Optimal. Leaf size=86 \[ \frac {c^3 n \log (x)}{3 b^3}-\frac {c^3 n \log (b+c x)}{3 b^3}+\frac {c^2 n}{3 b^2 x}-\frac {\log \left (d \left (b x+c x^2\right )^n\right )}{3 x^3}-\frac {c n}{6 b x^2}-\frac {n}{9 x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2525, 77} \[ \frac {c^2 n}{3 b^2 x}+\frac {c^3 n \log (x)}{3 b^3}-\frac {c^3 n \log (b+c x)}{3 b^3}-\frac {\log \left (d \left (b x+c x^2\right )^n\right )}{3 x^3}-\frac {c n}{6 b x^2}-\frac {n}{9 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 77
Rule 2525
Rubi steps
\begin {align*} \int \frac {\log \left (d \left (b x+c x^2\right )^n\right )}{x^4} \, dx &=-\frac {\log \left (d \left (b x+c x^2\right )^n\right )}{3 x^3}+\frac {1}{3} n \int \frac {b+2 c x}{x^4 (b+c x)} \, dx\\ &=-\frac {\log \left (d \left (b x+c x^2\right )^n\right )}{3 x^3}+\frac {1}{3} n \int \left (\frac {1}{x^4}+\frac {c}{b x^3}-\frac {c^2}{b^2 x^2}+\frac {c^3}{b^3 x}-\frac {c^4}{b^3 (b+c x)}\right ) \, dx\\ &=-\frac {n}{9 x^3}-\frac {c n}{6 b x^2}+\frac {c^2 n}{3 b^2 x}+\frac {c^3 n \log (x)}{3 b^3}-\frac {c^3 n \log (b+c x)}{3 b^3}-\frac {\log \left (d \left (b x+c x^2\right )^n\right )}{3 x^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 77, normalized size = 0.90 \[ \frac {1}{3} n \left (\frac {c^3 \log (x)}{b^3}-\frac {c^3 \log (b+c x)}{b^3}+\frac {c^2}{b^2 x}-\frac {c}{2 b x^2}-\frac {1}{3 x^3}\right )-\frac {\log \left (d (x (b+c x))^n\right )}{3 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.46, size = 82, normalized size = 0.95 \[ -\frac {6 \, c^{3} n x^{3} \log \left (c x + b\right ) - 6 \, c^{3} n x^{3} \log \relax (x) - 6 \, b c^{2} n x^{2} + 3 \, b^{2} c n x + 6 \, b^{3} n \log \left (c x^{2} + b x\right ) + 2 \, b^{3} n + 6 \, b^{3} \log \relax (d)}{18 \, b^{3} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 80, normalized size = 0.93 \[ -\frac {c^{3} n \log \left (c x + b\right )}{3 \, b^{3}} + \frac {c^{3} n \log \relax (x)}{3 \, b^{3}} - \frac {n \log \left (c x^{2} + b x\right )}{3 \, x^{3}} + \frac {6 \, c^{2} n x^{2} - 3 \, b c n x - 2 \, b^{2} n - 6 \, b^{2} \log \relax (d)}{18 \, b^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (d \left (c \,x^{2}+b x \right )^{n}\right )}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.68, size = 75, normalized size = 0.87 \[ -\frac {1}{18} \, n {\left (\frac {6 \, c^{3} \log \left (c x + b\right )}{b^{3}} - \frac {6 \, c^{3} \log \relax (x)}{b^{3}} - \frac {6 \, c^{2} x^{2} - 3 \, b c x - 2 \, b^{2}}{b^{2} x^{3}}\right )} - \frac {\log \left ({\left (c x^{2} + b x\right )}^{n} d\right )}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.46, size = 68, normalized size = 0.79 \[ -\frac {\ln \left (d\,{\left (c\,x^2+b\,x\right )}^n\right )}{3\,x^3}-\frac {\frac {n}{3}-\frac {c^2\,n\,x^2}{b^2}+\frac {c\,n\,x}{2\,b}}{3\,x^3}-\frac {2\,c^3\,n\,\mathrm {atanh}\left (\frac {2\,c\,x}{b}+1\right )}{3\,b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 13.29, size = 133, normalized size = 1.55 \[ \begin {cases} - \frac {n \log {\left (b x + c x^{2} \right )}}{3 x^{3}} - \frac {n}{9 x^{3}} - \frac {\log {\relax (d )}}{3 x^{3}} - \frac {c n}{6 b x^{2}} + \frac {c^{2} n}{3 b^{2} x} - \frac {2 c^{3} n \log {\left (b + c x \right )}}{3 b^{3}} + \frac {c^{3} n \log {\left (b x + c x^{2} \right )}}{3 b^{3}} & \text {for}\: b \neq 0 \\- \frac {n \log {\relax (c )}}{3 x^{3}} - \frac {2 n \log {\relax (x )}}{3 x^{3}} - \frac {2 n}{9 x^{3}} - \frac {\log {\relax (d )}}{3 x^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________