3.5.39 \(\int \frac {4 b+a x^3}{\sqrt [4]{b+a x^3} (-b-a x^3+x^4)} \, dx\) [439]

Optimal. Leaf size=35 \[ 2 \text {ArcTan}\left (\frac {\sqrt [4]{b+a x^3}}{x}\right )-2 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{b+a x^3}}\right ) \]

[Out]

2*arctan((a*x^3+b)^(1/4)/x)-2*arctanh(x/(a*x^3+b)^(1/4))

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Rubi [F]
time = 0.47, 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 {4 b+a x^3}{\sqrt [4]{b+a x^3} \left (-b-a x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-4*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]

Rubi steps

\begin {align*} \int \frac {4 b+a x^3}{\sqrt [4]{b+a x^3} \left (-b-a x^3+x^4\right )} \, dx &=\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 (a \int \frac {x^3}{\sqrt [4]{b+a x^3} \left (b+a x^3-x^4\right )} \, dx\right )-(4 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 [A]
time = 0.53, size = 35, normalized size = 1.00 \begin {gather*} 2 \text {ArcTan}\left (\frac {\sqrt [4]{b+a x^3}}{x}\right )-2 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{b+a x^3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.16, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{3}+4 b}{\left (a \,x^{3}+b \right )^{\frac {1}{4}} \left (-a \,x^{3}+x^{4}-b \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-1)] Timed out
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^3+4*b)/(a*x^3+b)^(1/4)/(-a*x^3+x^4-b),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)] Timed out
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**3+4*b)/(a*x**3+b)**(1/4)/(-a*x**3+x**4-b),x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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