Optimal. Leaf size=126 \[ \frac{a^2 (6 A b-5 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 b^{7/2}}+\frac{x^{3/2} \sqrt{a+b x} (6 A b-5 a B)}{12 b^2}-\frac{a \sqrt{x} \sqrt{a+b x} (6 A b-5 a B)}{8 b^3}+\frac{B x^{5/2} \sqrt{a+b x}}{3 b} \]
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Rubi [A] time = 0.0494739, 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} \[ \frac{a^2 (6 A b-5 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 b^{7/2}}+\frac{x^{3/2} \sqrt{a+b x} (6 A b-5 a B)}{12 b^2}-\frac{a \sqrt{x} \sqrt{a+b x} (6 A b-5 a B)}{8 b^3}+\frac{B x^{5/2} \sqrt{a+b x}}{3 b} \]
Antiderivative was successfully verified.
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Rule 80
Rule 50
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{x^{3/2} (A+B x)}{\sqrt{a+b x}} \, dx &=\frac{B x^{5/2} \sqrt{a+b x}}{3 b}+\frac{\left (3 A b-\frac{5 a B}{2}\right ) \int \frac{x^{3/2}}{\sqrt{a+b x}} \, dx}{3 b}\\ &=\frac{(6 A b-5 a B) x^{3/2} \sqrt{a+b x}}{12 b^2}+\frac{B x^{5/2} \sqrt{a+b x}}{3 b}-\frac{(a (6 A b-5 a B)) \int \frac{\sqrt{x}}{\sqrt{a+b x}} \, dx}{8 b^2}\\ &=-\frac{a (6 A b-5 a B) \sqrt{x} \sqrt{a+b x}}{8 b^3}+\frac{(6 A b-5 a B) x^{3/2} \sqrt{a+b x}}{12 b^2}+\frac{B x^{5/2} \sqrt{a+b x}}{3 b}+\frac{\left (a^2 (6 A b-5 a B)\right ) \int \frac{1}{\sqrt{x} \sqrt{a+b x}} \, dx}{16 b^3}\\ &=-\frac{a (6 A b-5 a B) \sqrt{x} \sqrt{a+b x}}{8 b^3}+\frac{(6 A b-5 a B) x^{3/2} \sqrt{a+b x}}{12 b^2}+\frac{B x^{5/2} \sqrt{a+b x}}{3 b}+\frac{\left (a^2 (6 A b-5 a B)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\sqrt{x}\right )}{8 b^3}\\ &=-\frac{a (6 A b-5 a B) \sqrt{x} \sqrt{a+b x}}{8 b^3}+\frac{(6 A b-5 a B) x^{3/2} \sqrt{a+b x}}{12 b^2}+\frac{B x^{5/2} \sqrt{a+b x}}{3 b}+\frac{\left (a^2 (6 A b-5 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^3}\\ &=-\frac{a (6 A b-5 a B) \sqrt{x} \sqrt{a+b x}}{8 b^3}+\frac{(6 A b-5 a B) x^{3/2} \sqrt{a+b x}}{12 b^2}+\frac{B x^{5/2} \sqrt{a+b x}}{3 b}+\frac{a^2 (6 A b-5 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 b^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.0775625, size = 113, normalized size = 0.9 \[ \frac{\sqrt{b} \sqrt{x} (a+b x) \left (15 a^2 B-2 a b (9 A+5 B x)+4 b^2 x (3 A+2 B x)\right )-3 a^{5/2} \sqrt{\frac{b x}{a}+1} (5 a B-6 A b) \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{24 b^{7/2} \sqrt{a+b x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 176, normalized size = 1.4 \begin{align*}{\frac{1}{48}\sqrt{x}\sqrt{bx+a} \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-20\,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-36\,A\sqrt{x \left ( bx+a \right ) }{b}^{3/2}a-15\,B\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){a}^{3}+30\,B\sqrt{x \left ( bx+a \right ) }\sqrt{b}{a}^{2} \right ){b}^{-{\frac{7}{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 = 2.70251, size = 502, normalized size = 3.98 \begin{align*} \left [-\frac{3 \,{\left (5 \, 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} + 15 \, B a^{2} b - 18 \, A a b^{2} - 2 \,{\left (5 \, B a b^{2} - 6 \, A b^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{48 \, b^{4}}, \frac{3 \,{\left (5 \, 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} + 15 \, B a^{2} b - 18 \, A a b^{2} - 2 \,{\left (5 \, B a b^{2} - 6 \, A b^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{24 \, b^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 37.33, size = 245, normalized size = 1.94 \begin{align*} - \frac{3 A a^{\frac{3}{2}} \sqrt{x}}{4 b^{2} \sqrt{1 + \frac{b x}{a}}} - \frac{A \sqrt{a} x^{\frac{3}{2}}}{4 b \sqrt{1 + \frac{b x}{a}}} + \frac{3 A a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 b^{\frac{5}{2}}} + \frac{A x^{\frac{5}{2}}}{2 \sqrt{a} \sqrt{1 + \frac{b x}{a}}} + \frac{5 B a^{\frac{5}{2}} \sqrt{x}}{8 b^{3} \sqrt{1 + \frac{b x}{a}}} + \frac{5 B a^{\frac{3}{2}} x^{\frac{3}{2}}}{24 b^{2} \sqrt{1 + \frac{b x}{a}}} - \frac{B \sqrt{a} x^{\frac{5}{2}}}{12 b \sqrt{1 + \frac{b x}{a}}} - \frac{5 B a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{8 b^{\frac{7}{2}}} + \frac{B x^{\frac{7}{2}}}{3 \sqrt{a} \sqrt{1 + \frac{b x}{a}}} \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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