Optimal. Leaf size=159 \[ \frac{5 a^3 (8 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{64 b^{3/2}}+\frac{5 a^2 \sqrt{x} \sqrt{a+b x} (8 A b-a B)}{64 b}+\frac{\sqrt{x} (a+b x)^{5/2} (8 A b-a B)}{24 b}+\frac{5 a \sqrt{x} (a+b x)^{3/2} (8 A b-a B)}{96 b}+\frac{B \sqrt{x} (a+b x)^{7/2}}{4 b} \]
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Rubi [A] time = 0.065605, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{5 a^3 (8 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{64 b^{3/2}}+\frac{5 a^2 \sqrt{x} \sqrt{a+b x} (8 A b-a B)}{64 b}+\frac{\sqrt{x} (a+b x)^{5/2} (8 A b-a B)}{24 b}+\frac{5 a \sqrt{x} (a+b x)^{3/2} (8 A b-a B)}{96 b}+\frac{B \sqrt{x} (a+b x)^{7/2}}{4 b} \]
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)^{5/2} (A+B x)}{\sqrt{x}} \, dx &=\frac{B \sqrt{x} (a+b x)^{7/2}}{4 b}+\frac{\left (4 A b-\frac{a B}{2}\right ) \int \frac{(a+b x)^{5/2}}{\sqrt{x}} \, dx}{4 b}\\ &=\frac{(8 A b-a B) \sqrt{x} (a+b x)^{5/2}}{24 b}+\frac{B \sqrt{x} (a+b x)^{7/2}}{4 b}+\frac{(5 a (8 A b-a B)) \int \frac{(a+b x)^{3/2}}{\sqrt{x}} \, dx}{48 b}\\ &=\frac{5 a (8 A b-a B) \sqrt{x} (a+b x)^{3/2}}{96 b}+\frac{(8 A b-a B) \sqrt{x} (a+b x)^{5/2}}{24 b}+\frac{B \sqrt{x} (a+b x)^{7/2}}{4 b}+\frac{\left (5 a^2 (8 A b-a B)\right ) \int \frac{\sqrt{a+b x}}{\sqrt{x}} \, dx}{64 b}\\ &=\frac{5 a^2 (8 A b-a B) \sqrt{x} \sqrt{a+b x}}{64 b}+\frac{5 a (8 A b-a B) \sqrt{x} (a+b x)^{3/2}}{96 b}+\frac{(8 A b-a B) \sqrt{x} (a+b x)^{5/2}}{24 b}+\frac{B \sqrt{x} (a+b x)^{7/2}}{4 b}+\frac{\left (5 a^3 (8 A b-a B)\right ) \int \frac{1}{\sqrt{x} \sqrt{a+b x}} \, dx}{128 b}\\ &=\frac{5 a^2 (8 A b-a B) \sqrt{x} \sqrt{a+b x}}{64 b}+\frac{5 a (8 A b-a B) \sqrt{x} (a+b x)^{3/2}}{96 b}+\frac{(8 A b-a B) \sqrt{x} (a+b x)^{5/2}}{24 b}+\frac{B \sqrt{x} (a+b x)^{7/2}}{4 b}+\frac{\left (5 a^3 (8 A b-a B)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\sqrt{x}\right )}{64 b}\\ &=\frac{5 a^2 (8 A b-a B) \sqrt{x} \sqrt{a+b x}}{64 b}+\frac{5 a (8 A b-a B) \sqrt{x} (a+b x)^{3/2}}{96 b}+\frac{(8 A b-a B) \sqrt{x} (a+b x)^{5/2}}{24 b}+\frac{B \sqrt{x} (a+b x)^{7/2}}{4 b}+\frac{\left (5 a^3 (8 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 )}{64 b}\\ &=\frac{5 a^2 (8 A b-a B) \sqrt{x} \sqrt{a+b x}}{64 b}+\frac{5 a (8 A b-a B) \sqrt{x} (a+b x)^{3/2}}{96 b}+\frac{(8 A b-a B) \sqrt{x} (a+b x)^{5/2}}{24 b}+\frac{B \sqrt{x} (a+b x)^{7/2}}{4 b}+\frac{5 a^3 (8 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{64 b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.232588, size = 126, normalized size = 0.79 \[ \frac{\sqrt{a+b x} \left (\sqrt{b} \sqrt{x} \left (2 a^2 b (132 A+59 B x)+15 a^3 B+8 a b^2 x (26 A+17 B x)+16 b^3 x^2 (4 A+3 B x)\right )-\frac{15 a^{5/2} (a B-8 A b) \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{\frac{b x}{a}+1}}\right )}{192 b^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 218, normalized size = 1.4 \begin{align*}{\frac{1}{384}\sqrt{bx+a}\sqrt{x} \left ( 96\,B{x}^{3}{b}^{7/2}\sqrt{x \left ( bx+a \right ) }+128\,A{x}^{2}{b}^{7/2}\sqrt{x \left ( bx+a \right ) }+272\,B{x}^{2}a{b}^{5/2}\sqrt{x \left ( bx+a \right ) }+416\,A\sqrt{x \left ( bx+a \right ) }{b}^{5/2}xa+236\,B\sqrt{x \left ( bx+a \right ) }{b}^{3/2}x{a}^{2}+120\,A\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){a}^{3}b+528\,A\sqrt{x \left ( bx+a \right ) }{b}^{3/2}{a}^{2}-15\,B\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){a}^{4}+30\,B\sqrt{x \left ( bx+a \right ) }\sqrt{b}{a}^{3} \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 = 2.61389, size = 616, normalized size = 3.87 \begin{align*} \left [-\frac{15 \,{\left (B a^{4} - 8 \, A a^{3} b\right )} \sqrt{b} \log \left (2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x} + a\right ) - 2 \,{\left (48 \, B b^{4} x^{3} + 15 \, B a^{3} b + 264 \, A a^{2} b^{2} + 8 \,{\left (17 \, B a b^{3} + 8 \, A b^{4}\right )} x^{2} + 2 \,{\left (59 \, B a^{2} b^{2} + 104 \, A a b^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{384 \, b^{2}}, \frac{15 \,{\left (B a^{4} - 8 \, A a^{3} b\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) +{\left (48 \, B b^{4} x^{3} + 15 \, B a^{3} b + 264 \, A a^{2} b^{2} + 8 \,{\left (17 \, B a b^{3} + 8 \, A b^{4}\right )} x^{2} + 2 \,{\left (59 \, B a^{2} b^{2} + 104 \, A a b^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{192 \, b^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 130.176, size = 262, normalized size = 1.65 \begin{align*} A \left (\frac{11 a^{\frac{5}{2}} \sqrt{x} \sqrt{1 + \frac{b x}{a}}}{8} + \frac{13 a^{\frac{3}{2}} b x^{\frac{3}{2}} \sqrt{1 + \frac{b x}{a}}}{12} + \frac{\sqrt{a} b^{2} x^{\frac{5}{2}} \sqrt{1 + \frac{b x}{a}}}{3} + \frac{5 a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{8 \sqrt{b}}\right ) + B \left (\frac{5 a^{\frac{7}{2}} \sqrt{x}}{64 b \sqrt{1 + \frac{b x}{a}}} + \frac{133 a^{\frac{5}{2}} x^{\frac{3}{2}}}{192 \sqrt{1 + \frac{b x}{a}}} + \frac{127 a^{\frac{3}{2}} b x^{\frac{5}{2}}}{96 \sqrt{1 + \frac{b x}{a}}} + \frac{23 \sqrt{a} b^{2} x^{\frac{7}{2}}}{24 \sqrt{1 + \frac{b x}{a}}} - \frac{5 a^{4} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{64 b^{\frac{3}{2}}} + \frac{b^{3} x^{\frac{9}{2}}}{4 \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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