Optimal. Leaf size=288 \[ -\frac {5 b^2 e^3 n^2 \sqrt {x}}{6 d^3}+\frac {b^2 e^2 n^2 x}{6 d^2}+\frac {5 b^2 e^4 n^2 \log \left (d+\frac {e}{\sqrt {x}}\right )}{6 d^4}+\frac {b e^3 n \left (d+\frac {e}{\sqrt {x}}\right ) \sqrt {x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{d^4}-\frac {b e^2 n x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 d^2}+\frac {b e n x^{3/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{3 d}+\frac {b e^4 n \log \left (1-\frac {d}{d+\frac {e}{\sqrt {x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{d^4}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2+\frac {11 b^2 e^4 n^2 \log (x)}{12 d^4}-\frac {b^2 e^4 n^2 \text {Li}_2\left (\frac {d}{d+\frac {e}{\sqrt {x}}}\right )}{d^4} \]
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Rubi [A]
time = 0.37, antiderivative size = 288, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 10, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {2504, 2445,
2458, 2389, 2379, 2438, 2351, 31, 2356, 46} \begin {gather*} -\frac {b^2 e^4 n^2 \text {PolyLog}\left (2,\frac {d}{d+\frac {e}{\sqrt {x}}}\right )}{d^4}+\frac {b e^4 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^4}+\frac {b e^3 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^4}-\frac {b e^2 n x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 d^2}+\frac {b e n x^{3/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{3 d}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2+\frac {5 b^2 e^4 n^2 \log \left (d+\frac {e}{\sqrt {x}}\right )}{6 d^4}+\frac {11 b^2 e^4 n^2 \log (x)}{12 d^4}-\frac {5 b^2 e^3 n^2 \sqrt {x}}{6 d^3}+\frac {b^2 e^2 n^2 x}{6 d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 46
Rule 2351
Rule 2356
Rule 2379
Rule 2389
Rule 2438
Rule 2445
Rule 2458
Rule 2504
Rubi steps
\begin {align*} \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \, dx &=-\left (2 \text {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^5} \, dx,x,\frac {1}{\sqrt {x}}\right )\right )\\ &=\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2-(b e n) \text {Subst}\left (\int \frac {a+b \log \left (c (d+e x)^n\right )}{x^4 (d+e x)} \, dx,x,\frac {1}{\sqrt {x}}\right )\\ &=\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2-(b n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^4} \, dx,x,d+\frac {e}{\sqrt {x}}\right )\\ &=\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2-\frac {(b n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^4} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{d}+\frac {(b e n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^3} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{d}\\ &=\frac {b e n x^{3/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{3 d}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2+\frac {(b e n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^3} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{d^2}-\frac {\left (b e^2 n\right ) \text {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 )}{d^2}-\frac {\left (b^2 e n^2\right ) \text {Subst}\left (\int \frac {1}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^3} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{3 d}\\ &=-\frac {b e^2 n x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 d^2}+\frac {b e n x^{3/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{3 d}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2-\frac {\left (b e^2 n\right ) \text {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^3}+\frac {\left (b e^3 n\right ) \text {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^3}-\frac {\left (b^2 e n^2\right ) \text {Subst}\left (\int \left (-\frac {e^3}{d (d-x)^3}-\frac {e^3}{d^2 (d-x)^2}-\frac {e^3}{d^3 (d-x)}-\frac {e^3}{d^3 x}\right ) \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{3 d}+\frac {\left (b^2 e^2 n^2\right ) \text {Subst}\left (\int \frac {1}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{2 d^2}\\ &=-\frac {b^2 e^3 n^2 \sqrt {x}}{3 d^3}+\frac {b^2 e^2 n^2 x}{6 d^2}+\frac {b^2 e^4 n^2 \log \left (d+\frac {e}{\sqrt {x}}\right )}{3 d^4}+\frac {b e^3 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^4}-\frac {b e^2 n x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 d^2}+\frac {b e n x^{3/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{3 d}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2+\frac {b^2 e^4 n^2 \log (x)}{6 d^4}+\frac {\left (b e^3 n\right ) \text {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^4}-\frac {\left (b e^4 n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{d^4}+\frac {\left (b^2 e^2 n^2\right ) \text {Subst}\left (\int \left (\frac {e^2}{d (d-x)^2}+\frac {e^2}{d^2 (d-x)}+\frac {e^2}{d^2 x}\right ) \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{2 d^2}-\frac {\left (b^2 e^3 n^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{d^4}\\ &=-\frac {5 b^2 e^3 n^2 \sqrt {x}}{6 d^3}+\frac {b^2 e^2 n^2 x}{6 d^2}+\frac {5 b^2 e^4 n^2 \log \left (d+\frac {e}{\sqrt {x}}\right )}{6 d^4}+\frac {b e^3 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^4}-\frac {b e^2 n x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 d^2}+\frac {b e n x^{3/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{3 d}-\frac {e^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 d^4}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2+\frac {b e^4 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^4}+\frac {11 b^2 e^4 n^2 \log (x)}{12 d^4}-\frac {\left (b^2 e^4 n^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{d}\right )}{x} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{d^4}\\ &=-\frac {5 b^2 e^3 n^2 \sqrt {x}}{6 d^3}+\frac {b^2 e^2 n^2 x}{6 d^2}+\frac {5 b^2 e^4 n^2 \log \left (d+\frac {e}{\sqrt {x}}\right )}{6 d^4}+\frac {b e^3 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^4}-\frac {b e^2 n x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 d^2}+\frac {b e n x^{3/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{3 d}-\frac {e^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 d^4}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2+\frac {b e^4 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^4}+\frac {11 b^2 e^4 n^2 \log (x)}{12 d^4}+\frac {b^2 e^4 n^2 \text {Li}_2\left (1+\frac {e}{d \sqrt {x}}\right )}{d^4}\\ \end {align*}
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Mathematica [A]
time = 0.37, size = 307, normalized size = 1.07 \begin {gather*} \frac {3 d^4 x^2 \left (a-b n \log \left (d+\frac {e}{\sqrt {x}}\right )+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2+b n \left (a-b n \log \left (d+\frac {e}{\sqrt {x}}\right )+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \left (d e \sqrt {x} \left (6 e^2-3 d e \sqrt {x}+2 d^2 x\right )-6 \left (e^4-d^4 x^2\right ) \log \left (d+\frac {e}{\sqrt {x}}\right )+6 e^4 \log \left (\frac {e}{\sqrt {x}}\right )\right )-b^2 n^2 \left (3 \left (e^4-d^4 x^2\right ) \log ^2\left (d+\frac {e}{\sqrt {x}}\right )+e^2 \left (5 d e \sqrt {x}-d^2 x+11 e^2 \log \left (-\frac {e}{d \sqrt {x}}\right )\right )-e \log \left (d+\frac {e}{\sqrt {x}}\right ) \left (11 e^3+6 d e^2 \sqrt {x}-3 d^2 e x+2 d^3 x^{3/2}+6 e^3 \log \left (-\frac {e}{d \sqrt {x}}\right )\right )-6 e^4 \text {Li}_2\left (1+\frac {e}{d \sqrt {x}}\right )\right )}{6 d^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int x \left (a +b \ln \left (c \left (d +\frac {e}{\sqrt {x}}\right )^{n}\right )\right )^{2}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x \left (a + b \log {\left (c \left (d + \frac {e}{\sqrt {x}}\right )^{n} \right )}\right )^{2}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x\,{\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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