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

Optimal. Leaf size=47 \[ \frac {2 \sqrt {x^5-x}}{x^4-1}-2 \sqrt {a} \tanh ^{-1}\left (\frac {x}{\sqrt {a} \sqrt {x^5-x}}\right ) \]

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Rubi [F]  time = 2.23, 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)^(3/4)*(1 + (-1)^(1/4)*x)^2)/x)]*Sqrt[(I*(1 - x^4))/x^2]*El
lipticF[ArcSin[Sqrt[((-1)^(3/4)*(Sqrt[2] - 2*(-1)^(1/4)*x + I*Sqrt[2]*x^2))/x]/2], -2*(1 - Sqrt[2])])/(4*Sqrt[
2 + Sqrt[2]]*a*(1 + (-1)^(1/4)*x)*Sqrt[-x + x^5]) + (3*x^2*Sqrt[((-1)^(3/4)*(1 - (-1)^(1/4)*x)^2)/x]*Sqrt[(I*(
1 - x^4))/x^2]*EllipticF[ArcSin[Sqrt[-(((-1)^(3/4)*(Sqrt[2] + 2*(-1)^(1/4)*x + I*Sqrt[2]*x^2))/x)]/2], -2*(1 -
Sqrt[2])])/(4*Sqrt[2 + Sqrt[2]]*a*(1 - (-1)^(1/4)*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 (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 {\left (3 (-1)^{3/4} \sqrt {x} \sqrt {-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1-\sqrt [4]{-1} x^2}{\sqrt {-1+x^8}} \, dx,x,\sqrt {x}\right )}{4 a \sqrt {-x+x^5}}+\frac {\left (3 (-1)^{3/4} \sqrt {x} \sqrt {-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1+\sqrt [4]{-1} x^2}{\sqrt {-1+x^8}} \, dx,x,\sqrt {x}\right )}{4 a \sqrt {-x+x^5}}\\ &=-\frac {3 x^2}{2 a \sqrt {-x+x^5}}+\frac {3 x^2 \sqrt {-\frac {(-1)^{3/4} \left (1+\sqrt [4]{-1} x\right )^2}{x}} \sqrt {\frac {i \left (1-x^4\right )}{x^2}} F\left (\sin ^{-1}\left (\frac {1}{2} \sqrt {\frac {(-1)^{3/4} \left (\sqrt {2}-2 \sqrt [4]{-1} x+i \sqrt {2} x^2\right )}{x}}\right )|-2 \left (1-\sqrt {2}\right )\right )}{4 \sqrt {2+\sqrt {2}} a \left (1+\sqrt [4]{-1} x\right ) \sqrt {-x+x^5}}+\frac {3 x^2 \sqrt {\frac {(-1)^{3/4} \left (1-\sqrt [4]{-1} x\right )^2}{x}} \sqrt {\frac {i \left (1-x^4\right )}{x^2}} F\left (\sin ^{-1}\left (\frac {1}{2} \sqrt {-\frac {(-1)^{3/4} \left (\sqrt {2}+2 \sqrt [4]{-1} x+i \sqrt {2} x^2\right )}{x}}\right )|-2 \left (1-\sqrt {2}\right )\right )}{4 \sqrt {2+\sqrt {2}} a \left (1-\sqrt [4]{-1} x\right ) \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*}

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Mathematica [F]  time = 1.49, 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]

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IntegrateAlgebraic [A]  time = 2.02, size = 47, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {-x+x^5}}{-1+x^4}-2 \sqrt {a} \tanh ^{-1}\left (\frac {x}{\sqrt {a} \sqrt {-x+x^5}}\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[x/(Sqrt[a]*Sqrt[-x + x^5])]

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fricas [A]  time = 0.52, size = 193, normalized size = 4.11 \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)]

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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)

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maple [F]  time = 0.03, 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)

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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)

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mupad [B]  time = 0.79, size = 61, normalized size = 1.30 \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^5 - x)^(1/2)*(x^4 - 1)*(a + x - a*x^4)),x)

[Out]

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

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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

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