Optimal. Leaf size=331 \[ -\frac{a^2 3^{-q} x^{3 m} \left (c x^n\right )^{-\frac{3 m}{n}} \log ^q\left (c x^n\right ) \left (-\frac{m \log \left (c x^n\right )}{n}\right )^{-q} (a e m-b d n q) \text{Gamma}\left (q+1,-\frac{3 m \log \left (c x^n\right )}{n}\right )}{m n q}-\frac{b^2 x^m \left (c x^n\right )^{-\frac{m}{n}} \log ^{3 q}\left (c x^n\right ) \left (-\frac{m \log \left (c x^n\right )}{n}\right )^{-3 q} (a e m-b d n q) \text{Gamma}\left (3 q+1,-\frac{m \log \left (c x^n\right )}{n}\right )}{m n q}-\frac{3 a b 2^{-2 q-1} x^{2 m} \left (c x^n\right )^{-\frac{2 m}{n}} \log ^{2 q}\left (c x^n\right ) \left (-\frac{m \log \left (c x^n\right )}{n}\right )^{-2 q} (a e m-b d n q) \text{Gamma}\left (2 q+1,-\frac{2 m \log \left (c x^n\right )}{n}\right )}{m n q}-\frac{a^3 x^{4 m} (a e m-b d n q)}{4 b m n q}+\frac{e \left (a x^m+b \log ^q\left (c x^n\right )\right )^4}{4 b n q} \]
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Rubi [A] time = 0.503742, antiderivative size = 331, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2545, 6742, 2310, 2181} \[ -\frac{a^2 3^{-q} x^{3 m} \left (c x^n\right )^{-\frac{3 m}{n}} \log ^q\left (c x^n\right ) \left (-\frac{m \log \left (c x^n\right )}{n}\right )^{-q} (a e m-b d n q) \text{Gamma}\left (q+1,-\frac{3 m \log \left (c x^n\right )}{n}\right )}{m n q}-\frac{b^2 x^m \left (c x^n\right )^{-\frac{m}{n}} \log ^{3 q}\left (c x^n\right ) \left (-\frac{m \log \left (c x^n\right )}{n}\right )^{-3 q} (a e m-b d n q) \text{Gamma}\left (3 q+1,-\frac{m \log \left (c x^n\right )}{n}\right )}{m n q}-\frac{3 a b 2^{-2 q-1} x^{2 m} \left (c x^n\right )^{-\frac{2 m}{n}} \log ^{2 q}\left (c x^n\right ) \left (-\frac{m \log \left (c x^n\right )}{n}\right )^{-2 q} (a e m-b d n q) \text{Gamma}\left (2 q+1,-\frac{2 m \log \left (c x^n\right )}{n}\right )}{m n q}-\frac{a^3 x^{4 m} (a e m-b d n q)}{4 b m n q}+\frac{e \left (a x^m+b \log ^q\left (c x^n\right )\right )^4}{4 b n q} \]
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 )^3}{x} \, dx &=\frac{e \left (a x^m+b \log ^q\left (c x^n\right )\right )^4}{4 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 )^3 \, dx\\ &=\frac{e \left (a x^m+b \log ^q\left (c x^n\right )\right )^4}{4 b n q}-\left (-d+\frac{a e m}{b n q}\right ) \int \left (a^3 x^{-1+4 m}+3 a^2 b x^{-1+3 m} \log ^q\left (c x^n\right )+3 a b^2 x^{-1+2 m} \log ^{2 q}\left (c x^n\right )+b^3 x^{-1+m} \log ^{3 q}\left (c x^n\right )\right ) \, dx\\ &=\frac{a^3 \left (d-\frac{a e m}{b n q}\right ) x^{4 m}}{4 m}+\frac{e \left (a x^m+b \log ^q\left (c x^n\right )\right )^4}{4 b n q}-\left (3 a^2 b \left (-d+\frac{a e m}{b n q}\right )\right ) \int x^{-1+3 m} \log ^q\left (c x^n\right ) \, dx-\left (b^3 \left (-d+\frac{a e m}{b n q}\right )\right ) \int x^{-1+m} \log ^{3 q}\left (c x^n\right ) \, dx-\frac{(3 a b (a e m-b d n q)) \int x^{-1+2 m} \log ^{2 q}\left (c x^n\right ) \, dx}{n q}\\ &=\frac{a^3 \left (d-\frac{a e m}{b n q}\right ) x^{4 m}}{4 m}+\frac{e \left (a x^m+b \log ^q\left (c x^n\right )\right )^4}{4 b n q}-\frac{\left (3 a^2 b \left (-d+\frac{a e m}{b n q}\right ) x^{3 m} \left (c x^n\right )^{-\frac{3 m}{n}}\right ) \operatorname{Subst}\left (\int e^{\frac{3 m x}{n}} x^q \, dx,x,\log \left (c x^n\right )\right )}{n}-\frac{\left (3 a b (a e m-b d n q) x^{2 m} \left (c x^n\right )^{-\frac{2 m}{n}}\right ) \operatorname{Subst}\left (\int e^{\frac{2 m x}{n}} x^{2 q} \, dx,x,\log \left (c x^n\right )\right )}{n^2 q}-\frac{\left (b^3 \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^{3 q} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{a^3 \left (d-\frac{a e m}{b n q}\right ) x^{4 m}}{4 m}-\frac{b^2 (a e m-b d n q) x^m \left (c x^n\right )^{-\frac{m}{n}} \Gamma \left (1+3 q,-\frac{m \log \left (c x^n\right )}{n}\right ) \log ^{3 q}\left (c x^n\right ) \left (-\frac{m \log \left (c x^n\right )}{n}\right )^{-3 q}}{m n q}-\frac{3\ 2^{-1-2 q} a b (a e m-b d n q) x^{2 m} \left (c x^n\right )^{-\frac{2 m}{n}} \Gamma \left (1+2 q,-\frac{2 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{3^{-q} a^2 (a e m-b d n q) x^{3 m} \left (c x^n\right )^{-\frac{3 m}{n}} \Gamma \left (1+q,-\frac{3 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 )^4}{4 b n q}\\ \end{align*}
Mathematica [A] time = 1.55066, size = 445, normalized size = 1.34 \[ \frac{3^{-q} 4^{-q-1} \left (c x^n\right )^{-\frac{3 m}{n}} \left (-\frac{m \log \left (c x^n\right )}{n}\right )^{-3 q} \left (\left (-\frac{m \log \left (c x^n\right )}{n}\right )^q \left (4^q \left (-\frac{m \log \left (c x^n\right )}{n}\right )^q \left (4 a^2 b d n q x^{3 m} \log ^q\left (c x^n\right ) \text{Gamma}\left (q+1,-\frac{3 m \log \left (c x^n\right )}{n}\right )-4 a^3 e m q x^{3 m} \log ^q\left (c x^n\right ) \text{Gamma}\left (q,-\frac{3 m \log \left (c x^n\right )}{n}\right )+3^q \left (c x^n\right )^{\frac{3 m}{n}} \left (-\frac{m \log \left (c x^n\right )}{n}\right )^q \left (a^3 d n q x^{4 m}+b^3 e m \log ^{4 q}\left (c x^n\right )\right )\right )-4 a^2 b e m 3^{q+1} q x^{2 m} \left (c x^n\right )^{m/n} \log ^{2 q}\left (c x^n\right ) \text{Gamma}\left (2 q,-\frac{2 m \log \left (c x^n\right )}{n}\right )+2 a b^2 d n 3^{q+1} q x^{2 m} \left (c x^n\right )^{m/n} \log ^{2 q}\left (c x^n\right ) \text{Gamma}\left (2 q+1,-\frac{2 m \log \left (c x^n\right )}{n}\right )\right )+a b^2 e m \left (-12^{q+1}\right ) q x^m \left (c x^n\right )^{\frac{2 m}{n}} \log ^{3 q}\left (c x^n\right ) \text{Gamma}\left (3 q,-\frac{m \log \left (c x^n\right )}{n}\right )+b^3 d n 3^q 4^{q+1} q x^m \left (c x^n\right )^{\frac{2 m}{n}} \log ^{3 q}\left (c x^n\right ) \text{Gamma}\left (3 q+1,-\frac{m \log \left (c x^n\right )}{n}\right )\right )}{m n q} \]
Antiderivative was successfully verified.
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Maple [F] time = 7.961, 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 ) ^{3}}{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^{3} e x^{3 \, m} \log \left (c x^{n}\right )^{q - 1} + a^{3} d x^{4 \, m} +{\left (b^{3} d x^{m} + b^{3} e \log \left (c x^{n}\right )^{q - 1}\right )} \log \left (c x^{n}\right )^{3 \, q} + 3 \,{\left (a b^{2} e x^{m} \log \left (c x^{n}\right )^{q - 1} + a b^{2} d x^{2 \, m}\right )} \log \left (c x^{n}\right )^{2 \, q} + 3 \,{\left (a^{2} b e x^{2 \, m} \log \left (c x^{n}\right )^{q - 1} + a^{2} b d x^{3 \, 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 )}^{3}{\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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