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

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

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Rubi [F]  time = 1.42, 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+5 x^4\right )}{\sqrt {x+x^5} \left (-1-x^4+a x^9\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(18*Sqrt[x]*Sqrt[1 + x^4]*Defer[Subst][Defer[Int][x^8/(Sqrt[1 + x^8]*(-1 - x^8 + a*x^18)), x], x, Sqrt[x]])/Sq
rt[x + x^5] + (10*Sqrt[x]*Sqrt[1 + x^4]*Defer[Subst][Defer[Int][x^16/(Sqrt[1 + x^8]*(-1 - x^8 + a*x^18)), x],
x, Sqrt[x]])/Sqrt[x + x^5]

Rubi steps

\begin {align*} \int \frac {x^4 \left (9+5 x^4\right )}{\sqrt {x+x^5} \left (-1-x^4+a x^9\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt {1+x^4}\right ) \int \frac {x^{7/2} \left (9+5 x^4\right )}{\sqrt {1+x^4} \left (-1-x^4+a x^9\right )} \, dx}{\sqrt {x+x^5}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^8 \left (9+5 x^8\right )}{\sqrt {1+x^8} \left (-1-x^8+a x^{18}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {9 x^8}{\sqrt {1+x^8} \left (-1-x^8+a x^{18}\right )}+\frac {5 x^{16}}{\sqrt {1+x^8} \left (-1-x^8+a x^{18}\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}}\\ &=\frac {\left (10 \sqrt {x} \sqrt {1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^{16}}{\sqrt {1+x^8} \left (-1-x^8+a x^{18}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}}+\frac {\left (18 \sqrt {x} \sqrt {1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^8}{\sqrt {1+x^8} \left (-1-x^8+a x^{18}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}}\\ \end {align*}

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Mathematica [F]  time = 0.27, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^4 \left (9+5 x^4\right )}{\sqrt {x+x^5} \left (-1-x^4+a x^9\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 4.56, size = 26, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {x+x^5}}{\sqrt {a} x^5}\right )}{\sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.67, size = 139, normalized size = 5.35 \begin {gather*} \left [\frac {\log \left (\frac {a^{2} x^{18} + 6 \, a x^{13} + 6 \, a x^{9} + x^{8} + 2 \, x^{4} - 4 \, {\left (a x^{13} + x^{8} + x^{4}\right )} \sqrt {x^{5} + x} \sqrt {a} + 1}{a^{2} x^{18} - 2 \, a x^{13} - 2 \, a x^{9} + x^{8} + 2 \, x^{4} + 1}\right )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {{\left (a x^{9} + x^{4} + 1\right )} \sqrt {x^{5} + x} \sqrt {-a}}{2 \, {\left (a x^{9} + a x^{5}\right )}}\right )}{a}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*log((a^2*x^18 + 6*a*x^13 + 6*a*x^9 + x^8 + 2*x^4 - 4*(a*x^13 + x^8 + x^4)*sqrt(x^5 + x)*sqrt(a) + 1)/(a^2
*x^18 - 2*a*x^13 - 2*a*x^9 + x^8 + 2*x^4 + 1))/sqrt(a), sqrt(-a)*arctan(1/2*(a*x^9 + x^4 + 1)*sqrt(x^5 + x)*sq
rt(-a)/(a*x^9 + a*x^5))/a]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.93, size = 46, normalized size = 1.77 \begin {gather*} \frac {\ln \left (\frac {a\,x^9+x^4-2\,\sqrt {a}\,x^4\,\sqrt {x^5+x}+1}{-4\,a\,x^9+4\,x^4+4}\right )}{\sqrt {a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log((a*x^9 + x^4 - 2*a^(1/2)*x^4*(x + x^5)^(1/2) + 1)/(4*x^4 - 4*a*x^9 + 4))/a^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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