Optimal. Leaf size=75 \[ -\frac{b^2 p x^2}{8 a^2}+\frac{b^3 p x}{4 a^3}-\frac{b^4 p \log (a x+b)}{4 a^4}+\frac{1}{4} x^4 \log \left (c \left (a+\frac{b}{x}\right )^p\right )+\frac{b p x^3}{12 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.040069, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {2455, 263, 43} \[ -\frac{b^2 p x^2}{8 a^2}+\frac{b^3 p x}{4 a^3}-\frac{b^4 p \log (a x+b)}{4 a^4}+\frac{1}{4} x^4 \log \left (c \left (a+\frac{b}{x}\right )^p\right )+\frac{b p x^3}{12 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2455
Rule 263
Rule 43
Rubi steps
\begin{align*} \int x^3 \log \left (c \left (a+\frac{b}{x}\right )^p\right ) \, dx &=\frac{1}{4} x^4 \log \left (c \left (a+\frac{b}{x}\right )^p\right )+\frac{1}{4} (b p) \int \frac{x^2}{a+\frac{b}{x}} \, dx\\ &=\frac{1}{4} x^4 \log \left (c \left (a+\frac{b}{x}\right )^p\right )+\frac{1}{4} (b p) \int \frac{x^3}{b+a x} \, dx\\ &=\frac{1}{4} x^4 \log \left (c \left (a+\frac{b}{x}\right )^p\right )+\frac{1}{4} (b p) \int \left (\frac{b^2}{a^3}-\frac{b x}{a^2}+\frac{x^2}{a}-\frac{b^3}{a^3 (b+a x)}\right ) \, dx\\ &=\frac{b^3 p x}{4 a^3}-\frac{b^2 p x^2}{8 a^2}+\frac{b p x^3}{12 a}+\frac{1}{4} x^4 \log \left (c \left (a+\frac{b}{x}\right )^p\right )-\frac{b^4 p \log (b+a x)}{4 a^4}\\ \end{align*}
Mathematica [A] time = 0.0319432, size = 74, normalized size = 0.99 \[ \frac{a b p x \left (2 a^2 x^2-3 a b x+6 b^2\right )+6 a^4 x^4 \log \left (c \left (a+\frac{b}{x}\right )^p\right )-6 b^4 p \log \left (a+\frac{b}{x}\right )-6 b^4 p \log (x)}{24 a^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.236, size = 0, normalized size = 0. \begin{align*} \int{x}^{3}\ln \left ( c \left ( a+{\frac{b}{x}} \right ) ^{p} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.20977, size = 86, normalized size = 1.15 \begin{align*} \frac{1}{4} \, x^{4} \log \left ({\left (a + \frac{b}{x}\right )}^{p} c\right ) - \frac{1}{24} \, b p{\left (\frac{6 \, b^{3} \log \left (a x + b\right )}{a^{4}} - \frac{2 \, a^{2} x^{3} - 3 \, a b x^{2} + 6 \, b^{2} x}{a^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.28267, size = 177, normalized size = 2.36 \begin{align*} \frac{6 \, a^{4} p x^{4} \log \left (\frac{a x + b}{x}\right ) + 6 \, a^{4} x^{4} \log \left (c\right ) + 2 \, a^{3} b p x^{3} - 3 \, a^{2} b^{2} p x^{2} + 6 \, a b^{3} p x - 6 \, b^{4} p \log \left (a x + b\right )}{24 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 16.4616, size = 109, normalized size = 1.45 \begin{align*} \begin{cases} \frac{p x^{4} \log{\left (a + \frac{b}{x} \right )}}{4} + \frac{x^{4} \log{\left (c \right )}}{4} + \frac{b p x^{3}}{12 a} - \frac{b^{2} p x^{2}}{8 a^{2}} + \frac{b^{3} p x}{4 a^{3}} - \frac{b^{4} p \log{\left (a x + b \right )}}{4 a^{4}} & \text{for}\: a \neq 0 \\\frac{p x^{4} \log{\left (b \right )}}{4} - \frac{p x^{4} \log{\left (x \right )}}{4} + \frac{p x^{4}}{16} + \frac{x^{4} \log{\left (c \right )}}{4} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.29949, size = 101, normalized size = 1.35 \begin{align*} \frac{1}{4} \, p x^{4} \log \left (a x + b\right ) - \frac{1}{4} \, p x^{4} \log \left (x\right ) + \frac{1}{4} \, x^{4} \log \left (c\right ) + \frac{b p x^{3}}{12 \, a} - \frac{b^{2} p x^{2}}{8 \, a^{2}} + \frac{b^{3} p x}{4 \, a^{3}} - \frac{b^{4} p \log \left (a x + b\right )}{4 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]