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

Optimal. Leaf size=23 \[ \text {Int}\left (\frac {a+b \log \left (c x^n\right )}{d+e x^r},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.02, 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 = {} \[ \int \frac {a+b \log \left (c x^n\right )}{d+e x^r} \, dx \]

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*}

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Mathematica [A]  time = 0.09, size = 69, normalized size = 3.00 \[ \frac {x \left (\, _2F_1\left (1,\frac {1}{r};1+\frac {1}{r};-\frac {e x^r}{d}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \, _3F_2\left (1,\frac {1}{r},\frac {1}{r};1+\frac {1}{r},1+\frac {1}{r};-\frac {e x^r}{d}\right )\right )}{d} \]

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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fricas [A]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \log \left (c x^{n}\right ) + a}{e x^{r} + d}, x\right ) \]

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

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)

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maple [A]  time = 0.64, size = 0, normalized size = 0.00 \[ \int \frac {b \ln \left (c \,x^{n}\right )+a}{e \,x^{r}+d}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*x^n))/(d + e*x^r),x)

[Out]

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

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

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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