Optimal. Leaf size=99 \[ \frac{b^3 n x^2}{10 c^3}-\frac{b^2 n x^3}{15 c^2}-\frac{b^4 n x}{5 c^4}+\frac{b^5 n \log (b+c x)}{5 c^5}+\frac{1}{5} x^5 \log \left (d \left (b x+c x^2\right )^n\right )+\frac{b n x^4}{20 c}-\frac{2 n x^5}{25} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0731365, antiderivative size = 99, normalized size of antiderivative = 1., 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} \[ \frac{b^3 n x^2}{10 c^3}-\frac{b^2 n x^3}{15 c^2}-\frac{b^4 n x}{5 c^4}+\frac{b^5 n \log (b+c x)}{5 c^5}+\frac{1}{5} x^5 \log \left (d \left (b x+c x^2\right )^n\right )+\frac{b n x^4}{20 c}-\frac{2 n x^5}{25} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2525
Rule 77
Rubi steps
\begin{align*} \int x^4 \log \left (d \left (b x+c x^2\right )^n\right ) \, dx &=\frac{1}{5} x^5 \log \left (d \left (b x+c x^2\right )^n\right )-\frac{1}{5} n \int \frac{x^4 (b+2 c x)}{b+c x} \, dx\\ &=\frac{1}{5} x^5 \log \left (d \left (b x+c x^2\right )^n\right )-\frac{1}{5} n \int \left (\frac{b^4}{c^4}-\frac{b^3 x}{c^3}+\frac{b^2 x^2}{c^2}-\frac{b x^3}{c}+2 x^4-\frac{b^5}{c^4 (b+c x)}\right ) \, dx\\ &=-\frac{b^4 n x}{5 c^4}+\frac{b^3 n x^2}{10 c^3}-\frac{b^2 n x^3}{15 c^2}+\frac{b n x^4}{20 c}-\frac{2 n x^5}{25}+\frac{b^5 n \log (b+c x)}{5 c^5}+\frac{1}{5} x^5 \log \left (d \left (b x+c x^2\right )^n\right )\\ \end{align*}
Mathematica [A] time = 0.0484504, size = 85, normalized size = 0.86 \[ \frac{c n x \left (-20 b^2 c^2 x^2+30 b^3 c x-60 b^4+15 b c^3 x^3-24 c^4 x^4\right )+60 b^5 n \log (b+c x)+60 c^5 x^5 \log \left (d (x (b+c x))^n\right )}{300 c^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.038, size = 0, normalized size = 0. \begin{align*} \int{x}^{4}\ln \left ( d \left ( c{x}^{2}+bx \right ) ^{n} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.04288, size = 117, normalized size = 1.18 \begin{align*} \frac{1}{5} \, x^{5} \log \left ({\left (c x^{2} + b x\right )}^{n} d\right ) + \frac{1}{300} \, n{\left (\frac{60 \, b^{5} \log \left (c x + b\right )}{c^{5}} - \frac{24 \, c^{4} x^{5} - 15 \, b c^{3} x^{4} + 20 \, b^{2} c^{2} x^{3} - 30 \, b^{3} c x^{2} + 60 \, b^{4} x}{c^{4}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.6118, size = 232, normalized size = 2.34 \begin{align*} \frac{60 \, c^{5} n x^{5} \log \left (c x^{2} + b x\right ) - 24 \, c^{5} n x^{5} + 60 \, c^{5} x^{5} \log \left (d\right ) + 15 \, b c^{4} n x^{4} - 20 \, b^{2} c^{3} n x^{3} + 30 \, b^{3} c^{2} n x^{2} - 60 \, b^{4} c n x + 60 \, b^{5} n \log \left (c x + b\right )}{300 \, c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 14.2238, size = 134, normalized size = 1.35 \begin{align*} \begin{cases} \frac{b^{5} n \log{\left (b + c x \right )}}{5 c^{5}} - \frac{b^{4} n x}{5 c^{4}} + \frac{b^{3} n x^{2}}{10 c^{3}} - \frac{b^{2} n x^{3}}{15 c^{2}} + \frac{b n x^{4}}{20 c} + \frac{n x^{5} \log{\left (b x + c x^{2} \right )}}{5} - \frac{2 n x^{5}}{25} + \frac{x^{5} \log{\left (d \right )}}{5} & \text{for}\: c \neq 0 \\\frac{n x^{5} \log{\left (b \right )}}{5} + \frac{n x^{5} \log{\left (x \right )}}{5} - \frac{n x^{5}}{25} + \frac{x^{5} \log{\left (d \right )}}{5} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.30019, size = 120, normalized size = 1.21 \begin{align*} \frac{1}{5} \, n x^{5} \log \left (c x^{2} + b x\right ) - \frac{1}{25} \,{\left (2 \, n - 5 \, \log \left (d\right )\right )} x^{5} + \frac{b n x^{4}}{20 \, c} - \frac{b^{2} n x^{3}}{15 \, c^{2}} + \frac{b^{3} n x^{2}}{10 \, c^{3}} - \frac{b^{4} n x}{5 \, c^{4}} + \frac{b^{5} n \log \left (c x + b\right )}{5 \, c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]