Optimal. Leaf size=116 \[ -\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{\sqrt{x} (a+b x)}-\frac{2 a A \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^{3/2} (a+b x)}+\frac{2 b B \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]
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Rubi [A] time = 0.0435198, antiderivative size = 116, 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} \[ -\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{\sqrt{x} (a+b x)}-\frac{2 a A \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^{3/2} (a+b x)}+\frac{2 b B \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]
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}}{x^{5/2}} \, 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)}{x^{5/2}} \, dx}{a b+b^2 x}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (\frac{a A b}{x^{5/2}}+\frac{b (A b+a B)}{x^{3/2}}+\frac{b^2 B}{\sqrt{x}}\right ) \, dx}{a b+b^2 x}\\ &=-\frac{2 a A \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^{3/2} (a+b x)}-\frac{2 (A b+a B) \sqrt{a^2+2 a b x+b^2 x^2}}{\sqrt{x} (a+b x)}+\frac{2 b B \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}\\ \end{align*}
Mathematica [A] time = 0.028561, size = 46, normalized size = 0.4 \[ -\frac{2 \sqrt{(a+b x)^2} (a (A+3 B x)+3 b x (A-B x))}{3 x^{3/2} (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 43, normalized size = 0.4 \begin{align*} -{\frac{-6\,Bb{x}^{2}+6\,Abx+6\,aBx+2\,aA}{3\,bx+3\,a}\sqrt{ \left ( bx+a \right ) ^{2}}{x}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12713, size = 45, normalized size = 0.39 \begin{align*} \frac{2 \,{\left (b x^{2} - a x\right )} B}{x^{\frac{3}{2}}} - \frac{2 \,{\left (3 \, b x^{2} + a x\right )} A}{3 \, x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53443, size = 66, normalized size = 0.57 \begin{align*} \frac{2 \,{\left (3 \, B b x^{2} - A a - 3 \,{\left (B a + A b\right )} x\right )}}{3 \, x^{\frac{3}{2}}} \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 = 1.17982, size = 69, normalized size = 0.59 \begin{align*} 2 \, B b \sqrt{x} \mathrm{sgn}\left (b x + a\right ) - \frac{2 \,{\left (3 \, B a x \mathrm{sgn}\left (b x + a\right ) + 3 \, A b x \mathrm{sgn}\left (b x + a\right ) + A a \mathrm{sgn}\left (b x + a\right )\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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