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

Optimal. Leaf size=43 \[ \frac {2 \sqrt {x^6+x}}{x^5+1}-2 \sqrt {a} \tanh ^{-1}\left (\frac {x}{\sqrt {a} \sqrt {x^6+x}}\right ) \]

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Rubi [F]  time = 1.75, 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+4 x^6}{\left (1+x^5\right ) \left (a-x+a x^5\right ) \sqrt {x+x^6}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(8*x^2)/(5*a*Sqrt[x + x^6]) + (16*x^2*Sqrt[1 + x^5]*Hypergeometric2F1[3/10, 1/2, 13/10, -x^5])/(15*a*Sqrt[x +
x^6]) - (10*Sqrt[x]*Sqrt[1 + x^5]*Defer[Subst][Defer[Int][x^2/((1 + x^10)^(3/2)*(a - x^2 + a*x^10)), x], x, Sq
rt[x]])/Sqrt[x + x^6] + (8*Sqrt[x]*Sqrt[1 + x^5]*Defer[Subst][Defer[Int][x^4/((1 + x^10)^(3/2)*(a - x^2 + a*x^
10)), x], x, Sqrt[x]])/(a*Sqrt[x + x^6])

Rubi steps

\begin {align*} \int \frac {-x+4 x^6}{\left (1+x^5\right ) \left (a-x+a x^5\right ) \sqrt {x+x^6}} \, dx &=\int \frac {x \left (-1+4 x^5\right )}{\left (1+x^5\right ) \left (a-x+a x^5\right ) \sqrt {x+x^6}} \, dx\\ &=\frac {\left (\sqrt {x} \sqrt {1+x^5}\right ) \int \frac {\sqrt {x} \left (-1+4 x^5\right )}{\left (1+x^5\right )^{3/2} \left (a-x+a x^5\right )} \, dx}{\sqrt {x+x^6}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (-1+4 x^{10}\right )}{\left (1+x^{10}\right )^{3/2} \left (a-x^2+a 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 x^2}{a \left (1+x^{10}\right )^{3/2}}+\frac {x^2 \left (-5 a+4 x^2\right )}{a \left (1+x^{10}\right )^{3/2} \left (a-x^2+a 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 {x^2 \left (-5 a+4 x^2\right )}{\left (1+x^{10}\right )^{3/2} \left (a-x^2+a x^{10}\right )} \, dx,x,\sqrt {x}\right )}{a \sqrt {x+x^6}}+\frac {\left (8 \sqrt {x} \sqrt {1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (1+x^{10}\right )^{3/2}} \, dx,x,\sqrt {x}\right )}{a \sqrt {x+x^6}}\\ &=\frac {8 x^2}{5 a \sqrt {x+x^6}}+\frac {\left (2 \sqrt {x} \sqrt {1+x^5}\right ) \operatorname {Subst}\left (\int \left (-\frac {5 a x^2}{\left (1+x^{10}\right )^{3/2} \left (a-x^2+a x^{10}\right )}+\frac {4 x^4}{\left (1+x^{10}\right )^{3/2} \left (a-x^2+a x^{10}\right )}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {x+x^6}}+\frac {\left (16 \sqrt {x} \sqrt {1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+x^{10}}} \, dx,x,\sqrt {x}\right )}{5 a \sqrt {x+x^6}}\\ &=\frac {8 x^2}{5 a \sqrt {x+x^6}}+\frac {16 x^2 \sqrt {1+x^5} \, _2F_1\left (\frac {3}{10},\frac {1}{2};\frac {13}{10};-x^5\right )}{15 a \sqrt {x+x^6}}-\frac {\left (10 \sqrt {x} \sqrt {1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (1+x^{10}\right )^{3/2} \left (a-x^2+a 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 {x^4}{\left (1+x^{10}\right )^{3/2} \left (a-x^2+a x^{10}\right )} \, dx,x,\sqrt {x}\right )}{a \sqrt {x+x^6}}\\ \end {align*}

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Mathematica [F]  time = 0.64, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-x+4 x^6}{\left (1+x^5\right ) \left (a-x+a x^5\right ) \sqrt {x+x^6}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 2.70, size = 43, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {x+x^6}}{1+x^5}-2 \sqrt {a} \tanh ^{-1}\left (\frac {x}{\sqrt {a} \sqrt {x+x^6}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[x + x^6])/(1 + x^5) - 2*Sqrt[a]*ArcTanh[x/(Sqrt[a]*Sqrt[x + x^6])]

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fricas [A]  time = 0.65, size = 172, normalized size = 4.00 \begin {gather*} \left [\frac {{\left (x^{5} + 1\right )} \sqrt {a} \log \left (-\frac {a^{2} x^{10} + 2 \, a^{2} x^{5} + 6 \, a x^{6} - 4 \, {\left (a x^{5} + a + x\right )} \sqrt {x^{6} + x} \sqrt {a} + a^{2} + 6 \, a x + x^{2}}{a^{2} x^{10} + 2 \, a^{2} x^{5} - 2 \, a x^{6} + a^{2} - 2 \, a x + x^{2}}\right ) + 4 \, \sqrt {x^{6} + x}}{2 \, {\left (x^{5} + 1\right )}}, \frac {{\left (x^{5} + 1\right )} \sqrt {-a} \arctan \left (\frac {2 \, \sqrt {x^{6} + x} \sqrt {-a}}{a x^{5} + a + x}\right ) + 2 \, \sqrt {x^{6} + x}}{x^{5} + 1}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {4 x^{6}-x}{\left (x^{5}+1\right ) \left (a \,x^{5}+a -x \right ) \sqrt {x^{6}+x}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((4*x^6-x)/(x^5+1)/(a*x^5+a-x)/(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^{6} - x}{{\left (a x^{5} + a - x\right )} \sqrt {x^{6} + x} {\left (x^{5} + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int -\frac {x-4\,x^6}{\left (x^5+1\right )\,\sqrt {x^6+x}\,\left (a\,x^5-x+a\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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