Optimal. Leaf size=142 \[ \frac {2 b d \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac {b e x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c}-\frac {e \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2}-\frac {d^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 e}+\frac {(d+e x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 e}-2 b^2 d x-\frac {1}{4} b^2 e x^2 \]
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Rubi [A] time = 0.31, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {4743, 4763, 4641, 4677, 8, 4707, 30} \[ \frac {2 b d \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac {b e x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c}-\frac {e \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2}-\frac {d^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 e}+\frac {(d+e x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 e}-2 b^2 d x-\frac {1}{4} b^2 e x^2 \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 4641
Rule 4677
Rule 4707
Rule 4743
Rule 4763
Rubi steps
\begin {align*} \int (d+e x) \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac {(d+e x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 e}-\frac {(b c) \int \frac {(d+e x)^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{e}\\ &=\frac {(d+e x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 e}-\frac {(b c) \int \left (\frac {d^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}+\frac {2 d e x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}+\frac {e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}\right ) \, dx}{e}\\ &=\frac {(d+e x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 e}-(2 b c d) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx-\frac {\left (b c d^2\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{e}-(b c e) \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {2 b d \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac {b e x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c}-\frac {d^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 e}+\frac {(d+e x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 e}-\left (2 b^2 d\right ) \int 1 \, dx-\frac {1}{2} \left (b^2 e\right ) \int x \, dx-\frac {(b e) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{2 c}\\ &=-2 b^2 d x-\frac {1}{4} b^2 e x^2+\frac {2 b d \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac {b e x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c}-\frac {d^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 e}-\frac {e \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2}+\frac {(d+e x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 e}\\ \end {align*}
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Mathematica [A] time = 0.38, size = 147, normalized size = 1.04 \[ \frac {c \left (2 a^2 c x (2 d+e x)+2 a b \sqrt {1-c^2 x^2} (4 d+e x)-b^2 c x (8 d+e x)\right )+2 b \sin ^{-1}(c x) \left (4 a c^2 d x+a e \left (2 c^2 x^2-1\right )+b c \sqrt {1-c^2 x^2} (4 d+e x)\right )+b^2 \sin ^{-1}(c x)^2 \left (4 c^2 d x+e \left (2 c^2 x^2-1\right )\right )}{4 c^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 156, normalized size = 1.10 \[ \frac {{\left (2 \, a^{2} - b^{2}\right )} c^{2} e x^{2} + 4 \, {\left (a^{2} - 2 \, b^{2}\right )} c^{2} d x + {\left (2 \, b^{2} c^{2} e x^{2} + 4 \, b^{2} c^{2} d x - b^{2} e\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (2 \, a b c^{2} e x^{2} + 4 \, a b c^{2} d x - a b e\right )} \arcsin \left (c x\right ) + 2 \, {\left (a b c e x + 4 \, a b c d + {\left (b^{2} c e x + 4 \, b^{2} c d\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{4 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.43, size = 253, normalized size = 1.78 \[ b^{2} d x \arcsin \left (c x\right )^{2} + 2 \, a b d x \arcsin \left (c x\right ) + \frac {\sqrt {-c^{2} x^{2} + 1} b^{2} x \arcsin \left (c x\right ) e}{2 \, c} + a^{2} d x - 2 \, b^{2} d x + \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} \arcsin \left (c x\right )^{2} e}{2 \, c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d \arcsin \left (c x\right )}{c} + \frac {\sqrt {-c^{2} x^{2} + 1} a b x e}{2 \, c} + \frac {{\left (c^{2} x^{2} - 1\right )} a b \arcsin \left (c x\right ) e}{c^{2}} + \frac {b^{2} \arcsin \left (c x\right )^{2} e}{4 \, c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a b d}{c} + \frac {{\left (c^{2} x^{2} - 1\right )} a^{2} e}{2 \, c^{2}} - \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} e}{4 \, c^{2}} + \frac {a b \arcsin \left (c x\right ) e}{2 \, c^{2}} - \frac {b^{2} e}{8 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 198, normalized size = 1.39 \[ \frac {\frac {a^{2} \left (\frac {1}{2} x^{2} c^{2} e +c^{2} d x \right )}{c}+\frac {b^{2} \left (\frac {e \left (2 \arcsin \left (c x \right )^{2} c^{2} x^{2}+2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x -\arcsin \left (c x \right )^{2}-c^{2} x^{2}\right )}{4}+d c \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )\right )}{c}+\frac {2 a b \left (\frac {\arcsin \left (c x \right ) c^{2} x^{2} e}{2}+\arcsin \left (c x \right ) d \,c^{2} x -\frac {e \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{2}+d c \sqrt {-c^{2} x^{2}+1}\right )}{c}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ b^{2} d x \arcsin \left (c x\right )^{2} + \frac {1}{2} \, a^{2} e x^{2} + \frac {1}{2} \, {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} a b e + \frac {1}{2} \, {\left (x^{2} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, c \int \frac {\sqrt {c x + 1} \sqrt {-c x + 1} x^{2} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{c^{2} x^{2} - 1}\,{d x}\right )} b^{2} e - 2 \, b^{2} d {\left (x - \frac {\sqrt {-c^{2} x^{2} + 1} \arcsin \left (c x\right )}{c}\right )} + a^{2} d x + \frac {2 \, {\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} a b d}{c} \]
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 \[ \int {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\left (d+e\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.67, size = 233, normalized size = 1.64 \[ \begin {cases} a^{2} d x + \frac {a^{2} e x^{2}}{2} + 2 a b d x \operatorname {asin}{\left (c x \right )} + a b e x^{2} \operatorname {asin}{\left (c x \right )} + \frac {2 a b d \sqrt {- c^{2} x^{2} + 1}}{c} + \frac {a b e x \sqrt {- c^{2} x^{2} + 1}}{2 c} - \frac {a b e \operatorname {asin}{\left (c x \right )}}{2 c^{2}} + b^{2} d x \operatorname {asin}^{2}{\left (c x \right )} - 2 b^{2} d x + \frac {b^{2} e x^{2} \operatorname {asin}^{2}{\left (c x \right )}}{2} - \frac {b^{2} e x^{2}}{4} + \frac {2 b^{2} d \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{c} + \frac {b^{2} e x \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{2 c} - \frac {b^{2} e \operatorname {asin}^{2}{\left (c x \right )}}{4 c^{2}} & \text {for}\: c \neq 0 \\a^{2} \left (d x + \frac {e x^{2}}{2}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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