3.3.48 \(\int \frac {-1+4 x^5}{(1-a x+x^5) \sqrt {x+x^6}} \, dx\)

Optimal. Leaf size=24 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {x^6+x}}\right )}{\sqrt {a}} \]

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Rubi [F]  time = 1.07, 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 {-1+4 x^5}{\left (1-a x+x^5\right ) \sqrt {x+x^6}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-1 + 4*x^5)/((1 - a*x + x^5)*Sqrt[x + x^6]),x]

[Out]

(8*x*Sqrt[1 + x^5]*Hypergeometric2F1[1/10, 1/2, 11/10, -x^5])/Sqrt[x + x^6] + (10*Sqrt[x]*Sqrt[1 + x^5]*Defer[
Subst][Defer[Int][1/((-1 + a*x^2 - x^10)*Sqrt[1 + x^10]), x], x, Sqrt[x]])/Sqrt[x + x^6] - (8*a*Sqrt[x]*Sqrt[1
+ x^5]*Defer[Subst][Defer[Int][x^2/((-1 + a*x^2 - x^10)*Sqrt[1 + x^10]), x], x, Sqrt[x]])/Sqrt[x + x^6]

Rubi steps

\begin {align*} \int \frac {-1+4 x^5}{\left (1-a x+x^5\right ) \sqrt {x+x^6}} \, dx &=\frac {\left (\sqrt {x} \sqrt {1+x^5}\right ) \int \frac {-1+4 x^5}{\sqrt {x} \sqrt {1+x^5} \left (1-a x+x^5\right )} \, dx}{\sqrt {x+x^6}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {1+x^5}\right ) \operatorname {Subst}\left (\int \frac {-1+4 x^{10}}{\sqrt {1+x^{10}} \left (1-a x^2+x^{10}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^6}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {1+x^5}\right ) \operatorname {Subst}\left (\int \left (\frac {4}{\sqrt {1+x^{10}}}-\frac {5-4 a x^2}{\sqrt {1+x^{10}} \left (1-a x^2+x^{10}\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^6}}\\ &=-\frac {\left (2 \sqrt {x} \sqrt {1+x^5}\right ) \operatorname {Subst}\left (\int \frac {5-4 a x^2}{\sqrt {1+x^{10}} \left (1-a x^2+x^{10}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^6}}+\frac {\left (8 \sqrt {x} \sqrt {1+x^5}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^{10}}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^6}}\\ &=\frac {8 x \sqrt {1+x^5} \, _2F_1\left (\frac {1}{10},\frac {1}{2};\frac {11}{10};-x^5\right )}{\sqrt {x+x^6}}-\frac {\left (2 \sqrt {x} \sqrt {1+x^5}\right ) \operatorname {Subst}\left (\int \left (-\frac {5}{\left (-1+a x^2-x^{10}\right ) \sqrt {1+x^{10}}}+\frac {4 a x^2}{\left (-1+a x^2-x^{10}\right ) \sqrt {1+x^{10}}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^6}}\\ &=\frac {8 x \sqrt {1+x^5} \, _2F_1\left (\frac {1}{10},\frac {1}{2};\frac {11}{10};-x^5\right )}{\sqrt {x+x^6}}+\frac {\left (10 \sqrt {x} \sqrt {1+x^5}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+a x^2-x^{10}\right ) \sqrt {1+x^{10}}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^6}}-\frac {\left (8 a \sqrt {x} \sqrt {1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-1+a x^2-x^{10}\right ) \sqrt {1+x^{10}}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^6}}\\ \end {align*}

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Mathematica [F]  time = 0.47, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-1+4 x^5}{\left (1-a x+x^5\right ) \sqrt {x+x^6}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(-1 + 4*x^5)/((1 - a*x + x^5)*Sqrt[x + x^6]),x]

[Out]

Integrate[(-1 + 4*x^5)/((1 - a*x + x^5)*Sqrt[x + x^6]), x]

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IntegrateAlgebraic [A]  time = 2.67, size = 24, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {x+x^6}}\right )}{\sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-1 + 4*x^5)/((1 - a*x + x^5)*Sqrt[x + x^6]),x]

[Out]

(-2*ArcTanh[(Sqrt[a]*x)/Sqrt[x + x^6]])/Sqrt[a]

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fricas [A]  time = 0.69, size = 119, normalized size = 4.96 \begin {gather*} \left [\frac {\log \left (-\frac {x^{10} + 6 \, a x^{6} + 2 \, x^{5} + a^{2} x^{2} - 4 \, \sqrt {x^{6} + x} {\left (x^{5} + a x + 1\right )} \sqrt {a} + 6 \, a x + 1}{x^{10} - 2 \, a x^{6} + 2 \, x^{5} + a^{2} x^{2} - 2 \, a x + 1}\right )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {2 \, \sqrt {x^{6} + x} \sqrt {-a}}{x^{5} + a x + 1}\right )}{a}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^5-1)/(x^5-a*x+1)/(x^6+x)^(1/2),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {4 \, x^{5} - 1}{\sqrt {x^{6} + x} {\left (x^{5} - a x + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^5-1)/(x^5-a*x+1)/(x^6+x)^(1/2),x, algorithm="giac")

[Out]

integrate((4*x^5 - 1)/(sqrt(x^6 + x)*(x^5 - a*x + 1)), x)

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {4 x^{5}-1}{\left (x^{5}-a x +1\right ) \sqrt {x^{6}+x}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^5-1)/(x^5-a*x+1)/(x^6+x)^(1/2),x)

[Out]

int((4*x^5-1)/(x^5-a*x+1)/(x^6+x)^(1/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {4 \, x^{5} - 1}{\sqrt {x^{6} + x} {\left (x^{5} - a x + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^5-1)/(x^5-a*x+1)/(x^6+x)^(1/2),x, algorithm="maxima")

[Out]

integrate((4*x^5 - 1)/(sqrt(x^6 + x)*(x^5 - a*x + 1)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {4\,x^5-1}{\sqrt {x^6+x}\,\left (x^5-a\,x+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^5 - 1)/((x + x^6)^(1/2)*(x^5 - a*x + 1)),x)

[Out]

int((4*x^5 - 1)/((x + x^6)^(1/2)*(x^5 - a*x + 1)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**5-1)/(x**5-a*x+1)/(x**6+x)**(1/2),x)

[Out]

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

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