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

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

[Out]

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

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Rubi [A]  time = 0.06, 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 = {} \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 \left (d+e x^r\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.20, size = 135, normalized size = 5.19 \[ \frac {-b n (r+1) \left (d+e x^r\right ) \, _3F_2\left (1,-\frac {1}{r},-\frac {1}{r};1-\frac {1}{r},1-\frac {1}{r};-\frac {e x^r}{d}\right )-\left (d+e x^r\right ) \, _2F_1\left (1,-\frac {1}{r};\frac {r-1}{r};-\frac {e x^r}{d}\right ) \left (a r+a+b (r+1) \log \left (c x^n\right )-b n\right )+d \left (a+b \log \left (c x^n\right )\right )}{d^2 r x \left (d+e x^r\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-(b*n*(1 + r)*(d + e*x^r)*HypergeometricPFQ[{1, -r^(-1), -r^(-1)}, {1 - r^(-1), 1 - r^(-1)}, -((e*x^r)/d)]) +
d*(a + b*Log[c*x^n]) - (d + e*x^r)*Hypergeometric2F1[1, -r^(-1), (-1 + r)/r, -((e*x^r)/d)]*(a - b*n + a*r + b
*(1 + r)*Log[c*x^n]))/(d^2*r*x*(d + e*x^r))

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fricas [A]  time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \log \left (c x^{n}\right ) + a}{e^{2} x^{2} x^{2 \, r} + 2 \, d e x^{2} x^{r} + d^{2} x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.79, size = 0, normalized size = 0.00 \[ \int \frac {b \ln \left (c \,x^{n}\right )+a}{\left (e \,x^{r}+d \right )^{2} x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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