Optimal. Leaf size=167 \[ \frac{5 a^2 (6 A b-7 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 b^{9/2}}-\frac{x^{5/2} \sqrt{a+b x} (6 A b-7 a B)}{3 a b^2}+\frac{5 x^{3/2} \sqrt{a+b x} (6 A b-7 a B)}{12 b^3}-\frac{5 a \sqrt{x} \sqrt{a+b x} (6 A b-7 a B)}{8 b^4}+\frac{2 x^{7/2} (A b-a B)}{a b \sqrt{a+b x}} \]
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Rubi [A] time = 0.0690089, antiderivative size = 167, 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} \[ \frac{5 a^2 (6 A b-7 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 b^{9/2}}-\frac{x^{5/2} \sqrt{a+b x} (6 A b-7 a B)}{3 a b^2}+\frac{5 x^{3/2} \sqrt{a+b x} (6 A b-7 a B)}{12 b^3}-\frac{5 a \sqrt{x} \sqrt{a+b x} (6 A b-7 a B)}{8 b^4}+\frac{2 x^{7/2} (A b-a B)}{a b \sqrt{a+b x}} \]
Antiderivative was successfully verified.
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Rule 78
Rule 50
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{x^{5/2} (A+B x)}{(a+b x)^{3/2}} \, dx &=\frac{2 (A b-a B) x^{7/2}}{a b \sqrt{a+b x}}-\frac{\left (2 \left (3 A b-\frac{7 a B}{2}\right )\right ) \int \frac{x^{5/2}}{\sqrt{a+b x}} \, dx}{a b}\\ &=\frac{2 (A b-a B) x^{7/2}}{a b \sqrt{a+b x}}-\frac{(6 A b-7 a B) x^{5/2} \sqrt{a+b x}}{3 a b^2}+\frac{(5 (6 A b-7 a B)) \int \frac{x^{3/2}}{\sqrt{a+b x}} \, dx}{6 b^2}\\ &=\frac{2 (A b-a B) x^{7/2}}{a b \sqrt{a+b x}}+\frac{5 (6 A b-7 a B) x^{3/2} \sqrt{a+b x}}{12 b^3}-\frac{(6 A b-7 a B) x^{5/2} \sqrt{a+b x}}{3 a b^2}-\frac{(5 a (6 A b-7 a B)) \int \frac{\sqrt{x}}{\sqrt{a+b x}} \, dx}{8 b^3}\\ &=\frac{2 (A b-a B) x^{7/2}}{a b \sqrt{a+b x}}-\frac{5 a (6 A b-7 a B) \sqrt{x} \sqrt{a+b x}}{8 b^4}+\frac{5 (6 A b-7 a B) x^{3/2} \sqrt{a+b x}}{12 b^3}-\frac{(6 A b-7 a B) x^{5/2} \sqrt{a+b x}}{3 a b^2}+\frac{\left (5 a^2 (6 A b-7 a B)\right ) \int \frac{1}{\sqrt{x} \sqrt{a+b x}} \, dx}{16 b^4}\\ &=\frac{2 (A b-a B) x^{7/2}}{a b \sqrt{a+b x}}-\frac{5 a (6 A b-7 a B) \sqrt{x} \sqrt{a+b x}}{8 b^4}+\frac{5 (6 A b-7 a B) x^{3/2} \sqrt{a+b x}}{12 b^3}-\frac{(6 A b-7 a B) x^{5/2} \sqrt{a+b x}}{3 a b^2}+\frac{\left (5 a^2 (6 A b-7 a B)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\sqrt{x}\right )}{8 b^4}\\ &=\frac{2 (A b-a B) x^{7/2}}{a b \sqrt{a+b x}}-\frac{5 a (6 A b-7 a B) \sqrt{x} \sqrt{a+b x}}{8 b^4}+\frac{5 (6 A b-7 a B) x^{3/2} \sqrt{a+b x}}{12 b^3}-\frac{(6 A b-7 a B) x^{5/2} \sqrt{a+b x}}{3 a b^2}+\frac{\left (5 a^2 (6 A b-7 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^4}\\ &=\frac{2 (A b-a B) x^{7/2}}{a b \sqrt{a+b x}}-\frac{5 a (6 A b-7 a B) \sqrt{x} \sqrt{a+b x}}{8 b^4}+\frac{5 (6 A b-7 a B) x^{3/2} \sqrt{a+b x}}{12 b^3}-\frac{(6 A b-7 a B) x^{5/2} \sqrt{a+b x}}{3 a b^2}+\frac{5 a^2 (6 A b-7 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 b^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.167263, size = 140, normalized size = 0.84 \[ \frac{\frac{(a+b x) (7 a B-6 A b) \left (b x \sqrt{\frac{b x}{a}+1} \left (15 a^2-10 a b x+8 b^2 x^2\right )-15 a^{5/2} \sqrt{b} \sqrt{x} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )\right )}{3 \sqrt{\frac{b x}{a}+1}}+16 b^4 x^4 (A b-a B)}{8 a b^5 \sqrt{x} \sqrt{a+b x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.016, size = 288, normalized size = 1.7 \begin{align*}{\frac{1}{48} \left ( 16\,B{x}^{3}{b}^{7/2}\sqrt{x \left ( bx+a \right ) }+24\,A{x}^{2}{b}^{7/2}\sqrt{x \left ( bx+a \right ) }-28\,B{x}^{2}a{b}^{5/2}\sqrt{x \left ( bx+a \right ) }+90\,A\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) x{a}^{2}{b}^{2}-60\,A\sqrt{x \left ( bx+a \right ) }{b}^{5/2}xa-105\,B\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) x{a}^{3}b+70\,B\sqrt{x \left ( bx+a \right ) }{b}^{3/2}x{a}^{2}+90\,A\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){a}^{3}b-180\,A\sqrt{x \left ( bx+a \right ) }{b}^{3/2}{a}^{2}-105\,B\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){a}^{4}+210\,B\sqrt{x \left ( bx+a \right ) }\sqrt{b}{a}^{3} \right ) \sqrt{x}{b}^{-{\frac{9}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}{\frac{1}{\sqrt{bx+a}}}} \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 = 2.82028, size = 718, normalized size = 4.3 \begin{align*} \left [-\frac{15 \,{\left (7 \, B a^{4} - 6 \, A a^{3} b +{\left (7 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} x\right )} \sqrt{b} \log \left (2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x} + a\right ) - 2 \,{\left (8 \, B b^{4} x^{3} + 105 \, B a^{3} b - 90 \, A a^{2} b^{2} - 2 \,{\left (7 \, B a b^{3} - 6 \, A b^{4}\right )} x^{2} + 5 \,{\left (7 \, B a^{2} b^{2} - 6 \, A a b^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{48 \,{\left (b^{6} x + a b^{5}\right )}}, \frac{15 \,{\left (7 \, B a^{4} - 6 \, A a^{3} b +{\left (7 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} x\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) +{\left (8 \, B b^{4} x^{3} + 105 \, B a^{3} b - 90 \, A a^{2} b^{2} - 2 \,{\left (7 \, B a b^{3} - 6 \, A b^{4}\right )} x^{2} + 5 \,{\left (7 \, B a^{2} b^{2} - 6 \, A a b^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{24 \,{\left (b^{6} x + a b^{5}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [A] time = 82.7259, size = 293, normalized size = 1.75 \begin{align*} \frac{1}{24} \, \sqrt{{\left (b x + a\right )} b - a b} \sqrt{b x + a}{\left (2 \,{\left (b x + a\right )}{\left (\frac{4 \,{\left (b x + a\right )} B{\left | b \right |}}{b^{6}} - \frac{19 \, B a b^{17}{\left | b \right |} - 6 \, A b^{18}{\left | b \right |}}{b^{23}}\right )} + \frac{3 \,{\left (29 \, B a^{2} b^{17}{\left | b \right |} - 18 \, A a b^{18}{\left | b \right |}\right )}}{b^{23}}\right )} + \frac{5 \,{\left (7 \, B a^{3} \sqrt{b}{\left | b \right |} - 6 \, A a^{2} b^{\frac{3}{2}}{\left | b \right |}\right )} \log \left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2}\right )}{16 \, b^{6}} + \frac{4 \,{\left (B a^{4} \sqrt{b}{\left | b \right |} - A a^{3} b^{\frac{3}{2}}{\left | b \right |}\right )}}{{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )} b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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