Optimal. Leaf size=233 \[ \frac{(e+f x)^4 \left (a+b \cot ^{-1}(c+d x)\right )}{4 f}+\frac{b f x \left (-\left (1-6 c^2\right ) f^2-12 c d e f+6 d^2 e^2\right )}{4 d^3}+\frac{b \left (-6 \left (1-c^2\right ) d^2 e^2 f^2+4 c \left (3-c^2\right ) d e f^3+\left (c^4-6 c^2+1\right ) f^4-4 c d^3 e^3 f+d^4 e^4\right ) \tan ^{-1}(c+d x)}{4 d^4 f}+\frac{b f^2 (c+d x)^2 (d e-c f)}{2 d^4}+\frac{b (d e-c f) (-c f+d e+f) (d e-(c+1) f) \log \left ((c+d x)^2+1\right )}{2 d^4}+\frac{b f^3 (c+d x)^3}{12 d^4} \]
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Rubi [A] time = 0.356664, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5048, 4863, 702, 635, 203, 260} \[ \frac{(e+f x)^4 \left (a+b \cot ^{-1}(c+d x)\right )}{4 f}+\frac{b f x \left (-\left (1-6 c^2\right ) f^2-12 c d e f+6 d^2 e^2\right )}{4 d^3}+\frac{b \left (-6 \left (1-c^2\right ) d^2 e^2 f^2+4 c \left (3-c^2\right ) d e f^3+\left (c^4-6 c^2+1\right ) f^4-4 c d^3 e^3 f+d^4 e^4\right ) \tan ^{-1}(c+d x)}{4 d^4 f}+\frac{b f^2 (c+d x)^2 (d e-c f)}{2 d^4}+\frac{b (d e-c f) (-c f+d e+f) (d e-(c+1) f) \log \left ((c+d x)^2+1\right )}{2 d^4}+\frac{b f^3 (c+d x)^3}{12 d^4} \]
Antiderivative was successfully verified.
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Rule 5048
Rule 4863
Rule 702
Rule 635
Rule 203
Rule 260
Rubi steps
\begin{align*} \int (e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \left (\frac{d e-c f}{d}+\frac{f x}{d}\right )^3 \left (a+b \cot ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac{(e+f x)^4 \left (a+b \cot ^{-1}(c+d x)\right )}{4 f}+\frac{b \operatorname{Subst}\left (\int \frac{\left (\frac{d e-c f}{d}+\frac{f x}{d}\right )^4}{1+x^2} \, dx,x,c+d x\right )}{4 f}\\ &=\frac{(e+f x)^4 \left (a+b \cot ^{-1}(c+d x)\right )}{4 f}+\frac{b \operatorname{Subst}\left (\int \left (\frac{f^2 \left (6 d^2 e^2-12 c d e f-\left (1-6 c^2\right ) f^2\right )}{d^4}+\frac{4 f^3 (d e-c f) x}{d^4}+\frac{f^4 x^2}{d^4}+\frac{d^4 e^4-4 c d^3 e^3 f-6 \left (1-c^2\right ) d^2 e^2 f^2+4 c \left (3-c^2\right ) d e f^3+\left (1-6 c^2+c^4\right ) f^4+4 f (d e-c f) (d e-f-c f) (d e+f-c f) x}{d^4 \left (1+x^2\right )}\right ) \, dx,x,c+d x\right )}{4 f}\\ &=\frac{b f \left (6 d^2 e^2-12 c d e f-\left (1-6 c^2\right ) f^2\right ) x}{4 d^3}+\frac{b f^2 (d e-c f) (c+d x)^2}{2 d^4}+\frac{b f^3 (c+d x)^3}{12 d^4}+\frac{(e+f x)^4 \left (a+b \cot ^{-1}(c+d x)\right )}{4 f}+\frac{b \operatorname{Subst}\left (\int \frac{d^4 e^4-4 c d^3 e^3 f-6 \left (1-c^2\right ) d^2 e^2 f^2+4 c \left (3-c^2\right ) d e f^3+\left (1-6 c^2+c^4\right ) f^4+4 f (d e-c f) (d e-f-c f) (d e+f-c f) x}{1+x^2} \, dx,x,c+d x\right )}{4 d^4 f}\\ &=\frac{b f \left (6 d^2 e^2-12 c d e f-\left (1-6 c^2\right ) f^2\right ) x}{4 d^3}+\frac{b f^2 (d e-c f) (c+d x)^2}{2 d^4}+\frac{b f^3 (c+d x)^3}{12 d^4}+\frac{(e+f x)^4 \left (a+b \cot ^{-1}(c+d x)\right )}{4 f}+\frac{(b (d e-c f) (d e+f-c f) (d e-(1+c) f)) \operatorname{Subst}\left (\int \frac{x}{1+x^2} \, dx,x,c+d x\right )}{d^4}+\frac{\left (b \left (d^4 e^4-4 c d^3 e^3 f-6 \left (1-c^2\right ) d^2 e^2 f^2+4 c \left (3-c^2\right ) d e f^3+\left (1-6 c^2+c^4\right ) f^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,c+d x\right )}{4 d^4 f}\\ &=\frac{b f \left (6 d^2 e^2-12 c d e f-\left (1-6 c^2\right ) f^2\right ) x}{4 d^3}+\frac{b f^2 (d e-c f) (c+d x)^2}{2 d^4}+\frac{b f^3 (c+d x)^3}{12 d^4}+\frac{(e+f x)^4 \left (a+b \cot ^{-1}(c+d x)\right )}{4 f}+\frac{b \left (d^4 e^4-4 c d^3 e^3 f-6 \left (1-c^2\right ) d^2 e^2 f^2+4 c \left (3-c^2\right ) d e f^3+\left (1-6 c^2+c^4\right ) f^4\right ) \tan ^{-1}(c+d x)}{4 d^4 f}+\frac{b (d e-c f) (d e+f-c f) (d e-(1+c) f) \log \left (1+(c+d x)^2\right )}{2 d^4}\\ \end{align*}
Mathematica [C] time = 0.266223, size = 157, normalized size = 0.67 \[ \frac{(e+f x)^4 \left (a+b \cot ^{-1}(c+d x)\right )+\frac{b \left (6 d f^2 x \left (\left (6 c^2-1\right ) f^2-12 c d e f+6 d^2 e^2\right )+12 f^3 (c+d x)^2 (d e-c f)-3 i (d e-(c-i) f)^4 \log (-c-d x+i)+3 i (d e-(c+i) f)^4 \log (c+d x+i)+2 f^4 (c+d x)^3\right )}{6 d^4}}{4 f} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.052, size = 526, normalized size = 2.3 \begin{align*}{\frac{b{f}^{3}\ln \left ( 1+ \left ( dx+c \right ) ^{2} \right ) c}{2\,{d}^{4}}}-{\frac{3\,bf\arctan \left ( dx+c \right ){e}^{2}}{2\,{d}^{2}}}-{\frac{3\,b{f}^{3}\arctan \left ( dx+c \right ){c}^{2}}{2\,{d}^{4}}}-{\frac{b\arctan \left ( dx+c \right ) c{e}^{3}}{d}}+{\frac{3\,bf{\rm arccot} \left (dx+c\right ){e}^{2}{x}^{2}}{2}}+{\frac{3\,b{f}^{3}{c}^{2}x}{4\,{d}^{3}}}+{\frac{3\,bf{e}^{2}x}{2\,d}}-{\frac{b{f}^{3}c{x}^{2}}{4\,{d}^{2}}}+{\frac{b{f}^{2}e{x}^{2}}{2\,d}}-{\frac{b{f}^{3}\ln \left ( 1+ \left ( dx+c \right ) ^{2} \right ){c}^{3}}{2\,{d}^{4}}}+{\frac{b{f}^{3}\arctan \left ( dx+c \right ){c}^{4}}{4\,{d}^{4}}}-{\frac{b{f}^{2}\ln \left ( 1+ \left ( dx+c \right ) ^{2} \right ) e}{2\,{d}^{3}}}+b{f}^{2}{\rm arccot} \left (dx+c\right )e{x}^{3}+{\frac{3\,bfc{e}^{2}}{2\,{d}^{2}}}-{\frac{5\,b{f}^{2}{c}^{2}e}{2\,{d}^{3}}}+ax{e}^{3}+{\frac{a{f}^{3}{x}^{4}}{4}}+{\frac{13\,b{f}^{3}{c}^{3}}{12\,{d}^{4}}}+{\frac{a{e}^{4}}{4\,f}}-{\frac{b{f}^{3}c}{4\,{d}^{4}}}-{\frac{b{f}^{3}x}{4\,{d}^{3}}}+a{f}^{2}{x}^{3}e+{\frac{3\,af{x}^{2}{e}^{2}}{2}}+{\frac{b{f}^{3}{x}^{3}}{12\,d}}+{\frac{b{\rm arccot} \left (dx+c\right ){e}^{4}}{4\,f}}+{\frac{b{f}^{3}{\rm arccot} \left (dx+c\right ){x}^{4}}{4}}+{\frac{b\arctan \left ( dx+c \right ){e}^{4}}{4\,f}}+{\rm arccot} \left (dx+c\right )xb{e}^{3}+{\frac{b\ln \left ( 1+ \left ( dx+c \right ) ^{2} \right ){e}^{3}}{2\,d}}+{\frac{b{f}^{3}\arctan \left ( dx+c \right ) }{4\,{d}^{4}}}+3\,{\frac{b{f}^{2}\arctan \left ( dx+c \right ) ce}{{d}^{3}}}+{\frac{3\,b{f}^{2}\ln \left ( 1+ \left ( dx+c \right ) ^{2} \right ){c}^{2}e}{2\,{d}^{3}}}-{\frac{3\,bf\ln \left ( 1+ \left ( dx+c \right ) ^{2} \right ) c{e}^{2}}{2\,{d}^{2}}}+{\frac{3\,bf\arctan \left ( dx+c \right ){c}^{2}{e}^{2}}{2\,{d}^{2}}}-{\frac{b{f}^{2}\arctan \left ( dx+c \right ){c}^{3}e}{{d}^{3}}}-2\,{\frac{b{f}^{2}cex}{{d}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49634, size = 460, normalized size = 1.97 \begin{align*} \frac{1}{4} \, a f^{3} x^{4} + a e f^{2} x^{3} + \frac{3}{2} \, a e^{2} f x^{2} + \frac{3}{2} \,{\left (x^{2} \operatorname{arccot}\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 e^{2} f + \frac{1}{2} \,{\left (2 \, x^{3} \operatorname{arccot}\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 e f^{2} + \frac{1}{12} \,{\left (3 \, x^{4} \operatorname{arccot}\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 f^{3} + a e^{3} x + \frac{{\left (2 \,{\left (d x + c\right )} \operatorname{arccot}\left (d x + c\right ) + \log \left ({\left (d x + c\right )}^{2} + 1\right )\right )} b e^{3}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.46891, size = 694, normalized size = 2.98 \begin{align*} \frac{3 \, a d^{4} f^{3} x^{4} +{\left (12 \, a d^{4} e f^{2} + b d^{3} f^{3}\right )} x^{3} + 3 \,{\left (6 \, a d^{4} e^{2} f + 2 \, b d^{3} e f^{2} - b c d^{2} f^{3}\right )} x^{2} + 3 \,{\left (4 \, a d^{4} e^{3} + 6 \, b d^{3} e^{2} f - 8 \, b c d^{2} e f^{2} +{\left (3 \, b c^{2} - b\right )} d f^{3}\right )} x + 3 \,{\left (b d^{4} f^{3} x^{4} + 4 \, b d^{4} e f^{2} x^{3} + 6 \, b d^{4} e^{2} f x^{2} + 4 \, b d^{4} e^{3} x\right )} \operatorname{arccot}\left (d x + c\right ) - 3 \,{\left (4 \, b c d^{3} e^{3} - 6 \,{\left (b c^{2} - b\right )} d^{2} e^{2} f + 4 \,{\left (b c^{3} - 3 \, b c\right )} d e f^{2} -{\left (b c^{4} - 6 \, b c^{2} + b\right )} f^{3}\right )} \arctan \left (d x + c\right ) + 6 \,{\left (b d^{3} e^{3} - 3 \, b c d^{2} e^{2} f +{\left (3 \, b c^{2} - b\right )} d e f^{2} -{\left (b c^{3} - b c\right )} f^{3}\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}{12 \, d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.92132, size = 627, normalized size = 2.69 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 3.22393, size = 1045, normalized size = 4.48 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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