Optimal. Leaf size=94 \[ \frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d+e x}}-\frac {4 b n \sqrt {d+e x}}{e^2}+\frac {8 b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{e^2} \]
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Rubi [A] time = 0.09, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {43, 2350, 12, 80, 63, 208} \[ \frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d+e x}}-\frac {4 b n \sqrt {d+e x}}{e^2}+\frac {8 b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{e^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 63
Rule 80
Rule 208
Rule 2350
Rubi steps
\begin {align*} \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^{3/2}} \, dx &=\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^2}-(b n) \int \frac {2 (2 d+e x)}{e^2 x \sqrt {d+e x}} \, dx\\ &=\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {(2 b n) \int \frac {2 d+e x}{x \sqrt {d+e x}} \, dx}{e^2}\\ &=-\frac {4 b n \sqrt {d+e x}}{e^2}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {(4 b d n) \int \frac {1}{x \sqrt {d+e x}} \, dx}{e^2}\\ &=-\frac {4 b n \sqrt {d+e x}}{e^2}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {(8 b d n) \operatorname {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{e^3}\\ &=-\frac {4 b n \sqrt {d+e x}}{e^2}+\frac {8 b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{e^2}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^2}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 83, normalized size = 0.88 \[ \frac {2 \left (2 a d+a e x+b (2 d+e x) \log \left (c x^n\right )+4 b \sqrt {d} n \sqrt {d+e x} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )-2 b d n-2 b e n x\right )}{e^2 \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 223, normalized size = 2.37 \[ \left [\frac {2 \, {\left (2 \, {\left (b e n x + b d n\right )} \sqrt {d} \log \left (\frac {e x + 2 \, \sqrt {e x + d} \sqrt {d} + 2 \, d}{x}\right ) - {\left (2 \, b d n - 2 \, a d + {\left (2 \, b e n - a e\right )} x - {\left (b e x + 2 \, b d\right )} \log \relax (c) - {\left (b e n x + 2 \, b d n\right )} \log \relax (x)\right )} \sqrt {e x + d}\right )}}{e^{3} x + d e^{2}}, -\frac {2 \, {\left (4 \, {\left (b e n x + b d n\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {e x + d} \sqrt {-d}}{d}\right ) + {\left (2 \, b d n - 2 \, a d + {\left (2 \, b e n - a e\right )} x - {\left (b e x + 2 \, b d\right )} \log \relax (c) - {\left (b e n x + 2 \, b d n\right )} \log \relax (x)\right )} \sqrt {e x + d}\right )}}{e^{3} x + d e^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 105, normalized size = 1.12 \[ -\frac {8 \, b d n \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right ) e^{\left (-2\right )}}{\sqrt {-d}} + \frac {2 \, {\left ({\left (x e + d\right )} b n \log \left (x e\right ) + b d n \log \left (x e\right ) - 3 \, {\left (x e + d\right )} b n - b d n + {\left (x e + d\right )} b \log \relax (c) + b d \log \relax (c) + {\left (x e + d\right )} a + a d\right )} e^{\left (-2\right )}}{\sqrt {x e + d}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.37, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right ) x}{\left (e x +d \right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.42, size = 112, normalized size = 1.19 \[ -4 \, b n {\left (\frac {\sqrt {d} \log \left (\frac {\sqrt {e x + d} - \sqrt {d}}{\sqrt {e x + d} + \sqrt {d}}\right )}{e^{2}} + \frac {\sqrt {e x + d}}{e^{2}}\right )} + 2 \, b {\left (\frac {\sqrt {e x + d}}{e^{2}} + \frac {d}{\sqrt {e x + d} e^{2}}\right )} \log \left (c x^{n}\right ) + 2 \, a {\left (\frac {\sqrt {e x + d}}{e^{2}} + \frac {d}{\sqrt {e x + d} e^{2}}\right )} \]
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 \[ \int \frac {x\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (d+e\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 58.82, size = 153, normalized size = 1.63 \[ \frac {\frac {2 a d}{\sqrt {d + e x}} + 2 a \sqrt {d + e x} - 2 b d \left (\frac {2 n \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {- d}} \right )}}{\sqrt {- d}} - \frac {\log {\left (c \left (- \frac {d}{e} + \frac {d + e x}{e}\right )^{n} \right )}}{\sqrt {d + e x}}\right ) + 2 b \left (\sqrt {d + e x} \log {\left (c \left (- \frac {d}{e} + \frac {d + e x}{e}\right )^{n} \right )} - \frac {2 n \left (\frac {d e \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {- d}} \right )}}{\sqrt {- d}} + e \sqrt {d + e x}\right )}{e}\right )}{e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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