Optimal. Leaf size=150 \[ d^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {2 b e \left (1-c^2 x^2\right )^{3/2} \left (5 c^2 d+3 e\right )}{45 c^5}+\frac {b e^2 \left (1-c^2 x^2\right )^{5/2}}{25 c^5}+\frac {b \sqrt {1-c^2 x^2} \left (15 c^4 d^2+10 c^2 d e+3 e^2\right )}{15 c^5} \]
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Rubi [A] time = 0.14, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {194, 4665, 12, 1247, 698} \[ d^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {b \sqrt {1-c^2 x^2} \left (15 c^4 d^2+10 c^2 d e+3 e^2\right )}{15 c^5}-\frac {2 b e \left (1-c^2 x^2\right )^{3/2} \left (5 c^2 d+3 e\right )}{45 c^5}+\frac {b e^2 \left (1-c^2 x^2\right )^{5/2}}{25 c^5} \]
Antiderivative was successfully verified.
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Rule 12
Rule 194
Rule 698
Rule 1247
Rule 4665
Rubi steps
\begin {align*} \int \left (d+e x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx &=d^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac {x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )}{15 \sqrt {1-c^2 x^2}} \, dx\\ &=d^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{15} (b c) \int \frac {x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=d^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{30} (b c) \operatorname {Subst}\left (\int \frac {15 d^2+10 d e x+3 e^2 x^2}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )\\ &=d^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{30} (b c) \operatorname {Subst}\left (\int \left (\frac {15 c^4 d^2+10 c^2 d e+3 e^2}{c^4 \sqrt {1-c^2 x}}-\frac {2 e \left (5 c^2 d+3 e\right ) \sqrt {1-c^2 x}}{c^4}+\frac {3 e^2 \left (1-c^2 x\right )^{3/2}}{c^4}\right ) \, dx,x,x^2\right )\\ &=\frac {b \left (15 c^4 d^2+10 c^2 d e+3 e^2\right ) \sqrt {1-c^2 x^2}}{15 c^5}-\frac {2 b e \left (5 c^2 d+3 e\right ) \left (1-c^2 x^2\right )^{3/2}}{45 c^5}+\frac {b e^2 \left (1-c^2 x^2\right )^{5/2}}{25 c^5}+d^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A] time = 0.17, size = 125, normalized size = 0.83 \[ \frac {1}{225} \left (15 a x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )+\frac {b \sqrt {1-c^2 x^2} \left (c^4 \left (225 d^2+50 d e x^2+9 e^2 x^4\right )+4 c^2 e \left (25 d+3 e x^2\right )+24 e^2\right )}{c^5}+15 b x \sin ^{-1}(c x) \left (15 d^2+10 d e x^2+3 e^2 x^4\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 151, normalized size = 1.01 \[ \frac {45 \, a c^{5} e^{2} x^{5} + 150 \, a c^{5} d e x^{3} + 225 \, a c^{5} d^{2} x + 15 \, {\left (3 \, b c^{5} e^{2} x^{5} + 10 \, b c^{5} d e x^{3} + 15 \, b c^{5} d^{2} x\right )} \arcsin \left (c x\right ) + {\left (9 \, b c^{4} e^{2} x^{4} + 225 \, b c^{4} d^{2} + 100 \, b c^{2} d e + 24 \, b e^{2} + 2 \, {\left (25 \, b c^{4} d e + 6 \, b c^{2} e^{2}\right )} x^{2}\right )} \sqrt {-c^{2} x^{2} + 1}}{225 \, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 263, normalized size = 1.75 \[ \frac {1}{5} \, a x^{5} e^{2} + \frac {2}{3} \, a d x^{3} e + b d^{2} x \arcsin \left (c x\right ) + a d^{2} x + \frac {2 \, {\left (c^{2} x^{2} - 1\right )} b d x \arcsin \left (c x\right ) e}{3 \, c^{2}} + \frac {2 \, b d x \arcsin \left (c x\right ) e}{3 \, c^{2}} + \frac {\sqrt {-c^{2} x^{2} + 1} b d^{2}}{c} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b x \arcsin \left (c x\right ) e^{2}}{5 \, c^{4}} - \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d e}{9 \, c^{3}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )} b x \arcsin \left (c x\right ) e^{2}}{5 \, c^{4}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b d e}{3 \, c^{3}} + \frac {b x \arcsin \left (c x\right ) e^{2}}{5 \, c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b e^{2}}{25 \, c^{5}} - \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b e^{2}}{15 \, c^{5}} + \frac {\sqrt {-c^{2} x^{2} + 1} b e^{2}}{5 \, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 209, normalized size = 1.39 \[ \frac {\frac {a \left (\frac {1}{5} e^{2} c^{5} x^{5}+\frac {2}{3} c^{5} e d \,x^{3}+d^{2} c^{5} x \right )}{c^{4}}+\frac {b \left (\frac {\arcsin \left (c x \right ) e^{2} c^{5} x^{5}}{5}+\frac {2 \arcsin \left (c x \right ) c^{5} e d \,x^{3}}{3}+\arcsin \left (c x \right ) d^{2} c^{5} x -\frac {e^{2} \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}-\frac {2 c^{2} e d \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{3}+d^{2} c^{4} \sqrt {-c^{2} x^{2}+1}\right )}{c^{4}}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 182, normalized size = 1.21 \[ \frac {1}{5} \, a e^{2} x^{5} + \frac {2}{3} \, a d e x^{3} + \frac {2}{9} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d e + \frac {1}{75} \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b e^{2} + a d^{2} x + \frac {{\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} b d^{2}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.27, size = 240, normalized size = 1.60 \[ \begin {cases} a d^{2} x + \frac {2 a d e x^{3}}{3} + \frac {a e^{2} x^{5}}{5} + b d^{2} x \operatorname {asin}{\left (c x \right )} + \frac {2 b d e x^{3} \operatorname {asin}{\left (c x \right )}}{3} + \frac {b e^{2} x^{5} \operatorname {asin}{\left (c x \right )}}{5} + \frac {b d^{2} \sqrt {- c^{2} x^{2} + 1}}{c} + \frac {2 b d e x^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c} + \frac {b e^{2} x^{4} \sqrt {- c^{2} x^{2} + 1}}{25 c} + \frac {4 b d e \sqrt {- c^{2} x^{2} + 1}}{9 c^{3}} + \frac {4 b e^{2} x^{2} \sqrt {- c^{2} x^{2} + 1}}{75 c^{3}} + \frac {8 b e^{2} \sqrt {- c^{2} x^{2} + 1}}{75 c^{5}} & \text {for}\: c \neq 0 \\a \left (d^{2} x + \frac {2 d e x^{3}}{3} + \frac {e^{2} x^{5}}{5}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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