3.10.39 \(\int \frac {(-2 b+a x^6) (b+a x^6)^{3/4}}{x^4 (b-c x^4+a x^6)} \, dx\)

Optimal. Leaf size=71 \[ -c^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a x^6+b}}\right )-c^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a x^6+b}}\right )+\frac {2 \left (a x^6+b\right )^{3/4}}{3 x^3} \]

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Rubi [F]  time = 1.75, 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 {\left (-2 b+a x^6\right ) \left (b+a x^6\right )^{3/4}}{x^4 \left (b-c 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)^(3/4))/(x^4*(b - c*x^4 + a*x^6)),x]

[Out]

(-2*a*x^3)/(b + a*x^6)^(1/4) + (2*(b + a*x^6)^(3/4))/(3*x^3) + (2*Sqrt[a]*Sqrt[b]*(1 + (a*x^6)/b)^(1/4)*Ellipt
icE[ArcTan[(Sqrt[a]*x^3)/Sqrt[b]]/2, 2])/(b + a*x^6)^(1/4) + 2*c*Defer[Int][(b + a*x^6)^(3/4)/(-b + c*x^4 - a*
x^6), x] + 3*a*Defer[Int][(x^2*(b + a*x^6)^(3/4))/(b - c*x^4 + a*x^6), x]

Rubi steps

\begin {align*} \int \frac {\left (-2 b+a x^6\right ) \left (b+a x^6\right )^{3/4}}{x^4 \left (b-c x^4+a x^6\right )} \, dx &=\int \left (-\frac {2 \left (b+a x^6\right )^{3/4}}{x^4}+\frac {\left (2 c-3 a x^2\right ) \left (b+a x^6\right )^{3/4}}{-b+c x^4-a x^6}\right ) \, dx\\ &=-\left (2 \int \frac {\left (b+a x^6\right )^{3/4}}{x^4} \, dx\right )+\int \frac {\left (2 c-3 a x^2\right ) \left (b+a x^6\right )^{3/4}}{-b+c x^4-a x^6} \, dx\\ &=-\left (\frac {2}{3} \operatorname {Subst}\left (\int \frac {\left (b+a x^2\right )^{3/4}}{x^2} \, dx,x,x^3\right )\right )+\int \left (\frac {2 c \left (b+a x^6\right )^{3/4}}{-b+c x^4-a x^6}+\frac {3 a x^2 \left (b+a x^6\right )^{3/4}}{b-c x^4+a x^6}\right ) \, dx\\ &=\frac {2 \left (b+a x^6\right )^{3/4}}{3 x^3}-a \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{b+a x^2}} \, dx,x,x^3\right )+(3 a) \int \frac {x^2 \left (b+a x^6\right )^{3/4}}{b-c x^4+a x^6} \, dx+(2 c) \int \frac {\left (b+a x^6\right )^{3/4}}{-b+c x^4-a x^6} \, dx\\ &=\frac {2 \left (b+a x^6\right )^{3/4}}{3 x^3}+(3 a) \int \frac {x^2 \left (b+a x^6\right )^{3/4}}{b-c x^4+a x^6} \, dx+(2 c) \int \frac {\left (b+a x^6\right )^{3/4}}{-b+c x^4-a x^6} \, dx-\frac {\left (a \sqrt [4]{1+\frac {a x^6}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+\frac {a x^2}{b}}} \, dx,x,x^3\right )}{\sqrt [4]{b+a x^6}}\\ &=-\frac {2 a x^3}{\sqrt [4]{b+a x^6}}+\frac {2 \left (b+a x^6\right )^{3/4}}{3 x^3}+(3 a) \int \frac {x^2 \left (b+a x^6\right )^{3/4}}{b-c x^4+a x^6} \, dx+(2 c) \int \frac {\left (b+a x^6\right )^{3/4}}{-b+c x^4-a x^6} \, dx+\frac {\left (a \sqrt [4]{1+\frac {a x^6}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {a x^2}{b}\right )^{5/4}} \, dx,x,x^3\right )}{\sqrt [4]{b+a x^6}}\\ &=-\frac {2 a x^3}{\sqrt [4]{b+a x^6}}+\frac {2 \left (b+a x^6\right )^{3/4}}{3 x^3}+\frac {2 \sqrt {a} \sqrt {b} \sqrt [4]{1+\frac {a x^6}{b}} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {a} x^3}{\sqrt {b}}\right )\right |2\right )}{\sqrt [4]{b+a x^6}}+(3 a) \int \frac {x^2 \left (b+a x^6\right )^{3/4}}{b-c x^4+a x^6} \, dx+(2 c) \int \frac {\left (b+a x^6\right )^{3/4}}{-b+c x^4-a x^6} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.47, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-2 b+a x^6\right ) \left (b+a x^6\right )^{3/4}}{x^4 \left (b-c 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)^(3/4))/(x^4*(b - c*x^4 + a*x^6)),x]

[Out]

Integrate[((-2*b + a*x^6)*(b + a*x^6)^(3/4))/(x^4*(b - c*x^4 + a*x^6)), x]

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IntegrateAlgebraic [A]  time = 3.47, size = 71, normalized size = 1.00 \begin {gather*} \frac {2 \left (b+a x^6\right )^{3/4}}{3 x^3}-c^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{b+a x^6}}\right )-c^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{b+a x^6}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-2*b + a*x^6)*(b + a*x^6)^(3/4))/(x^4*(b - c*x^4 + a*x^6)),x]

[Out]

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

[Out]

integrate((a*x^6 + b)^(3/4)*(a*x^6 - 2*b)/((a*x^6 - c*x^4 + b)*x^4), x)

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{6}-2 b \right ) \left (a \,x^{6}+b \right )^{\frac {3}{4}}}{x^{4} \left (a \,x^{6}-c \,x^{4}+b \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^6-2*b)*(a*x^6+b)^(3/4)/x^4/(a*x^6-c*x^4+b),x)

[Out]

int((a*x^6-2*b)*(a*x^6+b)^(3/4)/x^4/(a*x^6-c*x^4+b),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x^{6} + b\right )}^{\frac {3}{4}} {\left (a x^{6} - 2 \, b\right )}}{{\left (a x^{6} - c x^{4} + b\right )} x^{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)^(3/4)/x^4/(a*x^6-c*x^4+b),x, algorithm="maxima")

[Out]

integrate((a*x^6 + b)^(3/4)*(a*x^6 - 2*b)/((a*x^6 - c*x^4 + b)*x^4), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {{\left (a\,x^6+b\right )}^{3/4}\,\left (2\,b-a\,x^6\right )}{x^4\,\left (a\,x^6-c\,x^4+b\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((b + a*x^6)^(3/4)*(2*b - a*x^6))/(x^4*(b + a*x^6 - c*x^4)),x)

[Out]

int(-((b + a*x^6)^(3/4)*(2*b - a*x^6))/(x^4*(b + a*x^6 - c*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)**(3/4)/x**4/(a*x**6-c*x**4+b),x)

[Out]

Timed out

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