Optimal. Leaf size=144 \[ \frac{5 a^2 (a B+6 A b) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 \sqrt{b}}+\frac{\sqrt{x} (a+b x)^{5/2} (a B+6 A b)}{3 a}+\frac{5}{12} \sqrt{x} (a+b x)^{3/2} (a B+6 A b)+\frac{5}{8} a \sqrt{x} \sqrt{a+b x} (a B+6 A b)-\frac{2 A (a+b x)^{7/2}}{a \sqrt{x}} \]
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Rubi [A] time = 0.0626256, antiderivative size = 144, 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 = {78, 50, 63, 217, 206} \[ \frac{5 a^2 (a B+6 A b) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 \sqrt{b}}+\frac{\sqrt{x} (a+b x)^{5/2} (a B+6 A b)}{3 a}+\frac{5}{12} \sqrt{x} (a+b x)^{3/2} (a B+6 A b)+\frac{5}{8} a \sqrt{x} \sqrt{a+b x} (a B+6 A b)-\frac{2 A (a+b x)^{7/2}}{a \sqrt{x}} \]
Antiderivative was successfully verified.
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Rule 78
Rule 50
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{(a+b x)^{5/2} (A+B x)}{x^{3/2}} \, dx &=-\frac{2 A (a+b x)^{7/2}}{a \sqrt{x}}+\frac{\left (2 \left (3 A b+\frac{a B}{2}\right )\right ) \int \frac{(a+b x)^{5/2}}{\sqrt{x}} \, dx}{a}\\ &=\frac{(6 A b+a B) \sqrt{x} (a+b x)^{5/2}}{3 a}-\frac{2 A (a+b x)^{7/2}}{a \sqrt{x}}+\frac{1}{6} (5 (6 A b+a B)) \int \frac{(a+b x)^{3/2}}{\sqrt{x}} \, dx\\ &=\frac{5}{12} (6 A b+a B) \sqrt{x} (a+b x)^{3/2}+\frac{(6 A b+a B) \sqrt{x} (a+b x)^{5/2}}{3 a}-\frac{2 A (a+b x)^{7/2}}{a \sqrt{x}}+\frac{1}{8} (5 a (6 A b+a B)) \int \frac{\sqrt{a+b x}}{\sqrt{x}} \, dx\\ &=\frac{5}{8} a (6 A b+a B) \sqrt{x} \sqrt{a+b x}+\frac{5}{12} (6 A b+a B) \sqrt{x} (a+b x)^{3/2}+\frac{(6 A b+a B) \sqrt{x} (a+b x)^{5/2}}{3 a}-\frac{2 A (a+b x)^{7/2}}{a \sqrt{x}}+\frac{1}{16} \left (5 a^2 (6 A b+a B)\right ) \int \frac{1}{\sqrt{x} \sqrt{a+b x}} \, dx\\ &=\frac{5}{8} a (6 A b+a B) \sqrt{x} \sqrt{a+b x}+\frac{5}{12} (6 A b+a B) \sqrt{x} (a+b x)^{3/2}+\frac{(6 A b+a B) \sqrt{x} (a+b x)^{5/2}}{3 a}-\frac{2 A (a+b x)^{7/2}}{a \sqrt{x}}+\frac{1}{8} \left (5 a^2 (6 A b+a B)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\sqrt{x}\right )\\ &=\frac{5}{8} a (6 A b+a B) \sqrt{x} \sqrt{a+b x}+\frac{5}{12} (6 A b+a B) \sqrt{x} (a+b x)^{3/2}+\frac{(6 A b+a B) \sqrt{x} (a+b x)^{5/2}}{3 a}-\frac{2 A (a+b x)^{7/2}}{a \sqrt{x}}+\frac{1}{8} \left (5 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 )\\ &=\frac{5}{8} a (6 A b+a B) \sqrt{x} \sqrt{a+b x}+\frac{5}{12} (6 A b+a B) \sqrt{x} (a+b x)^{3/2}+\frac{(6 A b+a B) \sqrt{x} (a+b x)^{5/2}}{3 a}-\frac{2 A (a+b x)^{7/2}}{a \sqrt{x}}+\frac{5 a^2 (6 A b+a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 \sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.232181, size = 111, normalized size = 0.77 \[ \frac{1}{24} \sqrt{a+b x} \left (\frac{a^2 (33 B x-48 A)+2 a b x (27 A+13 B x)+4 b^2 x^2 (3 A+2 B x)}{\sqrt{x}}+\frac{15 a^{3/2} (a B+6 A b) \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{b} \sqrt{\frac{b x}{a}+1}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 202, normalized size = 1.4 \begin{align*}{\frac{1}{48}\sqrt{bx+a} \left ( 16\,B\sqrt{x \left ( bx+a \right ) }{b}^{5/2}{x}^{3}+24\,A\sqrt{x \left ( bx+a \right ) }{b}^{5/2}{x}^{2}+52\,B\sqrt{x \left ( bx+a \right ) }{b}^{3/2}{x}^{2}a+90\,Ab\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) x{a}^{2}+108\,Axa{b}^{3/2}\sqrt{x \left ( bx+a \right ) }+15\,B\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) x{a}^{3}+66\,Bx{a}^{2}\sqrt{x \left ( bx+a \right ) }\sqrt{b}-96\,A{a}^{2}\sqrt{x \left ( bx+a \right ) }\sqrt{b} \right ){\frac{1}{\sqrt{x}}}{\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 = 2.74425, size = 570, normalized size = 3.96 \begin{align*} \left [\frac{15 \,{\left (B a^{3} + 6 \, A a^{2} b\right )} \sqrt{b} x \log \left (2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x} + a\right ) + 2 \,{\left (8 \, B b^{3} x^{3} - 48 \, A a^{2} b + 2 \,{\left (13 \, B a b^{2} + 6 \, A b^{3}\right )} x^{2} + 3 \,{\left (11 \, B a^{2} b + 18 \, A a b^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{48 \, b x}, -\frac{15 \,{\left (B a^{3} + 6 \, A a^{2} b\right )} \sqrt{-b} x \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) -{\left (8 \, B b^{3} x^{3} - 48 \, A a^{2} b + 2 \,{\left (13 \, B a b^{2} + 6 \, A b^{3}\right )} x^{2} + 3 \,{\left (11 \, B a^{2} b + 18 \, A a b^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{24 \, b x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 140.612, size = 233, normalized size = 1.62 \begin{align*} A \left (- \frac{2 a^{\frac{5}{2}}}{\sqrt{x} \sqrt{1 + \frac{b x}{a}}} + \frac{a^{\frac{3}{2}} b \sqrt{x}}{4 \sqrt{1 + \frac{b x}{a}}} + \frac{11 \sqrt{a} b^{2} x^{\frac{3}{2}}}{4 \sqrt{1 + \frac{b x}{a}}} + \frac{15 a^{2} \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4} + \frac{b^{3} x^{\frac{5}{2}}}{2 \sqrt{a} \sqrt{1 + \frac{b x}{a}}}\right ) + B \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 ) \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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