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

Optimal. Leaf size=75 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x \left (a x^6-b\right )^{3/4}}{b-a x^6}\right )}{\sqrt [4]{2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x \left (a x^6-b\right )^{3/4}}{b-a x^6}\right )}{\sqrt [4]{2}} \]

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Rubi [F]  time = 1.09, 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 {2 b+a x^6}{\sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(x*(1 - (a*x^6)/b)^(1/4)*Hypergeometric2F1[1/6, 1/4, 7/6, (a*x^6)/b])/(-b + a*x^6)^(1/4) - 3*b*Defer[Int][1/((
b + 2*x^4 - a*x^6)*(-b + a*x^6)^(1/4)), x] + 2*Defer[Int][x^4/((-b + a*x^6)^(1/4)*(-b - 2*x^4 + a*x^6)), x]

Rubi steps

\begin {align*} \int \frac {2 b+a x^6}{\sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )} \, dx &=\int \left (\frac {1}{\sqrt [4]{-b+a x^6}}+\frac {3 b+2 x^4}{\sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )}\right ) \, dx\\ &=\int \frac {1}{\sqrt [4]{-b+a x^6}} \, dx+\int \frac {3 b+2 x^4}{\sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )} \, dx\\ &=\frac {\sqrt [4]{1-\frac {a x^6}{b}} \int \frac {1}{\sqrt [4]{1-\frac {a x^6}{b}}} \, dx}{\sqrt [4]{-b+a x^6}}+\int \left (-\frac {3 b}{\left (b+2 x^4-a x^6\right ) \sqrt [4]{-b+a x^6}}+\frac {2 x^4}{\sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )}\right ) \, dx\\ &=\frac {x \sqrt [4]{1-\frac {a x^6}{b}} \, _2F_1\left (\frac {1}{6},\frac {1}{4};\frac {7}{6};\frac {a x^6}{b}\right )}{\sqrt [4]{-b+a x^6}}+2 \int \frac {x^4}{\sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )} \, dx-(3 b) \int \frac {1}{\left (b+2 x^4-a x^6\right ) \sqrt [4]{-b+a x^6}} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.30, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 b+a x^6}{\sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 5.20, size = 75, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x \left (-b+a x^6\right )^{3/4}}{b-a x^6}\right )}{\sqrt [4]{2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x \left (-b+a x^6\right )^{3/4}}{b-a x^6}\right )}{\sqrt [4]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[(2^(1/4)*x*(-b + a*x^6)^(3/4))/(b - a*x^6)]/2^(1/4) + ArcTanh[(2^(1/4)*x*(-b + a*x^6)^(3/4))/(b - a*x^6
)]/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((a*x^6+2*b)/(a*x^6-b)^(1/4)/(a*x^6-2*x^4-b),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 {a x^{6} + 2 \, b}{{\left (a x^{6} - 2 \, x^{4} - b\right )} {\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((a*x^6+2*b)/(a*x^6-b)^(1/4)/(a*x^6-2*x^4-b),x, algorithm="giac")

[Out]

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

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{6}+2 b}{\left (a \,x^{6}-b \right )^{\frac {1}{4}} \left (a \,x^{6}-2 x^{4}-b \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

integrate((a*x^6 + 2*b)/((a*x^6 - 2*x^4 - b)*(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 {a\,x^6+2\,b}{{\left (a\,x^6-b\right )}^{1/4}\,\left (-a\,x^6+2\,x^4+b\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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