3.149 \(\int \frac{(a+b \log (c x^n)) \log (d (e+f x^m)^k)}{x^3} \, dx\)

Optimal. Leaf size=28 \[ \text{Unintegrable}\left (\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{x^3},x\right ) \]

[Out]

Unintegrable[((a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k])/x^3, x]

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Rubi [A]  time = 0.0193067, 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 = {} \[ \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[((a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k])/x^3,x]

[Out]

Defer[Int][((a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k])/x^3, 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^3} \, 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^3} \, dx\\ \end{align*}

Mathematica [A]  time = 0.152721, size = 292, normalized size = 10.43 \[ \frac{b e k (m-2) m n \, _3F_2\left (1,-\frac{2}{m},-\frac{2}{m};1-\frac{2}{m},1-\frac{2}{m};-\frac{f x^m}{e}\right )+8 a e \log \left (d \left (e+f x^m\right )^k\right )-4 a e m \log \left (d \left (e+f x^m\right )^k\right )+4 a f k m x^m \, _2F_1\left (1,\frac{m-2}{m};2-\frac{2}{m};-\frac{f x^m}{e}\right )+8 b e \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )-4 b e m \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )+b e k (m-2) m \left (2 \log \left (c x^n\right )+n\right ) \, _2F_1\left (1,-\frac{2}{m};\frac{m-2}{m};-\frac{f x^m}{e}\right )-2 b e k m^2 \log \left (c x^n\right )+4 b e k m \log \left (c x^n\right )+4 b e n \log \left (d \left (e+f x^m\right )^k\right )-2 b e m n \log \left (d \left (e+f x^m\right )^k\right )-2 b e k m^2 n+4 b e k m n}{8 e (m-2) x^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k])/x^3,x]

[Out]

(4*b*e*k*m*n - 2*b*e*k*m^2*n + 4*a*f*k*m*x^m*Hypergeometric2F1[1, (-2 + m)/m, 2 - 2/m, -((f*x^m)/e)] + b*e*k*(
-2 + m)*m*n*HypergeometricPFQ[{1, -2/m, -2/m}, {1 - 2/m, 1 - 2/m}, -((f*x^m)/e)] + 4*b*e*k*m*Log[c*x^n] - 2*b*
e*k*m^2*Log[c*x^n] + b*e*k*(-2 + m)*m*Hypergeometric2F1[1, -2/m, (-2 + m)/m, -((f*x^m)/e)]*(n + 2*Log[c*x^n])
+ 8*a*e*Log[d*(e + f*x^m)^k] - 4*a*e*m*Log[d*(e + f*x^m)^k] + 4*b*e*n*Log[d*(e + f*x^m)^k] - 2*b*e*m*n*Log[d*(
e + f*x^m)^k] + 8*b*e*Log[c*x^n]*Log[d*(e + f*x^m)^k] - 4*b*e*m*Log[c*x^n]*Log[d*(e + f*x^m)^k])/(8*e*(-2 + m)
*x^2)

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Maple [A]  time = 0.088, 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}^{3}}}\, 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^3,x)

[Out]

int((a+b*ln(c*x^n))*ln(d*(e+f*x^m)^k)/x^3,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (b{\left (n + 2 \, \log \left (c\right )\right )} + 2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} \log \left ({\left (f x^{m} + e\right )}^{k}\right )}{4 \, x^{2}} + \int \frac{4 \, b e \log \left (c\right ) \log \left (d\right ) + 4 \, a e \log \left (d\right ) +{\left (2 \,{\left (f k m + 2 \, f \log \left (d\right )\right )} a +{\left (f k m n + 2 \,{\left (f k m + 2 \, f \log \left (d\right )\right )} \log \left (c\right )\right )} b\right )} x^{m} + 2 \,{\left ({\left (f k m + 2 \, f \log \left (d\right )\right )} b x^{m} + 2 \, b e \log \left (d\right )\right )} \log \left (x^{n}\right )}{4 \,{\left (f x^{3} x^{m} + e x^{3}\right )}}\,{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^3,x, algorithm="maxima")

[Out]

-1/4*(b*(n + 2*log(c)) + 2*b*log(x^n) + 2*a)*log((f*x^m + e)^k)/x^2 + integrate(1/4*(4*b*e*log(c)*log(d) + 4*a
*e*log(d) + (2*(f*k*m + 2*f*log(d))*a + (f*k*m*n + 2*(f*k*m + 2*f*log(d))*log(c))*b)*x^m + 2*((f*k*m + 2*f*log
(d))*b*x^m + 2*b*e*log(d))*log(x^n))/(f*x^3*x^m + e*x^3), 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^{3}}, 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^3,x, algorithm="fricas")

[Out]

integral((b*log(c*x^n) + a)*log((f*x^m + e)^k*d)/x^3, 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**3,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^{3}}\,{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^3,x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)*log((f*x^m + e)^k*d)/x^3, x)