3.411 \(\int \frac{a+b \log (c x^n)}{d+e x^r} \, dx\)

Optimal. Leaf size=22 \[ \text{Unintegrable}\left (\frac{a+b \log \left (c x^n\right )}{d+e x^r},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0180241, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{a+b \log \left (c x^n\right )}{d+e x^r} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*Log[c*x^n])/(d + e*x^r),x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.0787297, size = 69, normalized size = 3.14 \[ \frac{x \left (\, _2F_1\left (1,\frac{1}{r};1+\frac{1}{r};-\frac{e x^r}{d}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \, _3F_2\left (1,\frac{1}{r},\frac{1}{r};1+\frac{1}{r},1+\frac{1}{r};-\frac{e x^r}{d}\right )\right )}{d} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.642, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c{x}^{n} \right ) }{d+e{x}^{r}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \log \left (c x^{n}\right ) + a}{e x^{r} + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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