Optimal. Leaf size=275 \[ \frac {1}{3} d^2 x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \text {sech}^{-1}(c x)\right )-\frac {b e^2 x^5 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{42 c^2}-\frac {b e x^3 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (84 c^2 d+25 e\right )}{840 c^4}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) \sin ^{-1}(c x)}{1680 c^7}-\frac {b x \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (280 c^4 d^2+252 c^2 d e+75 e^2\right )}{1680 c^6} \]
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Rubi [A] time = 0.23, antiderivative size = 275, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {270, 6301, 12, 1267, 459, 321, 216} \[ \frac {1}{3} d^2 x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \text {sech}^{-1}(c x)\right )-\frac {b x \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (280 c^4 d^2+252 c^2 d e+75 e^2\right )}{1680 c^6}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) \sin ^{-1}(c x)}{1680 c^7}-\frac {b e x^3 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (84 c^2 d+25 e\right )}{840 c^4}-\frac {b e^2 x^5 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{42 c^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 216
Rule 270
Rule 321
Rule 459
Rule 1267
Rule 6301
Rubi steps
\begin {align*} \int x^2 \left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx &=\frac {1}{3} d^2 x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \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 (35 d^2+42 d e x^2+15 e^2 x^4\right )}{105 \sqrt {1-c^2 x^2}} \, dx\\ &=\frac {1}{3} d^2 x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{105} \left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {x^2 \left (35 d^2+42 d e x^2+15 e^2 x^4\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {b e^2 x^5 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{42 c^2}+\frac {1}{3} d^2 x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \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 {x^2 \left (-210 c^2 d^2-3 e \left (84 c^2 d+25 e\right ) x^2\right )}{\sqrt {1-c^2 x^2}} \, dx}{630 c^2}\\ &=-\frac {b e \left (84 c^2 d+25 e\right ) x^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{840 c^4}-\frac {b e^2 x^5 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{42 c^2}+\frac {1}{3} d^2 x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \text {sech}^{-1}(c x)\right )+-\frac {\left (b \left (-840 c^4 d^2-9 e \left (84 c^2 d+25 e\right )\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{2520 c^4}\\ &=-\frac {b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) x \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{1680 c^6}-\frac {b e \left (84 c^2 d+25 e\right ) x^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{840 c^4}-\frac {b e^2 x^5 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{42 c^2}+\frac {1}{3} d^2 x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \text {sech}^{-1}(c x)\right )+-\frac {\left (b \left (-840 c^4 d^2-9 e \left (84 c^2 d+25 e\right )\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{5040 c^6}\\ &=-\frac {b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) x \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{1680 c^6}-\frac {b e \left (84 c^2 d+25 e\right ) x^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{840 c^4}-\frac {b e^2 x^5 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{42 c^2}+\frac {1}{3} d^2 x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sin ^{-1}(c x)}{1680 c^7}\\ \end {align*}
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Mathematica [C] time = 0.50, size = 207, normalized size = 0.75 \[ \frac {16 a c^7 x^3 \left (35 d^2+42 d e x^2+15 e^2 x^4\right )+16 b c^7 x^3 \text {sech}^{-1}(c x) \left (35 d^2+42 d e x^2+15 e^2 x^4\right )-b c x \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (8 c^4 \left (35 d^2+21 d e x^2+5 e^2 x^4\right )+2 c^2 e \left (126 d+25 e x^2\right )+75 e^2\right )+i b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) \log \left (2 \sqrt {\frac {1-c x}{c x+1}} (c x+1)-2 i c x\right )}{1680 c^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.82, size = 341, normalized size = 1.24 \[ \frac {240 \, a c^{7} e^{2} x^{7} + 672 \, a c^{7} d e x^{5} + 560 \, a c^{7} d^{2} x^{3} - 2 \, {\left (280 \, b c^{4} d^{2} + 252 \, b c^{2} d e + 75 \, b e^{2}\right )} \arctan \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{c x}\right ) - 16 \, {\left (35 \, b c^{7} d^{2} + 42 \, b c^{7} d e + 15 \, b c^{7} e^{2}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) + 16 \, {\left (15 \, b c^{7} e^{2} x^{7} + 42 \, b c^{7} d e x^{5} + 35 \, b c^{7} d^{2} x^{3} - 35 \, b c^{7} d^{2} - 42 \, b c^{7} d e - 15 \, b c^{7} e^{2}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (40 \, b c^{6} e^{2} x^{6} + 2 \, {\left (84 \, b c^{6} d e + 25 \, b c^{4} e^{2}\right )} x^{4} + {\left (280 \, b c^{6} d^{2} + 252 \, b c^{4} d e + 75 \, b c^{2} e^{2}\right )} x^{2}\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{1680 \, c^{7}} \]
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 \[ \int {\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 300, normalized size = 1.09 \[ \frac {\frac {a \left (\frac {1}{7} e^{2} c^{7} x^{7}+\frac {2}{5} c^{7} d e \,x^{5}+\frac {1}{3} x^{3} c^{7} d^{2}\right )}{c^{4}}+\frac {b \left (\frac {\mathrm {arcsech}\left (c x \right ) e^{2} c^{7} x^{7}}{7}+\frac {2 \,\mathrm {arcsech}\left (c x \right ) c^{7} d e \,x^{5}}{5}+\frac {\mathrm {arcsech}\left (c x \right ) c^{7} x^{3} d^{2}}{3}+\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (-40 c^{5} x^{5} e^{2} \sqrt {-c^{2} x^{2}+1}-168 c^{5} x^{3} d e \sqrt {-c^{2} x^{2}+1}-280 d^{2} c^{5} x \sqrt {-c^{2} x^{2}+1}+280 d^{2} c^{4} \arcsin \left (c x \right )-50 e^{2} c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-252 c^{3} d e x \sqrt {-c^{2} x^{2}+1}+252 c^{2} d e \arcsin \left (c x \right )-75 e^{2} c x \sqrt {-c^{2} x^{2}+1}+75 e^{2} \arcsin \left (c x \right )\right )}{1680 \sqrt {-c^{2} x^{2}+1}}\right )}{c^{4}}}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 328, normalized size = 1.19 \[ \frac {1}{7} \, a e^{2} x^{7} + \frac {2}{5} \, a d e x^{5} + \frac {1}{3} \, a d^{2} x^{3} + \frac {1}{6} \, {\left (2 \, x^{3} \operatorname {arsech}\left (c x\right ) - \frac {\frac {\sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{2}} + \frac {\arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )}{c^{2}}}{c}\right )} b d^{2} + \frac {1}{20} \, {\left (8 \, x^{5} \operatorname {arsech}\left (c x\right ) - \frac {\frac {3 \, {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 5 \, \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 2 \, c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{4}} + \frac {3 \, \arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )}{c^{4}}}{c}\right )} b d e + \frac {1}{336} \, {\left (48 \, x^{7} \operatorname {arsech}\left (c x\right ) - \frac {\frac {15 \, {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {5}{2}} + 40 \, {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 33 \, \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{3} + 3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{6}} + \frac {15 \, \arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )}{c^{6}}}{c}\right )} b e^{2} \]
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 \[ \int x^2\,{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
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 \[ \int x^{2} \left (a + b \operatorname {asech}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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