Optimal. Leaf size=118 \[ \frac{2 x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{3 (a+b x)}+\frac{2 a A \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{2 b B x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)} \]
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Rubi [A] time = 0.042182, antiderivative size = 118, 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} \[ \frac{2 x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{3 (a+b x)}+\frac{2 a A \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{2 b B x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 770
Rule 76
Rubi steps
\begin{align*} \int \frac{(A+B x) \sqrt{a^2+2 a b x+b^2 x^2}}{\sqrt{x}} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{\left (a b+b^2 x\right ) (A+B x)}{\sqrt{x}} \, dx}{a b+b^2 x}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (\frac{a A b}{\sqrt{x}}+b (A b+a B) \sqrt{x}+b^2 B x^{3/2}\right ) \, dx}{a b+b^2 x}\\ &=\frac{2 a A \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{2 (A b+a B) x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac{2 b B x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0297499, size = 49, normalized size = 0.42 \[ \frac{2 \sqrt{x} \sqrt{(a+b x)^2} (5 a (3 A+B x)+b x (5 A+3 B x))}{15 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 44, normalized size = 0.4 \begin{align*}{\frac{6\,Bb{x}^{2}+10\,Abx+10\,aBx+30\,aA}{15\,bx+15\,a}\sqrt{x}\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 = 1.0606, size = 46, normalized size = 0.39 \begin{align*} \frac{2}{15} \,{\left (3 \, b x^{2} + 5 \, a x\right )} B \sqrt{x} + \frac{2 \,{\left (b x^{2} + 3 \, a x\right )} A}{3 \, \sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52014, size = 72, normalized size = 0.61 \begin{align*} \frac{2}{15} \,{\left (3 \, B b x^{2} + 15 \, A a + 5 \,{\left (B a + A b\right )} x\right )} \sqrt{x} \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{\left (A + B x\right ) \sqrt{\left (a + b x\right )^{2}}}{\sqrt{x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15125, size = 72, normalized size = 0.61 \begin{align*} \frac{2}{5} \, B b x^{\frac{5}{2}} \mathrm{sgn}\left (b x + a\right ) + \frac{2}{3} \, B a x^{\frac{3}{2}} \mathrm{sgn}\left (b x + a\right ) + \frac{2}{3} \, A b x^{\frac{3}{2}} \mathrm{sgn}\left (b x + a\right ) + 2 \, A a \sqrt{x} \mathrm{sgn}\left (b x + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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