Optimal. Leaf size=80 \[ -\frac {3 a \sqrt {1-(a+b x)^2}}{4 b^2}+\frac {x \sqrt {1-(a+b x)^2}}{4 b}-\frac {\left (1+2 a^2\right ) \text {ArcSin}(a+b x)}{4 b^2}+\frac {1}{2} x^2 \text {ArcSin}(a+b x) \]
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Rubi [A]
time = 0.05, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {4889, 4827, 757,
655, 222} \begin {gather*} -\frac {\left (2 a^2+1\right ) \text {ArcSin}(a+b x)}{4 b^2}+\frac {1}{2} x^2 \text {ArcSin}(a+b x)-\frac {3 a \sqrt {1-(a+b x)^2}}{4 b^2}+\frac {x \sqrt {1-(a+b x)^2}}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 655
Rule 757
Rule 4827
Rule 4889
Rubi steps
\begin {align*} \int x \sin ^{-1}(a+b x) \, dx &=\frac {\text {Subst}\left (\int \left (-\frac {a}{b}+\frac {x}{b}\right ) \sin ^{-1}(x) \, dx,x,a+b x\right )}{b}\\ &=\frac {1}{2} x^2 \sin ^{-1}(a+b x)-\frac {1}{2} \text {Subst}\left (\int \frac {\left (-\frac {a}{b}+\frac {x}{b}\right )^2}{\sqrt {1-x^2}} \, dx,x,a+b x\right )\\ &=\frac {x \sqrt {1-(a+b x)^2}}{4 b}+\frac {1}{2} x^2 \sin ^{-1}(a+b x)+\frac {1}{4} \text {Subst}\left (\int \frac {-\frac {1+2 a^2}{b^2}+\frac {3 a x}{b^2}}{\sqrt {1-x^2}} \, dx,x,a+b x\right )\\ &=-\frac {3 a \sqrt {1-(a+b x)^2}}{4 b^2}+\frac {x \sqrt {1-(a+b x)^2}}{4 b}+\frac {1}{2} x^2 \sin ^{-1}(a+b x)-\frac {\left (1+2 a^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{4 b^2}\\ &=-\frac {3 a \sqrt {1-(a+b x)^2}}{4 b^2}+\frac {x \sqrt {1-(a+b x)^2}}{4 b}-\frac {\left (1+2 a^2\right ) \sin ^{-1}(a+b x)}{4 b^2}+\frac {1}{2} x^2 \sin ^{-1}(a+b x)\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 62, normalized size = 0.78 \begin {gather*} \frac {(-3 a+b x) \sqrt {1-a^2-2 a b x-b^2 x^2}+\left (-1-2 a^2+2 b^2 x^2\right ) \text {ArcSin}(a+b x)}{4 b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.00, size = 79, normalized size = 0.99
method | result | size |
derivativedivides | \(\frac {\frac {\arcsin \left (b x +a \right ) \left (b x +a \right )^{2}}{2}-\arcsin \left (b x +a \right ) a \left (b x +a \right )-a \sqrt {1-\left (b x +a \right )^{2}}+\frac {\left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}}{4}-\frac {\arcsin \left (b x +a \right )}{4}}{b^{2}}\) | \(79\) |
default | \(\frac {\frac {\arcsin \left (b x +a \right ) \left (b x +a \right )^{2}}{2}-\arcsin \left (b x +a \right ) a \left (b x +a \right )-a \sqrt {1-\left (b x +a \right )^{2}}+\frac {\left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}}{4}-\frac {\arcsin \left (b x +a \right )}{4}}{b^{2}}\) | \(79\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 153 vs.
\(2 (68) = 136\).
time = 0.47, size = 153, normalized size = 1.91 \begin {gather*} \frac {1}{2} \, x^{2} \arcsin \left (b x + a\right ) + \frac {1}{4} \, b {\left (\frac {3 \, 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 {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} x}{b^{2}} - \frac {{\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 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a}{b^{3}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.61, size = 58, normalized size = 0.72 \begin {gather*} \frac {{\left (2 \, b^{2} x^{2} - 2 \, a^{2} - 1\right )} \arcsin \left (b x + a\right ) + \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x - 3 \, a\right )}}{4 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.11, size = 104, normalized size = 1.30 \begin {gather*} \begin {cases} - \frac {a^{2} \operatorname {asin}{\left (a + b x \right )}}{2 b^{2}} - \frac {3 a \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{4 b^{2}} + \frac {x^{2} \operatorname {asin}{\left (a + b x \right )}}{2} + \frac {x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{4 b} - \frac {\operatorname {asin}{\left (a + b x \right )}}{4 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{2} \operatorname {asin}{\left (a \right )}}{2} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 91, normalized size = 1.14 \begin {gather*} -\frac {{\left (b x + a\right )} a \arcsin \left (b x + a\right )}{b^{2}} + \frac {{\left ({\left (b x + a\right )}^{2} - 1\right )} \arcsin \left (b x + a\right )}{2 \, b^{2}} + \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (b x + a\right )}}{4 \, b^{2}} - \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} a}{b^{2}} + \frac {\arcsin \left (b x + a\right )}{4 \, b^{2}} \end {gather*}
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 \begin {gather*} \int x\,\mathrm {asin}\left (a+b\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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