Optimal. Leaf size=235 \[ -\frac {a^2 x^{3 m} (a e m-b d n q)}{3 b m n q}-\frac {a 2^{-q} x^{2 m} \left (c x^n\right )^{-\frac {2 m}{n}} \log ^q\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-q} (a e m-b d n q) \Gamma \left (q+1,-\frac {2 m \log \left (c x^n\right )}{n}\right )}{m n q}-\frac {b x^m \left (c x^n\right )^{-\frac {m}{n}} \log ^{2 q}\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-2 q} (a e m-b d n q) \Gamma \left (2 q+1,-\frac {m \log \left (c x^n\right )}{n}\right )}{m n q}+\frac {e \left (a x^m+b \log ^q\left (c x^n\right )\right )^3}{3 b n q} \]
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Rubi [A] time = 0.40, antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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 2181
Rule 2310
Rule 2545
Rule 6742
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*}
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Mathematica [A] time = 0.95, size = 298, normalized size = 1.27 \[ \frac {2^{-q} \left (c x^n\right )^{-\frac {2 m}{n}} \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-2 q} \left (\left (-\frac {m \log \left (c x^n\right )}{n}\right )^q \left (2^q \left (c x^n\right )^{\frac {2 m}{n}} \left (-\frac {m \log \left (c x^n\right )}{n}\right )^q \left (a^2 d n q x^{3 m}+b^2 e m \log ^{3 q}\left (c x^n\right )\right )-3 a^2 e m q x^{2 m} \log ^q\left (c x^n\right ) \Gamma \left (q,-\frac {2 m \log \left (c x^n\right )}{n}\right )+3 a b d n q x^{2 m} \log ^q\left (c x^n\right ) \Gamma \left (q+1,-\frac {2 m \log \left (c x^n\right )}{n}\right )\right )-3 a b e m 2^{q+1} q x^m \left (c x^n\right )^{m/n} \log ^{2 q}\left (c x^n\right ) \Gamma \left (2 q,-\frac {m \log \left (c x^n\right )}{n}\right )+3 b^2 d n 2^q q x^m \left (c x^n\right )^{m/n} \log ^{2 q}\left (c x^n\right ) \Gamma \left (2 q+1,-\frac {m \log \left (c x^n\right )}{n}\right )\right )}{3 m n q} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{2} e x^{2 \, m} \log \left (c x^{n}\right )^{q - 1} + a^{2} d x^{3 \, m} + {\left (b^{2} d x^{m} + b^{2} e \log \left (c x^{n}\right )^{q - 1}\right )} \log \left (c x^{n}\right )^{2 \, q} + 2 \, {\left (a b e x^{m} \log \left (c x^{n}\right )^{q - 1} + a b d x^{2 \, m}\right )} \log \left (c x^{n}\right )^{q}}{x}, x\right ) \]
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 \[ \int \frac {{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )}^{2} {\left (d x^{m} + e \log \left (c x^{n}\right )^{q - 1}\right )}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 111.81, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \,x^{m}+e \ln \left (c \,x^{n}\right )^{q -1}\right ) \left (a \,x^{m}+b \ln \left (c \,x^{n}\right )^{q}\right )^{2}}{x}\, dx \]
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 = -0.00 \[ \int \frac {{\left (a\,x^m+b\,{\ln \left (c\,x^n\right )}^q\right )}^2\,\left (d\,x^m+e\,{\ln \left (c\,x^n\right )}^{q-1}\right )}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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