3.8.15 \(\int \frac {x^3}{2 a+2 b+x^4} \, dx\) [715]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, 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 = {266} \begin {gather*} \frac {1}{4} \log \left (2 (a+b)+x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/(2*a + 2*b + x^4),x]

[Out]

Log[2*(a + b) + x^4]/4

Rule 266

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*}

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Mathematica [A]
time = 0.00, size = 15, normalized size = 1.07 \begin {gather*} \frac {1}{4} \log \left (2 a+2 b+x^4\right ) \end {gather*}

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 = 0.16, size = 14, normalized size = 1.00

method	result	size
derivativedivides	\(\frac {\ln \left (x^{4}+2 a +2 b \right )}{4}\)	\(14\)
default	\(\frac {\ln \left (x^{4}+2 a +2 b \right )}{4}\)	\(14\)
norman	\(\frac {\ln \left (x^{4}+2 a +2 b \right )}{4}\)	\(14\)
risch	\(\frac {\ln \left (x^{4}+2 a +2 b \right )}{4}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 13, normalized size = 0.93 \begin {gather*} \frac {1}{4} \, \log \left (x^{4} + 2 \, a + 2 \, b\right ) \end {gather*}

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 = 0.38, size = 13, normalized size = 0.93 \begin {gather*} \frac {1}{4} \, \log \left (x^{4} + 2 \, a + 2 \, b\right ) \end {gather*}

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 = 0.06, size = 12, normalized size = 0.86 \begin {gather*} \frac {\log {\left (2 a + 2 b + x^{4} \right )}}{4} \end {gather*}

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 = 0.97, size = 14, normalized size = 1.00 \begin {gather*} \frac {1}{4} \, \log \left ({\left | x^{4} + 2 \, a + 2 \, b \right |}\right ) \end {gather*}

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))

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Mupad [B]
time = 0.04, size = 13, normalized size = 0.93 \begin {gather*} \frac {\ln \left (x^4+2\,a+2\,b\right )}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(2*a + 2*b + x^4),x)

[Out]

log(2*a + 2*b + x^4)/4

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