3.8.91 \(\int \frac {x^5 (7 b+9 a x^2)}{\sqrt [4]{b x^3+a x^5} (-2+b x^7+a x^9)} \, dx\)

Optimal. Leaf size=60 \[ 2^{3/4} \tan ^{-1}\left (\frac {x \sqrt [4]{a x^5+b x^3}}{\sqrt [4]{2}}\right )-2^{3/4} \tanh ^{-1}\left (\frac {x \sqrt [4]{a x^5+b x^3}}{\sqrt [4]{2}}\right ) \]

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Rubi [F]  time = 2.38, 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^5 \left (7 b+9 a x^2\right )}{\sqrt [4]{b x^3+a x^5} \left (-2+b x^7+a x^9\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x^5*(7*b + 9*a*x^2))/((b*x^3 + a*x^5)^(1/4)*(-2 + b*x^7 + a*x^9)),x]

[Out]

(28*b*x^(3/4)*(b + a*x^2)^(1/4)*Defer[Subst][Defer[Int][x^20/((b + a*x^8)^(1/4)*(-2 + b*x^28 + a*x^36)), x], x
, x^(1/4)])/(b*x^3 + a*x^5)^(1/4) + (36*a*x^(3/4)*(b + a*x^2)^(1/4)*Defer[Subst][Defer[Int][x^28/((b + a*x^8)^
(1/4)*(-2 + b*x^28 + a*x^36)), x], x, x^(1/4)])/(b*x^3 + a*x^5)^(1/4)

Rubi steps

\begin {align*} \int \frac {x^5 \left (7 b+9 a x^2\right )}{\sqrt [4]{b x^3+a x^5} \left (-2+b x^7+a x^9\right )} \, dx &=\frac {\left (x^{3/4} \sqrt [4]{b+a x^2}\right ) \int \frac {x^{17/4} \left (7 b+9 a x^2\right )}{\sqrt [4]{b+a x^2} \left (-2+b x^7+a x^9\right )} \, dx}{\sqrt [4]{b x^3+a x^5}}\\ &=\frac {\left (4 x^{3/4} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {x^{20} \left (7 b+9 a x^8\right )}{\sqrt [4]{b+a x^8} \left (-2+b x^{28}+a x^{36}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x^3+a x^5}}\\ &=\frac {\left (4 x^{3/4} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {7 b x^{20}}{\sqrt [4]{b+a x^8} \left (-2+b x^{28}+a x^{36}\right )}+\frac {9 a x^{28}}{\sqrt [4]{b+a x^8} \left (-2+b x^{28}+a x^{36}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x^3+a x^5}}\\ &=\frac {\left (36 a x^{3/4} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {x^{28}}{\sqrt [4]{b+a x^8} \left (-2+b x^{28}+a x^{36}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x^3+a x^5}}+\frac {\left (28 b x^{3/4} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {x^{20}}{\sqrt [4]{b+a x^8} \left (-2+b x^{28}+a x^{36}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x^3+a x^5}}\\ \end {align*}

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Mathematica [F]  time = 0.29, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^5 \left (7 b+9 a x^2\right )}{\sqrt [4]{b x^3+a x^5} \left (-2+b x^7+a x^9\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(x^5*(7*b + 9*a*x^2))/((b*x^3 + a*x^5)^(1/4)*(-2 + b*x^7 + a*x^9)),x]

[Out]

Integrate[(x^5*(7*b + 9*a*x^2))/((b*x^3 + a*x^5)^(1/4)*(-2 + b*x^7 + a*x^9)), x]

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IntegrateAlgebraic [A]  time = 15.87, size = 60, normalized size = 1.00 \begin {gather*} 2^{3/4} \tan ^{-1}\left (\frac {x \sqrt [4]{b x^3+a x^5}}{\sqrt [4]{2}}\right )-2^{3/4} \tanh ^{-1}\left (\frac {x \sqrt [4]{b x^3+a x^5}}{\sqrt [4]{2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^5*(7*b + 9*a*x^2))/((b*x^3 + a*x^5)^(1/4)*(-2 + b*x^7 + a*x^9)),x]

[Out]

2^(3/4)*ArcTan[(x*(b*x^3 + a*x^5)^(1/4))/2^(1/4)] - 2^(3/4)*ArcTanh[(x*(b*x^3 + a*x^5)^(1/4))/2^(1/4)]

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fricas [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(x^5*(9*a*x^2+7*b)/(a*x^5+b*x^3)^(1/4)/(a*x^9+b*x^7-2),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (9 \, a x^{2} + 7 \, b\right )} x^{5}}{{\left (a x^{9} + b x^{7} - 2\right )} {\left (a x^{5} + b x^{3}\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(9*a*x^2+7*b)/(a*x^5+b*x^3)^(1/4)/(a*x^9+b*x^7-2),x, algorithm="giac")

[Out]

integrate((9*a*x^2 + 7*b)*x^5/((a*x^9 + b*x^7 - 2)*(a*x^5 + b*x^3)^(1/4)), x)

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x^{5} \left (9 a \,x^{2}+7 b \right )}{\left (a \,x^{5}+b \,x^{3}\right )^{\frac {1}{4}} \left (a \,x^{9}+b \,x^{7}-2\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(9*a*x^2+7*b)/(a*x^5+b*x^3)^(1/4)/(a*x^9+b*x^7-2),x)

[Out]

int(x^5*(9*a*x^2+7*b)/(a*x^5+b*x^3)^(1/4)/(a*x^9+b*x^7-2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (9 \, a x^{2} + 7 \, b\right )} x^{5}}{{\left (a x^{9} + b x^{7} - 2\right )} {\left (a x^{5} + b x^{3}\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(9*a*x^2+7*b)/(a*x^5+b*x^3)^(1/4)/(a*x^9+b*x^7-2),x, algorithm="maxima")

[Out]

integrate((9*a*x^2 + 7*b)*x^5/((a*x^9 + b*x^7 - 2)*(a*x^5 + b*x^3)^(1/4)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^5\,\left (9\,a\,x^2+7\,b\right )}{{\left (a\,x^5+b\,x^3\right )}^{1/4}\,\left (a\,x^9+b\,x^7-2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5*(7*b + 9*a*x^2))/((a*x^5 + b*x^3)^(1/4)*(a*x^9 + b*x^7 - 2)),x)

[Out]

int((x^5*(7*b + 9*a*x^2))/((a*x^5 + b*x^3)^(1/4)*(a*x^9 + b*x^7 - 2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{5} \left (9 a x^{2} + 7 b\right )}{\sqrt [4]{x^{3} \left (a x^{2} + b\right )} \left (a x^{9} + b x^{7} - 2\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(9*a*x**2+7*b)/(a*x**5+b*x**3)**(1/4)/(a*x**9+b*x**7-2),x)

[Out]

Integral(x**5*(9*a*x**2 + 7*b)/((x**3*(a*x**2 + b))**(1/4)*(a*x**9 + b*x**7 - 2)), x)

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