Optimal. Leaf size=255 \[ -\frac {16}{3} b d n \sqrt {d+e x}-\frac {4}{9} b n (d+e x)^{3/2}+\frac {16}{3} b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )+2 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2+2 d \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )+\frac {2}{3} (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )-2 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-4 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )-2 b d^{3/2} n \text {Li}_2\left (1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right ) \]
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Rubi [A]
time = 0.30, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps
used = 18, number of rules used = 11, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {2388, 65,
214, 2390, 12, 6131, 6055, 2449, 2352, 2356, 52} \begin {gather*} -2 b d^{3/2} n \text {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )-2 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {2}{3} (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )+2 d \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )+2 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2+\frac {16}{3} b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )-4 b d^{3/2} n \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )-\frac {4}{9} b n (d+e x)^{3/2}-\frac {16}{3} b d n \sqrt {d+e x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 52
Rule 65
Rule 214
Rule 2352
Rule 2356
Rule 2388
Rule 2390
Rule 2449
Rule 6055
Rule 6131
Rubi steps
\begin {align*} \int \frac {(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=d \int \frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx+e \int \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right ) \, dx\\ &=\frac {2}{3} (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )+d^2 \int \frac {a+b \log \left (c x^n\right )}{x \sqrt {d+e x}} \, dx+(d e) \int \frac {a+b \log \left (c x^n\right )}{\sqrt {d+e x}} \, dx-\frac {1}{3} (2 b n) \int \frac {(d+e x)^{3/2}}{x} \, dx\\ &=-\frac {4}{9} b n (d+e x)^{3/2}+2 d \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )+\frac {2}{3} (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )-2 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{3} (2 b d n) \int \frac {\sqrt {d+e x}}{x} \, dx-(2 b d n) \int \frac {\sqrt {d+e x}}{x} \, dx-\left (b d^2 n\right ) \int -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{\sqrt {d} x} \, dx\\ &=-\frac {16}{3} b d n \sqrt {d+e x}-\frac {4}{9} b n (d+e x)^{3/2}+2 d \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )+\frac {2}{3} (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )-2 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )+\left (2 b d^{3/2} n\right ) \int \frac {\tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{x} \, dx-\frac {1}{3} \left (2 b d^2 n\right ) \int \frac {1}{x \sqrt {d+e x}} \, dx-\left (2 b d^2 n\right ) \int \frac {1}{x \sqrt {d+e x}} \, dx\\ &=-\frac {16}{3} b d n \sqrt {d+e x}-\frac {4}{9} b n (d+e x)^{3/2}+2 d \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )+\frac {2}{3} (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )-2 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )+\left (4 b d^{3/2} n\right ) \text {Subst}\left (\int \frac {x \tanh ^{-1}\left (\frac {x}{\sqrt {d}}\right )}{-d+x^2} \, dx,x,\sqrt {d+e x}\right )-\frac {\left (4 b d^2 n\right ) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{3 e}-\frac {\left (4 b d^2 n\right ) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{e}\\ &=-\frac {16}{3} b d n \sqrt {d+e x}-\frac {4}{9} b n (d+e x)^{3/2}+\frac {16}{3} b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )+2 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2+2 d \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )+\frac {2}{3} (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )-2 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-(4 b d n) \text {Subst}\left (\int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {d}}\right )}{1-\frac {x}{\sqrt {d}}} \, dx,x,\sqrt {d+e x}\right )\\ &=-\frac {16}{3} b d n \sqrt {d+e x}-\frac {4}{9} b n (d+e x)^{3/2}+\frac {16}{3} b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )+2 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2+2 d \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )+\frac {2}{3} (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )-2 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-4 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )+(4 b d n) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1-\frac {x}{\sqrt {d}}}\right )}{1-\frac {x^2}{d}} \, dx,x,\sqrt {d+e x}\right )\\ &=-\frac {16}{3} b d n \sqrt {d+e x}-\frac {4}{9} b n (d+e x)^{3/2}+\frac {16}{3} b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )+2 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2+2 d \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )+\frac {2}{3} (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )-2 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-4 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )-\left (4 b d^{3/2} n\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\frac {\sqrt {d+e x}}{\sqrt {d}}}\right )\\ &=-\frac {16}{3} b d n \sqrt {d+e x}-\frac {4}{9} b n (d+e x)^{3/2}+\frac {16}{3} b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )+2 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2+2 d \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )+\frac {2}{3} (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )-2 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-4 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )-2 b d^{3/2} n \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {d+e x}}{\sqrt {d}}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 375, normalized size = 1.47 \begin {gather*} 2 a d \sqrt {d+e x}-\frac {4}{9} b n (d+e x)^{3/2}+\frac {16}{3} b d n \left (-\sqrt {d+e x}+\sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )+2 b d \sqrt {d+e x} \log \left (c x^n\right )+\frac {2}{3} (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )+d^{3/2} \left (a+b \log \left (c x^n\right )\right ) \log \left (\sqrt {d}-\sqrt {d+e x}\right )-d^{3/2} \left (a+b \log \left (c x^n\right )\right ) \log \left (\sqrt {d}+\sqrt {d+e x}\right )-\frac {1}{2} b d^{3/2} n \left (\log \left (\sqrt {d}-\sqrt {d+e x}\right ) \left (\log \left (\sqrt {d}-\sqrt {d+e x}\right )+2 \log \left (\frac {1}{2} \left (1+\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )\right )+2 \text {Li}_2\left (\frac {1}{2}-\frac {\sqrt {d+e x}}{2 \sqrt {d}}\right )\right )+\frac {1}{2} b d^{3/2} n \left (\log \left (\sqrt {d}+\sqrt {d+e x}\right ) \left (\log \left (\sqrt {d}+\sqrt {d+e x}\right )+2 \log \left (\frac {1}{2}-\frac {\sqrt {d+e x}}{2 \sqrt {d}}\right )\right )+2 \text {Li}_2\left (\frac {1}{2} \left (1+\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (e x +d \right )^{\frac {3}{2}} \left (a +b \ln \left (c \,x^{n}\right )\right )}{x}\, dx\]
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 \frac {\left (a+b\,\ln \left (c\,x^n\right )\right )\,{\left (d+e\,x\right )}^{3/2}}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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