3.21.44 \(\int \frac {x^4 (-4 b+a x^3)}{\sqrt [4]{-b+a x^3} (-b^2+2 a b x^3-a^2 x^6+x^8)} \, dx\)

Optimal. Leaf size=146 \[ \tan ^{-1}\left (\frac {\sqrt [4]{a x^3-b}}{x}\right )+\tanh ^{-1}\left (\frac {x \left (a x^3-b\right )^{3/4}}{b-a x^3}\right )-\frac {\tan ^{-1}\left (\frac {\frac {\sqrt {a x^3-b}}{\sqrt {2}}-\frac {x^2}{\sqrt {2}}}{x \sqrt [4]{a x^3-b}}\right )}{\sqrt {2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{a x^3-b}}{\sqrt {a x^3-b}+x^2}\right )}{\sqrt {2}} \]

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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^4 \left (-4 b+a x^3\right )}{\sqrt [4]{-b+a x^3} \left (-b^2+2 a b x^3-a^2 x^6+x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x^4*(-4*b + a*x^3))/((-b + a*x^3)^(1/4)*(-b^2 + 2*a*b*x^3 - a^2*x^6 + x^8)),x]

[Out]

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

Rubi steps

\begin {align*} \int \frac {x^4 \left (-4 b+a x^3\right )}{\sqrt [4]{-b+a x^3} \left (-b^2+2 a b x^3-a^2 x^6+x^8\right )} \, dx &=\int \left (\frac {4 b-a x^3}{2 \sqrt [4]{-b+a x^3} \left (b-a x^3-x^4\right )}+\frac {-4 b+a x^3}{2 \sqrt [4]{-b+a x^3} \left (b-a x^3+x^4\right )}\right ) \, dx\\ &=\frac {1}{2} \int \frac {4 b-a x^3}{\sqrt [4]{-b+a x^3} \left (b-a x^3-x^4\right )} \, dx+\frac {1}{2} \int \frac {-4 b+a x^3}{\sqrt [4]{-b+a x^3} \left (b-a x^3+x^4\right )} \, dx\\ &=\frac {1}{2} \int \left (-\frac {a x^3}{\sqrt [4]{-b+a x^3} \left (-b+a x^3-x^4\right )}-\frac {4 b}{\sqrt [4]{-b+a x^3} \left (b-a x^3+x^4\right )}\right ) \, dx+\frac {1}{2} \int \left (\frac {4 b}{\sqrt [4]{-b+a x^3} \left (b-a x^3-x^4\right )}+\frac {a x^3}{\sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )}\right ) \, dx\\ &=-\left (\frac {1}{2} a \int \frac {x^3}{\sqrt [4]{-b+a x^3} \left (-b+a x^3-x^4\right )} \, dx\right )+\frac {1}{2} a \int \frac {x^3}{\sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx+(2 b) \int \frac {1}{\sqrt [4]{-b+a x^3} \left (b-a x^3-x^4\right )} \, dx-(2 b) \int \frac {1}{\sqrt [4]{-b+a x^3} \left (b-a x^3+x^4\right )} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.51, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^4 \left (-4 b+a x^3\right )}{\sqrt [4]{-b+a x^3} \left (-b^2+2 a b x^3-a^2 x^6+x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(x^4*(-4*b + a*x^3))/((-b + a*x^3)^(1/4)*(-b^2 + 2*a*b*x^3 - a^2*x^6 + x^8)),x]

[Out]

Integrate[(x^4*(-4*b + a*x^3))/((-b + a*x^3)^(1/4)*(-b^2 + 2*a*b*x^3 - a^2*x^6 + x^8)), x]

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IntegrateAlgebraic [A]  time = 1.77, size = 146, normalized size = 1.00 \begin {gather*} \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x^3}}{x}\right )-\frac {\tan ^{-1}\left (\frac {-\frac {x^2}{\sqrt {2}}+\frac {\sqrt {-b+a x^3}}{\sqrt {2}}}{x \sqrt [4]{-b+a x^3}}\right )}{\sqrt {2}}+\tanh ^{-1}\left (\frac {x \left (-b+a x^3\right )^{3/4}}{b-a x^3}\right )+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{-b+a x^3}}{x^2+\sqrt {-b+a x^3}}\right )}{\sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^4*(-4*b + a*x^3))/((-b + a*x^3)^(1/4)*(-b^2 + 2*a*b*x^3 - a^2*x^6 + x^8)),x]

[Out]

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

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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^3-4*b)/(a*x^3-b)^(1/4)/(-a^2*x^6+x^8+2*a*b*x^3-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^{3} - 4 \, b\right )} x^{4}}{{\left (a^{2} x^{6} - x^{8} - 2 \, a b x^{3} + b^{2}\right )} {\left (a x^{3} - 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^3-4*b)/(a*x^3-b)^(1/4)/(-a^2*x^6+x^8+2*a*b*x^3-b^2),x, algorithm="giac")

[Out]

integrate(-(a*x^3 - 4*b)*x^4/((a^2*x^6 - x^8 - 2*a*b*x^3 + b^2)*(a*x^3 - b)^(1/4)), x)

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {x^{4} \left (a \,x^{3}-4 b \right )}{\left (a \,x^{3}-b \right )^{\frac {1}{4}} \left (-a^{2} x^{6}+x^{8}+2 a b \,x^{3}-b^{2}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a*x^3-4*b)/(a*x^3-b)^(1/4)/(-a^2*x^6+x^8+2*a*b*x^3-b^2),x)

[Out]

int(x^4*(a*x^3-4*b)/(a*x^3-b)^(1/4)/(-a^2*x^6+x^8+2*a*b*x^3-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^{3} - 4 \, b\right )} x^{4}}{{\left (a^{2} x^{6} - x^{8} - 2 \, a b x^{3} + b^{2}\right )} {\left (a x^{3} - 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^3-4*b)/(a*x^3-b)^(1/4)/(-a^2*x^6+x^8+2*a*b*x^3-b^2),x, algorithm="maxima")

[Out]

-integrate((a*x^3 - 4*b)*x^4/((a^2*x^6 - x^8 - 2*a*b*x^3 + b^2)*(a*x^3 - 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 (4\,b-a\,x^3\right )}{{\left (a\,x^3-b\right )}^{1/4}\,\left (a^2\,x^6-2\,a\,b\,x^3+b^2-x^8\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4*(4*b - a*x^3))/((a*x^3 - b)^(1/4)*(b^2 - x^8 + a^2*x^6 - 2*a*b*x^3)),x)

[Out]

-int(-(x^4*(4*b - a*x^3))/((a*x^3 - b)^(1/4)*(b^2 - x^8 + a^2*x^6 - 2*a*b*x^3)), 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**3-4*b)/(a*x**3-b)**(1/4)/(-a**2*x**6+x**8+2*a*b*x**3-b**2),x)

[Out]

Timed out

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