3.6.9 \(\int \frac {\sqrt {-1+x^5} (2+3 x^5)}{x^2 (-1-a x^2+x^5)} \, dx\)

Optimal. Leaf size=39 \[ \frac {2 \sqrt {x^5-1}}{x}-2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {x^5-1}}\right ) \]

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Rubi [F]  time = 0.79, 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 {\sqrt {-1+x^5} \left (2+3 x^5\right )}{x^2 \left (-1-a x^2+x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(2*Sqrt[-1 + x^5])/x - (5*x^4*Sqrt[1 - x^5]*Hypergeometric2F1[1/2, 4/5, 9/5, x^5])/(4*Sqrt[-1 + x^5]) + 2*a*De
fer[Int][Sqrt[-1 + x^5]/(1 + a*x^2 - x^5), x] + 5*Defer[Int][(x^3*Sqrt[-1 + x^5])/(-1 - a*x^2 + x^5), x]

Rubi steps

\begin {align*} \int \frac {\sqrt {-1+x^5} \left (2+3 x^5\right )}{x^2 \left (-1-a x^2+x^5\right )} \, dx &=\int \left (-\frac {2 \sqrt {-1+x^5}}{x^2}+\frac {\left (2 a-5 x^3\right ) \sqrt {-1+x^5}}{1+a x^2-x^5}\right ) \, dx\\ &=-\left (2 \int \frac {\sqrt {-1+x^5}}{x^2} \, dx\right )+\int \frac {\left (2 a-5 x^3\right ) \sqrt {-1+x^5}}{1+a x^2-x^5} \, dx\\ &=\frac {2 \sqrt {-1+x^5}}{x}-5 \int \frac {x^3}{\sqrt {-1+x^5}} \, dx+\int \left (\frac {2 a \sqrt {-1+x^5}}{1+a x^2-x^5}+\frac {5 x^3 \sqrt {-1+x^5}}{-1-a x^2+x^5}\right ) \, dx\\ &=\frac {2 \sqrt {-1+x^5}}{x}+5 \int \frac {x^3 \sqrt {-1+x^5}}{-1-a x^2+x^5} \, dx+(2 a) \int \frac {\sqrt {-1+x^5}}{1+a x^2-x^5} \, dx-\frac {\left (5 \sqrt {1-x^5}\right ) \int \frac {x^3}{\sqrt {1-x^5}} \, dx}{\sqrt {-1+x^5}}\\ &=\frac {2 \sqrt {-1+x^5}}{x}-\frac {5 x^4 \sqrt {1-x^5} \, _2F_1\left (\frac {1}{2},\frac {4}{5};\frac {9}{5};x^5\right )}{4 \sqrt {-1+x^5}}+5 \int \frac {x^3 \sqrt {-1+x^5}}{-1-a x^2+x^5} \, dx+(2 a) \int \frac {\sqrt {-1+x^5}}{1+a x^2-x^5} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.33, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {-1+x^5} \left (2+3 x^5\right )}{x^2 \left (-1-a x^2+x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 1.02, size = 39, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {-1+x^5}}{x}-2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {-1+x^5}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 2.96, size = 157, normalized size = 4.03 \begin {gather*} \left [\frac {\sqrt {a} x \log \left (-\frac {x^{10} + 6 \, a x^{7} + a^{2} x^{4} - 2 \, x^{5} - 6 \, a x^{2} - 4 \, {\left (x^{6} + a x^{3} - x\right )} \sqrt {x^{5} - 1} \sqrt {a} + 1}{x^{10} - 2 \, a x^{7} + a^{2} x^{4} - 2 \, x^{5} + 2 \, a x^{2} + 1}\right ) + 4 \, \sqrt {x^{5} - 1}}{2 \, x}, \frac {\sqrt {-a} x \arctan \left (\frac {2 \, \sqrt {x^{5} - 1} \sqrt {-a} x}{x^{5} + a x^{2} - 1}\right ) + 2 \, \sqrt {x^{5} - 1}}{x}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(sqrt(a)*x*log(-(x^10 + 6*a*x^7 + a^2*x^4 - 2*x^5 - 6*a*x^2 - 4*(x^6 + a*x^3 - x)*sqrt(x^5 - 1)*sqrt(a) +
1)/(x^10 - 2*a*x^7 + a^2*x^4 - 2*x^5 + 2*a*x^2 + 1)) + 4*sqrt(x^5 - 1))/x, (sqrt(-a)*x*arctan(2*sqrt(x^5 - 1)
*sqrt(-a)*x/(x^5 + a*x^2 - 1)) + 2*sqrt(x^5 - 1))/x]

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (3 \, x^{5} + 2\right )} \sqrt {x^{5} - 1}}{{\left (x^{5} - a x^{2} - 1\right )} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x^{5}-1}\, \left (3 x^{5}+2\right )}{x^{2} \left (x^{5}-a \,x^{2}-1\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (3 \, x^{5} + 2\right )} \sqrt {x^{5} - 1}}{{\left (x^{5} - a x^{2} - 1\right )} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.89, size = 56, normalized size = 1.44 \begin {gather*} \sqrt {a}\,\ln \left (\frac {a\,x^2+x^5-2\,\sqrt {a}\,x\,\sqrt {x^5-1}-1}{-x^5+a\,x^2+1}\right )+\frac {2\,\sqrt {x^5-1}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\left (x - 1\right ) \left (x^{4} + x^{3} + x^{2} + x + 1\right )} \left (3 x^{5} + 2\right )}{x^{2} \left (- a x^{2} + x^{5} - 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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