3.5.12 \(\int \frac {a+b \log (c x^n)}{x^2 (d+e x^r)} \, dx\) [412]

Optimal. Leaf size=26 \[ \text {Int}\left (\frac {a+b \log \left (c x^n\right )}{x^2 \left (d+e x^r\right )},x\right ) \]

[Out]

Unintegrable((a+b*ln(c*x^n))/x^2/(d+e*x^r),x)

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Rubi [A]
time = 0.04, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {a+b \log \left (c x^n\right )}{x^2 \left (d+e x^r\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(a + b*Log[c*x^n])/(x^2*(d + e*x^r)),x]

[Out]

Defer[Int][(a + b*Log[c*x^n])/(x^2*(d + e*x^r)), x]

Rubi steps

\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{x^2 \left (d+e x^r\right )} \, dx &=\int \frac {a+b \log \left (c x^n\right )}{x^2 \left (d+e x^r\right )} \, dx\\ \end {align*}

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Mathematica [A] Leaf count is larger than twice the leaf count of optimal. \(83\) vs. \(2(26)=52\).
time = 0.06, size = 83, normalized size = 3.19 \begin {gather*} -\frac {b n \, _3F_2\left (1,-\frac {1}{r},-\frac {1}{r};1-\frac {1}{r},1-\frac {1}{r};-\frac {e x^r}{d}\right )+\, _2F_1\left (1,-\frac {1}{r};\frac {-1+r}{r};-\frac {e x^r}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{d x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*x^n])/(x^2*(d + e*x^r)),x]

[Out]

-((b*n*HypergeometricPFQ[{1, -r^(-1), -r^(-1)}, {1 - r^(-1), 1 - r^(-1)}, -((e*x^r)/d)] + Hypergeometric2F1[1,
-r^(-1), (-1 + r)/r, -((e*x^r)/d)]*(a + b*Log[c*x^n]))/(d*x))

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Maple [A]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \,x^{n}\right )}{x^{2} \left (d +e \,x^{r}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))/x^2/(d+e*x^r),x)

[Out]

int((a+b*ln(c*x^n))/x^2/(d+e*x^r),x)

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/x^2/(d+e*x^r),x, algorithm="maxima")

[Out]

integrate((b*log(c*x^n) + a)/((e*x^r + d)*x^2), x)

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/x^2/(d+e*x^r),x, algorithm="fricas")

[Out]

integral((b*log(c*x^n) + a)/(x^2*x^r*e + d*x^2), x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b \log {\left (c x^{n} \right )}}{x^{2} \left (d + e x^{r}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))/x**2/(d+e*x**r),x)

[Out]

Integral((a + b*log(c*x**n))/(x**2*(d + e*x**r)), x)

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/x^2/(d+e*x^r),x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)/((x^r*e + d)*x^2), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {a+b\,\ln \left (c\,x^n\right )}{x^2\,\left (d+e\,x^r\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*x^n))/(x^2*(d + e*x^r)),x)

[Out]

int((a + b*log(c*x^n))/(x^2*(d + e*x^r)), x)

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