Optimal. Leaf size=69 \[ \frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (A b-a B)}{2 b^2}+\frac{B \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 b^2} \]
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Rubi [A] time = 0.0217286, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {640, 609} \[ \frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (A b-a B)}{2 b^2}+\frac{B \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 b^2} \]
Antiderivative was successfully verified.
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Rule 640
Rule 609
Rubi steps
\begin{align*} \int (A+B x) \sqrt{a^2+2 a b x+b^2 x^2} \, dx &=\frac{B \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 b^2}+\frac{\left (2 A b^2-2 a b B\right ) \int \sqrt{a^2+2 a b x+b^2 x^2} \, dx}{2 b^2}\\ &=\frac{(A b-a B) (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{2 b^2}+\frac{B \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 b^2}\\ \end{align*}
Mathematica [A] time = 0.0157749, size = 45, normalized size = 0.65 \[ \frac{x \sqrt{(a+b x)^2} (3 a (2 A+B x)+b x (3 A+2 B x))}{6 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 42, normalized size = 0.6 \begin{align*}{\frac{x \left ( 2\,bB{x}^{2}+3\,Abx+3\,aBx+6\,aA \right ) }{6\,bx+6\,a}\sqrt{ \left ( bx+a \right ) ^{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 = 1.31996, size = 58, normalized size = 0.84 \begin{align*} \frac{1}{3} \, B b x^{3} + A a x + \frac{1}{2} \,{\left (B a + A b\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.119367, size = 26, normalized size = 0.38 \begin{align*} A a x + \frac{B b x^{3}}{3} + x^{2} \left (\frac{A b}{2} + \frac{B a}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15978, size = 100, normalized size = 1.45 \begin{align*} \frac{1}{3} \, B b x^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{2} \, B a x^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{2} \, A b x^{2} \mathrm{sgn}\left (b x + a\right ) + A a x \mathrm{sgn}\left (b x + a\right ) - \frac{{\left (B a^{3} - 3 \, A a^{2} b\right )} \mathrm{sgn}\left (b x + a\right )}{6 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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