Optimal. Leaf size=130 \[ -\frac{\left (2 a^2-2 a+1\right ) \sqrt{-a-b x+1} \sqrt{a+b x+1}}{2 b^3}+\frac{\left (2 a^2-2 a+1\right ) \sin ^{-1}(a+b x)}{2 b^3}-\frac{x \sqrt{-a-b x+1} (a+b x+1)^{3/2}}{3 b^2}-\frac{(1-4 a) \sqrt{-a-b x+1} (a+b x+1)^{3/2}}{6 b^3} \]
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Rubi [A] time = 0.166757, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {6163, 90, 80, 50, 53, 619, 216} \[ -\frac{\left (2 a^2-2 a+1\right ) \sqrt{-a-b x+1} \sqrt{a+b x+1}}{2 b^3}+\frac{\left (2 a^2-2 a+1\right ) \sin ^{-1}(a+b x)}{2 b^3}-\frac{x \sqrt{-a-b x+1} (a+b x+1)^{3/2}}{3 b^2}-\frac{(1-4 a) \sqrt{-a-b x+1} (a+b x+1)^{3/2}}{6 b^3} \]
Antiderivative was successfully verified.
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Rule 6163
Rule 90
Rule 80
Rule 50
Rule 53
Rule 619
Rule 216
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 \sqrt{1-a-b x} (1+a+b x)^{3/2}}{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) \sqrt{1-a-b x} (1+a+b x)^{3/2}}{6 b^3}-\frac{x \sqrt{1-a-b x} (1+a+b x)^{3/2}}{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) \sqrt{1-a-b x} (1+a+b x)^{3/2}}{6 b^3}-\frac{x \sqrt{1-a-b x} (1+a+b x)^{3/2}}{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) \sqrt{1-a-b x} (1+a+b x)^{3/2}}{6 b^3}-\frac{x \sqrt{1-a-b x} (1+a+b x)^{3/2}}{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) \sqrt{1-a-b x} (1+a+b x)^{3/2}}{6 b^3}-\frac{x \sqrt{1-a-b x} (1+a+b x)^{3/2}}{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) \sqrt{1-a-b x} (1+a+b x)^{3/2}}{6 b^3}-\frac{x \sqrt{1-a-b x} (1+a+b x)^{3/2}}{3 b^2}+\frac{\left (1-2 a+2 a^2\right ) \sin ^{-1}(a+b x)}{2 b^3}\\ \end{align*}
Mathematica [A] time = 0.174695, size = 159, normalized size = 1.22 \[ \frac{-\sqrt{b} \sqrt{-a^2-2 a b x-b^2 x^2+1} \left (2 a^2-a (2 b x+9)+2 b^2 x^2+3 b x+4\right )+6 \left (2 a^2+1\right ) \sqrt{-b} \sinh ^{-1}\left (\frac{\sqrt{-b} \sqrt{-a-b x+1}}{\sqrt{2} \sqrt{b}}\right )+12 a \sqrt{-b} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{-a-b x+1}}{\sqrt{2} \sqrt{-b}}\right )}{6 b^{7/2}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.062, size = 315, normalized size = 2.4 \begin{align*} -{\frac{{x}^{2}}{3\,b}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}+{\frac{ax}{3\,{b}^{2}}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}-{\frac{{a}^{2}}{3\,{b}^{3}}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}-{\frac{a}{{b}^{2}}\arctan \left ({\sqrt{{b}^{2}} \left ( x+{\frac{a}{b}} \right ){\frac{1}{\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}} \right ){\frac{1}{\sqrt{{b}^{2}}}}}-{\frac{2}{3\,{b}^{3}}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}-{\frac{x}{2\,{b}^{2}}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}+{\frac{3\,a}{2\,{b}^{3}}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}+{\frac{{a}^{2}}{{b}^{2}}\arctan \left ({\sqrt{{b}^{2}} \left ( x+{\frac{a}{b}} \right ){\frac{1}{\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}} \right ){\frac{1}{\sqrt{{b}^{2}}}}}+{\frac{1}{2\,{b}^{2}}\arctan \left ({\sqrt{{b}^{2}} \left ( x+{\frac{a}{b}} \right ){\frac{1}{\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}} \right ){\frac{1}{\sqrt{{b}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.16545, size = 267, normalized size = 2.05 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \left (a + b x + 1\right )}{\sqrt{- \left (a + b x - 1\right ) \left (a + b x + 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17233, size = 131, normalized size = 1.01 \begin{align*} -\frac{1}{6} \, \sqrt{-{\left (b x + a\right )}^{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}\left (b\right )}{2 \, b^{2}{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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