Optimal. Leaf size=169 \[ \frac{2 x^{5/2} (4 A b-7 a B)}{3 a b^2 \sqrt{a+b x}}-\frac{5 x^{3/2} \sqrt{a+b x} (4 A b-7 a B)}{6 a b^3}+\frac{5 \sqrt{x} \sqrt{a+b x} (4 A b-7 a B)}{4 b^4}-\frac{5 a (4 A b-7 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{4 b^{9/2}}+\frac{2 x^{7/2} (A b-a B)}{3 a b (a+b x)^{3/2}} \]
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Rubi [A] time = 0.0693003, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {78, 47, 50, 63, 217, 206} \[ \frac{2 x^{5/2} (4 A b-7 a B)}{3 a b^2 \sqrt{a+b x}}-\frac{5 x^{3/2} \sqrt{a+b x} (4 A b-7 a B)}{6 a b^3}+\frac{5 \sqrt{x} \sqrt{a+b x} (4 A b-7 a B)}{4 b^4}-\frac{5 a (4 A b-7 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{4 b^{9/2}}+\frac{2 x^{7/2} (A b-a B)}{3 a b (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 78
Rule 47
Rule 50
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{x^{5/2} (A+B x)}{(a+b x)^{5/2}} \, dx &=\frac{2 (A b-a B) x^{7/2}}{3 a b (a+b x)^{3/2}}-\frac{\left (2 \left (2 A b-\frac{7 a B}{2}\right )\right ) \int \frac{x^{5/2}}{(a+b x)^{3/2}} \, dx}{3 a b}\\ &=\frac{2 (A b-a B) x^{7/2}}{3 a b (a+b x)^{3/2}}+\frac{2 (4 A b-7 a B) x^{5/2}}{3 a b^2 \sqrt{a+b x}}-\frac{(5 (4 A b-7 a B)) \int \frac{x^{3/2}}{\sqrt{a+b x}} \, dx}{3 a b^2}\\ &=\frac{2 (A b-a B) x^{7/2}}{3 a b (a+b x)^{3/2}}+\frac{2 (4 A b-7 a B) x^{5/2}}{3 a b^2 \sqrt{a+b x}}-\frac{5 (4 A b-7 a B) x^{3/2} \sqrt{a+b x}}{6 a b^3}+\frac{(5 (4 A b-7 a B)) \int \frac{\sqrt{x}}{\sqrt{a+b x}} \, dx}{4 b^3}\\ &=\frac{2 (A b-a B) x^{7/2}}{3 a b (a+b x)^{3/2}}+\frac{2 (4 A b-7 a B) x^{5/2}}{3 a b^2 \sqrt{a+b x}}+\frac{5 (4 A b-7 a B) \sqrt{x} \sqrt{a+b x}}{4 b^4}-\frac{5 (4 A b-7 a B) x^{3/2} \sqrt{a+b x}}{6 a b^3}-\frac{(5 a (4 A b-7 a B)) \int \frac{1}{\sqrt{x} \sqrt{a+b x}} \, dx}{8 b^4}\\ &=\frac{2 (A b-a B) x^{7/2}}{3 a b (a+b x)^{3/2}}+\frac{2 (4 A b-7 a B) x^{5/2}}{3 a b^2 \sqrt{a+b x}}+\frac{5 (4 A b-7 a B) \sqrt{x} \sqrt{a+b x}}{4 b^4}-\frac{5 (4 A b-7 a B) x^{3/2} \sqrt{a+b x}}{6 a b^3}-\frac{(5 a (4 A b-7 a B)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\sqrt{x}\right )}{4 b^4}\\ &=\frac{2 (A b-a B) x^{7/2}}{3 a b (a+b x)^{3/2}}+\frac{2 (4 A b-7 a B) x^{5/2}}{3 a b^2 \sqrt{a+b x}}+\frac{5 (4 A b-7 a B) \sqrt{x} \sqrt{a+b x}}{4 b^4}-\frac{5 (4 A b-7 a B) x^{3/2} \sqrt{a+b x}}{6 a b^3}-\frac{(5 a (4 A b-7 a B)) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{a+b x}}\right )}{4 b^4}\\ &=\frac{2 (A b-a B) x^{7/2}}{3 a b (a+b x)^{3/2}}+\frac{2 (4 A b-7 a B) x^{5/2}}{3 a b^2 \sqrt{a+b x}}+\frac{5 (4 A b-7 a B) \sqrt{x} \sqrt{a+b x}}{4 b^4}-\frac{5 (4 A b-7 a B) x^{3/2} \sqrt{a+b x}}{6 a b^3}-\frac{5 a (4 A b-7 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{4 b^{9/2}}\\ \end{align*}
Mathematica [C] time = 0.0544458, size = 80, normalized size = 0.47 \[ \frac{2 x^{7/2} \left ((a+b x) \sqrt{\frac{b x}{a}+1} (7 a B-4 A b) \, _2F_1\left (\frac{3}{2},\frac{7}{2};\frac{9}{2};-\frac{b x}{a}\right )+7 a (A b-a B)\right )}{21 a^2 b (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.014, size = 362, normalized size = 2.1 \begin{align*} -{\frac{1}{24} \left ( -12\,B{x}^{3}{b}^{7/2}\sqrt{x \left ( bx+a \right ) }+60\,A\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){x}^{2}a{b}^{3}-24\,A{x}^{2}{b}^{7/2}\sqrt{x \left ( bx+a \right ) }-105\,B\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){x}^{2}{a}^{2}{b}^{2}+42\,B{x}^{2}a{b}^{5/2}\sqrt{x \left ( bx+a \right ) }+120\,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}-160\,A\sqrt{x \left ( bx+a \right ) }{b}^{5/2}xa-210\,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+280\,B\sqrt{x \left ( bx+a \right ) }{b}^{3/2}x{a}^{2}+60\,A\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){a}^{3}b-120\,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 ) }}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \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 = 2.67229, size = 857, normalized size = 5.07 \begin{align*} \left [-\frac{15 \,{\left (7 \, B a^{4} - 4 \, A a^{3} b +{\left (7 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{2} + 2 \,{\left (7 \, B a^{3} b - 4 \, 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 (6 \, B b^{4} x^{3} - 105 \, B a^{3} b + 60 \, A a^{2} b^{2} - 3 \,{\left (7 \, B a b^{3} - 4 \, A b^{4}\right )} x^{2} - 20 \,{\left (7 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{24 \,{\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}}, -\frac{15 \,{\left (7 \, B a^{4} - 4 \, A a^{3} b +{\left (7 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{2} + 2 \,{\left (7 \, B a^{3} b - 4 \, A a^{2} b^{2}\right )} x\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) -{\left (6 \, B b^{4} x^{3} - 105 \, B a^{3} b + 60 \, A a^{2} b^{2} - 3 \,{\left (7 \, B a b^{3} - 4 \, A b^{4}\right )} x^{2} - 20 \,{\left (7 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{12 \,{\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} 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 [B] time = 89.7974, size = 467, normalized size = 2.76 \begin{align*} \frac{1}{4} \, \sqrt{{\left (b x + a\right )} b - a b} \sqrt{b x + a}{\left (\frac{2 \,{\left (b x + a\right )} B{\left | b \right |}}{b^{6}} - \frac{13 \, B a b^{11}{\left | b \right |} - 4 \, A b^{12}{\left | b \right |}}{b^{17}}\right )} - \frac{5 \,{\left (7 \, B a^{2} \sqrt{b}{\left | b \right |} - 4 \, A a 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 )}{8 \, b^{6}} - \frac{4 \,{\left (12 \, B a^{3}{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{4} \sqrt{b}{\left | b \right |} + 18 \, B a^{4}{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac{3}{2}}{\left | b \right |} - 9 \, A a^{2}{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac{3}{2}}{\left | b \right |} + 10 \, B a^{5} b^{\frac{5}{2}}{\left | b \right |} - 12 \, A a^{3}{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac{5}{2}}{\left | b \right |} - 7 \, A a^{4} b^{\frac{7}{2}}{\left | b \right |}\right )}}{3 \,{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3} b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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