Optimal. Leaf size=185 \[ \frac{1}{4} d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \tan ^{-1}(c x)\right )-\frac{b x^3 \left (6 c^4 d^2-8 c^2 d e+3 e^2\right )}{72 c^5}+\frac{b x \left (6 c^4 d^2-8 c^2 d e+3 e^2\right )}{24 c^7}-\frac{b \left (6 c^4 d^2-8 c^2 d e+3 e^2\right ) \tan ^{-1}(c x)}{24 c^8}-\frac{b e x^5 \left (8 c^2 d-3 e\right )}{120 c^3}-\frac{b e^2 x^7}{56 c} \]
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Rubi [A] time = 0.193197, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {266, 43, 4976, 1261, 203} \[ \frac{1}{4} d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \tan ^{-1}(c x)\right )-\frac{b x^3 \left (6 c^4 d^2-8 c^2 d e+3 e^2\right )}{72 c^5}+\frac{b x \left (6 c^4 d^2-8 c^2 d e+3 e^2\right )}{24 c^7}-\frac{b \left (6 c^4 d^2-8 c^2 d e+3 e^2\right ) \tan ^{-1}(c x)}{24 c^8}-\frac{b e x^5 \left (8 c^2 d-3 e\right )}{120 c^3}-\frac{b e^2 x^7}{56 c} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rule 4976
Rule 1261
Rule 203
Rubi steps
\begin{align*} \int x^3 \left (d+e x^2\right )^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx &=\frac{1}{4} d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \tan ^{-1}(c x)\right )-(b c) \int \frac{x^4 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )}{24+24 c^2 x^2} \, dx\\ &=\frac{1}{4} d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \tan ^{-1}(c x)\right )-(b c) \int \left (-\frac{6 c^4 d^2-8 c^2 d e+3 e^2}{24 c^8}+\frac{\left (6 c^4 d^2-8 c^2 d e+3 e^2\right ) x^2}{24 c^6}+\frac{\left (8 c^2 d-3 e\right ) e x^4}{24 c^4}+\frac{e^2 x^6}{8 c^2}+\frac{6 c^4 d^2-8 c^2 d e+3 e^2}{c^8 \left (24+24 c^2 x^2\right )}\right ) \, dx\\ &=\frac{b \left (6 c^4 d^2-8 c^2 d e+3 e^2\right ) x}{24 c^7}-\frac{b \left (6 c^4 d^2-8 c^2 d e+3 e^2\right ) x^3}{72 c^5}-\frac{b \left (8 c^2 d-3 e\right ) e x^5}{120 c^3}-\frac{b e^2 x^7}{56 c}+\frac{1}{4} d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \tan ^{-1}(c x)\right )-\frac{\left (b \left (6 c^4 d^2-8 c^2 d e+3 e^2\right )\right ) \int \frac{1}{24+24 c^2 x^2} \, dx}{c^7}\\ &=\frac{b \left (6 c^4 d^2-8 c^2 d e+3 e^2\right ) x}{24 c^7}-\frac{b \left (6 c^4 d^2-8 c^2 d e+3 e^2\right ) x^3}{72 c^5}-\frac{b \left (8 c^2 d-3 e\right ) e x^5}{120 c^3}-\frac{b e^2 x^7}{56 c}-\frac{b \left (6 c^4 d^2-8 c^2 d e+3 e^2\right ) \tan ^{-1}(c x)}{24 c^8}+\frac{1}{4} d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \tan ^{-1}(c x)\right )\\ \end{align*}
Mathematica [A] time = 0.139875, size = 174, normalized size = 0.94 \[ \frac{105 a c^8 x^4 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )+b c x \left (-3 c^6 \left (70 d^2 x^2+56 d e x^4+15 e^2 x^6\right )+7 c^4 \left (90 d^2+40 d e x^2+9 e^2 x^4\right )-105 c^2 e \left (8 d+e x^2\right )+315 e^2\right )+105 b \tan ^{-1}(c x) \left (c^8 \left (6 d^2 x^4+8 d e x^6+3 e^2 x^8\right )-6 c^4 d^2+8 c^2 d e-3 e^2\right )}{2520 c^8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.038, size = 203, normalized size = 1.1 \begin{align*}{\frac{a{e}^{2}{x}^{8}}{8}}+{\frac{aed{x}^{6}}{3}}+{\frac{a{x}^{4}{d}^{2}}{4}}+{\frac{b\arctan \left ( cx \right ){e}^{2}{x}^{8}}{8}}+{\frac{b\arctan \left ( cx \right ) ed{x}^{6}}{3}}+{\frac{b\arctan \left ( cx \right ){d}^{2}{x}^{4}}{4}}-{\frac{b{e}^{2}{x}^{7}}{56\,c}}-{\frac{bed{x}^{5}}{15\,c}}-{\frac{b{d}^{2}{x}^{3}}{12\,c}}+{\frac{b{x}^{5}{e}^{2}}{40\,{c}^{3}}}+{\frac{b{x}^{3}de}{9\,{c}^{3}}}+{\frac{b{d}^{2}x}{4\,{c}^{3}}}-{\frac{b{x}^{3}{e}^{2}}{24\,{c}^{5}}}-{\frac{bedx}{3\,{c}^{5}}}+{\frac{bx{e}^{2}}{8\,{c}^{7}}}-{\frac{b{d}^{2}\arctan \left ( cx \right ) }{4\,{c}^{4}}}+{\frac{b\arctan \left ( cx \right ) ed}{3\,{c}^{6}}}-{\frac{b\arctan \left ( cx \right ){e}^{2}}{8\,{c}^{8}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.47338, size = 248, normalized size = 1.34 \begin{align*} \frac{1}{8} \, a e^{2} x^{8} + \frac{1}{3} \, a d e x^{6} + \frac{1}{4} \, a d^{2} x^{4} + \frac{1}{12} \,{\left (3 \, x^{4} \arctan \left (c x\right ) - c{\left (\frac{c^{2} x^{3} - 3 \, x}{c^{4}} + \frac{3 \, \arctan \left (c x\right )}{c^{5}}\right )}\right )} b d^{2} + \frac{1}{45} \,{\left (15 \, x^{6} \arctan \left (c x\right ) - c{\left (\frac{3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac{15 \, \arctan \left (c x\right )}{c^{7}}\right )}\right )} b d e + \frac{1}{840} \,{\left (105 \, x^{8} \arctan \left (c x\right ) - c{\left (\frac{15 \, c^{6} x^{7} - 21 \, c^{4} x^{5} + 35 \, c^{2} x^{3} - 105 \, x}{c^{8}} + \frac{105 \, \arctan \left (c x\right )}{c^{9}}\right )}\right )} b e^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77162, size = 455, normalized size = 2.46 \begin{align*} \frac{315 \, a c^{8} e^{2} x^{8} + 840 \, a c^{8} d e x^{6} - 45 \, b c^{7} e^{2} x^{7} + 630 \, a c^{8} d^{2} x^{4} - 21 \,{\left (8 \, b c^{7} d e - 3 \, b c^{5} e^{2}\right )} x^{5} - 35 \,{\left (6 \, b c^{7} d^{2} - 8 \, b c^{5} d e + 3 \, b c^{3} e^{2}\right )} x^{3} + 105 \,{\left (6 \, b c^{5} d^{2} - 8 \, b c^{3} d e + 3 \, b c e^{2}\right )} x + 105 \,{\left (3 \, b c^{8} e^{2} x^{8} + 8 \, b c^{8} d e x^{6} + 6 \, b c^{8} d^{2} x^{4} - 6 \, b c^{4} d^{2} + 8 \, b c^{2} d e - 3 \, b e^{2}\right )} \arctan \left (c x\right )}{2520 \, c^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.68769, size = 260, normalized size = 1.41 \begin{align*} \begin{cases} \frac{a d^{2} x^{4}}{4} + \frac{a d e x^{6}}{3} + \frac{a e^{2} x^{8}}{8} + \frac{b d^{2} x^{4} \operatorname{atan}{\left (c x \right )}}{4} + \frac{b d e x^{6} \operatorname{atan}{\left (c x \right )}}{3} + \frac{b e^{2} x^{8} \operatorname{atan}{\left (c x \right )}}{8} - \frac{b d^{2} x^{3}}{12 c} - \frac{b d e x^{5}}{15 c} - \frac{b e^{2} x^{7}}{56 c} + \frac{b d^{2} x}{4 c^{3}} + \frac{b d e x^{3}}{9 c^{3}} + \frac{b e^{2} x^{5}}{40 c^{3}} - \frac{b d^{2} \operatorname{atan}{\left (c x \right )}}{4 c^{4}} - \frac{b d e x}{3 c^{5}} - \frac{b e^{2} x^{3}}{24 c^{5}} + \frac{b d e \operatorname{atan}{\left (c x \right )}}{3 c^{6}} + \frac{b e^{2} x}{8 c^{7}} - \frac{b e^{2} \operatorname{atan}{\left (c x \right )}}{8 c^{8}} & \text{for}\: c \neq 0 \\a \left (\frac{d^{2} x^{4}}{4} + \frac{d e x^{6}}{3} + \frac{e^{2} x^{8}}{8}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.36381, size = 315, normalized size = 1.7 \begin{align*} \frac{315 \, b c^{8} x^{8} \arctan \left (c x\right ) e^{2} + 315 \, a c^{8} x^{8} e^{2} + 840 \, b c^{8} d x^{6} \arctan \left (c x\right ) e + 840 \, a c^{8} d x^{6} e + 630 \, b c^{8} d^{2} x^{4} \arctan \left (c x\right ) - 45 \, b c^{7} x^{7} e^{2} + 630 \, a c^{8} d^{2} x^{4} - 168 \, b c^{7} d x^{5} e - 210 \, b c^{7} d^{2} x^{3} + 63 \, b c^{5} x^{5} e^{2} + 280 \, b c^{5} d x^{3} e + 630 \, b c^{5} d^{2} x - 630 \, b c^{4} d^{2} \arctan \left (c x\right ) - 105 \, b c^{3} x^{3} e^{2} - 840 \, \pi b c^{2} d e \mathrm{sgn}\left (c\right ) \mathrm{sgn}\left (x\right ) - 840 \, b c^{3} d x e + 840 \, b c^{2} d \arctan \left (c x\right ) e + 315 \, b c x e^{2} - 315 \, b \arctan \left (c x\right ) e^{2}}{2520 \, c^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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