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

Optimal. Leaf size=39 \[ \frac {2 \sqrt {x^6+x}}{x}-2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {x^6+x}}\right ) \]

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Rubi [F]  time = 1.24, 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^5\right ) \left (-1+4 x^5\right )}{x \left (1-a x+x^5\right ) \sqrt {x+x^6}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(2*(1 + x^5))/Sqrt[x + x^6] - (10*x^5*Sqrt[1 + x^5]*Hypergeometric2F1[1/2, 9/10, 19/10, -x^5])/(9*Sqrt[x + x^6
]) + (2*a*Sqrt[x]*Sqrt[1 + x^5]*Defer[Subst][Defer[Int][Sqrt[1 + x^10]/(-1 + a*x^2 - x^10), x], x, Sqrt[x]])/S
qrt[x + x^6] + (10*Sqrt[x]*Sqrt[1 + x^5]*Defer[Subst][Defer[Int][(x^8*Sqrt[1 + x^10])/(1 - a*x^2 + x^10), x],
x, Sqrt[x]])/Sqrt[x + x^6]

Rubi steps

\begin {align*} \int \frac {\left (1+x^5\right ) \left (-1+4 x^5\right )}{x \left (1-a x+x^5\right ) \sqrt {x+x^6}} \, dx &=\frac {\left (\sqrt {x} \sqrt {1+x^5}\right ) \int \frac {\sqrt {1+x^5} \left (-1+4 x^5\right )}{x^{3/2} \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 {\sqrt {1+x^{10}} \left (-1+4 x^{10}\right )}{x^2 \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 {\sqrt {1+x^{10}}}{x^2}+\frac {\left (a-5 x^8\right ) \sqrt {1+x^{10}}}{-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 \frac {\sqrt {1+x^{10}}}{x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^6}}+\frac {\left (2 \sqrt {x} \sqrt {1+x^5}\right ) \operatorname {Subst}\left (\int \frac {\left (a-5 x^8\right ) \sqrt {1+x^{10}}}{-1+a x^2-x^{10}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^6}}\\ &=\frac {2 \left (1+x^5\right )}{\sqrt {x+x^6}}+\frac {\left (2 \sqrt {x} \sqrt {1+x^5}\right ) \operatorname {Subst}\left (\int \left (\frac {a \sqrt {1+x^{10}}}{-1+a x^2-x^{10}}+\frac {5 x^8 \sqrt {1+x^{10}}}{1-a x^2+x^{10}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^6}}-\frac {\left (10 \sqrt {x} \sqrt {1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^8}{\sqrt {1+x^{10}}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^6}}\\ &=\frac {2 \left (1+x^5\right )}{\sqrt {x+x^6}}-\frac {10 x^5 \sqrt {1+x^5} \, _2F_1\left (\frac {1}{2},\frac {9}{10};\frac {19}{10};-x^5\right )}{9 \sqrt {x+x^6}}+\frac {\left (10 \sqrt {x} \sqrt {1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^8 \sqrt {1+x^{10}}}{1-a x^2+x^{10}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^6}}+\frac {\left (2 a \sqrt {x} \sqrt {1+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^{10}}}{-1+a x^2-x^{10}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^6}}\\ \end {align*}

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Mathematica [F]  time = 0.65, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1+x^5\right ) \left (-1+4 x^5\right )}{x \left (1-a x+x^5\right ) \sqrt {x+x^6}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.59, size = 39, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {x+x^6}}{x}-2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {x+x^6}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 3.98, size = 146, normalized size = 3.74 \begin {gather*} \left [\frac {\sqrt {a} x \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 ) + 4 \, \sqrt {x^{6} + x}}{2 \, x}, \frac {\sqrt {-a} x \arctan \left (\frac {2 \, \sqrt {x^{6} + x} \sqrt {-a}}{x^{5} + a x + 1}\right ) + 2 \, \sqrt {x^{6} + x}}{x}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\left (x^5+1\right )\,\left (4\,x^5-1\right )}{x\,\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(((x^5 + 1)*(4*x^5 - 1))/(x*(x + x^6)^(1/2)*(x^5 - a*x + 1)),x)

[Out]

int(((x^5 + 1)*(4*x^5 - 1))/(x*(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 {\left (x + 1\right ) \left (4 x^{5} - 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )}{x \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((x**5+1)*(4*x**5-1)/x/(x**5-a*x+1)/(x**6+x)**(1/2),x)

[Out]

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

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