Optimal. Leaf size=235 \[ -\frac{b x^m \left (c x^n\right )^{-\frac{m}{n}} \log ^{2 q}\left (c x^n\right ) \left (-\frac{m \log \left (c x^n\right )}{n}\right )^{-2 q} (a e m-b d n q) \text{Gamma}\left (2 q+1,-\frac{m \log \left (c x^n\right )}{n}\right )}{m n q}-\frac{a 2^{-q} x^{2 m} \left (c x^n\right )^{-\frac{2 m}{n}} \log ^q\left (c x^n\right ) \left (-\frac{m \log \left (c x^n\right )}{n}\right )^{-q} (a e m-b d n q) \text{Gamma}\left (q+1,-\frac{2 m \log \left (c x^n\right )}{n}\right )}{m n q}-\frac{a^2 x^{3 m} (a e m-b d n q)}{3 b m n q}+\frac{e \left (a x^m+b \log ^q\left (c x^n\right )\right )^3}{3 b n q} \]
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Rubi [A] time = 0.395189, antiderivative size = 235, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2545, 6742, 2310, 2181} \[ -\frac{b x^m \left (c x^n\right )^{-\frac{m}{n}} \log ^{2 q}\left (c x^n\right ) \left (-\frac{m \log \left (c x^n\right )}{n}\right )^{-2 q} (a e m-b d n q) \text{Gamma}\left (2 q+1,-\frac{m \log \left (c x^n\right )}{n}\right )}{m n q}-\frac{a 2^{-q} x^{2 m} \left (c x^n\right )^{-\frac{2 m}{n}} \log ^q\left (c x^n\right ) \left (-\frac{m \log \left (c x^n\right )}{n}\right )^{-q} (a e m-b d n q) \text{Gamma}\left (q+1,-\frac{2 m \log \left (c x^n\right )}{n}\right )}{m n q}-\frac{a^2 x^{3 m} (a e m-b d n q)}{3 b m n q}+\frac{e \left (a x^m+b \log ^q\left (c x^n\right )\right )^3}{3 b n q} \]
Antiderivative was successfully verified.
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Rule 2545
Rule 6742
Rule 2310
Rule 2181
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 )^2}{x} \, dx &=\frac{e \left (a x^m+b \log ^q\left (c x^n\right )\right )^3}{3 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 )^2 \, dx\\ &=\frac{e \left (a x^m+b \log ^q\left (c x^n\right )\right )^3}{3 b n q}-\left (-d+\frac{a e m}{b n q}\right ) \int \left (a^2 x^{-1+3 m}+2 a b x^{-1+2 m} \log ^q\left (c x^n\right )+b^2 x^{-1+m} \log ^{2 q}\left (c x^n\right )\right ) \, dx\\ &=\frac{a^2 \left (d-\frac{a e m}{b n q}\right ) x^{3 m}}{3 m}+\frac{e \left (a x^m+b \log ^q\left (c x^n\right )\right )^3}{3 b n q}-\left (2 a b \left (-d+\frac{a e m}{b n q}\right )\right ) \int x^{-1+2 m} \log ^q\left (c x^n\right ) \, dx-\left (b^2 \left (-d+\frac{a e m}{b n q}\right )\right ) \int x^{-1+m} \log ^{2 q}\left (c x^n\right ) \, dx\\ &=\frac{a^2 \left (d-\frac{a e m}{b n q}\right ) x^{3 m}}{3 m}+\frac{e \left (a x^m+b \log ^q\left (c x^n\right )\right )^3}{3 b n q}-\frac{\left (2 a b \left (-d+\frac{a e m}{b n q}\right ) x^{2 m} \left (c x^n\right )^{-\frac{2 m}{n}}\right ) \operatorname{Subst}\left (\int e^{\frac{2 m x}{n}} x^q \, dx,x,\log \left (c x^n\right )\right )}{n}-\frac{\left (b^2 \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^{2 q} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{a^2 \left (d-\frac{a e m}{b n q}\right ) x^{3 m}}{3 m}-\frac{b (a e m-b d n q) x^m \left (c x^n\right )^{-\frac{m}{n}} \Gamma \left (1+2 q,-\frac{m \log \left (c x^n\right )}{n}\right ) \log ^{2 q}\left (c x^n\right ) \left (-\frac{m \log \left (c x^n\right )}{n}\right )^{-2 q}}{m n q}-\frac{2^{-q} a (a e m-b d n q) x^{2 m} \left (c x^n\right )^{-\frac{2 m}{n}} \Gamma \left (1+q,-\frac{2 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}}{m n q}+\frac{e \left (a x^m+b \log ^q\left (c x^n\right )\right )^3}{3 b n q}\\ \end{align*}
Mathematica [A] time = 0.927323, size = 298, normalized size = 1.27 \[ \frac{2^{-q} \left (c x^n\right )^{-\frac{2 m}{n}} \left (-\frac{m \log \left (c x^n\right )}{n}\right )^{-2 q} \left (\left (-\frac{m \log \left (c x^n\right )}{n}\right )^q \left (-3 a^2 e m q x^{2 m} \log ^q\left (c x^n\right ) \text{Gamma}\left (q,-\frac{2 m \log \left (c x^n\right )}{n}\right )+3 a b d n q x^{2 m} \log ^q\left (c x^n\right ) \text{Gamma}\left (q+1,-\frac{2 m \log \left (c x^n\right )}{n}\right )+2^q \left (c x^n\right )^{\frac{2 m}{n}} \left (-\frac{m \log \left (c x^n\right )}{n}\right )^q \left (a^2 d n q x^{3 m}+b^2 e m \log ^{3 q}\left (c x^n\right )\right )\right )-3 a b e m 2^{q+1} q x^m \left (c x^n\right )^{m/n} \log ^{2 q}\left (c x^n\right ) \text{Gamma}\left (2 q,-\frac{m \log \left (c x^n\right )}{n}\right )+3 b^2 d n 2^q q x^m \left (c x^n\right )^{m/n} \log ^{2 q}\left (c x^n\right ) \text{Gamma}\left (2 q+1,-\frac{m \log \left (c x^n\right )}{n}\right )\right )}{3 m n q} \]
Antiderivative was successfully verified.
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Maple [F] time = 17.253, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( d{x}^{m}+e \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{-1+q} \right ) \left ( a{x}^{m}+b \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{q} \right ) ^{2}}{x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{a^{2} e x^{2 \, m} \log \left (c x^{n}\right )^{q - 1} + a^{2} d x^{3 \, m} +{\left (b^{2} d x^{m} + b^{2} e \log \left (c x^{n}\right )^{q - 1}\right )} \log \left (c x^{n}\right )^{2 \, q} + 2 \,{\left (a b e x^{m} \log \left (c x^{n}\right )^{q - 1} + a b d x^{2 \, m}\right )} \log \left (c x^{n}\right )^{q}}{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )}^{2}{\left (d x^{m} + e \log \left (c x^{n}\right )^{q - 1}\right )}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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