Optimal. Leaf size=110 \[ -\frac {b n \sqrt {d+e x^2}}{d^2 x}+\frac {2 b \sqrt {e} n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{d^2}-\frac {a+b \log \left (c x^n\right )}{d x \sqrt {d+e x^2}}-\frac {2 e x \left (a+b \log \left (c x^n\right )\right )}{d^2 \sqrt {d+e x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.09, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {277, 197, 2392,
12, 462, 223, 212} \begin {gather*} -\frac {2 e x \left (a+b \log \left (c x^n\right )\right )}{d^2 \sqrt {d+e x^2}}-\frac {a+b \log \left (c x^n\right )}{d x \sqrt {d+e x^2}}-\frac {b n \sqrt {d+e x^2}}{d^2 x}+\frac {2 b \sqrt {e} n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{d^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 197
Rule 212
Rule 223
Rule 277
Rule 462
Rule 2392
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{x^2 \left (d+e x^2\right )^{3/2}} \, dx &=-\frac {a+b \log \left (c x^n\right )}{d x \sqrt {d+e x^2}}-\frac {2 e x \left (a+b \log \left (c x^n\right )\right )}{d^2 \sqrt {d+e x^2}}-(b n) \int \frac {-d-2 e x^2}{d^2 x^2 \sqrt {d+e x^2}} \, dx\\ &=-\frac {a+b \log \left (c x^n\right )}{d x \sqrt {d+e x^2}}-\frac {2 e x \left (a+b \log \left (c x^n\right )\right )}{d^2 \sqrt {d+e x^2}}-\frac {(b n) \int \frac {-d-2 e x^2}{x^2 \sqrt {d+e x^2}} \, dx}{d^2}\\ &=-\frac {b n \sqrt {d+e x^2}}{d^2 x}-\frac {a+b \log \left (c x^n\right )}{d x \sqrt {d+e x^2}}-\frac {2 e x \left (a+b \log \left (c x^n\right )\right )}{d^2 \sqrt {d+e x^2}}+\frac {(2 b e n) \int \frac {1}{\sqrt {d+e x^2}} \, dx}{d^2}\\ &=-\frac {b n \sqrt {d+e x^2}}{d^2 x}-\frac {a+b \log \left (c x^n\right )}{d x \sqrt {d+e x^2}}-\frac {2 e x \left (a+b \log \left (c x^n\right )\right )}{d^2 \sqrt {d+e x^2}}+\frac {(2 b e n) \text {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{d^2}\\ &=-\frac {b n \sqrt {d+e x^2}}{d^2 x}+\frac {2 b \sqrt {e} n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{d^2}-\frac {a+b \log \left (c x^n\right )}{d x \sqrt {d+e x^2}}-\frac {2 e x \left (a+b \log \left (c x^n\right )\right )}{d^2 \sqrt {d+e x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.08, size = 103, normalized size = 0.94 \begin {gather*} \frac {-a d-b d n-2 a e x^2-b e n x^2-b \left (d+2 e x^2\right ) \log \left (c x^n\right )+2 b \sqrt {e} n x \sqrt {d+e x^2} \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right )}{d^2 x \sqrt {d+e x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \,x^{n}\right )}{x^{2} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.39, size = 128, normalized size = 1.16 \begin {gather*} \frac {{\left (b n x^{3} e + b d n x\right )} e^{\frac {1}{2}} \log \left (-2 \, x^{2} e - 2 \, \sqrt {x^{2} e + d} x e^{\frac {1}{2}} - d\right ) - {\left ({\left (b n + 2 \, a\right )} x^{2} e + b d n + a d + {\left (2 \, b x^{2} e + b d\right )} \log \left (c\right ) + {\left (2 \, b n x^{2} e + b d n\right )} \log \left (x\right )\right )} \sqrt {x^{2} e + d}}{d^{2} x^{3} e + d^{3} x} \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 {a + b \log {\left (c x^{n} \right )}}{x^{2} \left (d + e x^{2}\right )^{\frac {3}{2}}}\, 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 {a+b\,\ln \left (c\,x^n\right )}{x^2\,{\left (e\,x^2+d\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________