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

Optimal. Leaf size=25 \[ \text{Unintegrable}\left (\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ),x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0050263, 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 = {} \[ \int \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx &=\int \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 = 0.169122, size = 165, normalized size = 6.6 \[ x \left (-b k m n \, _3F_2\left (1,\frac{1}{m},\frac{1}{m};1+\frac{1}{m},1+\frac{1}{m};-\frac{f x^m}{e}\right )+k m \, _2F_1\left (1,\frac{1}{m};1+\frac{1}{m};-\frac{f x^m}{e}\right ) \left (a+b \log \left (c x^n\right )-b n\right )+a \log \left (d \left (e+f x^m\right )^k\right )+b \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )-b n \log \left (d \left (e+f x^m\right )^k\right )-b k m n \log (x)+b k m n\right )-k m x \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )+b k m n x \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.086, size = 0, normalized size = 0. \begin{align*} \int \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((a+b*ln(c*x^n))*ln(d*(e+f*x^m)^k),x)

[Out]

int((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*}{\left (b x \log \left (x^{n}\right ) -{\left (b{\left (n - \log \left (c\right )\right )} - a\right )} x\right )} \log \left ({\left (f x^{m} + e\right )}^{k}\right ) + \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^{m} + e}\,{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, algorithm="maxima")

[Out]

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

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

[Out]

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

[Out]

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