Optimal. Leaf size=204 \[ d^2 x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \text {sech}^{-1}(c x)\right )-\frac {b e^2 x^3 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{20 c^2}-\frac {b e x \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (40 c^2 d+9 e\right )}{120 c^4}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (120 c^4 d^2+40 c^2 d e+9 e^2\right ) \sin ^{-1}(c x)}{120 c^5} \]
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Rubi [A] time = 0.13, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {194, 6291, 12, 1159, 388, 216} \[ d^2 x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (120 c^4 d^2+40 c^2 d e+9 e^2\right ) \sin ^{-1}(c x)}{120 c^5}-\frac {b e x \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (40 c^2 d+9 e\right )}{120 c^4}-\frac {b e^2 x^3 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{20 c^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 194
Rule 216
Rule 388
Rule 1159
Rule 6291
Rubi steps
\begin {align*} \int \left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx &=d^2 x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e^2 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 {15 d^2+10 d e x^2+3 e^2 x^4}{15 \sqrt {1-c^2 x^2}} \, dx\\ &=d^2 x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e^2 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 {15 d^2+10 d e x^2+3 e^2 x^4}{\sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {b e^2 x^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{20 c^2}+d^2 x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \text {sech}^{-1}(c x)\right )-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {-60 c^2 d^2-e \left (40 c^2 d+9 e\right ) x^2}{\sqrt {1-c^2 x^2}} \, dx}{60 c^2}\\ &=-\frac {b e \left (40 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^2 x^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{20 c^2}+d^2 x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \text {sech}^{-1}(c x)\right )-\frac {\left (b \left (-120 c^4 d^2-e \left (40 c^2 d+9 e\right )\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{120 c^4}\\ &=-\frac {b e \left (40 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^2 x^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{20 c^2}+d^2 x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {b \left (120 c^4 d^2+40 c^2 d e+9 e^2\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 = 0.35, size = 174, normalized size = 0.85 \[ \frac {8 a c^5 x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )+8 b c^5 x \text {sech}^{-1}(c x) \left (15 d^2+10 d e x^2+3 e^2 x^4\right )-b c e x \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (c^2 \left (40 d+6 e x^2\right )+9 e\right )+i b \left (120 c^4 d^2+40 c^2 d e+9 e^2\right ) \log \left (2 \sqrt {\frac {1-c x}{c x+1}} (c x+1)-2 i c x\right )}{120 c^5} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.81, size = 305, normalized size = 1.50 \[ \frac {24 \, a c^{5} e^{2} x^{5} + 80 \, a c^{5} d e x^{3} + 120 \, a c^{5} d^{2} x - 2 \, {\left (120 \, b c^{4} d^{2} + 40 \, b c^{2} d e + 9 \, b e^{2}\right )} \arctan \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{c x}\right ) - 8 \, {\left (15 \, b c^{5} d^{2} + 10 \, b c^{5} d e + 3 \, b c^{5} e^{2}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) + 8 \, {\left (3 \, b c^{5} e^{2} x^{5} + 10 \, b c^{5} d e x^{3} + 15 \, b c^{5} d^{2} x - 15 \, b c^{5} d^{2} - 10 \, b c^{5} d e - 3 \, b c^{5} e^{2}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (6 \, b c^{4} e^{2} x^{4} + {\left (40 \, b c^{4} d e + 9 \, b c^{2} e^{2}\right )} x^{2}\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{120 \, c^{5}} \]
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 \[ \int {\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 228, normalized size = 1.12 \[ \frac {\frac {a \left (\frac {1}{5} e^{2} c^{5} x^{5}+\frac {2}{3} c^{5} d e \,x^{3}+x \,c^{5} d^{2}\right )}{c^{4}}+\frac {b \left (\frac {\mathrm {arcsech}\left (c x \right ) e^{2} c^{5} x^{5}}{5}+\frac {2 \,\mathrm {arcsech}\left (c x \right ) c^{5} d e \,x^{3}}{3}+\mathrm {arcsech}\left (c x \right ) c^{5} x \,d^{2}+\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (120 d^{2} c^{4} \arcsin \left (c x \right )-6 e^{2} c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-40 c^{3} d e x \sqrt {-c^{2} x^{2}+1}+40 c^{2} d e \arcsin \left (c x \right )-9 e^{2} c x \sqrt {-c^{2} x^{2}+1}+9 e^{2} \arcsin \left (c x \right )\right )}{120 \sqrt {-c^{2} x^{2}+1}}\right )}{c^{4}}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 224, normalized size = 1.10 \[ \frac {1}{5} \, a e^{2} x^{5} + \frac {2}{3} \, a d e x^{3} + \frac {1}{3} \, {\left (2 \, x^{3} \operatorname {arsech}\left (c x\right ) - \frac {\frac {\sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{2}} + \frac {\arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )}{c^{2}}}{c}\right )} b d e + \frac {1}{40} \, {\left (8 \, x^{5} \operatorname {arsech}\left (c x\right ) - \frac {\frac {3 \, {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 5 \, \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 2 \, c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{4}} + \frac {3 \, \arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )}{c^{4}}}{c}\right )} b e^{2} + a d^{2} x + \frac {{\left (c x \operatorname {arsech}\left (c x\right ) - \arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )\right )} b d^{2}}{c} \]
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 \[ \int {\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \operatorname {asech}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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