Optimal. Leaf size=114 \[ \frac{x^3 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{3 (a+b x)}+\frac{a A x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac{b B x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)} \]
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Rubi [A] time = 0.0456135, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {770, 76} \[ \frac{x^3 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{3 (a+b x)}+\frac{a A x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac{b B x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 770
Rule 76
Rubi steps
\begin{align*} \int x (A+B x) \sqrt{a^2+2 a b x+b^2 x^2} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int x \left (a b+b^2 x\right ) (A+B x) \, dx}{a b+b^2 x}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (a A b x+b (A b+a B) x^2+b^2 B x^3\right ) \, dx}{a b+b^2 x}\\ &=\frac{a A x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac{(A b+a B) x^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac{b B x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0167732, size = 47, normalized size = 0.41 \[ \frac{x^2 \sqrt{(a+b x)^2} (a (6 A+4 B x)+b x (4 A+3 B x))}{12 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 44, normalized size = 0.4 \begin{align*}{\frac{{x}^{2} \left ( 3\,Bb{x}^{2}+4\,Abx+4\,aBx+6\,aA \right ) }{12\,bx+12\,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 = 1.2566, size = 66, normalized size = 0.58 \begin{align*} \frac{1}{4} \, B b x^{4} + \frac{1}{2} \, A a x^{2} + \frac{1}{3} \,{\left (B a + A b\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.122032, size = 29, normalized size = 0.25 \begin{align*} \frac{A a x^{2}}{2} + \frac{B b x^{4}}{4} + x^{3} \left (\frac{A b}{3} + \frac{B a}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15756, size = 104, normalized size = 0.91 \begin{align*} \frac{1}{4} \, B b x^{4} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{3} \, B a x^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{3} \, A b x^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{2} \, A a x^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{{\left (B a^{4} - 2 \, A a^{3} b\right )} \mathrm{sgn}\left (b x + a\right )}{12 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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