3.3.97 \(\int \frac {x^4 (9+4 x^5)}{\sqrt {x+x^6} (-1-x^5+a x^9)} \, dx\)

Optimal. Leaf size=26 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {x^6+x}}{\sqrt {a} x^5}\right )}{\sqrt {a}} \]

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Rubi [F]  time = 1.54, 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 \left (9+4 x^5\right )}{\sqrt {x+x^6} \left (-1-x^5+a x^9\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x^4*(9 + 4*x^5))/(Sqrt[x + x^6]*(-1 - x^5 + a*x^9)),x]

[Out]

(8*x*Sqrt[1 + x^5]*Hypergeometric2F1[1/10, 1/2, 11/10, -x^5])/(a*Sqrt[x + x^6]) + (8*Sqrt[x]*Sqrt[1 + x^5]*Def
er[Subst][Defer[Int][1/(Sqrt[1 + x^10]*(-1 - x^10 + a*x^18)), x], x, Sqrt[x]])/(a*Sqrt[x + x^6]) + (18*Sqrt[x]
*Sqrt[1 + x^5]*Defer[Subst][Defer[Int][x^8/(Sqrt[1 + x^10]*(-1 - x^10 + a*x^18)), x], x, Sqrt[x]])/Sqrt[x + x^
6] + (8*Sqrt[x]*Sqrt[1 + x^5]*Defer[Subst][Defer[Int][x^10/(Sqrt[1 + x^10]*(-1 - x^10 + a*x^18)), x], x, Sqrt[
x]])/(a*Sqrt[x + x^6])

Rubi steps

\begin {align*} \int \frac {x^4 \left (9+4 x^5\right )}{\sqrt {x+x^6} \left (-1-x^5+a x^9\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt {1+x^5}\right ) \int \frac {x^{7/2} \left (9+4 x^5\right )}{\sqrt {1+x^5} \left (-1-x^5+a x^9\right )} \, dx}{\sqrt {x+x^6}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^8 \left (9+4 x^{10}\right )}{\sqrt {1+x^{10}} \left (-1-x^{10}+a x^{18}\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}{a \sqrt {1+x^{10}}}+\frac {4+9 a x^8+4 x^{10}}{a \sqrt {1+x^{10}} \left (-1-x^{10}+a x^{18}\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 {4+9 a x^8+4 x^{10}}{\sqrt {1+x^{10}} \left (-1-x^{10}+a x^{18}\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 {1}{\sqrt {1+x^{10}}} \, dx,x,\sqrt {x}\right )}{a \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 )}{a \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}} \left (-1-x^{10}+a x^{18}\right )}+\frac {9 a x^8}{\sqrt {1+x^{10}} \left (-1-x^{10}+a x^{18}\right )}+\frac {4 x^{10}}{\sqrt {1+x^{10}} \left (-1-x^{10}+a x^{18}\right )}\right ) \, dx,x,\sqrt {x}\right )}{a \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 )}{a \sqrt {x+x^6}}+\frac {\left (18 \sqrt {x} \sqrt {1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^8}{\sqrt {1+x^{10}} \left (-1-x^{10}+a x^{18}\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}} \left (-1-x^{10}+a x^{18}\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^{10}}{\sqrt {1+x^{10}} \left (-1-x^{10}+a x^{18}\right )} \, dx,x,\sqrt {x}\right )}{a \sqrt {x+x^6}}\\ \end {align*}

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Mathematica [F]  time = 0.34, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^4 \left (9+4 x^5\right )}{\sqrt {x+x^6} \left (-1-x^5+a x^9\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(x^4*(9 + 4*x^5))/(Sqrt[x + x^6]*(-1 - x^5 + a*x^9)),x]

[Out]

Integrate[(x^4*(9 + 4*x^5))/(Sqrt[x + x^6]*(-1 - x^5 + a*x^9)), x]

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IntegrateAlgebraic [A]  time = 9.48, size = 26, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {x+x^6}}{\sqrt {a} x^5}\right )}{\sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^4*(9 + 4*x^5))/(Sqrt[x + x^6]*(-1 - x^5 + a*x^9)),x]

[Out]

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

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fricas [A]  time = 1.13, size = 132, normalized size = 5.08 \begin {gather*} \left [\frac {\log \left (-\frac {a^{2} x^{18} + 6 \, a x^{14} + 6 \, a x^{9} + x^{10} + 2 \, x^{5} - 4 \, {\left (a x^{13} + x^{9} + x^{4}\right )} \sqrt {x^{6} + x} \sqrt {a} + 1}{a^{2} x^{18} - 2 \, a x^{14} - 2 \, a x^{9} + x^{10} + 2 \, x^{5} + 1}\right )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {2 \, \sqrt {x^{6} + x} \sqrt {-a} x^{4}}{a x^{9} + x^{5} + 1}\right )}{a}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(4*x^5+9)/(x^6+x)^(1/2)/(a*x^9-x^5-1),x, algorithm="fricas")

[Out]

[1/2*log(-(a^2*x^18 + 6*a*x^14 + 6*a*x^9 + x^10 + 2*x^5 - 4*(a*x^13 + x^9 + x^4)*sqrt(x^6 + x)*sqrt(a) + 1)/(a
^2*x^18 - 2*a*x^14 - 2*a*x^9 + x^10 + 2*x^5 + 1))/sqrt(a), sqrt(-a)*arctan(2*sqrt(x^6 + x)*sqrt(-a)*x^4/(a*x^9
+ x^5 + 1))/a]

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (4 \, x^{5} + 9\right )} x^{4}}{{\left (a x^{9} - x^{5} - 1\right )} \sqrt {x^{6} + x}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(4*x^5+9)/(x^6+x)^(1/2)/(a*x^9-x^5-1),x, algorithm="giac")

[Out]

integrate((4*x^5 + 9)*x^4/((a*x^9 - x^5 - 1)*sqrt(x^6 + x)), x)

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x^{4} \left (4 x^{5}+9\right )}{\sqrt {x^{6}+x}\, \left (a \,x^{9}-x^{5}-1\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(4*x^5+9)/(x^6+x)^(1/2)/(a*x^9-x^5-1),x)

[Out]

int(x^4*(4*x^5+9)/(x^6+x)^(1/2)/(a*x^9-x^5-1),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (4 \, x^{5} + 9\right )} x^{4}}{{\left (a x^{9} - x^{5} - 1\right )} \sqrt {x^{6} + x}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(4*x^5+9)/(x^6+x)^(1/2)/(a*x^9-x^5-1),x, algorithm="maxima")

[Out]

integrate((4*x^5 + 9)*x^4/((a*x^9 - x^5 - 1)*sqrt(x^6 + x)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {x^4\,\left (4\,x^5+9\right )}{\sqrt {x^6+x}\,\left (-a\,x^9+x^5+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^4*(4*x^5 + 9))/((x + x^6)^(1/2)*(x^5 - a*x^9 + 1)),x)

[Out]

int(-(x^4*(4*x^5 + 9))/((x + x^6)^(1/2)*(x^5 - a*x^9 + 1)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} \left (4 x^{5} + 9\right )}{\sqrt {x \left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )} \left (a x^{9} - x^{5} - 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(4*x**5+9)/(x**6+x)**(1/2)/(a*x**9-x**5-1),x)

[Out]

Integral(x**4*(4*x**5 + 9)/(sqrt(x*(x + 1)*(x**4 - x**3 + x**2 - x + 1))*(a*x**9 - x**5 - 1)), x)

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