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

Optimal. Leaf size=43 \[ \frac {2 \sqrt {x^5-x}}{x}-2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {x^5-x}}\right ) \]

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Rubi [F]  time = 1.40, 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 {\left (-1+x^4\right ) \left (1+3 x^4\right )}{x \left (-1-a x+x^4\right ) \sqrt {-x+x^5}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(-2*(1 - x^4))/Sqrt[-x + x^5] - (8*x^4*Sqrt[1 - x^4]*Hypergeometric2F1[1/2, 7/8, 15/8, x^4])/(7*Sqrt[-x + x^5]
) + (2*a*Sqrt[x]*Sqrt[-1 + x^4]*Defer[Subst][Defer[Int][Sqrt[-1 + x^8]/(1 + a*x^2 - x^8), x], x, Sqrt[x]])/Sqr
t[-x + x^5] + (8*Sqrt[x]*Sqrt[-1 + x^4]*Defer[Subst][Defer[Int][(x^6*Sqrt[-1 + x^8])/(-1 - a*x^2 + x^8), x], x
, Sqrt[x]])/Sqrt[-x + x^5]

Rubi steps

\begin {align*} \int \frac {\left (-1+x^4\right ) \left (1+3 x^4\right )}{x \left (-1-a x+x^4\right ) \sqrt {-x+x^5}} \, dx &=\frac {\left (\sqrt {x} \sqrt {-1+x^4}\right ) \int \frac {\sqrt {-1+x^4} \left (1+3 x^4\right )}{x^{3/2} \left (-1-a x+x^4\right )} \, dx}{\sqrt {-x+x^5}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-1+x^8} \left (1+3 x^8\right )}{x^2 \left (-1-a x^2+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 {\sqrt {-1+x^8}}{x^2}+\frac {\left (a-4 x^6\right ) \sqrt {-1+x^8}}{1+a x^2-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 \frac {\sqrt {-1+x^8}}{x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^5}}+\frac {\left (2 \sqrt {x} \sqrt {-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {\left (a-4 x^6\right ) \sqrt {-1+x^8}}{1+a x^2-x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^5}}\\ &=-\frac {2 \left (1-x^4\right )}{\sqrt {-x+x^5}}+\frac {\left (2 \sqrt {x} \sqrt {-1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {a \sqrt {-1+x^8}}{1+a x^2-x^8}+\frac {4 x^6 \sqrt {-1+x^8}}{-1-a x^2+x^8}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^5}}-\frac {\left (8 \sqrt {x} \sqrt {-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\sqrt {-1+x^8}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^5}}\\ &=-\frac {2 \left (1-x^4\right )}{\sqrt {-x+x^5}}-\frac {\left (8 \sqrt {x} \sqrt {1-x^4}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\sqrt {1-x^8}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^5}}+\frac {\left (8 \sqrt {x} \sqrt {-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^6 \sqrt {-1+x^8}}{-1-a x^2+x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^5}}+\frac {\left (2 a \sqrt {x} \sqrt {-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-1+x^8}}{1+a x^2-x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^5}}\\ &=-\frac {2 \left (1-x^4\right )}{\sqrt {-x+x^5}}-\frac {8 x^4 \sqrt {1-x^4} \, _2F_1\left (\frac {1}{2},\frac {7}{8};\frac {15}{8};x^4\right )}{7 \sqrt {-x+x^5}}+\frac {\left (8 \sqrt {x} \sqrt {-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^6 \sqrt {-1+x^8}}{-1-a x^2+x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^5}}+\frac {\left (2 a \sqrt {x} \sqrt {-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-1+x^8}}{1+a x^2-x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^5}}\\ \end {align*}

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Mathematica [F]  time = 0.67, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^4\right ) \left (1+3 x^4\right )}{x \left (-1-a x+x^4\right ) \sqrt {-x+x^5}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.34, size = 43, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {-x+x^5}}{x}-2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {-x+x^5}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 2.43, size = 154, normalized size = 3.58 \begin {gather*} \left [\frac {\sqrt {a} x \log \left (-\frac {x^{8} + 6 \, a x^{5} + a^{2} x^{2} - 2 \, x^{4} - 4 \, \sqrt {x^{5} - x} {\left (x^{4} + a x - 1\right )} \sqrt {a} - 6 \, a x + 1}{x^{8} - 2 \, a x^{5} + a^{2} x^{2} - 2 \, x^{4} + 2 \, a x + 1}\right ) + 4 \, \sqrt {x^{5} - x}}{2 \, x}, \frac {\sqrt {-a} x \arctan \left (\frac {2 \, \sqrt {x^{5} - x} \sqrt {-a}}{x^{4} + a x - 1}\right ) + 2 \, \sqrt {x^{5} - x}}{x}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((x^4-1)*(3*x^4+1)/x/(x^4-a*x-1)/(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 {{\left (3 \, x^{4} + 1\right )} {\left (x^{4} - 1\right )}}{\sqrt {x^{5} - x} {\left (x^{4} - a x - 1\right )} x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.57, size = 55, normalized size = 1.28 \begin {gather*} \sqrt {a}\,\ln \left (\frac {a\,x-2\,\sqrt {a}\,\sqrt {x^5-x}+x^4-1}{-x^4+a\,x+1}\right )+\frac {2\,\sqrt {x^5-x}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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