Integrand size = 24, antiderivative size = 677 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx=\frac {3^{-1-p} e^{-\frac {3 a}{b}} \Gamma \left (1+p,-\frac {3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )^{-p}}{c^3 e^6}-\frac {2^{1+p} 5^{-p} d e^{-\frac {5 a}{2 b}} \left (d+e \sqrt {x}\right )^5 \Gamma \left (1+p,-\frac {5 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )^{-p}}{e^6 \left (c \left (d+e \sqrt {x}\right )^2\right )^{5/2}}+\frac {5\ 2^{-p} d^2 e^{-\frac {2 a}{b}} \Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )^{-p}}{c^2 e^6}-\frac {5\ 2^{2+p} 3^{-1-p} d^3 e^{-\frac {3 a}{2 b}} \left (d+e \sqrt {x}\right )^3 \Gamma \left (1+p,-\frac {3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )^{-p}}{e^6 \left (c \left (d+e \sqrt {x}\right )^2\right )^{3/2}}+\frac {5 d^4 e^{-\frac {a}{b}} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )^{-p}}{c e^6}-\frac {2^{1+p} d^5 e^{-\frac {a}{2 b}} \left (d+e \sqrt {x}\right ) \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )^{-p}}{e^6 \sqrt {c \left (d+e \sqrt {x}\right )^2}} \]
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Time = 0.67 (sec) , antiderivative size = 677, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {2504, 2448, 2436, 2337, 2212, 2437, 2347} \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx=\frac {3^{-p-1} e^{-\frac {3 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )}{b}\right )}{c^3 e^6}+\frac {5 d^2 2^{-p} e^{-\frac {2 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )}{b}\right )}{c^2 e^6}-\frac {d^5 2^{p+1} e^{-\frac {a}{2 b}} \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{2 b}\right )}{e^6 \sqrt {c \left (d+e \sqrt {x}\right )^2}}+\frac {5 d^4 e^{-\frac {a}{b}} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )}{c e^6}-\frac {5 d^3 2^{p+2} 3^{-p-1} e^{-\frac {3 a}{2 b}} \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )}{2 b}\right )}{e^6 \left (c \left (d+e \sqrt {x}\right )^2\right )^{3/2}}-\frac {d 2^{p+1} 5^{-p} e^{-\frac {5 a}{2 b}} \left (d+e \sqrt {x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {5 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )}{2 b}\right )}{e^6 \left (c \left (d+e \sqrt {x}\right )^2\right )^{5/2}} \]
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Rule 2212
Rule 2337
Rule 2347
Rule 2436
Rule 2437
Rule 2448
Rule 2504
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int x^5 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (-\frac {d^5 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^5}+\frac {5 d^4 (d+e x) \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^5}-\frac {10 d^3 (d+e x)^2 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^5}+\frac {10 d^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^5}-\frac {5 d (d+e x)^4 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^5}+\frac {(d+e x)^5 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^5}\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {2 \text {Subst}\left (\int (d+e x)^5 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt {x}\right )}{e^5}-\frac {(10 d) \text {Subst}\left (\int (d+e x)^4 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt {x}\right )}{e^5}+\frac {\left (20 d^2\right ) \text {Subst}\left (\int (d+e x)^3 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt {x}\right )}{e^5}-\frac {\left (20 d^3\right ) \text {Subst}\left (\int (d+e x)^2 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt {x}\right )}{e^5}+\frac {\left (10 d^4\right ) \text {Subst}\left (\int (d+e x) \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt {x}\right )}{e^5}-\frac {\left (2 d^5\right ) \text {Subst}\left (\int \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt {x}\right )}{e^5} \\ & = \frac {2 \text {Subst}\left (\int x^5 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+e \sqrt {x}\right )}{e^6}-\frac {(10 d) \text {Subst}\left (\int x^4 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+e \sqrt {x}\right )}{e^6}+\frac {\left (20 d^2\right ) \text {Subst}\left (\int x^3 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+e \sqrt {x}\right )}{e^6}-\frac {\left (20 d^3\right ) \text {Subst}\left (\int x^2 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+e \sqrt {x}\right )}{e^6}+\frac {\left (10 d^4\right ) \text {Subst}\left (\int x \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+e \sqrt {x}\right )}{e^6}-\frac {\left (2 d^5\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+e \sqrt {x}\right )}{e^6} \\ & = \frac {\text {Subst}\left (\int e^{3 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )}{c^3 e^6}+\frac {\left (10 d^2\right ) \text {Subst}\left (\int e^{2 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )}{c^2 e^6}+\frac {\left (5 d^4\right ) \text {Subst}\left (\int e^x (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )}{c e^6}-\frac {\left (5 d \left (d+e \sqrt {x}\right )^5\right ) \text {Subst}\left (\int e^{5 x/2} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )}{e^6 \left (c \left (d+e \sqrt {x}\right )^2\right )^{5/2}}-\frac {\left (10 d^3 \left (d+e \sqrt {x}\right )^3\right ) \text {Subst}\left (\int e^{3 x/2} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )}{e^6 \left (c \left (d+e \sqrt {x}\right )^2\right )^{3/2}}-\frac {\left (d^5 \left (d+e \sqrt {x}\right )\right ) \text {Subst}\left (\int e^{x/2} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )}{e^6 \sqrt {c \left (d+e \sqrt {x}\right )^2}} \\ & = \frac {3^{-1-p} e^{-\frac {3 a}{b}} \Gamma \left (1+p,-\frac {3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )^{-p}}{c^3 e^6}-\frac {2^{1+p} 5^{-p} d e^{-\frac {5 a}{2 b}} \left (d+e \sqrt {x}\right )^5 \Gamma \left (1+p,-\frac {5 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )^{-p}}{e^6 \left (c \left (d+e \sqrt {x}\right )^2\right )^{5/2}}+\frac {5\ 2^{-p} d^2 e^{-\frac {2 a}{b}} \Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )^{-p}}{c^2 e^6}-\frac {5\ 2^{2+p} 3^{-1-p} d^3 e^{-\frac {3 a}{2 b}} \left (d+e \sqrt {x}\right )^3 \Gamma \left (1+p,-\frac {3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )^{-p}}{e^6 \left (c \left (d+e \sqrt {x}\right )^2\right )^{3/2}}+\frac {5 d^4 e^{-\frac {a}{b}} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )^{-p}}{c e^6}-\frac {2^{1+p} d^5 e^{-\frac {a}{2 b}} \left (d+e \sqrt {x}\right ) \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )^{-p}}{e^6 \sqrt {c \left (d+e \sqrt {x}\right )^2}} \\ \end{align*}
\[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx=\int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx \]
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\[\int x^{2} {\left (a +b \ln \left (c \left (d +e \sqrt {x}\right )^{2}\right )\right )}^{p}d x\]
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\[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{2} c\right ) + a\right )}^{p} x^{2} \,d x } \]
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Timed out. \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx=\text {Timed out} \]
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\[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{2} c\right ) + a\right )}^{p} x^{2} \,d x } \]
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\[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{2} c\right ) + a\right )}^{p} x^{2} \,d x } \]
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Timed out. \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx=\int x^2\,{\left (a+b\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^2\right )\right )}^p \,d x \]
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