3.715 \(\int \frac{x^3}{2 a+2 b+x^4} \, dx\)

Optimal. Leaf size=14 \[ \frac{1}{4} \log \left (2 (a+b)+x^4\right ) \]

[Out]

Log[2*(a + b) + x^4]/4

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Rubi [A]  time = 0.0034531, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {260} \[ \frac{1}{4} \log \left (2 (a+b)+x^4\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^3/(2*a + 2*b + x^4),x]

[Out]

Log[2*(a + b) + x^4]/4

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

\begin{align*} \int \frac{x^3}{2 a+2 b+x^4} \, dx &=\frac{1}{4} \log \left (2 (a+b)+x^4\right )\\ \end{align*}

Mathematica [A]  time = 0.0037697, size = 15, normalized size = 1.07 \[ \frac{1}{4} \log \left (2 a+2 b+x^4\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(2*a + 2*b + x^4),x]

[Out]

Log[2*a + 2*b + x^4]/4

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Maple [A]  time = 0.002, size = 14, normalized size = 1. \begin{align*}{\frac{\ln \left ({x}^{4}+2\,a+2\,b \right ) }{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*ln(x^4+2*a+2*b)

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Maxima [A]  time = 0.997749, size = 18, normalized size = 1.29 \begin{align*} \frac{1}{4} \, \log \left (x^{4} + 2 \, a + 2 \, b\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(x^4+2*a+2*b),x, algorithm="maxima")

[Out]

1/4*log(x^4 + 2*a + 2*b)

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Fricas [A]  time = 1.42453, size = 35, normalized size = 2.5 \begin{align*} \frac{1}{4} \, \log \left (x^{4} + 2 \, a + 2 \, b\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(x^4+2*a+2*b),x, algorithm="fricas")

[Out]

1/4*log(x^4 + 2*a + 2*b)

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Sympy [A]  time = 0.172592, size = 12, normalized size = 0.86 \begin{align*} \frac{\log{\left (2 a + 2 b + x^{4} \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(x**4+2*a+2*b),x)

[Out]

log(2*a + 2*b + x**4)/4

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Giac [A]  time = 1.10168, size = 19, normalized size = 1.36 \begin{align*} \frac{1}{4} \, \log \left ({\left | x^{4} + 2 \, a + 2 \, b \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(x^4+2*a+2*b),x, algorithm="giac")

[Out]

1/4*log(abs(x^4 + 2*a + 2*b))