Optimal. Leaf size=153 \[ \frac{x^{5/2} (A b-7 a B)}{12 a b^2 (a+b x)^2}+\frac{5 x^{3/2} (A b-7 a B)}{24 a b^3 (a+b x)}-\frac{5 \sqrt{x} (A b-7 a B)}{8 a b^4}+\frac{5 (A b-7 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{8 \sqrt{a} b^{9/2}}+\frac{x^{7/2} (A b-a B)}{3 a b (a+b x)^3} \]
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Rubi [A] time = 0.0654034, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {27, 78, 47, 50, 63, 205} \[ \frac{x^{5/2} (A b-7 a B)}{12 a b^2 (a+b x)^2}+\frac{5 x^{3/2} (A b-7 a B)}{24 a b^3 (a+b x)}-\frac{5 \sqrt{x} (A b-7 a B)}{8 a b^4}+\frac{5 (A b-7 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{8 \sqrt{a} b^{9/2}}+\frac{x^{7/2} (A b-a B)}{3 a b (a+b x)^3} \]
Antiderivative was successfully verified.
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Rule 27
Rule 78
Rule 47
Rule 50
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{x^{5/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac{x^{5/2} (A+B x)}{(a+b x)^4} \, dx\\ &=\frac{(A b-a B) x^{7/2}}{3 a b (a+b x)^3}-\frac{\left (\frac{A b}{2}-\frac{7 a B}{2}\right ) \int \frac{x^{5/2}}{(a+b x)^3} \, dx}{3 a b}\\ &=\frac{(A b-a B) x^{7/2}}{3 a b (a+b x)^3}+\frac{(A b-7 a B) x^{5/2}}{12 a b^2 (a+b x)^2}-\frac{(5 (A b-7 a B)) \int \frac{x^{3/2}}{(a+b x)^2} \, dx}{24 a b^2}\\ &=\frac{(A b-a B) x^{7/2}}{3 a b (a+b x)^3}+\frac{(A b-7 a B) x^{5/2}}{12 a b^2 (a+b x)^2}+\frac{5 (A b-7 a B) x^{3/2}}{24 a b^3 (a+b x)}-\frac{(5 (A b-7 a B)) \int \frac{\sqrt{x}}{a+b x} \, dx}{16 a b^3}\\ &=-\frac{5 (A b-7 a B) \sqrt{x}}{8 a b^4}+\frac{(A b-a B) x^{7/2}}{3 a b (a+b x)^3}+\frac{(A b-7 a B) x^{5/2}}{12 a b^2 (a+b x)^2}+\frac{5 (A b-7 a B) x^{3/2}}{24 a b^3 (a+b x)}+\frac{(5 (A b-7 a B)) \int \frac{1}{\sqrt{x} (a+b x)} \, dx}{16 b^4}\\ &=-\frac{5 (A b-7 a B) \sqrt{x}}{8 a b^4}+\frac{(A b-a B) x^{7/2}}{3 a b (a+b x)^3}+\frac{(A b-7 a B) x^{5/2}}{12 a b^2 (a+b x)^2}+\frac{5 (A b-7 a B) x^{3/2}}{24 a b^3 (a+b x)}+\frac{(5 (A b-7 a B)) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sqrt{x}\right )}{8 b^4}\\ &=-\frac{5 (A b-7 a B) \sqrt{x}}{8 a b^4}+\frac{(A b-a B) x^{7/2}}{3 a b (a+b x)^3}+\frac{(A b-7 a B) x^{5/2}}{12 a b^2 (a+b x)^2}+\frac{5 (A b-7 a B) x^{3/2}}{24 a b^3 (a+b x)}+\frac{5 (A b-7 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{8 \sqrt{a} b^{9/2}}\\ \end{align*}
Mathematica [C] time = 0.0287759, size = 61, normalized size = 0.4 \[ \frac{x^{7/2} \left (\frac{7 a^3 (A b-a B)}{(a+b x)^3}+(7 a B-A b) \, _2F_1\left (3,\frac{7}{2};\frac{9}{2};-\frac{b x}{a}\right )\right )}{21 a^4 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 163, normalized size = 1.1 \begin{align*} 2\,{\frac{B\sqrt{x}}{{b}^{4}}}+{\frac{29\,aB}{8\,{b}^{2} \left ( bx+a \right ) ^{3}}{x}^{{\frac{5}{2}}}}-{\frac{11\,A}{8\,b \left ( bx+a \right ) ^{3}}{x}^{{\frac{5}{2}}}}-{\frac{5\,aA}{3\,{b}^{2} \left ( bx+a \right ) ^{3}}{x}^{{\frac{3}{2}}}}+{\frac{17\,B{a}^{2}}{3\,{b}^{3} \left ( bx+a \right ) ^{3}}{x}^{{\frac{3}{2}}}}+{\frac{19\,B{a}^{3}}{8\,{b}^{4} \left ( bx+a \right ) ^{3}}\sqrt{x}}-{\frac{5\,A{a}^{2}}{8\,{b}^{3} \left ( bx+a \right ) ^{3}}\sqrt{x}}+{\frac{5\,A}{8\,{b}^{3}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{35\,aB}{8\,{b}^{4}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \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 = 1.96224, size = 933, normalized size = 6.1 \begin{align*} \left [\frac{15 \,{\left (7 \, B a^{4} - A a^{3} b +{\left (7 \, B a b^{3} - A b^{4}\right )} x^{3} + 3 \,{\left (7 \, B a^{2} b^{2} - A a b^{3}\right )} x^{2} + 3 \,{\left (7 \, B a^{3} b - A a^{2} b^{2}\right )} x\right )} \sqrt{-a b} \log \left (\frac{b x - a - 2 \, \sqrt{-a b} \sqrt{x}}{b x + a}\right ) + 2 \,{\left (48 \, B a b^{4} x^{3} + 105 \, B a^{4} b - 15 \, A a^{3} b^{2} + 33 \,{\left (7 \, B a^{2} b^{3} - A a b^{4}\right )} x^{2} + 40 \,{\left (7 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} x\right )} \sqrt{x}}{48 \,{\left (a b^{8} x^{3} + 3 \, a^{2} b^{7} x^{2} + 3 \, a^{3} b^{6} x + a^{4} b^{5}\right )}}, \frac{15 \,{\left (7 \, B a^{4} - A a^{3} b +{\left (7 \, B a b^{3} - A b^{4}\right )} x^{3} + 3 \,{\left (7 \, B a^{2} b^{2} - A a b^{3}\right )} x^{2} + 3 \,{\left (7 \, B a^{3} b - A a^{2} b^{2}\right )} x\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b}}{b \sqrt{x}}\right ) +{\left (48 \, B a b^{4} x^{3} + 105 \, B a^{4} b - 15 \, A a^{3} b^{2} + 33 \,{\left (7 \, B a^{2} b^{3} - A a b^{4}\right )} x^{2} + 40 \,{\left (7 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} x\right )} \sqrt{x}}{24 \,{\left (a b^{8} x^{3} + 3 \, a^{2} b^{7} x^{2} + 3 \, a^{3} b^{6} x + a^{4} 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 = 1.1365, size = 150, normalized size = 0.98 \begin{align*} \frac{2 \, B \sqrt{x}}{b^{4}} - \frac{5 \,{\left (7 \, B a - A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} b^{4}} + \frac{87 \, B a b^{2} x^{\frac{5}{2}} - 33 \, A b^{3} x^{\frac{5}{2}} + 136 \, B a^{2} b x^{\frac{3}{2}} - 40 \, A a b^{2} x^{\frac{3}{2}} + 57 \, B a^{3} \sqrt{x} - 15 \, A a^{2} b \sqrt{x}}{24 \,{\left (b x + a\right )}^{3} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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