3.651 \(\int \frac{1}{x^5 (a+c x^4)} \, dx\)

Optimal. Leaf size=35 \[ \frac{c \log \left (a+c x^4\right )}{4 a^2}-\frac{c \log (x)}{a^2}-\frac{1}{4 a x^4} \]

[Out]

-1/(4*a*x^4) - (c*Log[x])/a^2 + (c*Log[a + c*x^4])/(4*a^2)

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Rubi [A]  time = 0.0231348, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 44} \[ \frac{c \log \left (a+c x^4\right )}{4 a^2}-\frac{c \log (x)}{a^2}-\frac{1}{4 a x^4} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^5*(a + c*x^4)),x]

[Out]

-1/(4*a*x^4) - (c*Log[x])/a^2 + (c*Log[a + c*x^4])/(4*a^2)

Rule 266

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

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{x^5 \left (a+c x^4\right )} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{x^2 (a+c x)} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{1}{a x^2}-\frac{c}{a^2 x}+\frac{c^2}{a^2 (a+c x)}\right ) \, dx,x,x^4\right )\\ &=-\frac{1}{4 a x^4}-\frac{c \log (x)}{a^2}+\frac{c \log \left (a+c x^4\right )}{4 a^2}\\ \end{align*}

Mathematica [A]  time = 0.0066834, size = 35, normalized size = 1. \[ \frac{c \log \left (a+c x^4\right )}{4 a^2}-\frac{c \log (x)}{a^2}-\frac{1}{4 a x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*(a + c*x^4)),x]

[Out]

-1/(4*a*x^4) - (c*Log[x])/a^2 + (c*Log[a + c*x^4])/(4*a^2)

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Maple [A]  time = 0.005, size = 32, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,a{x}^{4}}}-{\frac{c\ln \left ( x \right ) }{{a}^{2}}}+{\frac{c\ln \left ( c{x}^{4}+a \right ) }{4\,{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^5/(c*x^4+a),x)

[Out]

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

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Maxima [A]  time = 0.969551, size = 45, normalized size = 1.29 \begin{align*} \frac{c \log \left (c x^{4} + a\right )}{4 \, a^{2}} - \frac{c \log \left (x^{4}\right )}{4 \, a^{2}} - \frac{1}{4 \, a x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(c*x^4+a),x, algorithm="maxima")

[Out]

1/4*c*log(c*x^4 + a)/a^2 - 1/4*c*log(x^4)/a^2 - 1/4/(a*x^4)

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Fricas [A]  time = 1.70416, size = 80, normalized size = 2.29 \begin{align*} \frac{c x^{4} \log \left (c x^{4} + a\right ) - 4 \, c x^{4} \log \left (x\right ) - a}{4 \, a^{2} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(c*x^4+a),x, algorithm="fricas")

[Out]

1/4*(c*x^4*log(c*x^4 + a) - 4*c*x^4*log(x) - a)/(a^2*x^4)

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Sympy [A]  time = 0.744159, size = 31, normalized size = 0.89 \begin{align*} - \frac{1}{4 a x^{4}} - \frac{c \log{\left (x \right )}}{a^{2}} + \frac{c \log{\left (\frac{a}{c} + x^{4} \right )}}{4 a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**5/(c*x**4+a),x)

[Out]

-1/(4*a*x**4) - c*log(x)/a**2 + c*log(a/c + x**4)/(4*a**2)

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Giac [A]  time = 1.62182, size = 58, normalized size = 1.66 \begin{align*} -\frac{c \log \left (x^{4}\right )}{4 \, a^{2}} + \frac{c \log \left ({\left | c x^{4} + a \right |}\right )}{4 \, a^{2}} + \frac{c x^{4} - a}{4 \, a^{2} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(c*x^4+a),x, algorithm="giac")

[Out]

-1/4*c*log(x^4)/a^2 + 1/4*c*log(abs(c*x^4 + a))/a^2 + 1/4*(c*x^4 - a)/(a^2*x^4)