Optimal. Leaf size=67 \[ \frac {\left (2 a^2+1\right ) \sin ^{-1}(a+b x)}{2 b^3}+\frac {3 a \sqrt {1-(a+b x)^2}}{2 b^3}-\frac {x \sqrt {1-(a+b x)^2}}{2 b^2} \]
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Rubi [A] time = 0.05, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {371, 743, 641, 216} \[ \frac {\left (2 a^2+1\right ) \sin ^{-1}(a+b x)}{2 b^3}+\frac {3 a \sqrt {1-(a+b x)^2}}{2 b^3}-\frac {x \sqrt {1-(a+b x)^2}}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 216
Rule 371
Rule 641
Rule 743
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {1-(a+b x)^2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(-a+x)^2}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{b^3}\\ &=-\frac {x \sqrt {1-(a+b x)^2}}{2 b^2}-\frac {\operatorname {Subst}\left (\int \frac {-1-2 a^2+3 a x}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{2 b^3}\\ &=\frac {3 a \sqrt {1-(a+b x)^2}}{2 b^3}-\frac {x \sqrt {1-(a+b x)^2}}{2 b^2}+\frac {\left (1+2 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{2 b^3}\\ &=\frac {3 a \sqrt {1-(a+b x)^2}}{2 b^3}-\frac {x \sqrt {1-(a+b x)^2}}{2 b^2}+\frac {\left (1+2 a^2\right ) \sin ^{-1}(a+b x)}{2 b^3}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 55, normalized size = 0.82 \[ \frac {\sqrt {-a^2-2 a b x-b^2 x^2+1} (3 a-b x)+\left (2 a^2+1\right ) \sin ^{-1}(a+b x)}{2 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 92, normalized size = 1.37 \[ -\frac {{\left (2 \, a^{2} + 1\right )} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x - 3 \, a\right )}}{2 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.49, size = 55, normalized size = 0.82 \[ -\frac {1}{2} \, \sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (\frac {x}{b^{2}} - \frac {3 \, a}{b^{3}}\right )} - \frac {{\left (2 \, a^{2} + 1\right )} \arcsin \left (-b x - a\right ) \mathrm {sgn}\relax (b)}{2 \, b^{2} {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 152, normalized size = 2.27 \[ \frac {a^{2} \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{\sqrt {b^{2}}\, b^{2}}-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, x}{2 b^{2}}+\frac {\arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{2 \sqrt {b^{2}}\, b^{2}}+\frac {3 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a}{2 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.46, size = 139, normalized size = 2.07 \[ -\frac {3 \, a^{2} \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{2 \, b^{3}} - \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} x}{2 \, b^{2}} + \frac {{\left (a^{2} - 1\right )} \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{2 \, b^{3}} + \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a}{2 \, b^{3}} \]
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 \[ \int \frac {x^2}{\sqrt {1-{\left (a+b\,x\right )}^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt {- \left (a + b x - 1\right ) \left (a + b x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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