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

Optimal. Leaf size=30 \[ -\text {Int}\left ((d x)^m \log \left (1-e x^q\right ) \left (a+b \log \left (c x^n\right )\right ),x\right ) \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [A]  time = 0.24, size = 266, normalized size = 8.87 \[ -\frac {x (d x)^m \left (-b n q \, _3F_2\left (1,\frac {m}{q}+\frac {1}{q},\frac {m}{q}+\frac {1}{q};\frac {m}{q}+\frac {1}{q}+1,\frac {m}{q}+\frac {1}{q}+1;e x^q\right )+q \, _2F_1\left (1,\frac {m+1}{q};\frac {m+q+1}{q};e x^q\right ) \left (a m+a+b (m+1) \log \left (c x^n\right )-b n\right )+a m^2 \log \left (1-e x^q\right )+2 a m \log \left (1-e x^q\right )+a \log \left (1-e x^q\right )-a m q-a q+b m^2 \log \left (c x^n\right ) \log \left (1-e x^q\right )+2 b m \log \left (c x^n\right ) \log \left (1-e x^q\right )+b \log \left (c x^n\right ) \log \left (1-e x^q\right )-b m q \log \left (c x^n\right )-b q \log \left (c x^n\right )-b m n \log \left (1-e x^q\right )-b n \log \left (1-e x^q\right )+2 b n q\right )}{(m+1)^3} \]

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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fricas [A]  time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\left (d x\right )^{m} b \log \left (c x^{n}\right ) \log \left (-e x^{q} + 1\right ) - \left (d x\right )^{m} a \log \left (-e x^{q} + 1\right ), x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -{\left (b \log \left (c x^{n}\right ) + a\right )} \left (d x\right )^{m} \log \left (-e x^{q} + 1\right )\,{d x} \]

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)

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maple [A]  time = 0.87, size = 844, normalized size = 28.13 \[ -\frac {\left (-\frac {\left (-m -q -1\right ) e q \,x^{m +q +1} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \Phi \left (e \,x^{q}, 1, \frac {m +q +1}{q}\right )}{\left (m +q +1\right ) \left (m +1\right )}+\frac {q \,x^{m +1} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (-e \,x^{q}+1\right )}{m +1}\right ) b \,x^{-m} \left (d x \right )^{m} \left (-e \right )^{-\frac {m}{q}-\frac {1}{q}} \ln \relax (c )}{q}-\frac {\left (-\frac {\left (-m -q -1\right ) e q \,x^{m +q +1} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \Phi \left (e \,x^{q}, 1, \frac {m +q +1}{q}\right )}{\left (m +q +1\right ) \left (m +1\right )}+\frac {q \,x^{m +1} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (-e \,x^{q}+1\right )}{m +1}\right ) a \,x^{-m} \left (d x \right )^{m} \left (-e \right )^{-\frac {m}{q}-\frac {1}{q}}}{q}+\left (-\frac {\left (-\frac {\left (-m -q -1\right ) e q \,x^{m +q} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \relax (x ) \Phi \left (e \,x^{q}, 1, \frac {m +q +1}{q}\right )}{\left (m +q +1\right ) \left (m +1\right )}+\frac {\left (-m -q -1\right ) e q \,x^{m +q} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \Phi \left (e \,x^{q}, 1, \frac {m +q +1}{q}\right )}{\left (m +q +1\right )^{2} \left (m +1\right )}+\frac {e q \,x^{m +q} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \Phi \left (e \,x^{q}, 1, \frac {m +q +1}{q}\right )}{\left (m +q +1\right ) \left (m +1\right )}+\frac {\left (-m -q -1\right ) e q \,x^{m +q} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \Phi \left (e \,x^{q}, 1, \frac {m +q +1}{q}\right )}{\left (m +q +1\right ) \left (m +1\right )^{2}}-\frac {\left (-m -q -1\right ) e \,x^{m +q} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (-e \right ) \Phi \left (e \,x^{q}, 1, \frac {m +q +1}{q}\right )}{\left (m +q +1\right ) \left (m +1\right )}+\frac {q \,x^{m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \relax (x ) \ln \left (-e \,x^{q}+1\right )}{m +1}+\frac {\left (-m -q -1\right ) e \,x^{m +q} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \Phi \left (e \,x^{q}, 2, \frac {m +q +1}{q}\right )}{\left (m +q +1\right ) \left (m +1\right )}-\frac {q \,x^{m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (-e \,x^{q}+1\right )}{\left (m +1\right )^{2}}+\frac {x^{m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (-e \right ) \ln \left (-e \,x^{q}+1\right )}{m +1}\right ) b n \,x^{-m} \left (d x \right )^{m} \left (-e \right )^{-\frac {m}{q}-\frac {1}{q}}}{q}+\frac {\left (-\frac {\left (-m -q -1\right ) e q \,x^{m +q} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \Phi \left (e \,x^{q}, 1, \frac {m +q +1}{q}\right )}{\left (m +q +1\right ) \left (m +1\right )}+\frac {q \,x^{m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (-e \,x^{q}+1\right )}{m +1}\right ) b n \,x^{-m} \left (d x \right )^{m} \left (-e \right )^{-\frac {m}{q}-\frac {1}{q}} \ln \left (-e \right )}{q^{2}}\right ) x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {{\left (b d^{m} {\left (m + 1\right )} x x^{m} \log \left (x^{n}\right ) + {\left (a d^{m} {\left (m + 1\right )} + {\left (d^{m} {\left (m + 1\right )} \log \relax (c) - d^{m} n\right )} b\right )} x x^{m}\right )} \log \left (-e x^{q} + 1\right )}{m^{2} + 2 \, m + 1} + \int \frac {{\left (m q + q\right )} b d^{m} e e^{\left (m \log \relax (x) + q \log \relax (x)\right )} \log \left (x^{n}\right ) + {\left ({\left (m q + q\right )} a d^{m} e - {\left (d^{m} e n q - {\left (m q + q\right )} d^{m} e \log \relax (c)\right )} b\right )} e^{\left (m \log \relax (x) + q \log \relax (x)\right )}}{{\left (m^{2} + 2 \, m + 1\right )} e x^{q} - m^{2} - 2 \, m - 1}\,{d x} \]

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]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int -\ln \left (1-e\,x^q\right )\,{\left (d\,x\right )}^m\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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