Optimal. Leaf size=148 \[ \frac {b n \left (d^2-e^2 x^2\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {b d^2 n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.20, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {2387, 2376,
272, 52, 65, 214} \begin {gather*} -\frac {\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}}+\frac {b n \left (d^2-e^2 x^2\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {b d^2 n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 52
Rule 65
Rule 214
Rule 272
Rule 2376
Rule 2387
Rubi steps
\begin {align*} \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx &=\frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\sqrt {1-\frac {e^2 x^2}{d^2}}} \, dx}{\sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (b d^2 n \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \int \frac {\sqrt {1-\frac {e^2 x^2}{d^2}}}{x} \, dx}{e^2 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (b d^2 n \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \text {Subst}\left (\int \frac {\sqrt {1-\frac {e^2 x}{d^2}}}{x} \, dx,x,x^2\right )}{2 e^2 \sqrt {d-e x} \sqrt {d+e x}}\\ &=\frac {b n \left (d^2-e^2 x^2\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (b d^2 n \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {e^2 x}{d^2}}} \, dx,x,x^2\right )}{2 e^2 \sqrt {d-e x} \sqrt {d+e x}}\\ &=\frac {b n \left (d^2-e^2 x^2\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (b d^4 n \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \text {Subst}\left (\int \frac {1}{\frac {d^2}{e^2}-\frac {d^2 x^2}{e^2}} \, dx,x,\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{e^4 \sqrt {d-e x} \sqrt {d+e x}}\\ &=\frac {b n \left (d^2-e^2 x^2\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {b d^2 n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.10, size = 113, normalized size = 0.76 \begin {gather*} \frac {b d n \log (x)}{e^2}-\frac {b n \sqrt {d-e x} \sqrt {d+e x} \log (x)}{e^2}-\frac {\sqrt {d-e x} \sqrt {d+e x} \left (a-b n+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )}{e^2}-\frac {b d n \log \left (d+\sqrt {d-e x} \sqrt {d+e x}\right )}{e^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {x \left (a +b \ln \left (c \,x^{n}\right )\right )}{\sqrt {-e x +d}\, \sqrt {e x +d}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.51, size = 98, normalized size = 0.66 \begin {gather*} -{\left (d \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-x^{2} e^{2} + d^{2}} d}{{\left | x \right |}}\right ) - \sqrt {-x^{2} e^{2} + d^{2}}\right )} b n e^{\left (-2\right )} - \sqrt {-x^{2} e^{2} + d^{2}} b e^{\left (-2\right )} \log \left (c x^{n}\right ) - \sqrt {-x^{2} e^{2} + d^{2}} a e^{\left (-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.38, size = 69, normalized size = 0.47 \begin {gather*} {\left (b d n \log \left (\frac {\sqrt {x e + d} \sqrt {-x e + d} - d}{x}\right ) - {\left (b n \log \left (x\right ) - b n + b \log \left (c\right ) + a\right )} \sqrt {x e + d} \sqrt {-x e + d}\right )} e^{\left (-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (a + b \log {\left (c x^{n} \right )}\right )}{\sqrt {d - e x} \sqrt {d + e x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[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 )}{\sqrt {d+e\,x}\,\sqrt {d-e\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________