Optimal. Leaf size=136 \[ \frac {x \left (\text {d1} x^{\text {q1}}\right )^{\text {m1}} \left (\text {d2} x^{\text {q2}}\right )^{\text {m2}} \left (a+b \log \left (c x^n\right )\right )^p e^{-\frac {a (\text {m1} \text {q1}+\text {m2} \text {q2}+1)}{b n}} \left (c x^n\right )^{-\frac {\text {m1} \text {q1}+\text {m2} \text {q2}+1}{n}} \left (-\frac {(\text {m1} \text {q1}+\text {m2} \text {q2}+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \Gamma \left (p+1,-\frac {(\text {m1} \text {q1}+\text {m2} \text {q2}+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{\text {m1} \text {q1}+\text {m2} \text {q2}+1} \]
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Rubi [A] time = 0.17, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 2310, 2181} \[ \frac {x \left (\text {d1} x^{\text {q1}}\right )^{\text {m1}} \left (\text {d2} x^{\text {q2}}\right )^{\text {m2}} \left (a+b \log \left (c x^n\right )\right )^p e^{-\frac {a (\text {m1} \text {q1}+\text {m2} \text {q2}+1)}{b n}} \left (c x^n\right )^{-\frac {\text {m1} \text {q1}+\text {m2} \text {q2}+1}{n}} \left (-\frac {(\text {m1} \text {q1}+\text {m2} \text {q2}+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \text {Gamma}\left (p+1,-\frac {(\text {m1} \text {q1}+\text {m2} \text {q2}+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{\text {m1} \text {q1}+\text {m2} \text {q2}+1} \]
Antiderivative was successfully verified.
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Rule 15
Rule 2181
Rule 2310
Rubi steps
\begin {align*} \int \left (\text {d1} x^{\text {q1}}\right )^{\text {m1}} \left (\text {d2} x^{\text {q2}}\right )^{\text {m2}} \left (a+b \log \left (c x^n\right )\right )^p \, dx &=\left (x^{-\text {m1} \text {q1}} \left (\text {d1} x^{\text {q1}}\right )^{\text {m1}}\right ) \int x^{\text {m1} \text {q1}} \left (\text {d2} x^{\text {q2}}\right )^{\text {m2}} \left (a+b \log \left (c x^n\right )\right )^p \, dx\\ &=\left (x^{-\text {m1} \text {q1}-\text {m2} \text {q2}} \left (\text {d1} x^{\text {q1}}\right )^{\text {m1}} \left (\text {d2} x^{\text {q2}}\right )^{\text {m2}}\right ) \int x^{\text {m1} \text {q1}+\text {m2} \text {q2}} \left (a+b \log \left (c x^n\right )\right )^p \, dx\\ &=\frac {\left (x \left (c x^n\right )^{-\frac {1+\text {m1} \text {q1}+\text {m2} \text {q2}}{n}} \left (\text {d1} x^{\text {q1}}\right )^{\text {m1}} \left (\text {d2} x^{\text {q2}}\right )^{\text {m2}}\right ) \operatorname {Subst}\left (\int e^{\frac {(1+\text {m1} \text {q1}+\text {m2} \text {q2}) x}{n}} (a+b x)^p \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {e^{-\frac {a (1+\text {m1} \text {q1}+\text {m2} \text {q2})}{b n}} x \left (c x^n\right )^{-\frac {1+\text {m1} \text {q1}+\text {m2} \text {q2}}{n}} \left (\text {d1} x^{\text {q1}}\right )^{\text {m1}} \left (\text {d2} x^{\text {q2}}\right )^{\text {m2}} \Gamma \left (1+p,-\frac {(1+\text {m1} \text {q1}+\text {m2} \text {q2}) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {(1+\text {m1} \text {q1}+\text {m2} \text {q2}) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{1+\text {m1} \text {q1}+\text {m2} \text {q2}}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 142, normalized size = 1.04 \[ \frac {\left (\text {d1} x^{\text {q1}}\right )^{\text {m1}} \left (\text {d2} x^{\text {q2}}\right )^{\text {m2}} x^{-\text {m1} \text {q1}-\text {m2} \text {q2}} \left (a+b \log \left (c x^n\right )\right )^p \exp \left (-\frac {(\text {m1} \text {q1}+\text {m2} \text {q2}+1) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{b n}\right ) \left (-\frac {(\text {m1} \text {q1}+\text {m2} \text {q2}+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \Gamma \left (p+1,-\frac {(\text {m1} \text {q1}+\text {m2} \text {q2}+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{\text {m1} \text {q1}+\text {m2} \text {q2}+1} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (d_{1} x^{q_{1}}\right )^{m_{1}} \left (d_{2} x^{q_{2}}\right )^{m_{2}} {\left (b \log \left (c x^{n}\right ) + a\right )}^{p}, x\right ) \]
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 \[ \int \left (d_{1} x^{q_{1}}\right )^{m_{1}} \left (d_{2} x^{q_{2}}\right )^{m_{2}} {\left (b \log \left (c x^{n}\right ) + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 24.09, size = 0, normalized size = 0.00 \[ \int \left (\mathit {d1} \,x^{\mathit {q1}}\right )^{\mathit {m1}} \left (\mathit {d2} \,x^{\mathit {q2}}\right )^{\mathit {m2}} \left (b \ln \left (c \,x^{n}\right )+a \right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d_{1} x^{q_{1}}\right )^{m_{1}} \left (d_{2} x^{q_{2}}\right )^{m_{2}} {\left (b \log \left (c x^{n}\right ) + a\right )}^{p}\,{d x} \]
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 \[ \int {\left (d_{1}\,x^{q_{1}}\right )}^{m_{1}}\,{\left (d_{2}\,x^{q_{2}}\right )}^{m_{2}}\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^p \,d x \]
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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