Optimal. Leaf size=210 \[ \frac {(f h-e i)^2 (a+b \log (c (e+f x)))^{1+p}}{b d f^3 (1+p)}+\frac {2^{-1-p} e^{-\frac {2 a}{b}} i^2 \Gamma \left (1+p,-\frac {2 (a+b \log (c (e+f x)))}{b}\right ) (a+b \log (c (e+f x)))^p \left (-\frac {a+b \log (c (e+f x))}{b}\right )^{-p}}{c^2 d f^3}+\frac {2 e^{-\frac {a}{b}} i (f h-e i) \Gamma \left (1+p,-\frac {a+b \log (c (e+f x))}{b}\right ) (a+b \log (c (e+f x)))^p \left (-\frac {a+b \log (c (e+f x))}{b}\right )^{-p}}{c d f^3} \]
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Rubi [A]
time = 0.34, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 8, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2458, 12,
2395, 2336, 2212, 2339, 30, 2346} \begin {gather*} \frac {i^2 2^{-p-1} e^{-\frac {2 a}{b}} (a+b \log (c (e+f x)))^p \left (-\frac {a+b \log (c (e+f x))}{b}\right )^{-p} \text {Gamma}\left (p+1,-\frac {2 (a+b \log (c (e+f x)))}{b}\right )}{c^2 d f^3}+\frac {2 i e^{-\frac {a}{b}} (f h-e i) (a+b \log (c (e+f x)))^p \left (-\frac {a+b \log (c (e+f x))}{b}\right )^{-p} \text {Gamma}\left (p+1,-\frac {a+b \log (c (e+f x))}{b}\right )}{c d f^3}+\frac {(f h-e i)^2 (a+b \log (c (e+f x)))^{p+1}}{b d f^3 (p+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 2212
Rule 2336
Rule 2339
Rule 2346
Rule 2395
Rule 2458
Rubi steps
\begin {align*} \int \frac {(h+211 x)^2 (a+b \log (c (e+f x)))^p}{d e+d f x} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (\frac {-211 e+f h}{f}+\frac {211 x}{f}\right )^2 (a+b \log (c x))^p}{d x} \, dx,x,e+f x\right )}{f}\\ &=\frac {\text {Subst}\left (\int \frac {\left (\frac {-211 e+f h}{f}+\frac {211 x}{f}\right )^2 (a+b \log (c x))^p}{x} \, dx,x,e+f x\right )}{d f}\\ &=\frac {\text {Subst}\left (\int \left (-\frac {422 (211 e-f h) (a+b \log (c x))^p}{f^2}+\frac {(211 e-f h)^2 (a+b \log (c x))^p}{f^2 x}+\frac {44521 x (a+b \log (c x))^p}{f^2}\right ) \, dx,x,e+f x\right )}{d f}\\ &=\frac {44521 \text {Subst}\left (\int x (a+b \log (c x))^p \, dx,x,e+f x\right )}{d f^3}-\frac {(422 (211 e-f h)) \text {Subst}\left (\int (a+b \log (c x))^p \, dx,x,e+f x\right )}{d f^3}+\frac {(211 e-f h)^2 \text {Subst}\left (\int \frac {(a+b \log (c x))^p}{x} \, dx,x,e+f x\right )}{d f^3}\\ &=\frac {44521 \text {Subst}\left (\int e^{2 x} (a+b x)^p \, dx,x,\log (c (e+f x))\right )}{c^2 d f^3}-\frac {(422 (211 e-f h)) \text {Subst}\left (\int e^x (a+b x)^p \, dx,x,\log (c (e+f x))\right )}{c d f^3}+\frac {(211 e-f h)^2 \text {Subst}\left (\int x^p \, dx,x,a+b \log (c (e+f x))\right )}{b d f^3}\\ &=\frac {(211 e-f h)^2 (a+b \log (c (e+f x)))^{1+p}}{b d f^3 (1+p)}+\frac {44521\ 2^{-1-p} e^{-\frac {2 a}{b}} \Gamma \left (1+p,-\frac {2 (a+b \log (c (e+f x)))}{b}\right ) (a+b \log (c (e+f x)))^p \left (-\frac {a+b \log (c (e+f x))}{b}\right )^{-p}}{c^2 d f^3}-\frac {422 e^{-\frac {a}{b}} (211 e-f h) \Gamma \left (1+p,-\frac {a+b \log (c (e+f x))}{b}\right ) (a+b \log (c (e+f x)))^p \left (-\frac {a+b \log (c (e+f x))}{b}\right )^{-p}}{c d f^3}\\ \end {align*}
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Mathematica [A]
time = 0.27, size = 189, normalized size = 0.90 \begin {gather*} \frac {2^{-1-p} e^{-\frac {2 a}{b}} (a+b \log (c (e+f x)))^p \left (-\frac {a+b \log (c (e+f x))}{b}\right )^{-p} \left (b i^2 (1+p) \Gamma \left (1+p,-\frac {2 (a+b \log (c (e+f x)))}{b}\right )+2^{1+p} c e^{a/b} (f h-e i) \left (2 b i (1+p) \Gamma \left (1+p,-\frac {a+b \log (c (e+f x))}{b}\right )-b c e^{a/b} (f h-e i) \left (-\frac {a+b \log (c (e+f x))}{b}\right )^{1+p}\right )\right )}{b c^2 d f^3 (1+p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.34, size = 0, normalized size = 0.00 \[\int \frac {\left (i x +h \right )^{2} \left (a +b \ln \left (c \left (f x +e \right )\right )\right )^{p}}{d f x +e d}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {h^{2} \left (a + b \log {\left (c e + c f x \right )}\right )^{p}}{e + f x}\, dx + \int \frac {i^{2} x^{2} \left (a + b \log {\left (c e + c f x \right )}\right )^{p}}{e + f x}\, dx + \int \frac {2 h i x \left (a + b \log {\left (c e + c f x \right )}\right )^{p}}{e + f x}\, dx}{d} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (h+i\,x\right )}^2\,{\left (a+b\,\ln \left (c\,\left (e+f\,x\right )\right )\right )}^p}{d\,e+d\,f\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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