Optimal. Leaf size=126 \[ \frac{a^2 (6 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 b^{3/2}}+\frac{\sqrt{x} (a+b x)^{3/2} (6 A b-a B)}{12 b}+\frac{a \sqrt{x} \sqrt{a+b x} (6 A b-a B)}{8 b}+\frac{B \sqrt{x} (a+b x)^{5/2}}{3 b} \]
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Rubi [A] time = 0.0538404, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {80, 50, 63, 217, 206} \[ \frac{a^2 (6 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 b^{3/2}}+\frac{\sqrt{x} (a+b x)^{3/2} (6 A b-a B)}{12 b}+\frac{a \sqrt{x} \sqrt{a+b x} (6 A b-a B)}{8 b}+\frac{B \sqrt{x} (a+b x)^{5/2}}{3 b} \]
Antiderivative was successfully verified.
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Rule 80
Rule 50
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{(a+b x)^{3/2} (A+B x)}{\sqrt{x}} \, dx &=\frac{B \sqrt{x} (a+b x)^{5/2}}{3 b}+\frac{\left (3 A b-\frac{a B}{2}\right ) \int \frac{(a+b x)^{3/2}}{\sqrt{x}} \, dx}{3 b}\\ &=\frac{(6 A b-a B) \sqrt{x} (a+b x)^{3/2}}{12 b}+\frac{B \sqrt{x} (a+b x)^{5/2}}{3 b}+\frac{(a (6 A b-a B)) \int \frac{\sqrt{a+b x}}{\sqrt{x}} \, dx}{8 b}\\ &=\frac{a (6 A b-a B) \sqrt{x} \sqrt{a+b x}}{8 b}+\frac{(6 A b-a B) \sqrt{x} (a+b x)^{3/2}}{12 b}+\frac{B \sqrt{x} (a+b x)^{5/2}}{3 b}+\frac{\left (a^2 (6 A b-a B)\right ) \int \frac{1}{\sqrt{x} \sqrt{a+b x}} \, dx}{16 b}\\ &=\frac{a (6 A b-a B) \sqrt{x} \sqrt{a+b x}}{8 b}+\frac{(6 A b-a B) \sqrt{x} (a+b x)^{3/2}}{12 b}+\frac{B \sqrt{x} (a+b x)^{5/2}}{3 b}+\frac{\left (a^2 (6 A b-a B)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\sqrt{x}\right )}{8 b}\\ &=\frac{a (6 A b-a B) \sqrt{x} \sqrt{a+b x}}{8 b}+\frac{(6 A b-a B) \sqrt{x} (a+b x)^{3/2}}{12 b}+\frac{B \sqrt{x} (a+b x)^{5/2}}{3 b}+\frac{\left (a^2 (6 A b-a B)\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{a+b x}}\right )}{8 b}\\ &=\frac{a (6 A b-a B) \sqrt{x} \sqrt{a+b x}}{8 b}+\frac{(6 A b-a B) \sqrt{x} (a+b x)^{3/2}}{12 b}+\frac{B \sqrt{x} (a+b x)^{5/2}}{3 b}+\frac{a^2 (6 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.230532, size = 107, normalized size = 0.85 \[ \frac{\sqrt{a+b x} \left (\sqrt{b} \sqrt{x} \left (3 a^2 B+2 a b (15 A+7 B x)+4 b^2 x (3 A+2 B x)\right )-\frac{3 a^{3/2} (a B-6 A b) \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{\frac{b x}{a}+1}}\right )}{24 b^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 176, normalized size = 1.4 \begin{align*}{\frac{1}{48}\sqrt{bx+a}\sqrt{x} \left ( 16\,B{x}^{2}{b}^{5/2}\sqrt{x \left ( bx+a \right ) }+24\,A\sqrt{x \left ( bx+a \right ) }{b}^{5/2}x+28\,B\sqrt{x \left ( bx+a \right ) }{b}^{3/2}xa+18\,A\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){a}^{2}b+60\,A\sqrt{x \left ( bx+a \right ) }{b}^{3/2}a-3\,B\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){a}^{3}+6\,B\sqrt{x \left ( bx+a \right ) }\sqrt{b}{a}^{2} \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}} \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 = 2.62616, size = 494, normalized size = 3.92 \begin{align*} \left [-\frac{3 \,{\left (B a^{3} - 6 \, A a^{2} b\right )} \sqrt{b} \log \left (2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x} + a\right ) - 2 \,{\left (8 \, B b^{3} x^{2} + 3 \, B a^{2} b + 30 \, A a b^{2} + 2 \,{\left (7 \, B a b^{2} + 6 \, A b^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{48 \, b^{2}}, \frac{3 \,{\left (B a^{3} - 6 \, A a^{2} b\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) +{\left (8 \, B b^{3} x^{2} + 3 \, B a^{2} b + 30 \, A a b^{2} + 2 \,{\left (7 \, B a b^{2} + 6 \, A b^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{24 \, b^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 67.7662, size = 204, normalized size = 1.62 \begin{align*} A \left (\frac{5 a^{\frac{3}{2}} \sqrt{x} \sqrt{1 + \frac{b x}{a}}}{4} + \frac{\sqrt{a} b x^{\frac{3}{2}} \sqrt{1 + \frac{b x}{a}}}{2} + \frac{3 a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 \sqrt{b}}\right ) + B \left (\frac{a^{\frac{5}{2}} \sqrt{x}}{8 b \sqrt{1 + \frac{b x}{a}}} + \frac{17 a^{\frac{3}{2}} x^{\frac{3}{2}}}{24 \sqrt{1 + \frac{b x}{a}}} + \frac{11 \sqrt{a} b x^{\frac{5}{2}}}{12 \sqrt{1 + \frac{b x}{a}}} - \frac{a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{8 b^{\frac{3}{2}}} + \frac{b^{2} x^{\frac{7}{2}}}{3 \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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