Optimal. Leaf size=52 \[ \frac{1}{2} \left (2 a^2-b^2\right ) \sinh ^{-1}(x)+\frac{3}{2} a b \sqrt{x^2+1}+\frac{1}{2} b \sqrt{x^2+1} (a+b x) \]
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Rubi [A] time = 0.0203178, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {743, 641, 215} \[ \frac{1}{2} \left (2 a^2-b^2\right ) \sinh ^{-1}(x)+\frac{3}{2} a b \sqrt{x^2+1}+\frac{1}{2} b \sqrt{x^2+1} (a+b x) \]
Antiderivative was successfully verified.
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Rule 743
Rule 641
Rule 215
Rubi steps
\begin{align*} \int \frac{(a+b x)^2}{\sqrt{1+x^2}} \, dx &=\frac{1}{2} b (a+b x) \sqrt{1+x^2}+\frac{1}{2} \int \frac{2 a^2-b^2+3 a b x}{\sqrt{1+x^2}} \, dx\\ &=\frac{3}{2} a b \sqrt{1+x^2}+\frac{1}{2} b (a+b x) \sqrt{1+x^2}+\frac{1}{2} \left (2 a^2-b^2\right ) \int \frac{1}{\sqrt{1+x^2}} \, dx\\ &=\frac{3}{2} a b \sqrt{1+x^2}+\frac{1}{2} b (a+b x) \sqrt{1+x^2}+\frac{1}{2} \left (2 a^2-b^2\right ) \sinh ^{-1}(x)\\ \end{align*}
Mathematica [A] time = 0.0259011, size = 36, normalized size = 0.69 \[ \left (a^2-\frac{b^2}{2}\right ) \sinh ^{-1}(x)+\frac{1}{2} b \sqrt{x^2+1} (4 a+b x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 38, normalized size = 0.7 \begin{align*}{b}^{2} \left ({\frac{x}{2}\sqrt{{x}^{2}+1}}-{\frac{{\it Arcsinh} \left ( x \right ) }{2}} \right ) +2\,ab\sqrt{{x}^{2}+1}+{a}^{2}{\it Arcsinh} \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.6898, size = 51, normalized size = 0.98 \begin{align*} \frac{1}{2} \, \sqrt{x^{2} + 1} b^{2} x + a^{2} \operatorname{arsinh}\left (x\right ) - \frac{1}{2} \, b^{2} \operatorname{arsinh}\left (x\right ) + 2 \, \sqrt{x^{2} + 1} a b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81245, size = 108, normalized size = 2.08 \begin{align*} -\frac{1}{2} \,{\left (2 \, a^{2} - b^{2}\right )} \log \left (-x + \sqrt{x^{2} + 1}\right ) + \frac{1}{2} \,{\left (b^{2} x + 4 \, a b\right )} \sqrt{x^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.251236, size = 42, normalized size = 0.81 \begin{align*} a^{2} \operatorname{asinh}{\left (x \right )} + 2 a b \sqrt{x^{2} + 1} + \frac{b^{2} x \sqrt{x^{2} + 1}}{2} - \frac{b^{2} \operatorname{asinh}{\left (x \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.50868, size = 61, normalized size = 1.17 \begin{align*} -\frac{1}{2} \,{\left (2 \, a^{2} - b^{2}\right )} \log \left (-x + \sqrt{x^{2} + 1}\right ) + \frac{1}{2} \,{\left (b^{2} x + 4 \, a b\right )} \sqrt{x^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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