\(\int \frac {x^3}{2 (a+b)+x^4} \, dx\) [716]

Optimal result

Integrand size = 15, antiderivative size = 14 \[ \int \frac {x^3}{2 (a+b)+x^4} \, dx=\frac {1}{4} \log \left (2 (a+b)+x^4\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {266} \[ \int \frac {x^3}{2 (a+b)+x^4} \, dx=\frac {1}{4} \log \left (2 (a+b)+x^4\right ) \]

[In]

Int[x^3/(2*(a + 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*} \text {integral}& = \frac {1}{4} \log \left (2 (a+b)+x^4\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07 \[ \int \frac {x^3}{2 (a+b)+x^4} \, dx=\frac {1}{4} \log \left (2 a+2 b+x^4\right ) \]

[In]

Integrate[x^3/(2*(a + b) + x^4),x]

[Out]

Log[2*a + 2*b + x^4]/4

Maple [A] (verified)

Time = 3.89 (sec) , antiderivative size = 14, normalized size of antiderivative = 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	\(\frac {\ln \left (x^{4}+2 a +2 b \right )}{4}\)	\(14\)
parallelrisch	\(\frac {\ln \left (x^{4}+2 a +2 b \right )}{4}\)	\(14\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93 \[ \int \frac {x^3}{2 (a+b)+x^4} \, dx=\frac {1}{4} \, \log \left (x^{4} + 2 \, a + 2 \, b\right ) \]

[In]

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

[Out]

1/4*log(x^4 + 2*a + 2*b)

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {x^3}{2 (a+b)+x^4} \, dx=\frac {\log {\left (2 a + 2 b + x^{4} \right )}}{4} \]

[In]

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

[Out]

log(2*a + 2*b + x**4)/4

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93 \[ \int \frac {x^3}{2 (a+b)+x^4} \, dx=\frac {1}{4} \, \log \left (x^{4} + 2 \, a + 2 \, b\right ) \]

[In]

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

[Out]

1/4*log(x^4 + 2*a + 2*b)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {x^3}{2 (a+b)+x^4} \, dx=\frac {1}{4} \, \log \left ({\left | x^{4} + 2 \, a + 2 \, b \right |}\right ) \]

[In]

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

[Out]

1/4*log(abs(x^4 + 2*a + 2*b))

Mupad [B] (verification not implemented)

Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93 \[ \int \frac {x^3}{2 (a+b)+x^4} \, dx=\frac {\ln \left (x^4+2\,a+2\,b\right )}{4} \]

[In]

int(x^3/(2*a + 2*b + x^4),x)

[Out]

log(2*a + 2*b + x^4)/4