Optimal. Leaf size=213 \[ \frac {e^{-\frac {a}{b}} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )}{c e^2}-\frac {d 2^{p+1} e^{-\frac {a}{2 b}} \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{2 b}\right )}{e^2 \sqrt {c \left (d+e \sqrt {x}\right )^2}} \]
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Rubi [A] time = 0.26, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {2451, 2401, 2389, 2300, 2181, 2390, 2310} \[ \frac {e^{-\frac {a}{b}} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )^{-p} \text {Gamma}\left (p+1,-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )}{c e^2}-\frac {d 2^{p+1} e^{-\frac {a}{2 b}} \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )^{-p} \text {Gamma}\left (p+1,-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{2 b}\right )}{e^2 \sqrt {c \left (d+e \sqrt {x}\right )^2}} \]
Antiderivative was successfully verified.
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Rule 2181
Rule 2300
Rule 2310
Rule 2389
Rule 2390
Rule 2401
Rule 2451
Rubi steps
\begin {align*} \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx &=2 \operatorname {Subst}\left (\int x \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (-\frac {d \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e}\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {2 \operatorname {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}-\frac {(2 d) \operatorname {Subst}\left (\int \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt {x}\right )}{e}\\ &=\frac {2 \operatorname {Subst}\left (\int x \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+e \sqrt {x}\right )}{e^2}-\frac {(2 d) \operatorname {Subst}\left (\int \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+e \sqrt {x}\right )}{e^2}\\ &=\frac {\operatorname {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^2}-\frac {\left (d \left (d+e \sqrt {x}\right )\right ) \operatorname {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^2 \sqrt {c \left (d+e \sqrt {x}\right )^2}}\\ &=\frac {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^2}-\frac {2^{1+p} d 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^2 \sqrt {c \left (d+e \sqrt {x}\right )^2}}\\ \end {align*}
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Mathematica [F] time = 0.12, size = 0, normalized size = 0.00 \[ \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \log \left (c e^{2} x + 2 \, c d e \sqrt {x} + c d^{2}\right ) + a\right )}^{p}, x\right ) \]
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 \[ \int {\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{2} c\right ) + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (\left (e \sqrt {x}+d \right )^{2} c \right )+a \right )^{p}\, dx \]
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 \[ \int {\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{2} c\right ) + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (a+b\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^2\right )\right )}^p \,d x \]
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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