Optimal. Leaf size=174 \[ \frac {1}{3} d x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \text {sech}^{-1}(c x)\right )-\frac {b e x^3 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{20 c^2}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (20 c^2 d+9 e\right ) \sin ^{-1}(c x)}{120 c^5}-\frac {b x \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (20 c^2 d+9 e\right )}{120 c^4} \]
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Rubi [A] time = 0.10, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {14, 6301, 12, 459, 321, 216} \[ \frac {1}{3} d x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \text {sech}^{-1}(c x)\right )-\frac {b x \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (20 c^2 d+9 e\right )}{120 c^4}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (20 c^2 d+9 e\right ) \sin ^{-1}(c x)}{120 c^5}-\frac {b e x^3 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{20 c^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 216
Rule 321
Rule 459
Rule 6301
Rubi steps
\begin {align*} \int x^2 \left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right ) \, dx &=\frac {1}{3} d x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e x^5 \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^2 \left (5 d+3 e x^2\right )}{15 \sqrt {1-c^2 x^2}} \, dx\\ &=\frac {1}{3} d x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{15} \left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {x^2 \left (5 d+3 e x^2\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {b e x^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{20 c^2}+\frac {1}{3} d x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{60} \left (b \left (20 d+\frac {9 e}{c^2}\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {b \left (20 c^2 d+9 e\right ) x \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{120 c^4}-\frac {b e x^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{20 c^2}+\frac {1}{3} d x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {\left (b \left (20 d+\frac {9 e}{c^2}\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{120 c^2}\\ &=-\frac {b \left (20 c^2 d+9 e\right ) x \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{120 c^4}-\frac {b e x^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{20 c^2}+\frac {1}{3} d x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {b \left (20 c^2 d+9 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sin ^{-1}(c x)}{120 c^5}\\ \end {align*}
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Mathematica [C] time = 0.22, size = 144, normalized size = 0.83 \[ \frac {8 a c^5 x^3 \left (5 d+3 e x^2\right )+8 b c^5 x^3 \text {sech}^{-1}(c x) \left (5 d+3 e x^2\right )-b c x \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (c^2 \left (20 d+6 e x^2\right )+9 e\right )+i b \left (20 c^2 d+9 e\right ) \log \left (2 \sqrt {\frac {1-c x}{c x+1}} (c x+1)-2 i c x\right )}{120 c^5} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.02, size = 238, normalized size = 1.37 \[ \frac {24 \, a c^{5} e x^{5} + 40 \, a c^{5} d x^{3} - 2 \, {\left (20 \, b c^{2} d + 9 \, b e\right )} \arctan \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{c x}\right ) - 8 \, {\left (5 \, b c^{5} d + 3 \, b c^{5} e\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) + 8 \, {\left (3 \, b c^{5} e x^{5} + 5 \, b c^{5} d x^{3} - 5 \, b c^{5} d - 3 \, b c^{5} e\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (6 \, b c^{4} e x^{4} + {\left (20 \, b c^{4} d + 9 \, b c^{2} e\right )} x^{2}\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{120 \, c^{5}} \]
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 \[ \int {\left (e x^{2} + d\right )} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 182, normalized size = 1.05 \[ \frac {\frac {a \left (\frac {1}{5} c^{5} x^{5} e +\frac {1}{3} c^{5} x^{3} d \right )}{c^{2}}+\frac {b \left (\frac {\mathrm {arcsech}\left (c x \right ) c^{5} x^{5} e}{5}+\frac {\mathrm {arcsech}\left (c x \right ) c^{5} x^{3} d}{3}+\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (-6 e \,c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-20 \sqrt {-c^{2} x^{2}+1}\, c^{3} x d +20 \arcsin \left (c x \right ) c^{2} d -9 e c x \sqrt {-c^{2} x^{2}+1}+9 e \arcsin \left (c x \right )\right )}{120 \sqrt {-c^{2} x^{2}+1}}\right )}{c^{2}}}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 182, normalized size = 1.05 \[ \frac {1}{5} \, a e x^{5} + \frac {1}{3} \, a d x^{3} + \frac {1}{6} \, {\left (2 \, x^{3} \operatorname {arsech}\left (c x\right ) - \frac {\frac {\sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{2}} + \frac {\arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )}{c^{2}}}{c}\right )} b d + \frac {1}{40} \, {\left (8 \, x^{5} \operatorname {arsech}\left (c x\right ) - \frac {\frac {3 \, {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 5 \, \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 2 \, c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{4}} + \frac {3 \, \arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )}{c^{4}}}{c}\right )} b e \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \left (a + b \operatorname {asech}{\left (c x \right )}\right ) \left (d + e x^{2}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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