Optimal. Leaf size=69 \[ -\frac{2 A (a+b x)^{3/2}}{3 a x^{3/2}}-\frac{2 B \sqrt{a+b x}}{\sqrt{x}}+2 \sqrt{b} B \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right ) \]
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Rubi [A] time = 0.0236461, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {78, 47, 63, 217, 206} \[ -\frac{2 A (a+b x)^{3/2}}{3 a x^{3/2}}-\frac{2 B \sqrt{a+b x}}{\sqrt{x}}+2 \sqrt{b} B \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right ) \]
Antiderivative was successfully verified.
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Rule 78
Rule 47
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x} (A+B x)}{x^{5/2}} \, dx &=-\frac{2 A (a+b x)^{3/2}}{3 a x^{3/2}}+B \int \frac{\sqrt{a+b x}}{x^{3/2}} \, dx\\ &=-\frac{2 B \sqrt{a+b x}}{\sqrt{x}}-\frac{2 A (a+b x)^{3/2}}{3 a x^{3/2}}+(b B) \int \frac{1}{\sqrt{x} \sqrt{a+b x}} \, dx\\ &=-\frac{2 B \sqrt{a+b x}}{\sqrt{x}}-\frac{2 A (a+b x)^{3/2}}{3 a x^{3/2}}+(2 b B) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\sqrt{x}\right )\\ &=-\frac{2 B \sqrt{a+b x}}{\sqrt{x}}-\frac{2 A (a+b x)^{3/2}}{3 a x^{3/2}}+(2 b B) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{a+b x}}\right )\\ &=-\frac{2 B \sqrt{a+b x}}{\sqrt{x}}-\frac{2 A (a+b x)^{3/2}}{3 a x^{3/2}}+2 \sqrt{b} B \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )\\ \end{align*}
Mathematica [C] time = 0.0728484, size = 85, normalized size = 1.23 \[ -\frac{2 (a+b x)^{3/2} (A b-a B)}{3 a b x^{3/2}}-\frac{2 a B \sqrt{a+b x} \, _2F_1\left (-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};-\frac{b x}{a}\right )}{3 b x^{3/2} \sqrt{\frac{b x}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 112, normalized size = 1.6 \begin{align*} -{\frac{1}{3\,a}\sqrt{bx+a} \left ( -3\,B\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){x}^{2}ab+2\,Ax{b}^{3/2}\sqrt{x \left ( bx+a \right ) }+6\,Bxa\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,Aa\sqrt{x \left ( bx+a \right ) }\sqrt{b} \right ){x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}{\frac{1}{\sqrt{b}}}} \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 = 2.60099, size = 354, normalized size = 5.13 \begin{align*} \left [\frac{3 \, B a \sqrt{b} x^{2} \log \left (2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x} + a\right ) - 2 \,{\left (A a +{\left (3 \, B a + A b\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{3 \, a x^{2}}, -\frac{2 \,{\left (3 \, B a \sqrt{-b} x^{2} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) +{\left (A a +{\left (3 \, B a + A b\right )} x\right )} \sqrt{b x + a} \sqrt{x}\right )}}{3 \, a x^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 26.6212, size = 114, normalized size = 1.65 \begin{align*} A \left (- \frac{2 \sqrt{b} \sqrt{\frac{a}{b x} + 1}}{3 x} - \frac{2 b^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}}{3 a}\right ) + B \left (- \frac{2 \sqrt{a}}{\sqrt{x} \sqrt{1 + \frac{b x}{a}}} + 2 \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )} - \frac{2 b \sqrt{x}}{\sqrt{a} \sqrt{1 + \frac{b x}{a}}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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