3.5.69 \(\int \frac {x (8 b+5 a x^3)}{\sqrt [4]{b+a x^3} (-b-a x^3+x^8)} \, dx\)

Optimal. Leaf size=37 \[ 2 \tan ^{-1}\left (\frac {\sqrt [4]{a x^3+b}}{x^2}\right )-2 \tanh ^{-1}\left (\frac {x^2}{\sqrt [4]{a x^3+b}}\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 0.92, 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 \left (8 b+5 a x^3\right )}{\sqrt [4]{b+a x^3} \left (-b-a x^3+x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x*(8*b + 5*a*x^3))/((b + a*x^3)^(1/4)*(-b - a*x^3 + x^8)),x]

[Out]

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

Rubi steps

\begin {align*} \int \frac {x \left (8 b+5 a x^3\right )}{\sqrt [4]{b+a x^3} \left (-b-a x^3+x^8\right )} \, dx &=\int \left (-\frac {8 b x}{\sqrt [4]{b+a x^3} \left (b+a x^3-x^8\right )}-\frac {5 a x^4}{\sqrt [4]{b+a x^3} \left (b+a x^3-x^8\right )}\right ) \, dx\\ &=-\left ((5 a) \int \frac {x^4}{\sqrt [4]{b+a x^3} \left (b+a x^3-x^8\right )} \, dx\right )-(8 b) \int \frac {x}{\sqrt [4]{b+a x^3} \left (b+a x^3-x^8\right )} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]  time = 0.24, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (8 b+5 a x^3\right )}{\sqrt [4]{b+a x^3} \left (-b-a x^3+x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(x*(8*b + 5*a*x^3))/((b + a*x^3)^(1/4)*(-b - a*x^3 + x^8)),x]

[Out]

Integrate[(x*(8*b + 5*a*x^3))/((b + a*x^3)^(1/4)*(-b - a*x^3 + x^8)), x]

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.85, size = 37, normalized size = 1.00 \begin {gather*} 2 \tan ^{-1}\left (\frac {\sqrt [4]{b+a x^3}}{x^2}\right )-2 \tanh ^{-1}\left (\frac {x^2}{\sqrt [4]{b+a x^3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x*(8*b + 5*a*x^3))/((b + a*x^3)^(1/4)*(-b - a*x^3 + x^8)),x]

[Out]

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

________________________________________________________________________________________

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*a*x^3+8*b)/(a*x^3+b)^(1/4)/(x^8-a*x^3-b),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (5 \, a x^{3} + 8 \, b\right )} x}{{\left (x^{8} - a x^{3} - b\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*(5*a*x^3+8*b)/(a*x^3+b)^(1/4)/(x^8-a*x^3-b),x, algorithm="giac")

[Out]

integrate((5*a*x^3 + 8*b)*x/((x^8 - a*x^3 - b)*(a*x^3 + b)^(1/4)), x)

________________________________________________________________________________________

maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x \left (5 a \,x^{3}+8 b \right )}{\left (a \,x^{3}+b \right )^{\frac {1}{4}} \left (x^{8}-a \,x^{3}-b \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(5*a*x^3+8*b)/(a*x^3+b)^(1/4)/(x^8-a*x^3-b),x)

[Out]

int(x*(5*a*x^3+8*b)/(a*x^3+b)^(1/4)/(x^8-a*x^3-b),x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (5 \, a x^{3} + 8 \, b\right )} x}{{\left (x^{8} - a x^{3} - b\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*(5*a*x^3+8*b)/(a*x^3+b)^(1/4)/(x^8-a*x^3-b),x, algorithm="maxima")

[Out]

integrate((5*a*x^3 + 8*b)*x/((x^8 - a*x^3 - b)*(a*x^3 + b)^(1/4)), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {x\,\left (5\,a\,x^3+8\,b\right )}{{\left (a\,x^3+b\right )}^{1/4}\,\left (-x^8+a\,x^3+b\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x*(8*b + 5*a*x^3))/((b + a*x^3)^(1/4)*(b + a*x^3 - x^8)),x)

[Out]

int(-(x*(8*b + 5*a*x^3))/((b + a*x^3)^(1/4)*(b + a*x^3 - x^8)), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (5 a x^{3} + 8 b\right )}{\sqrt [4]{a x^{3} + b} \left (- a x^{3} - b + x^{8}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(5*a*x**3+8*b)/(a*x**3+b)**(1/4)/(x**8-a*x**3-b),x)

[Out]

Integral(x*(5*a*x**3 + 8*b)/((a*x**3 + b)**(1/4)*(-a*x**3 - b + x**8)), x)

________________________________________________________________________________________