Optimal. Leaf size=278 \[ \frac{1}{4} d^2 x^4 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (1-c^2 x^2\right )^{3/2} \left (6 c^4 d^2+16 c^2 d e+9 e^2\right )}{72 c^8}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (6 c^4 d^2+8 c^2 d e+3 e^2\right )}{24 c^8}-\frac{b e \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (1-c^2 x^2\right )^{5/2} \left (8 c^2 d+9 e\right )}{120 c^8}+\frac{b e^2 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (1-c^2 x^2\right )^{7/2}}{56 c^8} \]
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Rubi [A] time = 0.237377, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {266, 43, 6301, 12, 1251, 771} \[ \frac{1}{4} d^2 x^4 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (1-c^2 x^2\right )^{3/2} \left (6 c^4 d^2+16 c^2 d e+9 e^2\right )}{72 c^8}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (6 c^4 d^2+8 c^2 d e+3 e^2\right )}{24 c^8}-\frac{b e \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (1-c^2 x^2\right )^{5/2} \left (8 c^2 d+9 e\right )}{120 c^8}+\frac{b e^2 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (1-c^2 x^2\right )^{7/2}}{56 c^8} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rule 6301
Rule 12
Rule 1251
Rule 771
Rubi steps
\begin{align*} \int x^3 \left (d+e x^2\right )^2 \left (a+b \text{sech}^{-1}(c x)\right ) \, dx &=\frac{1}{4} d^2 x^4 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \text{sech}^{-1}(c x)\right )+\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{x^3 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )}{24 \sqrt{1-c^2 x^2}} \, dx\\ &=\frac{1}{4} d^2 x^4 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{24} \left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{x^3 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )}{\sqrt{1-c^2 x^2}} \, dx\\ &=\frac{1}{4} d^2 x^4 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{48} \left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{x \left (6 d^2+8 d e x+3 e^2 x^2\right )}{\sqrt{1-c^2 x}} \, dx,x,x^2\right )\\ &=\frac{1}{4} d^2 x^4 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{48} \left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \left (\frac{6 c^4 d^2+8 c^2 d e+3 e^2}{c^6 \sqrt{1-c^2 x}}+\frac{\left (-6 c^4 d^2-16 c^2 d e-9 e^2\right ) \sqrt{1-c^2 x}}{c^6}+\frac{e \left (8 c^2 d+9 e\right ) \left (1-c^2 x\right )^{3/2}}{c^6}-\frac{3 e^2 \left (1-c^2 x\right )^{5/2}}{c^6}\right ) \, dx,x,x^2\right )\\ &=-\frac{b \left (6 c^4 d^2+8 c^2 d e+3 e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{24 c^8}+\frac{b \left (6 c^4 d^2+16 c^2 d e+9 e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \left (1-c^2 x^2\right )^{3/2}}{72 c^8}-\frac{b e \left (8 c^2 d+9 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \left (1-c^2 x^2\right )^{5/2}}{120 c^8}+\frac{b e^2 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \left (1-c^2 x^2\right )^{7/2}}{56 c^8}+\frac{1}{4} d^2 x^4 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \text{sech}^{-1}(c x)\right )\\ \end{align*}
Mathematica [A] time = 0.286706, size = 168, normalized size = 0.6 \[ \frac{1}{24} \left (6 a d^2 x^4+8 a d e x^6+3 a e^2 x^8-\frac{b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (3 c^6 \left (70 d^2 x^2+56 d e x^4+15 e^2 x^6\right )+c^4 \left (420 d^2+224 d e x^2+54 e^2 x^4\right )+8 c^2 e \left (56 d+9 e x^2\right )+144 e^2\right )}{105 c^8}+b x^4 \text{sech}^{-1}(c x) \left (6 d^2+8 d e x^2+3 e^2 x^4\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.181, size = 212, normalized size = 0.8 \begin{align*}{\frac{1}{{c}^{4}} \left ({\frac{a}{{c}^{4}} \left ({\frac{{e}^{2}{c}^{8}{x}^{8}}{8}}+{\frac{{c}^{8}de{x}^{6}}{3}}+{\frac{{x}^{4}{c}^{8}{d}^{2}}{4}} \right ) }+{\frac{b}{{c}^{4}} \left ({\frac{{\rm arcsech} \left (cx\right ){e}^{2}{c}^{8}{x}^{8}}{8}}+{\frac{{\rm arcsech} \left (cx\right ){c}^{8}de{x}^{6}}{3}}+{\frac{{\rm arcsech} \left (cx\right ){c}^{8}{x}^{4}{d}^{2}}{4}}-{\frac{cx \left ( 45\,{c}^{6}{e}^{2}{x}^{6}+168\,{c}^{6}de{x}^{4}+210\,{c}^{6}{d}^{2}{x}^{2}+54\,{c}^{4}{e}^{2}{x}^{4}+224\,{c}^{4}de{x}^{2}+420\,{d}^{2}{c}^{4}+72\,{c}^{2}{e}^{2}{x}^{2}+448\,{c}^{2}de+144\,{e}^{2} \right ) }{2520}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.994232, size = 331, normalized size = 1.19 \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} \operatorname{arsech}\left (c x\right ) + \frac{c^{2} x^{3}{\left (\frac{1}{c^{2} x^{2}} - 1\right )}^{\frac{3}{2}} - 3 \, x \sqrt{\frac{1}{c^{2} x^{2}} - 1}}{c^{3}}\right )} b d^{2} + \frac{1}{45} \,{\left (15 \, x^{6} \operatorname{arsech}\left (c x\right ) - \frac{3 \, c^{4} x^{5}{\left (\frac{1}{c^{2} x^{2}} - 1\right )}^{\frac{5}{2}} - 10 \, c^{2} x^{3}{\left (\frac{1}{c^{2} x^{2}} - 1\right )}^{\frac{3}{2}} + 15 \, x \sqrt{\frac{1}{c^{2} x^{2}} - 1}}{c^{5}}\right )} b d e + \frac{1}{280} \,{\left (35 \, x^{8} \operatorname{arsech}\left (c x\right ) + \frac{5 \, c^{6} x^{7}{\left (\frac{1}{c^{2} x^{2}} - 1\right )}^{\frac{7}{2}} - 21 \, c^{4} x^{5}{\left (\frac{1}{c^{2} x^{2}} - 1\right )}^{\frac{5}{2}} + 35 \, c^{2} x^{3}{\left (\frac{1}{c^{2} x^{2}} - 1\right )}^{\frac{3}{2}} - 35 \, x \sqrt{\frac{1}{c^{2} x^{2}} - 1}}{c^{7}}\right )} b e^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.08779, size = 509, normalized size = 1.83 \begin{align*} \frac{315 \, a c^{7} e^{2} x^{8} + 840 \, a c^{7} d e x^{6} + 630 \, a c^{7} d^{2} x^{4} + 105 \,{\left (3 \, b c^{7} e^{2} x^{8} + 8 \, b c^{7} d e x^{6} + 6 \, b c^{7} d^{2} x^{4}\right )} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) -{\left (45 \, b c^{6} e^{2} x^{7} + 6 \,{\left (28 \, b c^{6} d e + 9 \, b c^{4} e^{2}\right )} x^{5} + 2 \,{\left (105 \, b c^{6} d^{2} + 112 \, b c^{4} d e + 36 \, b c^{2} e^{2}\right )} x^{3} + 4 \,{\left (105 \, b c^{4} d^{2} + 112 \, b c^{2} d e + 36 \, b e^{2}\right )} x\right )} \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}}}{2520 \, c^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 51.1061, size = 332, normalized size = 1.19 \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{asech}{\left (c x \right )}}{4} + \frac{b d e x^{6} \operatorname{asech}{\left (c x \right )}}{3} + \frac{b e^{2} x^{8} \operatorname{asech}{\left (c x \right )}}{8} - \frac{b d^{2} x^{2} \sqrt{- c^{2} x^{2} + 1}}{12 c^{2}} - \frac{b d e x^{4} \sqrt{- c^{2} x^{2} + 1}}{15 c^{2}} - \frac{b e^{2} x^{6} \sqrt{- c^{2} x^{2} + 1}}{56 c^{2}} - \frac{b d^{2} \sqrt{- c^{2} x^{2} + 1}}{6 c^{4}} - \frac{4 b d e x^{2} \sqrt{- c^{2} x^{2} + 1}}{45 c^{4}} - \frac{3 b e^{2} x^{4} \sqrt{- c^{2} x^{2} + 1}}{140 c^{4}} - \frac{8 b d e \sqrt{- c^{2} x^{2} + 1}}{45 c^{6}} - \frac{b e^{2} x^{2} \sqrt{- c^{2} x^{2} + 1}}{35 c^{6}} - \frac{2 b e^{2} \sqrt{- c^{2} x^{2} + 1}}{35 c^{8}} & \text{for}\: c \neq 0 \\\left (a + \infty b\right ) \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x^{2} + d\right )}^{2}{\left (b \operatorname{arsech}\left (c x\right ) + a\right )} x^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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