Optimal. Leaf size=255 \[ -2 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {2}{3} (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )+2 d \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )-2 b d^{3/2} n \text {Li}_2\left (1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )+2 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2+\frac {16}{3} b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )-4 b d^{3/2} n \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )-\frac {4}{9} b n (d+e x)^{3/2}-\frac {16}{3} b d n \sqrt {d+e x} \]
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Rubi [A] time = 0.46, 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 = {2346, 63, 208, 2348, 12, 5984, 5918, 2402, 2315, 2319, 50} \[ -2 b d^{3/2} n \text {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )-2 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {2}{3} (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )+2 d \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )+2 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2+\frac {16}{3} b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )-4 b d^{3/2} n \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )-\frac {4}{9} b n (d+e x)^{3/2}-\frac {16}{3} b d n \sqrt {d+e x} \]
Antiderivative was successfully verified.
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Rule 12
Rule 50
Rule 63
Rule 208
Rule 2315
Rule 2319
Rule 2346
Rule 2348
Rule 2402
Rule 5918
Rule 5984
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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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) \operatorname {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) \operatorname {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 ) \operatorname {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 = 0.30, size = 375, normalized size = 1.47 \[ d^{3/2} \log \left (\sqrt {d}-\sqrt {d+e x}\right ) \left (a+b \log \left (c x^n\right )\right )-d^{3/2} \log \left (\sqrt {d+e x}+\sqrt {d}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {2}{3} (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )+2 a d \sqrt {d+e x}+2 b d \sqrt {d+e x} \log \left (c x^n\right )-\frac {1}{2} b d^{3/2} n \left (2 \text {Li}_2\left (\frac {1}{2}-\frac {\sqrt {d+e x}}{2 \sqrt {d}}\right )+\log \left (\sqrt {d}-\sqrt {d+e x}\right ) \left (\log \left (\sqrt {d}-\sqrt {d+e x}\right )+2 \log \left (\frac {1}{2} \left (\frac {\sqrt {d+e x}}{\sqrt {d}}+1\right )\right )\right )\right )+\frac {1}{2} b d^{3/2} n \left (2 \text {Li}_2\left (\frac {1}{2} \left (\frac {\sqrt {d+e x}}{\sqrt {d}}+1\right )\right )+\log \left (\sqrt {d+e x}+\sqrt {d}\right ) \left (\log \left (\sqrt {d+e x}+\sqrt {d}\right )+2 \log \left (\frac {1}{2}-\frac {\sqrt {d+e x}}{2 \sqrt {d}}\right )\right )\right )-\frac {4}{9} b n (d+e x)^{3/2}+\frac {16}{3} b d n \left (\sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )-\sqrt {d+e x}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b e x + b d\right )} \sqrt {e x + d} \log \left (c x^{n}\right ) + {\left (a e x + a d\right )} \sqrt {e x + d}}{x}, x\right ) \]
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 \[ \int \frac {{\left (e x + d\right )}^{\frac {3}{2}} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.36, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x +d \right )^{\frac {3}{2}} \left (b \ln \left (c \,x^{n}\right )+a \right )}{x}\, dx \]
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 \[ \frac {1}{3} \, {\left (3 \, d^{\frac {3}{2}} \log \left (\frac {\sqrt {e x + d} - \sqrt {d}}{\sqrt {e x + d} + \sqrt {d}}\right ) + 2 \, {\left (e x + d\right )}^{\frac {3}{2}} + 6 \, \sqrt {e x + d} d\right )} a + b \int \frac {{\left (e x \log \relax (c) + d \log \relax (c) + {\left (e x + d\right )} \log \left (x^{n}\right )\right )} \sqrt {e x + d}}{x}\,{d x} \]
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 \[ \int \frac {\left (a+b\,\ln \left (c\,x^n\right )\right )\,{\left (d+e\,x\right )}^{3/2}}{x} \,d x \]
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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