\(\int \frac {x^4}{(a+b x^5)^2} \, dx\) [1281]

Optimal result

Integrand size = 13, antiderivative size = 16 \[ \int \frac {x^4}{\left (a+b x^5\right )^2} \, dx=-\frac {1}{5 b \left (a+b x^5\right )} \]

[Out]

-1/5/b/(b*x^5+a)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {267} \[ \int \frac {x^4}{\left (a+b x^5\right )^2} \, dx=-\frac {1}{5 b \left (a+b x^5\right )} \]

[In]

Int[x^4/(a + b*x^5)^2,x]

[Out]

-1/5*1/(b*(a + b*x^5))

Rule 267

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

Rubi steps \begin{align*} \text {integral}& = -\frac {1}{5 b \left (a+b x^5\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {x^4}{\left (a+b x^5\right )^2} \, dx=-\frac {1}{5 b \left (a+b x^5\right )} \]

[In]

Integrate[x^4/(a + b*x^5)^2,x]

[Out]

-1/5*1/(b*(a + b*x^5))

Maple [A] (verified)

Time = 4.38 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94

method	result	size
gosper	\(-\frac {1}{5 b \left (b \,x^{5}+a \right )}\)	\(15\)
derivativedivides	\(-\frac {1}{5 b \left (b \,x^{5}+a \right )}\)	\(15\)
default	\(-\frac {1}{5 b \left (b \,x^{5}+a \right )}\)	\(15\)
norman	\(-\frac {1}{5 b \left (b \,x^{5}+a \right )}\)	\(15\)
risch	\(-\frac {1}{5 b \left (b \,x^{5}+a \right )}\)	\(15\)
parallelrisch	\(-\frac {1}{5 b \left (b \,x^{5}+a \right )}\)	\(15\)

[In]

int(x^4/(b*x^5+a)^2,x,method=_RETURNVERBOSE)

[Out]

-1/5/b/(b*x^5+a)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \frac {x^4}{\left (a+b x^5\right )^2} \, dx=-\frac {1}{5 \, {\left (b^{2} x^{5} + a b\right )}} \]

[In]

integrate(x^4/(b*x^5+a)^2,x, algorithm="fricas")

[Out]

-1/5/(b^2*x^5 + a*b)

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \frac {x^4}{\left (a+b x^5\right )^2} \, dx=- \frac {1}{5 a b + 5 b^{2} x^{5}} \]

[In]

integrate(x**4/(b*x**5+a)**2,x)

[Out]

-1/(5*a*b + 5*b**2*x**5)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {x^4}{\left (a+b x^5\right )^2} \, dx=-\frac {1}{5 \, {\left (b x^{5} + a\right )} b} \]

[In]

integrate(x^4/(b*x^5+a)^2,x, algorithm="maxima")

[Out]

-1/5/((b*x^5 + a)*b)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {x^4}{\left (a+b x^5\right )^2} \, dx=-\frac {1}{5 \, {\left (b x^{5} + a\right )} b} \]

[In]

integrate(x^4/(b*x^5+a)^2,x, algorithm="giac")

[Out]

-1/5/((b*x^5 + a)*b)

Mupad [B] (verification not implemented)

Time = 5.45 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {x^4}{\left (a+b x^5\right )^2} \, dx=-\frac {1}{5\,b\,\left (b\,x^5+a\right )} \]

[In]

int(x^4/(a + b*x^5)^2,x)

[Out]

-1/(5*b*(a + b*x^5))