Optimal. Leaf size=139 \[ x^m \left (c x^n\right )^{-\frac {m}{n}} \log ^q\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-q} \left (\frac {b d}{m}-\frac {a e}{n q}\right ) \Gamma \left (q+1,-\frac {m \log \left (c x^n\right )}{n}\right )+\frac {e \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}{2 b n q}-\frac {a x^{2 m} (a e m-b d n q)}{2 b m n q} \]
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Rubi [A] time = 0.17, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2545, 14, 2310, 2181} \[ x^m \left (c x^n\right )^{-\frac {m}{n}} \log ^q\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-q} \left (\frac {b d}{m}-\frac {a e}{n q}\right ) \text {Gamma}\left (q+1,-\frac {m \log \left (c x^n\right )}{n}\right )+\frac {e \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}{2 b n q}-\frac {a x^{2 m} (a e m-b d n q)}{2 b m n q} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2181
Rule 2310
Rule 2545
Rubi steps
\begin {align*} \int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )}{x} \, dx &=\frac {e \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}{2 b n q}-\left (-d+\frac {a e m}{b n q}\right ) \int x^{-1+m} \left (a x^m+b \log ^q\left (c x^n\right )\right ) \, dx\\ &=\frac {e \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}{2 b n q}-\left (-d+\frac {a e m}{b n q}\right ) \int \left (a x^{-1+2 m}+b x^{-1+m} \log ^q\left (c x^n\right )\right ) \, dx\\ &=\frac {a \left (d-\frac {a e m}{b n q}\right ) x^{2 m}}{2 m}+\frac {e \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}{2 b n q}-\left (b \left (-d+\frac {a e m}{b n q}\right )\right ) \int x^{-1+m} \log ^q\left (c x^n\right ) \, dx\\ &=\frac {a \left (d-\frac {a e m}{b n q}\right ) x^{2 m}}{2 m}+\frac {e \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}{2 b n q}-\frac {\left (b \left (-d+\frac {a e m}{b n q}\right ) x^m \left (c x^n\right )^{-\frac {m}{n}}\right ) \operatorname {Subst}\left (\int e^{\frac {m x}{n}} x^q \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {a \left (d-\frac {a e m}{b n q}\right ) x^{2 m}}{2 m}+\left (\frac {b d}{m}-\frac {a e}{n q}\right ) x^m \left (c x^n\right )^{-\frac {m}{n}} \Gamma \left (1+q,-\frac {m \log \left (c x^n\right )}{n}\right ) \log ^q\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-q}+\frac {e \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}{2 b n q}\\ \end {align*}
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Mathematica [A] time = 0.49, size = 157, normalized size = 1.13 \[ \frac {\left (c x^n\right )^{-\frac {m}{n}} \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-q} \left (\left (c x^n\right )^{m/n} \left (-\frac {m \log \left (c x^n\right )}{n}\right )^q \left (a d n q x^{2 m}+b e m \log ^{2 q}\left (c x^n\right )\right )-2 a e m q x^m \log ^q\left (c x^n\right ) \Gamma \left (q,-\frac {m \log \left (c x^n\right )}{n}\right )+2 b d n q x^m \log ^q\left (c x^n\right ) \Gamma \left (q+1,-\frac {m \log \left (c x^n\right )}{n}\right )\right )}{2 m n q} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a e x^{m} \log \left (c x^{n}\right )^{q - 1} + a d x^{2 \, m} + {\left (b d x^{m} + b e \log \left (c x^{n}\right )^{q - 1}\right )} \log \left (c x^{n}\right )^{q}}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )} {\left (d x^{m} + e \log \left (c x^{n}\right )^{q - 1}\right )}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 73.29, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \,x^{m}+e \ln \left (c \,x^{n}\right )^{q -1}\right ) \left (a \,x^{m}+b \ln \left (c \,x^{n}\right )^{q}\right )}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (a\,x^m+b\,{\ln \left (c\,x^n\right )}^q\right )\,\left (d\,x^m+e\,{\ln \left (c\,x^n\right )}^{q-1}\right )}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a x^{m} + b \log {\left (c x^{n} \right )}^{q}\right ) \left (d x^{m} + \frac {e \log {\left (c x^{n} \right )}^{q}}{\log {\left (c x^{n} \right )}}\right )}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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