3.220 \(\int -(d x)^m (a+b \log (c x^n)) \log (1-e x^q) \, dx\)

Optimal. Leaf size=29 \[ -\text{Unintegrable}\left ((d x)^m \log \left (1-e x^q\right ) \left (a+b \log \left (c x^n\right )\right ),x\right ) \]

[Out]

-Unintegrable[(d*x)^m*(a + b*Log[c*x^n])*Log[1 - e*x^q], x]

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Rubi [A]  time = 0.0215918, 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 = {} \[ \int -(d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Int[-((d*x)^m*(a + b*Log[c*x^n])*Log[1 - e*x^q]),x]

[Out]

-Defer[Int][(d*x)^m*(a + b*Log[c*x^n])*Log[1 - e*x^q], x]

Rubi steps

\begin{align*} \int -(d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ) \, dx &=-\int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ) \, dx\\ \end{align*}

Mathematica [A]  time = 0.232955, size = 266, normalized size = 9.17 \[ -\frac{x (d x)^m \left (-b n q \, _3F_2\left (1,\frac{m}{q}+\frac{1}{q},\frac{m}{q}+\frac{1}{q};\frac{m}{q}+\frac{1}{q}+1,\frac{m}{q}+\frac{1}{q}+1;e x^q\right )+q \, _2F_1\left (1,\frac{m+1}{q};\frac{m+q+1}{q};e x^q\right ) \left (a m+a+b (m+1) \log \left (c x^n\right )-b n\right )+a m^2 \log \left (1-e x^q\right )+2 a m \log \left (1-e x^q\right )+a \log \left (1-e x^q\right )-a m q-a q+b m^2 \log \left (c x^n\right ) \log \left (1-e x^q\right )+2 b m \log \left (c x^n\right ) \log \left (1-e x^q\right )+b \log \left (c x^n\right ) \log \left (1-e x^q\right )-b m q \log \left (c x^n\right )-b q \log \left (c x^n\right )-b m n \log \left (1-e x^q\right )-b n \log \left (1-e x^q\right )+2 b n q\right )}{(m+1)^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[-((d*x)^m*(a + b*Log[c*x^n])*Log[1 - e*x^q]),x]

[Out]

-((x*(d*x)^m*(-(a*q) - a*m*q + 2*b*n*q - b*n*q*HypergeometricPFQ[{1, q^(-1) + m/q, q^(-1) + m/q}, {1 + q^(-1)
+ m/q, 1 + q^(-1) + m/q}, e*x^q] - b*q*Log[c*x^n] - b*m*q*Log[c*x^n] + q*Hypergeometric2F1[1, (1 + m)/q, (1 +
m + q)/q, e*x^q]*(a + a*m - b*n + b*(1 + m)*Log[c*x^n]) + a*Log[1 - e*x^q] + 2*a*m*Log[1 - e*x^q] + a*m^2*Log[
1 - e*x^q] - b*n*Log[1 - e*x^q] - b*m*n*Log[1 - e*x^q] + b*Log[c*x^n]*Log[1 - e*x^q] + 2*b*m*Log[c*x^n]*Log[1
- e*x^q] + b*m^2*Log[c*x^n]*Log[1 - e*x^q]))/(1 + m)^3)

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Maple [A]  time = 0.468, size = 844, normalized size = 29.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(d*x)^m*(a+b*ln(c*x^n))*ln(1-e*x^q),x)

[Out]

-(d*x)^m*x^(-m)*(-e)^(-1/q*m-1/q)*a/q*(q*x^(1+m)*(-e)^(1/q*m+1/q)/(1+m)*ln(1-e*x^q)-q/(1+m+q)*x^(1+m+q)*e*(-e)
^(1/q*m+1/q)*(-q-m-1)/(1+m)*LerchPhi(e*x^q,1,(1+m+q)/q))-(d*x)^m*x^(-m)*(-e)^(-1/q*m-1/q)*b*ln(c)/q*(q*x^(1+m)
*(-e)^(1/q*m+1/q)/(1+m)*ln(1-e*x^q)-q/(1+m+q)*x^(1+m+q)*e*(-e)^(1/q*m+1/q)*(-q-m-1)/(1+m)*LerchPhi(e*x^q,1,(1+
m+q)/q))+(ln(-e)/q^2*(-e)^(-1/q*m-1/q)*(d*x)^m*x^(-m)*b*n*(q*x^m*(-e)^(1/q*m+1/q)/(1+m)*ln(1-e*x^q)-q/(1+m+q)*
x^(q+m)*e*(-e)^(1/q*m+1/q)*(-q-m-1)/(1+m)*LerchPhi(e*x^q,1,(1+m+q)/q))-(-e)^(-1/q*m-1/q)*(d*x)^m*x^(-m)*b*n/q*
(q*ln(x)*x^m*(-e)^(1/q*m+1/q)/(1+m)*ln(1-e*x^q)+ln(-e)*x^m*(-e)^(1/q*m+1/q)/(1+m)*ln(1-e*x^q)-q*x^m*(-e)^(1/q*
m+1/q)/(1+m)^2*ln(1-e*x^q)+q/(1+m+q)^2*x^(q+m)*e*(-e)^(1/q*m+1/q)*(-q-m-1)/(1+m)*LerchPhi(e*x^q,1,(1+m+q)/q)-q
/(1+m+q)*x^(q+m)*e*ln(x)*(-e)^(1/q*m+1/q)*(-q-m-1)/(1+m)*LerchPhi(e*x^q,1,(1+m+q)/q)-1/(1+m+q)*x^(q+m)*e*ln(-e
)*(-e)^(1/q*m+1/q)*(-q-m-1)/(1+m)*LerchPhi(e*x^q,1,(1+m+q)/q)+q/(1+m+q)*x^(q+m)*e*(-e)^(1/q*m+1/q)/(1+m)*Lerch
Phi(e*x^q,1,(1+m+q)/q)+q/(1+m+q)*x^(q+m)*e*(-e)^(1/q*m+1/q)*(-q-m-1)/(1+m)^2*LerchPhi(e*x^q,1,(1+m+q)/q)+1/(1+
m+q)*x^(q+m)*e*(-e)^(1/q*m+1/q)*(-q-m-1)/(1+m)*LerchPhi(e*x^q,2,(1+m+q)/q)))*x

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-(d*x)^m*(a+b*log(c*x^n))*log(1-e*x^q),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\left (d x\right )^{m} b \log \left (c x^{n}\right ) \log \left (-e x^{q} + 1\right ) - \left (d x\right )^{m} a \log \left (-e x^{q} + 1\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-(d*x)^m*(a+b*log(c*x^n))*log(1-e*x^q),x, algorithm="fricas")

[Out]

integral(-(d*x)^m*b*log(c*x^n)*log(-e*x^q + 1) - (d*x)^m*a*log(-e*x^q + 1), 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(-(d*x)**m*(a+b*ln(c*x**n))*ln(1-e*x**q),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 )} \left (d x\right )^{m} \log \left (-e x^{q} + 1\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-(d*x)^m*(a+b*log(c*x^n))*log(1-e*x^q),x, algorithm="giac")

[Out]

integrate(-(b*log(c*x^n) + a)*(d*x)^m*log(-e*x^q + 1), x)