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

Optimal. Leaf size=49 \[ \frac {2 \sqrt {x^5+x}}{x^4+1}-2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {x^5+x}}{\sqrt {a} \left (x^4+1\right )}\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 2.00, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-x+3 x^5}{\left (1+x^4\right ) \left (a-x+a x^4\right ) \sqrt {x+x^5}} \, dx \end {gather*}

Verification is not applicable to the result.

[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*} \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 \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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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}} F\left (\sin ^{-1}\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}} F\left (\sin ^{-1}\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 ) \operatorname {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 ) \operatorname {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]  time = 0.64, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-x+3 x^5}{\left (1+x^4\right ) \left (a-x+a x^4\right ) \sqrt {x+x^5}} \, dx \end {gather*}

Verification is not applicable to the result.

[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]

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 1.87, size = 49, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {x+x^5}}{1+x^4}-2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {x+x^5}}{\sqrt {a} \left (1+x^4\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.52, size = 180, normalized size = 3.67 \begin {gather*} \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 ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)]

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 \, x^{5} - x}{{\left (a x^{4} + a - x\right )} \sqrt {x^{5} + x} {\left (x^{4} + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {3 x^{5}-x}{\left (x^{4}+1\right ) \left (a \,x^{4}+a -x \right ) \sqrt {x^{5}+x}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 \, x^{5} - x}{{\left (a x^{4} + a - x\right )} \sqrt {x^{5} + x} {\left (x^{4} + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

mupad [B]  time = 0.77, size = 55, normalized size = 1.12 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________