3.145 \(\int x (a+b \log (c x^n)) \log (d (e+f x^m)^k) \, dx\)

Optimal. Leaf size=26 \[ \text{Unintegrable}\left (x \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ),x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0114566, 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 = {} \[ \int x \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.170561, size = 292, normalized size = 11.23 \[ -\frac{x^2 \left (b e k m (m+2) n \, _3F_2\left (1,\frac{2}{m},\frac{2}{m};1+\frac{2}{m},1+\frac{2}{m};-\frac{f x^m}{e}\right )-8 a e \log \left (d \left (e+f x^m\right )^k\right )-4 a e m \log \left (d \left (e+f x^m\right )^k\right )+4 a f k m x^m \, _2F_1\left (1,\frac{m+2}{m};2+\frac{2}{m};-\frac{f x^m}{e}\right )-8 b e \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )-4 b e m \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )+b e k m (m+2) \left (n-2 \log \left (c x^n\right )\right ) \, _2F_1\left (1,\frac{2}{m};\frac{m+2}{m};-\frac{f x^m}{e}\right )+2 b e k m^2 \log \left (c x^n\right )+4 b e k m \log \left (c x^n\right )+4 b e n \log \left (d \left (e+f x^m\right )^k\right )+2 b e m n \log \left (d \left (e+f x^m\right )^k\right )-2 b e k m^2 n-4 b e k m n\right )}{8 e (m+2)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(x^2*(-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*m*(2 + m)*n*HypergeometricPFQ[{1, 2/m, 2/m}, {1 + 2/m, 1 + 2/m}, -((f*x^m)/e)] + b*e*k*m*(2 + m)*Hypergeo
metric2F1[1, 2/m, (2 + m)/m, -((f*x^m)/e)]*(n - 2*Log[c*x^n]) + 4*b*e*k*m*Log[c*x^n] + 2*b*e*k*m^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)
)

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Maple [A]  time = 0.084, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \ln \left ( d \left ( e+f{x}^{m} \right ) ^{k} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{4} \,{\left (2 \, b x^{2} \log \left (x^{n}\right ) -{\left (b{\left (n - 2 \, \log \left (c\right )\right )} - 2 \, a\right )} x^{2}\right )} \log \left ({\left (f x^{m} + e\right )}^{k}\right ) + \int -\frac{{\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 x^{m} - 4 \,{\left (b e \log \left (c\right ) \log \left (d\right ) + a e \log \left (d\right )\right )} x + 2 \,{\left ({\left (f k m - 2 \, f \log \left (d\right )\right )} b x x^{m} - 2 \, b e x \log \left (d\right )\right )} \log \left (x^{n}\right )}{4 \,{\left (f x^{m} + e\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*b*x^2*log(x^n) - (b*(n - 2*log(c)) - 2*a)*x^2)*log((f*x^m + e)^k) + integrate(-1/4*((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*x^m - 4*(b*e*log(c)*log(d) + a*e*log(d))*x + 2*((f*k*m
- 2*f*log(d))*b*x*x^m - 2*b*e*x*log(d))*log(x^n))/(f*x^m + e), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x*log(c*x^n) + a*x)*log((f*x^m + e)^k*d), 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(x*(a+b*ln(c*x**n))*ln(d*(e+f*x**m)**k),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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