Integrand size = 22, antiderivative size = 730 \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \, dx=\frac {2^{-2-3 p} e^{-\frac {8 a}{b}} \Gamma \left (1+p,-\frac {8 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c^8 e^8}-\frac {2\ 7^{-p} d e^{-\frac {7 a}{b}} \Gamma \left (1+p,-\frac {7 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c^7 e^8}+\frac {7\ 6^{-p} d^2 e^{-\frac {6 a}{b}} \Gamma \left (1+p,-\frac {6 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c^6 e^8}-\frac {14\ 5^{-p} d^3 e^{-\frac {5 a}{b}} \Gamma \left (1+p,-\frac {5 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c^5 e^8}+\frac {35\ 2^{-1-2 p} d^4 e^{-\frac {4 a}{b}} \Gamma \left (1+p,-\frac {4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c^4 e^8}-\frac {14\ 3^{-p} d^5 e^{-\frac {3 a}{b}} \Gamma \left (1+p,-\frac {3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c^3 e^8}+\frac {7\ 2^{-p} d^6 e^{-\frac {2 a}{b}} \Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c^2 e^8}-\frac {2 d^7 e^{-\frac {a}{b}} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c e^8} \]
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Time = 0.90 (sec) , antiderivative size = 730, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {2504, 2448, 2436, 2336, 2212, 2437, 2346} \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \, dx=\frac {2^{-3 p-2} e^{-\frac {8 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {8 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right )}{c^8 e^8}-\frac {2 d 7^{-p} e^{-\frac {7 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {7 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right )}{c^7 e^8}+\frac {7 d^2 6^{-p} e^{-\frac {6 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {6 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right )}{c^6 e^8}-\frac {14 d^3 5^{-p} e^{-\frac {5 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {5 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right )}{c^5 e^8}+\frac {35 d^4 2^{-2 p-1} e^{-\frac {4 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right )}{c^4 e^8}-\frac {14 d^5 3^{-p} e^{-\frac {3 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right )}{c^3 e^8}+\frac {7 d^6 2^{-p} e^{-\frac {2 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right )}{c^2 e^8}-\frac {2 d^7 e^{-\frac {a}{b}} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )}{c e^8} \]
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Rule 2212
Rule 2336
Rule 2346
Rule 2436
Rule 2437
Rule 2448
Rule 2504
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int x^7 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (-\frac {d^7 (a+b \log (c (d+e x)))^p}{e^7}+\frac {7 d^6 (d+e x) (a+b \log (c (d+e x)))^p}{e^7}-\frac {21 d^5 (d+e x)^2 (a+b \log (c (d+e x)))^p}{e^7}+\frac {35 d^4 (d+e x)^3 (a+b \log (c (d+e x)))^p}{e^7}-\frac {35 d^3 (d+e x)^4 (a+b \log (c (d+e x)))^p}{e^7}+\frac {21 d^2 (d+e x)^5 (a+b \log (c (d+e x)))^p}{e^7}-\frac {7 d (d+e x)^6 (a+b \log (c (d+e x)))^p}{e^7}+\frac {(d+e x)^7 (a+b \log (c (d+e x)))^p}{e^7}\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {2 \text {Subst}\left (\int (d+e x)^7 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt {x}\right )}{e^7}-\frac {(14 d) \text {Subst}\left (\int (d+e x)^6 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt {x}\right )}{e^7}+\frac {\left (42 d^2\right ) \text {Subst}\left (\int (d+e x)^5 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt {x}\right )}{e^7}-\frac {\left (70 d^3\right ) \text {Subst}\left (\int (d+e x)^4 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt {x}\right )}{e^7}+\frac {\left (70 d^4\right ) \text {Subst}\left (\int (d+e x)^3 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt {x}\right )}{e^7}-\frac {\left (42 d^5\right ) \text {Subst}\left (\int (d+e x)^2 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt {x}\right )}{e^7}+\frac {\left (14 d^6\right ) \text {Subst}\left (\int (d+e x) (a+b \log (c (d+e x)))^p \, dx,x,\sqrt {x}\right )}{e^7}-\frac {\left (2 d^7\right ) \text {Subst}\left (\int (a+b \log (c (d+e x)))^p \, dx,x,\sqrt {x}\right )}{e^7} \\ & = \frac {2 \text {Subst}\left (\int x^7 (a+b \log (c x))^p \, dx,x,d+e \sqrt {x}\right )}{e^8}-\frac {(14 d) \text {Subst}\left (\int x^6 (a+b \log (c x))^p \, dx,x,d+e \sqrt {x}\right )}{e^8}+\frac {\left (42 d^2\right ) \text {Subst}\left (\int x^5 (a+b \log (c x))^p \, dx,x,d+e \sqrt {x}\right )}{e^8}-\frac {\left (70 d^3\right ) \text {Subst}\left (\int x^4 (a+b \log (c x))^p \, dx,x,d+e \sqrt {x}\right )}{e^8}+\frac {\left (70 d^4\right ) \text {Subst}\left (\int x^3 (a+b \log (c x))^p \, dx,x,d+e \sqrt {x}\right )}{e^8}-\frac {\left (42 d^5\right ) \text {Subst}\left (\int x^2 (a+b \log (c x))^p \, dx,x,d+e \sqrt {x}\right )}{e^8}+\frac {\left (14 d^6\right ) \text {Subst}\left (\int x (a+b \log (c x))^p \, dx,x,d+e \sqrt {x}\right )}{e^8}-\frac {\left (2 d^7\right ) \text {Subst}\left (\int (a+b \log (c x))^p \, dx,x,d+e \sqrt {x}\right )}{e^8} \\ & = \frac {2 \text {Subst}\left (\int e^{8 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{c^8 e^8}-\frac {(14 d) \text {Subst}\left (\int e^{7 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{c^7 e^8}+\frac {\left (42 d^2\right ) \text {Subst}\left (\int e^{6 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{c^6 e^8}-\frac {\left (70 d^3\right ) \text {Subst}\left (\int e^{5 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{c^5 e^8}+\frac {\left (70 d^4\right ) \text {Subst}\left (\int e^{4 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{c^4 e^8}-\frac {\left (42 d^5\right ) \text {Subst}\left (\int e^{3 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{c^3 e^8}+\frac {\left (14 d^6\right ) \text {Subst}\left (\int e^{2 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{c^2 e^8}-\frac {\left (2 d^7\right ) \text {Subst}\left (\int e^x (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{c e^8} \\ & = \frac {2^{-2-3 p} e^{-\frac {8 a}{b}} \Gamma \left (1+p,-\frac {8 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c^8 e^8}-\frac {2\ 7^{-p} d e^{-\frac {7 a}{b}} \Gamma \left (1+p,-\frac {7 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c^7 e^8}+\frac {7\ 6^{-p} d^2 e^{-\frac {6 a}{b}} \Gamma \left (1+p,-\frac {6 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c^6 e^8}-\frac {14\ 5^{-p} d^3 e^{-\frac {5 a}{b}} \Gamma \left (1+p,-\frac {5 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c^5 e^8}+\frac {35\ 2^{-1-2 p} d^4 e^{-\frac {4 a}{b}} \Gamma \left (1+p,-\frac {4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c^4 e^8}-\frac {14\ 3^{-p} d^5 e^{-\frac {3 a}{b}} \Gamma \left (1+p,-\frac {3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c^3 e^8}+\frac {7\ 2^{-p} d^6 e^{-\frac {2 a}{b}} \Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c^2 e^8}-\frac {2 d^7 e^{-\frac {a}{b}} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c e^8} \\ \end{align*}
\[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \, dx=\int x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \, dx \]
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\[\int x^{3} \left (a +b \ln \left (c \left (d +e \sqrt {x}\right )\right )\right )^{p}d x\]
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\[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e \sqrt {x} + d\right )} c\right ) + a\right )}^{p} x^{3} \,d x } \]
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Timed out. \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \, dx=\text {Timed out} \]
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\[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e \sqrt {x} + d\right )} c\right ) + a\right )}^{p} x^{3} \,d x } \]
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\[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e \sqrt {x} + d\right )} c\right ) + a\right )}^{p} x^{3} \,d x } \]
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Timed out. \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \, dx=\int x^3\,{\left (a+b\,\ln \left (c\,\left (d+e\,\sqrt {x}\right )\right )\right )}^p \,d x \]
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