Optimal. Leaf size=278 \[ \frac {1}{4} d^2 x^4 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \text {sech}^{-1}(c x)\right )-\frac {b e \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (1-c^2 x^2\right )^{5/2} \left (8 c^2 d+9 e\right )}{120 c^8}+\frac {b e^2 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (1-c^2 x^2\right )^{7/2}}{56 c^8}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2} \left (6 c^4 d^2+16 c^2 d e+9 e^2\right )}{72 c^8}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (6 c^4 d^2+8 c^2 d e+3 e^2\right )}{24 c^8} \]
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Rubi [A] time = 0.24, antiderivative size = 278, normalized size of antiderivative = 1.00, 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 12
Rule 43
Rule 266
Rule 771
Rule 1251
Rule 6301
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*}
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Mathematica [A] time = 0.31, size = 168, normalized size = 0.60 \[ \frac {1}{24} \left (6 a d^2 x^4+8 a d e x^6+3 a e^2 x^8-\frac {b \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (3 c^6 \left (70 d^2 x^2+56 d e x^4+15 e^2 x^6\right )+c^4 \left (420 d^2+224 d e x^2+54 e^2 x^4\right )+8 c^2 e \left (56 d+9 e x^2\right )+144 e^2\right )}{105 c^8}+b x^4 \text {sech}^{-1}(c x) \left (6 d^2+8 d e x^2+3 e^2 x^4\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.08, size = 227, normalized size = 0.82 \[ \frac {315 \, a c^{7} e^{2} x^{8} + 840 \, a c^{7} d e x^{6} + 630 \, a c^{7} d^{2} x^{4} + 105 \, {\left (3 \, b c^{7} e^{2} x^{8} + 8 \, b c^{7} d e x^{6} + 6 \, b c^{7} d^{2} x^{4}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (45 \, b c^{6} e^{2} x^{7} + 6 \, {\left (28 \, b c^{6} d e + 9 \, b c^{4} e^{2}\right )} x^{5} + 2 \, {\left (105 \, b c^{6} d^{2} + 112 \, b c^{4} d e + 36 \, b c^{2} e^{2}\right )} x^{3} + 4 \, {\left (105 \, b c^{4} d^{2} + 112 \, b c^{2} d e + 36 \, b e^{2}\right )} x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{2520 \, c^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 212, normalized size = 0.76 \[ \frac {\frac {a \left (\frac {1}{8} e^{2} c^{8} x^{8}+\frac {1}{3} c^{8} d e \,x^{6}+\frac {1}{4} c^{8} d^{2} x^{4}\right )}{c^{4}}+\frac {b \left (\frac {\mathrm {arcsech}\left (c x \right ) e^{2} c^{8} x^{8}}{8}+\frac {\mathrm {arcsech}\left (c x \right ) c^{8} d e \,x^{6}}{3}+\frac {\mathrm {arcsech}\left (c x \right ) c^{8} x^{4} d^{2}}{4}-\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (45 c^{6} e^{2} x^{6}+168 c^{6} d e \,x^{4}+210 c^{6} d^{2} x^{2}+54 c^{4} e^{2} x^{4}+224 c^{4} d e \,x^{2}+420 d^{2} c^{4}+72 c^{2} e^{2} x^{2}+448 c^{2} d e +144 e^{2}\right )}{2520}\right )}{c^{4}}}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 245, normalized size = 0.88 \[ \frac {1}{8} \, a e^{2} x^{8} + \frac {1}{3} \, a d e x^{6} + \frac {1}{4} \, a d^{2} x^{4} + \frac {1}{12} \, {\left (3 \, x^{4} \operatorname {arsech}\left (c x\right ) + \frac {c^{2} x^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} - 3 \, x \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{3}}\right )} b d^{2} + \frac {1}{45} \, {\left (15 \, x^{6} \operatorname {arsech}\left (c x\right ) - \frac {3 \, c^{4} x^{5} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {5}{2}} - 10 \, c^{2} x^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 15 \, x \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{5}}\right )} b d e + \frac {1}{280} \, {\left (35 \, x^{8} \operatorname {arsech}\left (c x\right ) + \frac {5 \, c^{6} x^{7} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {7}{2}} - 21 \, c^{4} x^{5} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {5}{2}} + 35 \, c^{2} x^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} - 35 \, x \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{7}}\right )} b e^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^3\,{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 16.46, size = 332, normalized size = 1.19 \[ \begin {cases} \frac {a d^{2} x^{4}}{4} + \frac {a d e x^{6}}{3} + \frac {a e^{2} x^{8}}{8} + \frac {b d^{2} x^{4} \operatorname {asech}{\left (c x \right )}}{4} + \frac {b d e x^{6} \operatorname {asech}{\left (c x \right )}}{3} + \frac {b e^{2} x^{8} \operatorname {asech}{\left (c x \right )}}{8} - \frac {b d^{2} x^{2} \sqrt {- c^{2} x^{2} + 1}}{12 c^{2}} - \frac {b d e x^{4} \sqrt {- c^{2} x^{2} + 1}}{15 c^{2}} - \frac {b e^{2} x^{6} \sqrt {- c^{2} x^{2} + 1}}{56 c^{2}} - \frac {b d^{2} \sqrt {- c^{2} x^{2} + 1}}{6 c^{4}} - \frac {4 b d e x^{2} \sqrt {- c^{2} x^{2} + 1}}{45 c^{4}} - \frac {3 b e^{2} x^{4} \sqrt {- c^{2} x^{2} + 1}}{140 c^{4}} - \frac {8 b d e \sqrt {- c^{2} x^{2} + 1}}{45 c^{6}} - \frac {b e^{2} x^{2} \sqrt {- c^{2} x^{2} + 1}}{35 c^{6}} - \frac {2 b e^{2} \sqrt {- c^{2} x^{2} + 1}}{35 c^{8}} & \text {for}\: c \neq 0 \\\left (a + \infty b\right ) \left (\frac {d^{2} x^{4}}{4} + \frac {d e x^{6}}{3} + \frac {e^{2} x^{8}}{8}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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