Optimal. Leaf size=105 \[ \frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^2}{2 d}+\frac {b e \sqrt {1-(c+d x)^2} (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )}{2 d}-\frac {e \left (a+b \sin ^{-1}(c+d x)\right )^2}{4 d}-\frac {b^2 e (c+d x)^2}{4 d} \]
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Rubi [A] time = 0.15, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4805, 12, 4627, 4707, 4641, 30} \[ \frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^2}{2 d}+\frac {b e \sqrt {1-(c+d x)^2} (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )}{2 d}-\frac {e \left (a+b \sin ^{-1}(c+d x)\right )^2}{4 d}-\frac {b^2 e (c+d x)^2}{4 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 4627
Rule 4641
Rule 4707
Rule 4805
Rubi steps
\begin {align*} \int (c e+d e x) \left (a+b \sin ^{-1}(c+d x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int e x \left (a+b \sin ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {e \operatorname {Subst}\left (\int x \left (a+b \sin ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^2}{2 d}-\frac {(b e) \operatorname {Subst}\left (\int \frac {x^2 \left (a+b \sin ^{-1}(x)\right )}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{d}\\ &=\frac {b e (c+d x) \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{2 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^2}{2 d}-\frac {(b e) \operatorname {Subst}\left (\int \frac {a+b \sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{2 d}-\frac {\left (b^2 e\right ) \operatorname {Subst}(\int x \, dx,x,c+d x)}{2 d}\\ &=-\frac {b^2 e (c+d x)^2}{4 d}+\frac {b e (c+d x) \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{2 d}-\frac {e \left (a+b \sin ^{-1}(c+d x)\right )^2}{4 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^2}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 86, normalized size = 0.82 \[ -\frac {e \left (-2 (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^2-2 b \sqrt {1-(c+d x)^2} (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )+\left (a+b \sin ^{-1}(c+d x)\right )^2+b^2 (c+d x)^2\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 188, normalized size = 1.79 \[ \frac {{\left (2 \, a^{2} - b^{2}\right )} d^{2} e x^{2} + 2 \, {\left (2 \, a^{2} - b^{2}\right )} c d e x + {\left (2 \, b^{2} d^{2} e x^{2} + 4 \, b^{2} c d e x + {\left (2 \, b^{2} c^{2} - b^{2}\right )} e\right )} \arcsin \left (d x + c\right )^{2} + 2 \, {\left (2 \, a b d^{2} e x^{2} + 4 \, a b c d e x + {\left (2 \, a b c^{2} - a b\right )} e\right )} \arcsin \left (d x + c\right ) + 2 \, {\left (a b d e x + a b c e + {\left (b^{2} d e x + b^{2} c e\right )} \arcsin \left (d x + c\right )\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.53, size = 193, normalized size = 1.84 \[ \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )} b^{2} \arcsin \left (d x + c\right )^{2} e}{2 \, d} + \frac {\sqrt {-{\left (d x + c\right )}^{2} + 1} {\left (d x + c\right )} b^{2} \arcsin \left (d x + c\right ) e}{2 \, d} + \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )} a b \arcsin \left (d x + c\right ) e}{d} + \frac {b^{2} \arcsin \left (d x + c\right )^{2} e}{4 \, d} + \frac {\sqrt {-{\left (d x + c\right )}^{2} + 1} {\left (d x + c\right )} a b e}{2 \, d} + \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )} a^{2} e}{2 \, d} - \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )} b^{2} e}{4 \, d} + \frac {a b \arcsin \left (d x + c\right ) e}{2 \, d} - \frac {b^{2} e}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 146, normalized size = 1.39 \[ \frac {\frac {e \left (d x +c \right )^{2} a^{2}}{2}+e \,b^{2} \left (\frac {\arcsin \left (d x +c \right )^{2} \left (\left (d x +c \right )^{2}-1\right )}{2}+\frac {\arcsin \left (d x +c \right ) \left (\left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}+\arcsin \left (d x +c \right )\right )}{2}-\frac {\arcsin \left (d x +c \right )^{2}}{4}-\frac {\left (d x +c \right )^{2}}{4}\right )+2 e a b \left (\frac {\arcsin \left (d x +c \right ) \left (d x +c \right )^{2}}{2}+\frac {\left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{4}-\frac {\arcsin \left (d x +c \right )}{4}\right )}{d} \]
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 \[ \frac {1}{2} \, a^{2} d e x^{2} + \frac {1}{2} \, {\left (2 \, x^{2} \arcsin \left (d x + c\right ) + d {\left (\frac {3 \, c^{2} \arcsin \left (-\frac {d^{2} x + c d}{\sqrt {c^{2} d^{2} - {\left (c^{2} - 1\right )} d^{2}}}\right )}{d^{3}} + \frac {\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} x}{d^{2}} - \frac {{\left (c^{2} - 1\right )} \arcsin \left (-\frac {d^{2} x + c d}{\sqrt {c^{2} d^{2} - {\left (c^{2} - 1\right )} d^{2}}}\right )}{d^{3}} - \frac {3 \, \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} c}{d^{3}}\right )}\right )} a b d e + a^{2} c e x + \frac {2 \, {\left ({\left (d x + c\right )} \arcsin \left (d x + c\right ) + \sqrt {-{\left (d x + c\right )}^{2} + 1}\right )} a b c e}{d} + \frac {1}{2} \, {\left (b^{2} d e x^{2} + 2 \, b^{2} c e x\right )} \arctan \left (d x + c, \sqrt {d x + c + 1} \sqrt {-d x - c + 1}\right )^{2} + \int \frac {{\left (b^{2} d^{2} e x^{2} + 2 \, b^{2} c d e x\right )} \sqrt {d x + c + 1} \sqrt {-d x - c + 1} \arctan \left (d x + c, \sqrt {d x + c + 1} \sqrt {-d x - c + 1}\right )}{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\,{d x} \]
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 \[ \int \left (c\,e+d\,e\,x\right )\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.82, size = 335, normalized size = 3.19 \[ \begin {cases} a^{2} c e x + \frac {a^{2} d e x^{2}}{2} + \frac {a b c^{2} e \operatorname {asin}{\left (c + d x \right )}}{d} + 2 a b c e x \operatorname {asin}{\left (c + d x \right )} + \frac {a b c e \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{2 d} + a b d e x^{2} \operatorname {asin}{\left (c + d x \right )} + \frac {a b e x \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{2} - \frac {a b e \operatorname {asin}{\left (c + d x \right )}}{2 d} + \frac {b^{2} c^{2} e \operatorname {asin}^{2}{\left (c + d x \right )}}{2 d} + b^{2} c e x \operatorname {asin}^{2}{\left (c + d x \right )} - \frac {b^{2} c e x}{2} + \frac {b^{2} c e \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}{\left (c + d x \right )}}{2 d} + \frac {b^{2} d e x^{2} \operatorname {asin}^{2}{\left (c + d x \right )}}{2} - \frac {b^{2} d e x^{2}}{4} + \frac {b^{2} e x \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}{\left (c + d x \right )}}{2} - \frac {b^{2} e \operatorname {asin}^{2}{\left (c + d x \right )}}{4 d} & \text {for}\: d \neq 0 \\c e x \left (a + b \operatorname {asin}{\relax (c )}\right )^{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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