Optimal. Leaf size=94 \[ \frac {x^2 \sqrt {1-(a+b x)^2}}{9 b}+\frac {\left (4+11 a^2-5 a b x\right ) \sqrt {1-(a+b x)^2}}{18 b^3}+\frac {a \left (3+2 a^2\right ) \text {ArcSin}(a+b x)}{6 b^3}+\frac {1}{3} x^3 \text {ArcSin}(a+b x) \]
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Rubi [A]
time = 0.08, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4889, 4827,
757, 794, 222} \begin {gather*} \frac {a \left (2 a^2+3\right ) \text {ArcSin}(a+b x)}{6 b^3}+\frac {\left (11 a^2-5 a b x+4\right ) \sqrt {1-(a+b x)^2}}{18 b^3}+\frac {1}{3} x^3 \text {ArcSin}(a+b x)+\frac {x^2 \sqrt {1-(a+b x)^2}}{9 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 757
Rule 794
Rule 4827
Rule 4889
Rubi steps
\begin {align*} \int x^2 \sin ^{-1}(a+b x) \, dx &=\frac {\text {Subst}\left (\int \left (-\frac {a}{b}+\frac {x}{b}\right )^2 \sin ^{-1}(x) \, dx,x,a+b x\right )}{b}\\ &=\frac {1}{3} x^3 \sin ^{-1}(a+b x)-\frac {1}{3} \text {Subst}\left (\int \frac {\left (-\frac {a}{b}+\frac {x}{b}\right )^3}{\sqrt {1-x^2}} \, dx,x,a+b x\right )\\ &=\frac {x^2 \sqrt {1-(a+b x)^2}}{9 b}+\frac {1}{3} x^3 \sin ^{-1}(a+b x)+\frac {1}{9} \text {Subst}\left (\int \frac {\left (-\frac {2+3 a^2}{b^2}+\frac {5 a x}{b^2}\right ) \left (-\frac {a}{b}+\frac {x}{b}\right )}{\sqrt {1-x^2}} \, dx,x,a+b x\right )\\ &=\frac {x^2 \sqrt {1-(a+b x)^2}}{9 b}+\frac {\left (4+11 a^2-5 a b x\right ) \sqrt {1-(a+b x)^2}}{18 b^3}+\frac {1}{3} x^3 \sin ^{-1}(a+b x)+\frac {\left (a \left (3+2 a^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{6 b^3}\\ &=\frac {x^2 \sqrt {1-(a+b x)^2}}{9 b}+\frac {\left (4+11 a^2-5 a b x\right ) \sqrt {1-(a+b x)^2}}{18 b^3}+\frac {a \left (3+2 a^2\right ) \sin ^{-1}(a+b x)}{6 b^3}+\frac {1}{3} x^3 \sin ^{-1}(a+b x)\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 77, normalized size = 0.82 \begin {gather*} \frac {\sqrt {1-a^2-2 a b x-b^2 x^2} \left (4+11 a^2-5 a b x+2 b^2 x^2\right )+\left (9 a+6 a^3+6 b^3 x^3\right ) \text {ArcSin}(a+b x)}{18 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.00, size = 137, normalized size = 1.46
method | result | size |
derivativedivides | \(\frac {\arcsin \left (b x +a \right ) a^{2} \left (b x +a \right )-\arcsin \left (b x +a \right ) a \left (b x +a \right )^{2}+\frac {\arcsin \left (b x +a \right ) \left (b x +a \right )^{3}}{3}+a^{2} \sqrt {1-\left (b x +a \right )^{2}}+a \left (-\frac {\left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}}{2}+\frac {\arcsin \left (b x +a \right )}{2}\right )+\frac {\left (b x +a \right )^{2} \sqrt {1-\left (b x +a \right )^{2}}}{9}+\frac {2 \sqrt {1-\left (b x +a \right )^{2}}}{9}}{b^{3}}\) | \(137\) |
default | \(\frac {\arcsin \left (b x +a \right ) a^{2} \left (b x +a \right )-\arcsin \left (b x +a \right ) a \left (b x +a \right )^{2}+\frac {\arcsin \left (b x +a \right ) \left (b x +a \right )^{3}}{3}+a^{2} \sqrt {1-\left (b x +a \right )^{2}}+a \left (-\frac {\left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}}{2}+\frac {\arcsin \left (b x +a \right )}{2}\right )+\frac {\left (b x +a \right )^{2} \sqrt {1-\left (b x +a \right )^{2}}}{9}+\frac {2 \sqrt {1-\left (b x +a \right )^{2}}}{9}}{b^{3}}\) | \(137\) |
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. 220 vs.
\(2 (82) = 164\).
time = 0.47, size = 220, normalized size = 2.34 \begin {gather*} \frac {1}{3} \, x^{3} \arcsin \left (b x + a\right ) + \frac {1}{18} \, b {\left (\frac {2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} x^{2}}{b^{2}} - \frac {15 \, a^{3} \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{b^{4}} - \frac {5 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a x}{b^{3}} + \frac {9 \, {\left (a^{2} - 1\right )} a \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{b^{4}} + \frac {15 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a^{2}}{b^{4}} - \frac {4 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a^{2} - 1\right )}}{b^{4}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.07, size = 74, normalized size = 0.79 \begin {gather*} \frac {3 \, {\left (2 \, b^{3} x^{3} + 2 \, a^{3} + 3 \, a\right )} \arcsin \left (b x + a\right ) + {\left (2 \, b^{2} x^{2} - 5 \, a b x + 11 \, a^{2} + 4\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{18 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 170 vs.
\(2 (83) = 166\).
time = 0.20, size = 170, normalized size = 1.81 \begin {gather*} \begin {cases} \frac {a^{3} \operatorname {asin}{\left (a + b x \right )}}{3 b^{3}} + \frac {11 a^{2} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{18 b^{3}} - \frac {5 a x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{18 b^{2}} + \frac {a \operatorname {asin}{\left (a + b x \right )}}{2 b^{3}} + \frac {x^{3} \operatorname {asin}{\left (a + b x \right )}}{3} + \frac {x^{2} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{9 b} + \frac {2 \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{9 b^{3}} & \text {for}\: b \neq 0 \\\frac {x^{3} \operatorname {asin}{\left (a \right )}}{3} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 173 vs.
\(2 (82) = 164\).
time = 0.41, size = 173, normalized size = 1.84 \begin {gather*} \frac {{\left (b x + a\right )} a^{2} \arcsin \left (b x + a\right )}{b^{3}} + \frac {{\left ({\left (b x + a\right )}^{2} - 1\right )} {\left (b x + a\right )} \arcsin \left (b x + a\right )}{3 \, b^{3}} - \frac {{\left ({\left (b x + a\right )}^{2} - 1\right )} a \arcsin \left (b x + a\right )}{b^{3}} - \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (b x + a\right )} a}{2 \, b^{3}} + \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} a^{2}}{b^{3}} + \frac {{\left (b x + a\right )} \arcsin \left (b x + a\right )}{3 \, b^{3}} - \frac {a \arcsin \left (b x + a\right )}{2 \, b^{3}} - \frac {{\left (-{\left (b x + a\right )}^{2} + 1\right )}^{\frac {3}{2}}}{9 \, b^{3}} + \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1}}{3 \, b^{3}} \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^2\,\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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