Optimal. Leaf size=94 \[ -\frac {4 b n \sqrt {d+e x}}{e^2}+\frac {8 b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{e^2}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, 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 = {45, 2392, 12,
81, 65, 214} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 45
Rule 65
Rule 81
Rule 214
Rule 2392
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) \text {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*}
________________________________________________________________________________________
Mathematica [A]
time = 0.05, size = 83, normalized size = 0.88 \begin {gather*} \frac {2 \left (2 a d-2 b d n+a e x-2 b e n x+4 b \sqrt {d} n \sqrt {d+e x} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )+b (2 d+e x) \log \left (c x^n\right )\right )}{e^2 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x \left (a +b \ln \left (c \,x^{n}\right )\right )}{\left (e x +d \right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.51, size = 113, normalized size = 1.20 \begin {gather*} -4 \, {\left (\sqrt {d} e^{\left (-2\right )} \log \left (\frac {\sqrt {x e + d} - \sqrt {d}}{\sqrt {x e + d} + \sqrt {d}}\right ) + \sqrt {x e + d} e^{\left (-2\right )}\right )} b n + 2 \, {\left (\sqrt {x e + d} e^{\left (-2\right )} + \frac {d e^{\left (-2\right )}}{\sqrt {x e + d}}\right )} b \log \left (c x^{n}\right ) + 2 \, {\left (\sqrt {x e + d} e^{\left (-2\right )} + \frac {d e^{\left (-2\right )}}{\sqrt {x e + d}}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.39, size = 230, normalized size = 2.45 \begin {gather*} \left [\frac {2 \, {\left (2 \, {\left (b n x e + b d n\right )} \sqrt {d} \log \left (\frac {x e + 2 \, \sqrt {x e + d} \sqrt {d} + 2 \, d}{x}\right ) - {\left (2 \, b d n + {\left (2 \, b n - a\right )} x e - 2 \, a d - {\left (b x e + 2 \, b d\right )} \log \left (c\right ) - {\left (b n x e + 2 \, b d n\right )} \log \left (x\right )\right )} \sqrt {x e + d}\right )}}{x e^{3} + d e^{2}}, -\frac {2 \, {\left (4 \, {\left (b n x e + b d n\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {x e + d} \sqrt {-d}}{d}\right ) + {\left (2 \, b d n + {\left (2 \, b n - a\right )} x e - 2 \, a d - {\left (b x e + 2 \, b d\right )} \log \left (c\right ) - {\left (b n x e + 2 \, b d n\right )} \log \left (x\right )\right )} \sqrt {x e + d}\right )}}{x e^{3} + d e^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 30.22, size = 153, normalized size = 1.63 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 2.31, size = 105, normalized size = 1.12 \begin {gather*} -\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 \left (c\right ) + b d \log \left (c\right ) + {\left (x e + d\right )} a + a d\right )} e^{\left (-2\right )}}{\sqrt {x e + d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (d+e\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________