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

Optimal. Leaf size=27 \[ \text{Unintegrable}\left (\frac{(f x)^m \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0688923, 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 = {} \[ \int \frac{(f x)^m \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.387871, size = 177, normalized size = 6.56 \[ \frac{x (f x)^m \left (b n (m-r+1) \left (d+e x^r\right ) \, _3F_2\left (1,\frac{m}{r}+\frac{1}{r},\frac{m}{r}+\frac{1}{r};\frac{m}{r}+\frac{1}{r}+1,\frac{m}{r}+\frac{1}{r}+1;-\frac{e x^r}{d}\right )-(m+1) \left (\left (d+e x^r\right ) \, _2F_1\left (1,\frac{m+1}{r};\frac{m+r+1}{r};-\frac{e x^r}{d}\right ) \left (a (m-r+1)+b (m-r+1) \log \left (c x^n\right )+b n\right )-d (m+1) \left (a+b \log \left (c x^n\right )\right )\right )\right )}{d^2 (m+1)^2 r \left (d+e x^r\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 1.058, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx \right ) ^{m} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{ \left ( d+e{x}^{r} \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (c x^{n}\right ) + a\right )} \left (f x\right )^{m}}{{\left (e x^{r} + d\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\left (f x\right )^{m} b \log \left (c x^{n}\right ) + \left (f x\right )^{m} a}{e^{2} x^{2 \, r} + 2 \, d e x^{r} + d^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (c x^{n}\right ) + a\right )} \left (f x\right )^{m}}{{\left (e x^{r} + d\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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