Optimal. Leaf size=152 \[ \frac {2 b e^2 n \log \left (1-\frac {d}{d+\frac {e}{\sqrt {x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{d^2}+\frac {2 b e n \sqrt {x} \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{d^2}+x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2-\frac {2 b^2 e^2 n^2 \text {Li}_2\left (\frac {d}{d+\frac {e}{\sqrt {x}}}\right )}{d^2}+\frac {b^2 e^2 n^2 \log (x)}{d^2} \]
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Rubi [A] time = 0.35, antiderivative size = 174, normalized size of antiderivative = 1.14, number of steps used = 11, number of rules used = 11, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {2451, 2454, 2398, 2411, 2347, 2344, 2301, 2317, 2391, 2314, 31} \[ \frac {2 b^2 e^2 n^2 \text {PolyLog}\left (2,\frac {e}{d \sqrt {x}}+1\right )}{d^2}-\frac {e^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{d^2}+\frac {2 b e^2 n \log \left (-\frac {e}{d \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{d^2}+\frac {2 b e n \sqrt {x} \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{d^2}+x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2+\frac {b^2 e^2 n^2 \log (x)}{d^2} \]
Antiderivative was successfully verified.
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Rule 31
Rule 2301
Rule 2314
Rule 2317
Rule 2344
Rule 2347
Rule 2391
Rule 2398
Rule 2411
Rule 2451
Rule 2454
Rubi steps
\begin {align*} \int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \, dx &=2 \operatorname {Subst}\left (\int x \left (a+b \log \left (c \left (d+\frac {e}{x}\right )^n\right )\right )^2 \, dx,x,\sqrt {x}\right )\\ &=-\left (2 \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3} \, dx,x,\frac {1}{\sqrt {x}}\right )\right )\\ &=x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2-(2 b e n) \operatorname {Subst}\left (\int \frac {a+b \log \left (c (d+e x)^n\right )}{x^2 (d+e x)} \, dx,x,\frac {1}{\sqrt {x}}\right )\\ &=x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2-(2 b n) \operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+\frac {e}{\sqrt {x}}\right )\\ &=x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2-\frac {(2 b n) \operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{d}+\frac {(2 b e n) \operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{d}\\ &=\frac {2 b e n \left (d+\frac {e}{\sqrt {x}}\right ) \sqrt {x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{d^2}+x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2+\frac {(2 b e n) \operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{d^2}-\frac {\left (2 b e^2 n\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{d^2}-\frac {\left (2 b^2 e n^2\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{d^2}\\ &=\frac {2 b e n \left (d+\frac {e}{\sqrt {x}}\right ) \sqrt {x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{d^2}-\frac {e^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{d^2}+x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2+\frac {2 b e^2 n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \log \left (-\frac {e}{d \sqrt {x}}\right )}{d^2}+\frac {b^2 e^2 n^2 \log (x)}{d^2}-\frac {\left (2 b^2 e^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{d}\right )}{x} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{d^2}\\ &=\frac {2 b e n \left (d+\frac {e}{\sqrt {x}}\right ) \sqrt {x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{d^2}-\frac {e^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{d^2}+x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2+\frac {2 b e^2 n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \log \left (-\frac {e}{d \sqrt {x}}\right )}{d^2}+\frac {b^2 e^2 n^2 \log (x)}{d^2}+\frac {2 b^2 e^2 n^2 \text {Li}_2\left (1+\frac {e}{d \sqrt {x}}\right )}{d^2}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 170, normalized size = 1.12 \[ \frac {b e n \left (-2 e \log \left (d \sqrt {x}+e\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )+2 a d \sqrt {x}+2 b d \sqrt {x} \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )+b e n \left (\log \left (d \sqrt {x}+e\right ) \left (\log \left (d \sqrt {x}+e\right )-2 \log \left (-\frac {d \sqrt {x}}{e}\right )\right )-2 \text {Li}_2\left (\frac {\sqrt {x} d}{e}+1\right )\right )+b e n \left (2 \log \left (d+\frac {e}{\sqrt {x}}\right )+\log (x)\right )\right )}{d^2}+x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{2} \log \left (c \left (\frac {d x + e \sqrt {x}}{x}\right )^{n}\right )^{2} + 2 \, a b \log \left (c \left (\frac {d x + e \sqrt {x}}{x}\right )^{n}\right ) + a^{2}, x\right ) \]
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 \[ \int {\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right ) + a\right )}^{2}\,{d x} \]
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 \[ \int \left (b \ln \left (c \left (d +\frac {e}{\sqrt {x}}\right )^{n}\right )+a \right )^{2}\, dx \]
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 \[ -2 \, {\left (e n {\left (\frac {e \log \left (d \sqrt {x} + e\right )}{d^{2}} - \frac {\sqrt {x}}{d}\right )} - x \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right )\right )} a b + {\left (n^{2} x \log \left (d \sqrt {x} + e\right )^{2} - \int -\frac {d x \log \relax (c)^{2} + e \sqrt {x} \log \relax (c)^{2} - {\left (d n x - 2 \, d x \log \relax (c) - 2 \, e \sqrt {x} \log \relax (c) + 2 \, {\left (d x + e \sqrt {x}\right )} \log \left (x^{\frac {1}{2} \, n}\right )\right )} n \log \left (d \sqrt {x} + e\right ) + {\left (d x + e \sqrt {x}\right )} \log \left (x^{\frac {1}{2} \, n}\right )^{2} - 2 \, {\left (d x \log \relax (c) + e \sqrt {x} \log \relax (c)\right )} \log \left (x^{\frac {1}{2} \, n}\right )}{d x + e \sqrt {x}}\,{d x}\right )} b^{2} + a^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )\right )}^2 \,d x \]
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 \[ \int \left (a + b \log {\left (c \left (d + \frac {e}{\sqrt {x}}\right )^{n} \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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