Optimal. Leaf size=120 \[ \frac{(f x+g)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 f}-\frac{b n x (d f-e g)^2}{3 e^2}+\frac{b n (d f-e g)^3 \log (d+e x)}{3 e^3 f}+\frac{b n (f x+g)^2 (d f-e g)}{6 e f}-\frac{b n (f x+g)^3}{9 f} \]
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Rubi [A] time = 0.113123, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2412, 2395, 43} \[ \frac{(f x+g)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 f}-\frac{b n x (d f-e g)^2}{3 e^2}+\frac{b n (d f-e g)^3 \log (d+e x)}{3 e^3 f}+\frac{b n (f x+g)^2 (d f-e g)}{6 e f}-\frac{b n (f x+g)^3}{9 f} \]
Antiderivative was successfully verified.
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Rule 2412
Rule 2395
Rule 43
Rubi steps
\begin{align*} \int \left (f+\frac{g}{x}\right )^2 x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx &=\int (g+f x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx\\ &=\frac{(g+f x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 f}-\frac{(b e n) \int \frac{(g+f x)^3}{d+e x} \, dx}{3 f}\\ &=\frac{(g+f x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 f}-\frac{(b e n) \int \left (\frac{f (-d f+e g)^2}{e^3}+\frac{(-d f+e g)^3}{e^3 (d+e x)}+\frac{f (-d f+e g) (g+f x)}{e^2}+\frac{f (g+f x)^2}{e}\right ) \, dx}{3 f}\\ &=-\frac{b (d f-e g)^2 n x}{3 e^2}+\frac{b (d f-e g) n (g+f x)^2}{6 e f}-\frac{b n (g+f x)^3}{9 f}+\frac{b (d f-e g)^3 n \log (d+e x)}{3 e^3 f}+\frac{(g+f x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 f}\\ \end{align*}
Mathematica [A] time = 0.140217, size = 150, normalized size = 1.25 \[ \frac{e \left (x \left (6 a e^2 \left (f^2 x^2+3 f g x+3 g^2\right )-b n \left (6 d^2 f^2-3 d e f (f x+6 g)+e^2 \left (2 f^2 x^2+9 f g x+18 g^2\right )\right )\right )+6 b e \left (3 d g^2+e x \left (f^2 x^2+3 f g x+3 g^2\right )\right ) \log \left (c (d+e x)^n\right )\right )+6 b d^2 f n (d f-3 e g) \log (d+e x)}{18 e^3} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.465, size = 585, normalized size = 4.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10652, size = 252, normalized size = 2.1 \begin{align*} \frac{1}{3} \, b f^{2} x^{3} \log \left ({\left (e x + d\right )}^{n} c\right ) + \frac{1}{3} \, a f^{2} x^{3} - b e g^{2} n{\left (\frac{x}{e} - \frac{d \log \left (e x + d\right )}{e^{2}}\right )} + \frac{1}{18} \, b e f^{2} n{\left (\frac{6 \, d^{3} \log \left (e x + d\right )}{e^{4}} - \frac{2 \, e^{2} x^{3} - 3 \, d e x^{2} + 6 \, d^{2} x}{e^{3}}\right )} - \frac{1}{2} \, b e f g n{\left (\frac{2 \, d^{2} \log \left (e x + d\right )}{e^{3}} + \frac{e x^{2} - 2 \, d x}{e^{2}}\right )} + b f g x^{2} \log \left ({\left (e x + d\right )}^{n} c\right ) + a f g x^{2} + b g^{2} x \log \left ({\left (e x + d\right )}^{n} c\right ) + a g^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.90533, size = 474, normalized size = 3.95 \begin{align*} -\frac{2 \,{\left (b e^{3} f^{2} n - 3 \, a e^{3} f^{2}\right )} x^{3} - 3 \,{\left (6 \, a e^{3} f g +{\left (b d e^{2} f^{2} - 3 \, b e^{3} f g\right )} n\right )} x^{2} - 6 \,{\left (3 \, a e^{3} g^{2} -{\left (b d^{2} e f^{2} - 3 \, b d e^{2} f g + 3 \, b e^{3} g^{2}\right )} n\right )} x - 6 \,{\left (b e^{3} f^{2} n x^{3} + 3 \, b e^{3} f g n x^{2} + 3 \, b e^{3} g^{2} n x +{\left (b d^{3} f^{2} - 3 \, b d^{2} e f g + 3 \, b d e^{2} g^{2}\right )} n\right )} \log \left (e x + d\right ) - 6 \,{\left (b e^{3} f^{2} x^{3} + 3 \, b e^{3} f g x^{2} + 3 \, b e^{3} g^{2} x\right )} \log \left (c\right )}{18 \, e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 11.0701, size = 277, normalized size = 2.31 \begin{align*} \begin{cases} \frac{a f^{2} x^{3}}{3} + a f g x^{2} + a g^{2} x + \frac{b d^{3} f^{2} n \log{\left (d + e x \right )}}{3 e^{3}} - \frac{b d^{2} f^{2} n x}{3 e^{2}} - \frac{b d^{2} f g n \log{\left (d + e x \right )}}{e^{2}} + \frac{b d f^{2} n x^{2}}{6 e} + \frac{b d f g n x}{e} + \frac{b d g^{2} n \log{\left (d + e x \right )}}{e} + \frac{b f^{2} n x^{3} \log{\left (d + e x \right )}}{3} - \frac{b f^{2} n x^{3}}{9} + \frac{b f^{2} x^{3} \log{\left (c \right )}}{3} + b f g n x^{2} \log{\left (d + e x \right )} - \frac{b f g n x^{2}}{2} + b f g x^{2} \log{\left (c \right )} + b g^{2} n x \log{\left (d + e x \right )} - b g^{2} n x + b g^{2} x \log{\left (c \right )} & \text{for}\: e \neq 0 \\\left (a + b \log{\left (c d^{n} \right )}\right ) \left (\frac{f^{2} x^{3}}{3} + f g x^{2} + g^{2} x\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.29719, size = 581, normalized size = 4.84 \begin{align*} \frac{1}{3} \,{\left (x e + d\right )}^{3} b f^{2} n e^{\left (-3\right )} \log \left (x e + d\right ) -{\left (x e + d\right )}^{2} b d f^{2} n e^{\left (-3\right )} \log \left (x e + d\right ) +{\left (x e + d\right )} b d^{2} f^{2} n e^{\left (-3\right )} \log \left (x e + d\right ) - \frac{1}{9} \,{\left (x e + d\right )}^{3} b f^{2} n e^{\left (-3\right )} + \frac{1}{2} \,{\left (x e + d\right )}^{2} b d f^{2} n e^{\left (-3\right )} -{\left (x e + d\right )} b d^{2} f^{2} n e^{\left (-3\right )} +{\left (x e + d\right )}^{2} b f g n e^{\left (-2\right )} \log \left (x e + d\right ) - 2 \,{\left (x e + d\right )} b d f g n e^{\left (-2\right )} \log \left (x e + d\right ) + \frac{1}{3} \,{\left (x e + d\right )}^{3} b f^{2} e^{\left (-3\right )} \log \left (c\right ) -{\left (x e + d\right )}^{2} b d f^{2} e^{\left (-3\right )} \log \left (c\right ) +{\left (x e + d\right )} b d^{2} f^{2} e^{\left (-3\right )} \log \left (c\right ) - \frac{1}{2} \,{\left (x e + d\right )}^{2} b f g n e^{\left (-2\right )} + 2 \,{\left (x e + d\right )} b d f g n e^{\left (-2\right )} + \frac{1}{3} \,{\left (x e + d\right )}^{3} a f^{2} e^{\left (-3\right )} -{\left (x e + d\right )}^{2} a d f^{2} e^{\left (-3\right )} +{\left (x e + d\right )} a d^{2} f^{2} e^{\left (-3\right )} +{\left (x e + d\right )} b g^{2} n e^{\left (-1\right )} \log \left (x e + d\right ) +{\left (x e + d\right )}^{2} b f g e^{\left (-2\right )} \log \left (c\right ) - 2 \,{\left (x e + d\right )} b d f g e^{\left (-2\right )} \log \left (c\right ) -{\left (x e + d\right )} b g^{2} n e^{\left (-1\right )} +{\left (x e + d\right )}^{2} a f g e^{\left (-2\right )} - 2 \,{\left (x e + d\right )} a d f g e^{\left (-2\right )} +{\left (x e + d\right )} b g^{2} e^{\left (-1\right )} \log \left (c\right ) +{\left (x e + d\right )} a g^{2} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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