Optimal. Leaf size=216 \[ \frac {d 2^{p+1} e^{-\frac {a}{2 b}} \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,\frac {-a-b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )}{2 b}\right )}{e^2 \sqrt {c \left (d+\frac {e}{\sqrt {x}}\right )^2}}-\frac {e^{-\frac {a}{b}} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )}{b}\right )}{c e^2} \]
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Rubi [A] time = 0.29, antiderivative size = 213, normalized size of antiderivative = 0.99, number of steps used = 9, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {2454, 2401, 2389, 2300, 2181, 2390, 2310} \[ \frac {d 2^{p+1} e^{-\frac {a}{2 b}} \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )}{b}\right )^{-p} \text {Gamma}\left (p+1,-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )}{2 b}\right )}{e^2 \sqrt {c \left (d+\frac {e}{\sqrt {x}}\right )^2}}-\frac {e^{-\frac {a}{b}} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )}{b}\right )^{-p} \text {Gamma}\left (p+1,-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )}{b}\right )}{c e^2} \]
Antiderivative was successfully verified.
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Rule 2181
Rule 2300
Rule 2310
Rule 2389
Rule 2390
Rule 2401
Rule 2454
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{x^2} \, dx &=-\left (2 \operatorname {Subst}\left (\int x \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\frac {1}{\sqrt {x}}\right )\right )\\ &=-\left (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,\frac {1}{\sqrt {x}}\right )\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,\frac {1}{\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,\frac {1}{\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+\frac {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+\frac {e}{\sqrt {x}}\right )}{e^2}\\ &=-\frac {\operatorname {Subst}\left (\int e^x (a+b x)^p \, dx,x,\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )}{c e^2}+\frac {\left (d \left (d+\frac {e}{\sqrt {x}}\right )\right ) \operatorname {Subst}\left (\int e^{x/2} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )}{e^2 \sqrt {c \left (d+\frac {e}{\sqrt {x}}\right )^2}}\\ &=-\frac {e^{-\frac {a}{b}} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )}{b}\right )^{-p}}{c e^2}+\frac {2^{1+p} d e^{-\frac {a}{2 b}} \left (d+\frac {e}{\sqrt {x}}\right ) \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )}{b}\right )^{-p}}{e^2 \sqrt {c \left (d+\frac {e}{\sqrt {x}}\right )^2}}\\ \end {align*}
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Mathematica [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{x^2} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b \log \left (\frac {c d^{2} x + 2 \, c d e \sqrt {x} + c e^{2}}{x}\right ) + a\right )}^{p}}{x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{2}\right ) + a\right )}^{p}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (\left (d +\frac {e}{\sqrt {x}}\right )^{2} c \right )+a \right )^{p}}{x^{2}}\, dx \]
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 \[ \int \frac {{\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{2}\right ) + a\right )}^{p}}{x^{2}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^2\right )\right )}^p}{x^2} \,d x \]
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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