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

Optimal. Leaf size=28 \[ \text{Unintegrable}\left (\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{x^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0195598, 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 = {} \[ \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.164843, size = 282, normalized size = 10.07 \[ \frac{b e k (m-1) m n \, _3F_2\left (1,-\frac{1}{m},-\frac{1}{m};1-\frac{1}{m},1-\frac{1}{m};-\frac{f x^m}{e}\right )+a e \log \left (d \left (e+f x^m\right )^k\right )-a e m \log \left (d \left (e+f x^m\right )^k\right )+a f k m x^m \, _2F_1\left (1,\frac{m-1}{m};2-\frac{1}{m};-\frac{f x^m}{e}\right )+b e \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )-b e m \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )+b e k (m-1) m \left (\log \left (c x^n\right )+n\right ) \, _2F_1\left (1,-\frac{1}{m};\frac{m-1}{m};-\frac{f x^m}{e}\right )-b e k m^2 \log \left (c x^n\right )+b e k m \log \left (c x^n\right )+b e n \log \left (d \left (e+f x^m\right )^k\right )-b e m n \log \left (d \left (e+f x^m\right )^k\right )-2 b e k m^2 n+2 b e k m n}{e (m-1) x} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.087, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \ln \left ( d \left ( e+f{x}^{m} \right ) ^{k} \right ) }{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*log(c*x^n) + a)*log((f*x^m + e)^k*d)/x^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((a+b*ln(c*x**n))*ln(d*(e+f*x**m)**k)/x**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 )} \log \left ({\left (f x^{m} + e\right )}^{k} d\right )}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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