\(\int \frac {-x+3 x^5}{(1+x^4) (a-x+a x^4) \sqrt {x+x^5}} \, dx\) [626]

Optimal result

Integrand size = 38, antiderivative size = 49 \[ \int \frac {-x+3 x^5}{\left (1+x^4\right ) \left (a-x+a x^4\right ) \sqrt {x+x^5}} \, dx=\frac {2 \sqrt {x+x^5}}{1+x^4}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {x+x^5}}{\sqrt {a} \left (1+x^4\right )}\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

(3*x^2)/(2*a*Sqrt[x + x^5]) - (3*x^2*Sqrt[(1 + x)^2/x]*Sqrt[-((1 + x^4)/x^2)]*EllipticF[ArcSin[Sqrt[-((Sqrt[2]
- 2*x + Sqrt[2]*x^2)/x)]/2], -2*(1 - Sqrt[2])])/(4*Sqrt[2 + Sqrt[2]]*a*(1 + x)*Sqrt[x + x^5]) - (3*Sqrt[-((1
- x)^2/x)]*x^2*Sqrt[-((1 + x^4)/x^2)]*EllipticF[ArcSin[Sqrt[(Sqrt[2] + 2*x + Sqrt[2]*x^2)/x]/2], -2*(1 - Sqrt[
2])])/(4*Sqrt[2 + Sqrt[2]]*a*(1 - x)*Sqrt[x + x^5]) - (8*Sqrt[x]*Sqrt[1 + x^4]*Defer[Subst][Defer[Int][x^2/((1
+ x^8)^(3/2)*(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^4/((1 + x^8)^(3/2)*(a - x^2 + a*x^8)), x], x, Sqrt[x]])/(a*Sqrt[x + x^5])

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

Mathematica [F]

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

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.38 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.92

method	result	size
pseudoelliptic	\(\frac {-2 \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (x^{4}+1\right )}\, \sqrt {a}}{x}\right ) \sqrt {x \left (x^{4}+1\right )}+2 x}{\sqrt {x \left (x^{4}+1\right )}}\)	\(45\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 180, normalized size of antiderivative = 3.67 \[ \int \frac {-x+3 x^5}{\left (1+x^4\right ) \left (a-x+a x^4\right ) \sqrt {x+x^5}} \, dx=\left [\frac {{\left (x^{4} + 1\right )} \sqrt {a} \log \left (\frac {a^{2} x^{8} + 2 \, a^{2} x^{4} + 6 \, a x^{5} - 4 \, {\left (a x^{4} + a + x\right )} \sqrt {x^{5} + x} \sqrt {a} + a^{2} + 6 \, a x + x^{2}}{a^{2} x^{8} + 2 \, a^{2} x^{4} - 2 \, a x^{5} + a^{2} - 2 \, a x + x^{2}}\right ) + 4 \, \sqrt {x^{5} + x}}{2 \, {\left (x^{4} + 1\right )}}, \frac {{\left (x^{4} + 1\right )} \sqrt {-a} \arctan \left (\frac {{\left (a x^{4} + a + x\right )} \sqrt {x^{5} + x} \sqrt {-a}}{2 \, {\left (a x^{5} + a x\right )}}\right ) + 2 \, \sqrt {x^{5} + x}}{x^{4} + 1}\right ] \]

[In]

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

[Out]

[1/2*((x^4 + 1)*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)*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)) + 4*sqrt(x^5 + x))/(x^4 + 1), ((x^4 + 1)*sqrt
(-a)*arctan(1/2*(a*x^4 + a + x)*sqrt(x^5 + x)*sqrt(-a)/(a*x^5 + a*x)) + 2*sqrt(x^5 + x))/(x^4 + 1)]

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 6.13 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.12 \[ \int \frac {-x+3 x^5}{\left (1+x^4\right ) \left (a-x+a x^4\right ) \sqrt {x+x^5}} \, dx=\frac {2\,\sqrt {x^5+x}}{x^4+1}+\sqrt {a}\,\ln \left (\frac {a+x-2\,\sqrt {a}\,\sqrt {x^5+x}+a\,x^4}{a\,x^4-x+a}\right ) \]

[In]

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

[Out]

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