Optimal. Leaf size=82 \[ -\frac {a+b \log \left (c x^n\right )}{4 e \left (d+e x^2\right )^2}-\frac {b n \log \left (d+e x^2\right )}{8 d^2 e}+\frac {b n \log (x)}{4 d^2 e}+\frac {b n}{8 d e \left (d+e x^2\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2338, 266, 44} \[ -\frac {a+b \log \left (c x^n\right )}{4 e \left (d+e x^2\right )^2}-\frac {b n \log \left (d+e x^2\right )}{8 d^2 e}+\frac {b n \log (x)}{4 d^2 e}+\frac {b n}{8 d e \left (d+e x^2\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 44
Rule 266
Rule 2338
Rubi steps
\begin {align*} \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx &=-\frac {a+b \log \left (c x^n\right )}{4 e \left (d+e x^2\right )^2}+\frac {(b n) \int \frac {1}{x \left (d+e x^2\right )^2} \, dx}{4 e}\\ &=-\frac {a+b \log \left (c x^n\right )}{4 e \left (d+e x^2\right )^2}+\frac {(b n) \operatorname {Subst}\left (\int \frac {1}{x (d+e x)^2} \, dx,x,x^2\right )}{8 e}\\ &=-\frac {a+b \log \left (c x^n\right )}{4 e \left (d+e x^2\right )^2}+\frac {(b n) \operatorname {Subst}\left (\int \left (\frac {1}{d^2 x}-\frac {e}{d (d+e x)^2}-\frac {e}{d^2 (d+e x)}\right ) \, dx,x,x^2\right )}{8 e}\\ &=\frac {b n}{8 d e \left (d+e x^2\right )}+\frac {b n \log (x)}{4 d^2 e}-\frac {a+b \log \left (c x^n\right )}{4 e \left (d+e x^2\right )^2}-\frac {b n \log \left (d+e x^2\right )}{8 d^2 e}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 111, normalized size = 1.35 \[ \frac {-a-b \left (\log \left (c x^n\right )-n \log (x)\right )}{4 e \left (d+e x^2\right )^2}-\frac {b n \log \left (d+e x^2\right )}{8 d^2 e}+\frac {b n \log (x)}{4 d^2 e}+\frac {b n}{8 d e \left (d+e x^2\right )}-\frac {b n \log (x)}{4 e \left (d+e x^2\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.67, size = 118, normalized size = 1.44 \[ \frac {b d e n x^{2} + b d^{2} n - 2 \, b d^{2} \log \relax (c) - 2 \, a d^{2} - {\left (b e^{2} n x^{4} + 2 \, b d e n x^{2} + b d^{2} n\right )} \log \left (e x^{2} + d\right ) + 2 \, {\left (b e^{2} n x^{4} + 2 \, b d e n x^{2}\right )} \log \relax (x)}{8 \, {\left (d^{2} e^{3} x^{4} + 2 \, d^{3} e^{2} x^{2} + d^{4} e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.29, size = 136, normalized size = 1.66 \[ -\frac {b n x^{4} e^{2} \log \left (x^{2} e + d\right ) - 2 \, b n x^{4} e^{2} \log \relax (x) + 2 \, b d n x^{2} e \log \left (x^{2} e + d\right ) - 4 \, b d n x^{2} e \log \relax (x) - b d n x^{2} e + b d^{2} n \log \left (x^{2} e + d\right ) - b d^{2} n + 2 \, b d^{2} \log \relax (c) + 2 \, a d^{2}}{8 \, {\left (d^{2} x^{4} e^{3} + 2 \, d^{3} x^{2} e^{2} + d^{4} e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.21, size = 243, normalized size = 2.96 \[ -\frac {b \ln \left (x^{n}\right )}{4 \left (e \,x^{2}+d \right )^{2} e}-\frac {-2 b \,e^{2} n \,x^{4} \ln \relax (x )+b \,e^{2} n \,x^{4} \ln \left (e \,x^{2}+d \right )-4 b d e n \,x^{2} \ln \relax (x )+2 b d e n \,x^{2} \ln \left (e \,x^{2}+d \right )-i \pi b \,d^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i \pi b \,d^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b \,d^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b \,d^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-b d e n \,x^{2}-2 b \,d^{2} n \ln \relax (x )+b \,d^{2} n \ln \left (e \,x^{2}+d \right )-b \,d^{2} n +2 b \,d^{2} \ln \relax (c )+2 a \,d^{2}}{8 \left (e \,x^{2}+d \right )^{2} d^{2} e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.52, size = 109, normalized size = 1.33 \[ \frac {1}{8} \, b n {\left (\frac {1}{d e^{2} x^{2} + d^{2} e} - \frac {\log \left (e x^{2} + d\right )}{d^{2} e} + \frac {\log \left (x^{2}\right )}{d^{2} e}\right )} - \frac {b \log \left (c x^{n}\right )}{4 \, {\left (e^{3} x^{4} + 2 \, d e^{2} x^{2} + d^{2} e\right )}} - \frac {a}{4 \, {\left (e^{3} x^{4} + 2 \, d e^{2} x^{2} + d^{2} e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.68, size = 109, normalized size = 1.33 \[ \frac {\frac {b\,n}{2}-a+\frac {b\,e\,n\,x^2}{2\,d}}{4\,d^2\,e+8\,d\,e^2\,x^2+4\,e^3\,x^4}-\frac {b\,\ln \left (c\,x^n\right )}{4\,e\,\left (d^2+2\,d\,e\,x^2+e^2\,x^4\right )}-\frac {b\,n\,\ln \left (e\,x^2+d\right )}{8\,d^2\,e}+\frac {b\,n\,\ln \relax (x)}{4\,d^2\,e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________