Integrand size = 40, antiderivative size = 331 \[ \int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^3}{x} \, dx=-\frac {a^3 (a e m-b d n q) x^{4 m}}{4 b m n q}-\frac {b^2 (a e m-b d n q) x^m \left (c x^n\right )^{-\frac {m}{n}} \Gamma \left (1+3 q,-\frac {m \log \left (c x^n\right )}{n}\right ) \log ^{3 q}\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-3 q}}{m n q}-\frac {3\ 2^{-1-2 q} a b (a e m-b d n q) x^{2 m} \left (c x^n\right )^{-\frac {2 m}{n}} \Gamma \left (1+2 q,-\frac {2 m \log \left (c x^n\right )}{n}\right ) \log ^{2 q}\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-2 q}}{m n q}-\frac {3^{-q} a^2 (a e m-b d n q) x^{3 m} \left (c x^n\right )^{-\frac {3 m}{n}} \Gamma \left (1+q,-\frac {3 m \log \left (c x^n\right )}{n}\right ) \log ^q\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-q}}{m n q}+\frac {e \left (a x^m+b \log ^q\left (c x^n\right )\right )^4}{4 b n q} \]
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Time = 0.35 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2625, 6874, 2347, 2212} \[ \int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^3}{x} \, dx=-\frac {a^3 x^{4 m} (a e m-b d n q)}{4 b m n q}-\frac {a^2 3^{-q} x^{3 m} \left (c x^n\right )^{-\frac {3 m}{n}} \log ^q\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-q} (a e m-b d n q) \Gamma \left (q+1,-\frac {3 m \log \left (c x^n\right )}{n}\right )}{m n q}-\frac {b^2 x^m \left (c x^n\right )^{-\frac {m}{n}} \log ^{3 q}\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-3 q} (a e m-b d n q) \Gamma \left (3 q+1,-\frac {m \log \left (c x^n\right )}{n}\right )}{m n q}-\frac {3 a b 2^{-2 q-1} x^{2 m} \left (c x^n\right )^{-\frac {2 m}{n}} \log ^{2 q}\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-2 q} (a e m-b d n q) \Gamma \left (2 q+1,-\frac {2 m \log \left (c x^n\right )}{n}\right )}{m n q}+\frac {e \left (a x^m+b \log ^q\left (c x^n\right )\right )^4}{4 b n q} \]
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Rule 2212
Rule 2347
Rule 2625
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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 ) \text {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 ) \text {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*}
Time = 1.10 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.34 \[ \int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^3}{x} \, dx=\frac {3^{-q} 4^{-1-q} \left (c x^n\right )^{-\frac {3 m}{n}} \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-3 q} \left (-12^{1+q} a b^2 e m q x^m \left (c x^n\right )^{\frac {2 m}{n}} \Gamma \left (3 q,-\frac {m \log \left (c x^n\right )}{n}\right ) \log ^{3 q}\left (c x^n\right )+3^q 4^{1+q} b^3 d n q x^m \left (c x^n\right )^{\frac {2 m}{n}} \Gamma \left (1+3 q,-\frac {m \log \left (c x^n\right )}{n}\right ) \log ^{3 q}\left (c x^n\right )+\left (-\frac {m \log \left (c x^n\right )}{n}\right )^q \left (-4 3^{1+q} a^2 b e m q x^{2 m} \left (c x^n\right )^{m/n} \Gamma \left (2 q,-\frac {2 m \log \left (c x^n\right )}{n}\right ) \log ^{2 q}\left (c x^n\right )+2\ 3^{1+q} a b^2 d n q x^{2 m} \left (c x^n\right )^{m/n} \Gamma \left (1+2 q,-\frac {2 m \log \left (c x^n\right )}{n}\right ) \log ^{2 q}\left (c x^n\right )+4^q \left (-\frac {m \log \left (c x^n\right )}{n}\right )^q \left (-4 a^3 e m q x^{3 m} \Gamma \left (q,-\frac {3 m \log \left (c x^n\right )}{n}\right ) \log ^q\left (c x^n\right )+4 a^2 b d n q x^{3 m} \Gamma \left (1+q,-\frac {3 m \log \left (c x^n\right )}{n}\right ) \log ^q\left (c x^n\right )+3^q \left (c x^n\right )^{\frac {3 m}{n}} \left (-\frac {m \log \left (c x^n\right )}{n}\right )^q \left (a^3 d n q x^{4 m}+b^3 e m \log ^{4 q}\left (c x^n\right )\right )\right )\right )\right )}{m n q} \]
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\[\int \frac {\left (d \,x^{m}+e \ln \left (c \,x^{n}\right )^{-1+q}\right ) \left (a \,x^{m}+b \ln \left (c \,x^{n}\right )^{q}\right )^{3}}{x}d x\]
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\[ \int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^3}{x} \, dx=\int { \frac {{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )}^{3} {\left (d x^{m} + e \log \left (c x^{n}\right )^{q - 1}\right )}}{x} \,d x } \]
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Timed out. \[ \int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^3}{x} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^3}{x} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^3}{x} \, dx=\int { \frac {{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )}^{3} {\left (d x^{m} + e \log \left (c x^{n}\right )^{q - 1}\right )}}{x} \,d x } \]
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Timed out. \[ \int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^3}{x} \, dx=\int \frac {{\left (a\,x^m+b\,{\ln \left (c\,x^n\right )}^q\right )}^3\,\left (d\,x^m+e\,{\ln \left (c\,x^n\right )}^{q-1}\right )}{x} \,d x \]
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