Optimal. Leaf size=72 \[ \frac{e^3 (c+d x)^4 \left (a+b \tan ^{-1}(c+d x)\right )}{4 d}-\frac{b e^3 (c+d x)^3}{12 d}-\frac{b e^3 \tan ^{-1}(c+d x)}{4 d}+\frac{1}{4} b e^3 x \]
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Rubi [A] time = 0.250649, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {5043, 12, 4852, 302, 203} \[ \frac{e^3 (c+d x)^4 \left (a+b \tan ^{-1}(c+d x)\right )}{4 d}-\frac{b e^3 (c+d x)^3}{12 d}-\frac{b e^3 \tan ^{-1}(c+d x)}{4 d}+\frac{1}{4} b e^3 x \]
Antiderivative was successfully verified.
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Rule 5043
Rule 12
Rule 4852
Rule 302
Rule 203
Rubi steps
\begin{align*} \int (c e+d e x)^3 \left (a+b \tan ^{-1}(c+d x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int e^3 x^3 \left (a+b \tan ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac{e^3 \operatorname{Subst}\left (\int x^3 \left (a+b \tan ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac{e^3 (c+d x)^4 \left (a+b \tan ^{-1}(c+d x)\right )}{4 d}-\frac{\left (b e^3\right ) \operatorname{Subst}\left (\int \frac{x^4}{1+x^2} \, dx,x,c+d x\right )}{4 d}\\ &=\frac{e^3 (c+d x)^4 \left (a+b \tan ^{-1}(c+d x)\right )}{4 d}-\frac{\left (b e^3\right ) \operatorname{Subst}\left (\int \left (-1+x^2+\frac{1}{1+x^2}\right ) \, dx,x,c+d x\right )}{4 d}\\ &=\frac{1}{4} b e^3 x-\frac{b e^3 (c+d x)^3}{12 d}+\frac{e^3 (c+d x)^4 \left (a+b \tan ^{-1}(c+d x)\right )}{4 d}-\frac{\left (b e^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,c+d x\right )}{4 d}\\ &=\frac{1}{4} b e^3 x-\frac{b e^3 (c+d x)^3}{12 d}-\frac{b e^3 \tan ^{-1}(c+d x)}{4 d}+\frac{e^3 (c+d x)^4 \left (a+b \tan ^{-1}(c+d x)\right )}{4 d}\\ \end{align*}
Mathematica [A] time = 0.0595514, size = 56, normalized size = 0.78 \[ \frac{e^3 \left (\frac{1}{4} (c+d x)^4 \left (a+b \tan ^{-1}(c+d x)\right )-\frac{1}{4} b \left (\frac{1}{3} (c+d x)^3+\tan ^{-1}(c+d x)-d x\right )\right )}{d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.035, size = 225, normalized size = 3.1 \begin{align*}{\frac{{d}^{3}{x}^{4}a{e}^{3}}{4}}+{d}^{2}{x}^{3}ac{e}^{3}+{\frac{3\,d{x}^{2}a{c}^{2}{e}^{3}}{2}}+xa{c}^{3}{e}^{3}+{\frac{a{c}^{4}{e}^{3}}{4\,d}}+{\frac{{d}^{3}\arctan \left ( dx+c \right ){x}^{4}b{e}^{3}}{4}}+{d}^{2}\arctan \left ( dx+c \right ){x}^{3}bc{e}^{3}+{\frac{3\,d\arctan \left ( dx+c \right ){x}^{2}b{c}^{2}{e}^{3}}{2}}+\arctan \left ( dx+c \right ) xb{c}^{3}{e}^{3}+{\frac{b\arctan \left ( dx+c \right ){c}^{4}{e}^{3}}{4\,d}}-{\frac{{d}^{2}{x}^{3}b{e}^{3}}{12}}-{\frac{d{x}^{2}bc{e}^{3}}{4}}-{\frac{xb{c}^{2}{e}^{3}}{4}}-{\frac{b{c}^{3}{e}^{3}}{12\,d}}+{\frac{b{e}^{3}x}{4}}+{\frac{bc{e}^{3}}{4\,d}}-{\frac{{e}^{3}b\arctan \left ( dx+c \right ) }{4\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.51803, size = 500, normalized size = 6.94 \begin{align*} \frac{1}{4} \, a d^{3} e^{3} x^{4} + a c d^{2} e^{3} x^{3} + \frac{3}{2} \, a c^{2} d e^{3} x^{2} + \frac{3}{2} \,{\left (x^{2} \arctan \left (d x + c\right ) - d{\left (\frac{x}{d^{2}} + \frac{{\left (c^{2} - 1\right )} \arctan \left (\frac{d^{2} x + c d}{d}\right )}{d^{3}} - \frac{c \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}{d^{3}}\right )}\right )} b c^{2} d e^{3} + \frac{1}{2} \,{\left (2 \, x^{3} \arctan \left (d x + c\right ) - d{\left (\frac{d x^{2} - 4 \, c x}{d^{3}} - \frac{2 \,{\left (c^{3} - 3 \, c\right )} \arctan \left (\frac{d^{2} x + c d}{d}\right )}{d^{4}} + \frac{{\left (3 \, c^{2} - 1\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}{d^{4}}\right )}\right )} b c d^{2} e^{3} + \frac{1}{12} \,{\left (3 \, x^{4} \arctan \left (d x + c\right ) - d{\left (\frac{d^{2} x^{3} - 3 \, c d x^{2} + 3 \,{\left (3 \, c^{2} - 1\right )} x}{d^{4}} + \frac{3 \,{\left (c^{4} - 6 \, c^{2} + 1\right )} \arctan \left (\frac{d^{2} x + c d}{d}\right )}{d^{5}} - \frac{6 \,{\left (c^{3} - c\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}{d^{5}}\right )}\right )} b d^{3} e^{3} + a c^{3} e^{3} x + \frac{{\left (2 \,{\left (d x + c\right )} \arctan \left (d x + c\right ) - \log \left ({\left (d x + c\right )}^{2} + 1\right )\right )} b c^{3} e^{3}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.5769, size = 315, normalized size = 4.38 \begin{align*} \frac{3 \, a d^{4} e^{3} x^{4} +{\left (12 \, a c - b\right )} d^{3} e^{3} x^{3} + 3 \,{\left (6 \, a c^{2} - b c\right )} d^{2} e^{3} x^{2} + 3 \,{\left (4 \, a c^{3} - b c^{2} + b\right )} d e^{3} x + 3 \,{\left (b d^{4} e^{3} x^{4} + 4 \, b c d^{3} e^{3} x^{3} + 6 \, b c^{2} d^{2} e^{3} x^{2} + 4 \, b c^{3} d e^{3} x +{\left (b c^{4} - b\right )} e^{3}\right )} \arctan \left (d x + c\right )}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.43974, size = 231, normalized size = 3.21 \begin{align*} \begin{cases} a c^{3} e^{3} x + \frac{3 a c^{2} d e^{3} x^{2}}{2} + a c d^{2} e^{3} x^{3} + \frac{a d^{3} e^{3} x^{4}}{4} + \frac{b c^{4} e^{3} \operatorname{atan}{\left (c + d x \right )}}{4 d} + b c^{3} e^{3} x \operatorname{atan}{\left (c + d x \right )} + \frac{3 b c^{2} d e^{3} x^{2} \operatorname{atan}{\left (c + d x \right )}}{2} - \frac{b c^{2} e^{3} x}{4} + b c d^{2} e^{3} x^{3} \operatorname{atan}{\left (c + d x \right )} - \frac{b c d e^{3} x^{2}}{4} + \frac{b d^{3} e^{3} x^{4} \operatorname{atan}{\left (c + d x \right )}}{4} - \frac{b d^{2} e^{3} x^{3}}{12} + \frac{b e^{3} x}{4} - \frac{b e^{3} \operatorname{atan}{\left (c + d x \right )}}{4 d} & \text{for}\: d \neq 0 \\c^{3} e^{3} x \left (a + b \operatorname{atan}{\left (c \right )}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15081, size = 317, normalized size = 4.4 \begin{align*} \frac{6 \, b d^{4} x^{4} \arctan \left (d x + c\right ) e^{3} + 6 \, a d^{4} x^{4} e^{3} + 24 \, b c d^{3} x^{3} \arctan \left (d x + c\right ) e^{3} + 24 \, a c d^{3} x^{3} e^{3} + 36 \, b c^{2} d^{2} x^{2} \arctan \left (d x + c\right ) e^{3} + 36 \, a c^{2} d^{2} x^{2} e^{3} - 2 \, b d^{3} x^{3} e^{3} + 24 \, b c^{3} d x \arctan \left (d x + c\right ) e^{3} + 3 \, \pi b c^{4} e^{3} \mathrm{sgn}\left (d x + c\right ) - 3 \, \pi b c^{4} e^{3} + 24 \, a c^{3} d x e^{3} - 6 \, b c d^{2} x^{2} e^{3} - 6 \, b c^{4} \arctan \left (\frac{1}{d x + c}\right ) e^{3} - 6 \, b c^{2} d x e^{3} + 6 \, b d x e^{3} - 3 \, \pi b e^{3} \mathrm{sgn}\left (d x + c\right ) + 3 \, \pi b e^{3} + 6 \, b \arctan \left (\frac{1}{d x + c}\right ) e^{3}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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