Optimal. Leaf size=130 \[ \frac {\left (2 a^2+2 a+1\right ) \sqrt {a+b x+1} \sqrt {-a-b x+1}}{2 b^3}+\frac {\left (2 a^2+2 a+1\right ) \sin ^{-1}(a+b x)}{2 b^3}+\frac {(4 a+1) \sqrt {a+b x+1} (-a-b x+1)^{3/2}}{6 b^3}-\frac {x \sqrt {a+b x+1} (-a-b x+1)^{3/2}}{3 b^2} \]
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Rubi [A] time = 0.15, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6163, 90, 80, 50, 53, 619, 216} \[ \frac {\left (2 a^2+2 a+1\right ) \sqrt {a+b x+1} \sqrt {-a-b x+1}}{2 b^3}+\frac {\left (2 a^2+2 a+1\right ) \sin ^{-1}(a+b x)}{2 b^3}-\frac {x \sqrt {a+b x+1} (-a-b x+1)^{3/2}}{3 b^2}+\frac {(4 a+1) \sqrt {a+b x+1} (-a-b x+1)^{3/2}}{6 b^3} \]
Antiderivative was successfully verified.
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Rule 50
Rule 53
Rule 80
Rule 90
Rule 216
Rule 619
Rule 6163
Rubi steps
\begin {align*} \int e^{-\tanh ^{-1}(a+b x)} x^2 \, dx &=\int \frac {x^2 \sqrt {1-a-b x}}{\sqrt {1+a+b x}} \, dx\\ &=-\frac {x (1-a-b x)^{3/2} \sqrt {1+a+b x}}{3 b^2}-\frac {\int \frac {\sqrt {1-a-b x} \left (-1+a^2+(1+4 a) b x\right )}{\sqrt {1+a+b x}} \, dx}{3 b^2}\\ &=\frac {(1+4 a) (1-a-b x)^{3/2} \sqrt {1+a+b x}}{6 b^3}-\frac {x (1-a-b x)^{3/2} \sqrt {1+a+b x}}{3 b^2}+\frac {\left (1+2 a+2 a^2\right ) \int \frac {\sqrt {1-a-b x}}{\sqrt {1+a+b x}} \, dx}{2 b^2}\\ &=\frac {\left (1+2 a+2 a^2\right ) \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 b^3}+\frac {(1+4 a) (1-a-b x)^{3/2} \sqrt {1+a+b x}}{6 b^3}-\frac {x (1-a-b x)^{3/2} \sqrt {1+a+b x}}{3 b^2}+\frac {\left (1+2 a+2 a^2\right ) \int \frac {1}{\sqrt {1-a-b x} \sqrt {1+a+b x}} \, dx}{2 b^2}\\ &=\frac {\left (1+2 a+2 a^2\right ) \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 b^3}+\frac {(1+4 a) (1-a-b x)^{3/2} \sqrt {1+a+b x}}{6 b^3}-\frac {x (1-a-b x)^{3/2} \sqrt {1+a+b x}}{3 b^2}+\frac {\left (1+2 a+2 a^2\right ) \int \frac {1}{\sqrt {(1-a) (1+a)-2 a b x-b^2 x^2}} \, dx}{2 b^2}\\ &=\frac {\left (1+2 a+2 a^2\right ) \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 b^3}+\frac {(1+4 a) (1-a-b x)^{3/2} \sqrt {1+a+b x}}{6 b^3}-\frac {x (1-a-b x)^{3/2} \sqrt {1+a+b x}}{3 b^2}-\frac {\left (1+2 a+2 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{4 b^2}}} \, dx,x,-2 a b-2 b^2 x\right )}{4 b^4}\\ &=\frac {\left (1+2 a+2 a^2\right ) \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 b^3}+\frac {(1+4 a) (1-a-b x)^{3/2} \sqrt {1+a+b x}}{6 b^3}-\frac {x (1-a-b x)^{3/2} \sqrt {1+a+b x}}{3 b^2}+\frac {\left (1+2 a+2 a^2\right ) \sin ^{-1}(a+b x)}{2 b^3}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 126, normalized size = 0.97 \[ \frac {\left (2 a^2+2 a+1\right ) \sqrt {-b} \sinh ^{-1}\left (\frac {\sqrt {-b} \sqrt {-a-b x+1}}{\sqrt {2} \sqrt {b}}\right )}{b^{7/2}}-\frac {\sqrt {a+b x+1} \left (2 a^3+7 a^2+a (8 b x-5)+2 b^3 x^3-5 b^2 x^2+7 b x-4\right )}{6 b^3 \sqrt {-a-b x+1}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.48, size = 117, normalized size = 0.90 \[ -\frac {3 \, {\left (2 \, a^{2} + 2 \, a + 1\right )} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - {\left (2 \, b^{2} x^{2} - {\left (2 \, a + 3\right )} b x + 2 \, a^{2} + 9 \, a + 4\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{6 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.84, size = 106, normalized size = 0.82 \[ \frac {1}{6} \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (x {\left (\frac {2 \, x}{b} - \frac {2 \, a b^{3} + 3 \, b^{3}}{b^{5}}\right )} + \frac {2 \, a^{2} b^{2} + 9 \, a b^{2} + 4 \, b^{2}}{b^{5}}\right )} - \frac {{\left (2 \, a^{2} + 2 \, a + 1\right )} \arcsin \left (-b x - a\right ) \mathrm {sgn}\relax (b)}{2 \, b^{2} {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 535, normalized size = 4.12 \[ -\frac {\left (-b^{2} x^{2}-2 a b x -a^{2}+1\right )^{\frac {3}{2}}}{3 b^{3}}-\frac {a x \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b^{2}}-\frac {a^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b^{3}}-\frac {a \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{b^{2} \sqrt {b^{2}}}-\frac {x \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 b^{2}}-\frac {a \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 b^{3}}-\frac {\arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{2 b^{2} \sqrt {b^{2}}}+\frac {\sqrt {-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )}\, a^{2}}{b^{3}}+\frac {2 \sqrt {-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )}\, a}{b^{3}}+\frac {\sqrt {-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )}}{b^{3}}+\frac {\arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {1+a}{b}-\frac {1}{b}\right )}{\sqrt {-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )}}\right ) a^{2}}{b^{2} \sqrt {b^{2}}}+\frac {2 \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {1+a}{b}-\frac {1}{b}\right )}{\sqrt {-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )}}\right ) a}{b^{2} \sqrt {b^{2}}}+\frac {\arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {1+a}{b}-\frac {1}{b}\right )}{\sqrt {-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )}}\right )}{b^{2} \sqrt {b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 174, normalized size = 1.34 \[ -\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a x}{b^{2}} + \frac {a^{2} \arcsin \left (b x + a\right )}{b^{3}} - \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} x}{2 \, b^{2}} + \frac {a \arcsin \left (b x + a\right )}{b^{3}} - \frac {{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}}}{3 \, b^{3}} + \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a}{2 \, b^{3}} + \frac {\arcsin \left (b x + a\right )}{2 \, b^{3}} + \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{b^{3}} \]
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 \[ \int \frac {x^2\,\sqrt {1-{\left (a+b\,x\right )}^2}}{a+b\,x+1} \,d x \]
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 \[ \int \frac {x^{2} \sqrt {- \left (a + b x - 1\right ) \left (a + b x + 1\right )}}{a + b x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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