3.14.22 \(\int \frac {x^4 (2 b+a x^6)}{\sqrt [4]{-b+a x^6} (-b-x^4+a x^6)^2} \, dx\)

Optimal. Leaf size=95 \[ -\frac {1}{4} \tan ^{-1}\left (\frac {x \left (a x^6-b\right )^{3/4}}{b-a x^6}\right )-\frac {1}{4} \tanh ^{-1}\left (\frac {x \left (a x^6-b\right )^{3/4}}{b-a x^6}\right )-\frac {x \left (a x^6-b\right )^{3/4}}{2 \left (a x^6-b-x^4\right )} \]

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Rubi [F]  time = 2.94, 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 (2 b+a x^6\right )}{\sqrt [4]{-b+a x^6} \left (-b-x^4+a x^6\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x^4*(2*b + a*x^6))/((-b + a*x^6)^(1/4)*(-b - x^4 + a*x^6)^2),x]

[Out]

(b*Defer[Int][1/((b + x^4 - a*x^6)^2*(-b + a*x^6)^(1/4)), x])/a^2 + (b*Defer[Int][x^2/((b + x^4 - a*x^6)^2*(-b
+ a*x^6)^(1/4)), x])/a + (a^(-2) + 3*b)*Defer[Int][x^4/((b + x^4 - a*x^6)^2*(-b + a*x^6)^(1/4)), x] + Defer[I
nt][1/((-b + a*x^6)^(1/4)*(-b - x^4 + a*x^6)), x]/a^2 + Defer[Int][x^2/((-b + a*x^6)^(1/4)*(-b - x^4 + a*x^6))
, x]/a + Defer[Int][x^4/((-b + a*x^6)^(1/4)*(-b - x^4 + a*x^6)), x]

Rubi steps

\begin {align*} \int \frac {x^4 \left (2 b+a x^6\right )}{\sqrt [4]{-b+a x^6} \left (-b-x^4+a x^6\right )^2} \, dx &=\int \left (\frac {b+a b x^2+\left (1+3 a^2 b\right ) x^4}{a^2 \left (b+x^4-a x^6\right )^2 \sqrt [4]{-b+a x^6}}+\frac {1+a x^2+a^2 x^4}{a^2 \sqrt [4]{-b+a x^6} \left (-b-x^4+a x^6\right )}\right ) \, dx\\ &=\frac {\int \frac {b+a b x^2+\left (1+3 a^2 b\right ) x^4}{\left (b+x^4-a x^6\right )^2 \sqrt [4]{-b+a x^6}} \, dx}{a^2}+\frac {\int \frac {1+a x^2+a^2 x^4}{\sqrt [4]{-b+a x^6} \left (-b-x^4+a x^6\right )} \, dx}{a^2}\\ &=\frac {\int \left (\frac {b}{\left (b+x^4-a x^6\right )^2 \sqrt [4]{-b+a x^6}}+\frac {a b x^2}{\left (b+x^4-a x^6\right )^2 \sqrt [4]{-b+a x^6}}+\frac {\left (1+3 a^2 b\right ) x^4}{\left (b+x^4-a x^6\right )^2 \sqrt [4]{-b+a x^6}}\right ) \, dx}{a^2}+\frac {\int \left (\frac {1}{\sqrt [4]{-b+a x^6} \left (-b-x^4+a x^6\right )}+\frac {a x^2}{\sqrt [4]{-b+a x^6} \left (-b-x^4+a x^6\right )}+\frac {a^2 x^4}{\sqrt [4]{-b+a x^6} \left (-b-x^4+a x^6\right )}\right ) \, dx}{a^2}\\ &=\frac {\int \frac {1}{\sqrt [4]{-b+a x^6} \left (-b-x^4+a x^6\right )} \, dx}{a^2}+\frac {\int \frac {x^2}{\sqrt [4]{-b+a x^6} \left (-b-x^4+a x^6\right )} \, dx}{a}+\frac {b \int \frac {1}{\left (b+x^4-a x^6\right )^2 \sqrt [4]{-b+a x^6}} \, dx}{a^2}+\frac {b \int \frac {x^2}{\left (b+x^4-a x^6\right )^2 \sqrt [4]{-b+a x^6}} \, dx}{a}+\left (\frac {1}{a^2}+3 b\right ) \int \frac {x^4}{\left (b+x^4-a x^6\right )^2 \sqrt [4]{-b+a x^6}} \, dx+\int \frac {x^4}{\sqrt [4]{-b+a x^6} \left (-b-x^4+a x^6\right )} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.76, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^4 \left (2 b+a x^6\right )}{\sqrt [4]{-b+a x^6} \left (-b-x^4+a x^6\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(x^4*(2*b + a*x^6))/((-b + a*x^6)^(1/4)*(-b - x^4 + a*x^6)^2),x]

[Out]

Integrate[(x^4*(2*b + a*x^6))/((-b + a*x^6)^(1/4)*(-b - x^4 + a*x^6)^2), x]

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IntegrateAlgebraic [A]  time = 15.56, size = 95, normalized size = 1.00 \begin {gather*} -\frac {x \left (-b+a x^6\right )^{3/4}}{2 \left (-b-x^4+a x^6\right )}-\frac {1}{4} \tan ^{-1}\left (\frac {x \left (-b+a x^6\right )^{3/4}}{b-a x^6}\right )-\frac {1}{4} \tanh ^{-1}\left (\frac {x \left (-b+a x^6\right )^{3/4}}{b-a x^6}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^4*(2*b + a*x^6))/((-b + a*x^6)^(1/4)*(-b - x^4 + a*x^6)^2),x]

[Out]

-1/2*(x*(-b + a*x^6)^(3/4))/(-b - x^4 + a*x^6) - ArcTan[(x*(-b + a*x^6)^(3/4))/(b - a*x^6)]/4 - ArcTanh[(x*(-b
+ a*x^6)^(3/4))/(b - a*x^6)]/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^4*(a*x^6+2*b)/(a*x^6-b)^(1/4)/(a*x^6-x^4-b)^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 (a x^{6} + 2 \, b\right )} x^{4}}{{\left (a x^{6} - x^{4} - b\right )}^{2} {\left (a x^{6} - b\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*x^6 + 2*b)*x^4/((a*x^6 - x^4 - b)^2*(a*x^6 - b)^(1/4)), x)

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {x^{4} \left (a \,x^{6}+2 b \right )}{\left (a \,x^{6}-b \right )^{\frac {1}{4}} \left (a \,x^{6}-x^{4}-b \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x^{6} + 2 \, b\right )} x^{4}}{{\left (a x^{6} - x^{4} - b\right )}^{2} {\left (a x^{6} - b\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*x^6 + 2*b)*x^4/((a*x^6 - x^4 - b)^2*(a*x^6 - b)^(1/4)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^4\,\left (a\,x^6+2\,b\right )}{{\left (a\,x^6-b\right )}^{1/4}\,{\left (-a\,x^6+x^4+b\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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