Optimal. Leaf size=110 \[ \frac {(a+b x)^4}{16 b}-\frac {5 (a+b x)^2}{16 b}+\frac {\left (1-(a+b x)^2\right )^{3/2} (a+b x) \sin ^{-1}(a+b x)}{4 b}+\frac {3 \sqrt {1-(a+b x)^2} (a+b x) \sin ^{-1}(a+b x)}{8 b}+\frac {3 \sin ^{-1}(a+b x)^2}{16 b} \]
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Rubi [A] time = 0.11, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {4807, 4649, 4647, 4641, 30, 14} \[ \frac {(a+b x)^4}{16 b}-\frac {5 (a+b x)^2}{16 b}+\frac {\left (1-(a+b x)^2\right )^{3/2} (a+b x) \sin ^{-1}(a+b x)}{4 b}+\frac {3 \sqrt {1-(a+b x)^2} (a+b x) \sin ^{-1}(a+b x)}{8 b}+\frac {3 \sin ^{-1}(a+b x)^2}{16 b} \]
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 4641
Rule 4647
Rule 4649
Rule 4807
Rubi steps
\begin {align*} \int \left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \sin ^{-1}(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \left (1-x^2\right )^{3/2} \sin ^{-1}(x) \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2} \sin ^{-1}(a+b x)}{4 b}-\frac {\operatorname {Subst}\left (\int x \left (1-x^2\right ) \, dx,x,a+b x\right )}{4 b}+\frac {3 \operatorname {Subst}\left (\int \sqrt {1-x^2} \sin ^{-1}(x) \, dx,x,a+b x\right )}{4 b}\\ &=\frac {3 (a+b x) \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{8 b}+\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2} \sin ^{-1}(a+b x)}{4 b}-\frac {\operatorname {Subst}\left (\int \left (x-x^3\right ) \, dx,x,a+b x\right )}{4 b}-\frac {3 \operatorname {Subst}(\int x \, dx,x,a+b x)}{8 b}+\frac {3 \operatorname {Subst}\left (\int \frac {\sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{8 b}\\ &=-\frac {5 (a+b x)^2}{16 b}+\frac {(a+b x)^4}{16 b}+\frac {3 (a+b x) \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{8 b}+\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2} \sin ^{-1}(a+b x)}{4 b}+\frac {3 \sin ^{-1}(a+b x)^2}{16 b}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 129, normalized size = 1.17 \[ \frac {1}{16} \left (\left (6 a^2-5\right ) b x^2+2 a \left (2 a^2-5\right ) x-\frac {2 \sqrt {-a^2-2 a b x-b^2 x^2+1} \left (2 a^3+6 a^2 b x+6 a b^2 x^2-5 a+2 b^3 x^3-5 b x\right ) \sin ^{-1}(a+b x)}{b}+4 a b^2 x^3+\frac {3 \sin ^{-1}(a+b x)^2}{b}+b^3 x^4\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 125, normalized size = 1.14 \[ \frac {b^{4} x^{4} + 4 \, a b^{3} x^{3} + {\left (6 \, a^{2} - 5\right )} b^{2} x^{2} + 2 \, {\left (2 \, a^{3} - 5 \, a\right )} b x - 2 \, {\left (2 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 2 \, a^{3} + {\left (6 \, a^{2} - 5\right )} b x - 5 \, a\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} \arcsin \left (b x + a\right ) + 3 \, \arcsin \left (b x + a\right )^{2}}{16 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.44, size = 141, normalized size = 1.28 \[ \frac {{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}} {\left (b x + a\right )} \arcsin \left (b x + a\right )}{4 \, b} + \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )} \arcsin \left (b x + a\right )}{8 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )}^{2}}{16 \, b} + \frac {3 \, \arcsin \left (b x + a\right )^{2}}{16 \, b} - \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )}}{16 \, b} - \frac {15}{128 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.14, size = 277, normalized size = 2.52 \[ \frac {-4 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right ) x^{3} b^{3}+x^{4} b^{4}-12 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right ) x^{2} a \,b^{2}+4 x^{3} a \,b^{3}-12 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right ) x \,a^{2} b +6 x^{2} a^{2} b^{2}-4 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right ) a^{3}+4 x \,a^{3} b +10 \arcsin \left (b x +a \right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, x b -5 b^{2} x^{2}+a^{4}+10 \arcsin \left (b x +a \right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a -10 a b x +3 \arcsin \left (b x +a \right )^{2}-5 a^{2}+4}{16 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.98, size = 402, normalized size = 3.65 \[ \frac {1}{16} \, {\left (b^{2} x^{4} + 4 \, a b x^{3} + 6 \, a^{2} x^{2} + \frac {4 \, a^{3} x}{b} - 5 \, x^{2} - \frac {10 \, a x}{b} + \frac {6 \, \arcsin \left (b x + a\right ) \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{b^{2}} + \frac {3 \, \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )^{2}}{b^{2}}\right )} b + \frac {1}{8} \, {\left (2 \, {\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}} x + \frac {2 \, {\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}} a}{b} - \frac {3 \, {\left (a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} a^{2} \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{b^{3}} + \frac {3 \, {\left (a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} x}{b^{2}} + \frac {3 \, {\left (a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} {\left (a^{2} - 1\right )} \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{b^{3}} + \frac {3 \, {\left (a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a}{b^{3}}\right )} \arcsin \left (b x + a\right ) \]
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 \[ \int \mathrm {asin}\left (a+b\,x\right )\,{\left (-a^2-2\,a\,b\,x-b^2\,x^2+1\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.33, size = 298, normalized size = 2.71 \[ \begin {cases} \frac {a^{3} x}{4} - \frac {a^{3} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{4 b} + \frac {3 a^{2} b x^{2}}{8} - \frac {3 a^{2} x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{4} + \frac {a b^{2} x^{3}}{4} - \frac {3 a b x^{2} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{4} - \frac {5 a x}{8} + \frac {5 a \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{8 b} + \frac {b^{3} x^{4}}{16} - \frac {b^{2} x^{3} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{4} - \frac {5 b x^{2}}{16} + \frac {5 x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{8} + \frac {3 \operatorname {asin}^{2}{\left (a + b x \right )}}{16 b} & \text {for}\: b \neq 0 \\x \left (1 - a^{2}\right )^{\frac {3}{2}} \operatorname {asin}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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