3.7.51 \(\int \frac {1}{x^5 (a+c x^4)} \, dx\) [651]

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

[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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Rubi [A]
time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, 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 = {272, 46} \begin {gather*} \frac {c \log \left (a+c x^4\right )}{4 a^2}-\frac {c \log (x)}{a^2}-\frac {1}{4 a x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^5*(a + c*x^4)),x]

[Out]

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

Rule 46

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] && Lt
Q[m + n + 2, 0])

Rule 272

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]]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 35, normalized size = 1.00 \begin {gather*} -\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 {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*(a + c*x^4)),x]

[Out]

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

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Maple [A]
time = 0.15, size = 32, normalized size = 0.91

method	result	size
default	\(-\frac {1}{4 a \,x^{4}}-\frac {c \ln \left (x \right )}{a^{2}}+\frac {c \ln \left (x^{4} c +a \right )}{4 a^{2}}\)	\(32\)
norman	\(-\frac {1}{4 a \,x^{4}}-\frac {c \ln \left (x \right )}{a^{2}}+\frac {c \ln \left (x^{4} c +a \right )}{4 a^{2}}\)	\(32\)
risch	\(-\frac {1}{4 a \,x^{4}}-\frac {c \ln \left (x \right )}{a^{2}}+\frac {c \ln \left (-x^{4} c -a \right )}{4 a^{2}}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.30, size = 33, normalized size = 0.94 \begin {gather*} \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 {gather*}

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 = 0.36, size = 33, normalized size = 0.94 \begin {gather*} \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 {gather*}

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 = 0.18, size = 31, normalized size = 0.89 \begin {gather*} - \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 {gather*}

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 = 0.66, size = 43, normalized size = 1.23 \begin {gather*} -\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 {gather*}

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)

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Mupad [B]
time = 0.07, size = 31, normalized size = 0.89 \begin {gather*} \frac {c\,\ln \left (c\,x^4+a\right )}{4\,a^2}-\frac {1}{4\,a\,x^4}-\frac {c\,\ln \left (x\right )}{a^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^5*(a + c*x^4)),x)

[Out]

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

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