Optimal. Leaf size=120 \[ \frac{2 x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{5 (a+b x)}+\frac{2 a A x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac{2 b B x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)} \]
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Rubi [A] time = 0.0453983, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {770, 76} \[ \frac{2 x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{5 (a+b x)}+\frac{2 a A x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac{2 b B x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 770
Rule 76
Rubi steps
\begin{align*} \int \sqrt{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 \sqrt{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 \sqrt{x}+b (A b+a B) x^{3/2}+b^2 B x^{5/2}\right ) \, dx}{a b+b^2 x}\\ &=\frac{2 a A x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac{2 (A b+a B) x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{2 b B x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0260709, size = 51, normalized size = 0.42 \[ \frac{2 x^{3/2} \sqrt{(a+b x)^2} (7 a (5 A+3 B x)+3 b x (7 A+5 B x))}{105 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 44, normalized size = 0.4 \begin{align*}{\frac{30\,Bb{x}^{2}+42\,Abx+42\,aBx+70\,aA}{105\,bx+105\,a}{x}^{{\frac{3}{2}}}\sqrt{ \left ( bx+a \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03904, size = 47, normalized size = 0.39 \begin{align*} \frac{2}{35} \,{\left (5 \, b x^{2} + 7 \, a x\right )} B x^{\frac{3}{2}} + \frac{2}{15} \,{\left (3 \, b x^{2} + 5 \, a x\right )} A \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61372, size = 81, normalized size = 0.68 \begin{align*} \frac{2}{105} \,{\left (15 \, B b x^{3} + 35 \, A a x + 21 \,{\left (B a + A b\right )} x^{2}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17032, size = 72, normalized size = 0.6 \begin{align*} \frac{2}{7} \, B b x^{\frac{7}{2}} \mathrm{sgn}\left (b x + a\right ) + \frac{2}{5} \, B a x^{\frac{5}{2}} \mathrm{sgn}\left (b x + a\right ) + \frac{2}{5} \, A b x^{\frac{5}{2}} \mathrm{sgn}\left (b x + a\right ) + \frac{2}{3} \, A a x^{\frac{3}{2}} \mathrm{sgn}\left (b x + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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