Optimal. Leaf size=124 \[ \frac {(d+e x)^3 \left (a+b \sin ^{-1}(c x)\right )}{3 e}-\frac {b d \left (\frac {3 e^2}{c^2}+2 d^2\right ) \sin ^{-1}(c x)}{6 e}+\frac {b \sqrt {1-c^2 x^2} (d+e x)^2}{9 c}+\frac {b \sqrt {1-c^2 x^2} \left (4 \left (4 c^2 d^2+e^2\right )+5 c^2 d e x\right )}{18 c^3} \]
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Rubi [A] time = 0.10, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4743, 743, 780, 216} \[ \frac {(d+e x)^3 \left (a+b \sin ^{-1}(c x)\right )}{3 e}+\frac {b \sqrt {1-c^2 x^2} \left (4 \left (4 c^2 d^2+e^2\right )+5 c^2 d e x\right )}{18 c^3}-\frac {b d \left (\frac {3 e^2}{c^2}+2 d^2\right ) \sin ^{-1}(c x)}{6 e}+\frac {b \sqrt {1-c^2 x^2} (d+e x)^2}{9 c} \]
Antiderivative was successfully verified.
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Rule 216
Rule 743
Rule 780
Rule 4743
Rubi steps
\begin {align*} \int (d+e x)^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac {(d+e x)^3 \left (a+b \sin ^{-1}(c x)\right )}{3 e}-\frac {(b c) \int \frac {(d+e x)^3}{\sqrt {1-c^2 x^2}} \, dx}{3 e}\\ &=\frac {b (d+e x)^2 \sqrt {1-c^2 x^2}}{9 c}+\frac {(d+e x)^3 \left (a+b \sin ^{-1}(c x)\right )}{3 e}+\frac {b \int \frac {(d+e x) \left (-3 c^2 d^2-2 e^2-5 c^2 d e x\right )}{\sqrt {1-c^2 x^2}} \, dx}{9 c e}\\ &=\frac {b (d+e x)^2 \sqrt {1-c^2 x^2}}{9 c}+\frac {b \left (4 \left (4 c^2 d^2+e^2\right )+5 c^2 d e x\right ) \sqrt {1-c^2 x^2}}{18 c^3}+\frac {(d+e x)^3 \left (a+b \sin ^{-1}(c x)\right )}{3 e}-\frac {1}{6} \left (b d \left (\frac {2 c d^2}{e}+\frac {3 e}{c}\right )\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {b (d+e x)^2 \sqrt {1-c^2 x^2}}{9 c}+\frac {b \left (4 \left (4 c^2 d^2+e^2\right )+5 c^2 d e x\right ) \sqrt {1-c^2 x^2}}{18 c^3}-\frac {b d \left (2 d^2+\frac {3 e^2}{c^2}\right ) \sin ^{-1}(c x)}{6 e}+\frac {(d+e x)^3 \left (a+b \sin ^{-1}(c x)\right )}{3 e}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 121, normalized size = 0.98 \[ \frac {6 a c^3 x \left (3 d^2+3 d e x+e^2 x^2\right )+b \sqrt {1-c^2 x^2} \left (c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )+4 e^2\right )+3 b c \sin ^{-1}(c x) \left (6 c^2 d^2 x+3 d e \left (2 c^2 x^2-1\right )+2 c^2 e^2 x^3\right )}{18 c^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 135, normalized size = 1.09 \[ \frac {6 \, a c^{3} e^{2} x^{3} + 18 \, a c^{3} d e x^{2} + 18 \, a c^{3} d^{2} x + 3 \, {\left (2 \, b c^{3} e^{2} x^{3} + 6 \, b c^{3} d e x^{2} + 6 \, b c^{3} d^{2} x - 3 \, b c d e\right )} \arcsin \left (c x\right ) + {\left (2 \, b c^{2} e^{2} x^{2} + 9 \, b c^{2} d e x + 18 \, b c^{2} d^{2} + 4 \, b e^{2}\right )} \sqrt {-c^{2} x^{2} + 1}}{18 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 193, normalized size = 1.56 \[ b d^{2} x \arcsin \left (c x\right ) + \frac {1}{3} \, a x^{3} e^{2} + a d^{2} x + \frac {\sqrt {-c^{2} x^{2} + 1} b d x e}{2 \, c} + \frac {{\left (c^{2} x^{2} - 1\right )} b x \arcsin \left (c x\right ) e^{2}}{3 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} b d \arcsin \left (c x\right ) e}{c^{2}} + \frac {\sqrt {-c^{2} x^{2} + 1} b d^{2}}{c} + \frac {b x \arcsin \left (c x\right ) e^{2}}{3 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} a d e}{c^{2}} + \frac {b d \arcsin \left (c x\right ) e}{2 \, c^{2}} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b e^{2}}{9 \, c^{3}} + \frac {\sqrt {-c^{2} x^{2} + 1} b e^{2}}{3 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 193, normalized size = 1.56 \[ \frac {\frac {\left (c e x +d c \right )^{3} a}{3 c^{2} e}+\frac {b \left (\frac {\arcsin \left (c x \right ) e^{2} c^{3} x^{3}}{3}+e \arcsin \left (c x \right ) c^{3} x^{2} d +\arcsin \left (c x \right ) c^{3} x \,d^{2}+\frac {\arcsin \left (c x \right ) c^{3} d^{3}}{3 e}-\frac {e^{3} \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )+3 d c \,e^{2} \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )-3 d^{2} c^{2} e \sqrt {-c^{2} x^{2}+1}+c^{3} d^{3} \arcsin \left (c x \right )}{3 e}\right )}{c^{2}}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 150, normalized size = 1.21 \[ \frac {1}{3} \, a e^{2} x^{3} + a d e x^{2} + \frac {1}{2} \, {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} b d e + \frac {1}{9} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b e^{2} + a d^{2} x + \frac {{\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} b d^{2}}{c} \]
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 \[ \left \{\begin {array}{cl} b\,e^2\,\left (\frac {\sqrt {\frac {1}{c^2}-x^2}\,\left (\frac {2}{c^2}+x^2\right )}{9}+\frac {x^3\,\mathrm {asin}\left (c\,x\right )}{3}\right )+\frac {a\,x\,\left (3\,d^2+3\,d\,e\,x+e^2\,x^2\right )}{3}+\frac {b\,d^2\,\left (\sqrt {1-c^2\,x^2}+c\,x\,\mathrm {asin}\left (c\,x\right )\right )}{c}+\frac {2\,b\,d\,e\,\left (\frac {\mathrm {asin}\left (c\,x\right )\,\left (2\,c^2\,x^2-1\right )}{4}+\frac {c\,x\,\sqrt {1-c^2\,x^2}}{4}\right )}{c^2} & \text {\ if\ \ }0<c\\ \int \left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d+e\,x\right )}^2 \,d x & \text {\ if\ \ }\neg 0<c \end {array}\right . \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.63, size = 190, normalized size = 1.53 \[ \begin {cases} a d^{2} x + a d e x^{2} + \frac {a e^{2} x^{3}}{3} + b d^{2} x \operatorname {asin}{\left (c x \right )} + b d e x^{2} \operatorname {asin}{\left (c x \right )} + \frac {b e^{2} x^{3} \operatorname {asin}{\left (c x \right )}}{3} + \frac {b d^{2} \sqrt {- c^{2} x^{2} + 1}}{c} + \frac {b d e x \sqrt {- c^{2} x^{2} + 1}}{2 c} + \frac {b e^{2} x^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c} - \frac {b d e \operatorname {asin}{\left (c x \right )}}{2 c^{2}} + \frac {2 b e^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c^{3}} & \text {for}\: c \neq 0 \\a \left (d^{2} x + d e x^{2} + \frac {e^{2} x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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