [B] (verification not implemented)

Time = 6.16 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.18 \[ \int \frac {x^2 \left (-1+3 x^4\right )}{\left (1+x^4\right )^2 \left (a-x+a x^4\right ) \sqrt {x+x^5}} \, dx=a^{3/2}\,\ln \left (\frac {a+x-2\,\sqrt {a}\,\sqrt {x^5+x}+a\,x^4}{a\,x^4-x+a}\right )+\frac {2\,a\,\sqrt {x^5+x}}{x^4+1}+\frac {2\,x\,\sqrt {x^5+x}}{3\,{\left (x^4+1\right )}^2} \]

[In]

int((x^2*(3*x^4 - 1))/((x^4 + 1)^2*(x + x^5)^(1/2)*(a - x + a*x^4)),x)

[Out]

a^(3/2)*log((a + x - 2*a^(1/2)*(x + x^5)^(1/2) + a*x^4)/(a - x + a*x^4)) + (2*a*(x + x^5)^(1/2))/(x^4 + 1) + (
2*x*(x + x^5)^(1/2))/(3*(x^4 + 1)^2)