Optimal. Leaf size=178 \[ \frac{7 (3 A b-a B)}{8 a^3 b x^{3/2} (a+b x)}-\frac{35 (3 A b-a B)}{24 a^4 b x^{3/2}}+\frac{3 A b-a B}{4 a^2 b x^{3/2} (a+b x)^2}+\frac{35 (3 A b-a B)}{8 a^5 \sqrt{x}}+\frac{35 \sqrt{b} (3 A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{8 a^{11/2}}+\frac{A b-a B}{3 a b x^{3/2} (a+b x)^3} \]
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Rubi [A] time = 0.0793817, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {27, 78, 51, 63, 205} \[ \frac{7 (3 A b-a B)}{8 a^3 b x^{3/2} (a+b x)}-\frac{35 (3 A b-a B)}{24 a^4 b x^{3/2}}+\frac{3 A b-a B}{4 a^2 b x^{3/2} (a+b x)^2}+\frac{35 (3 A b-a B)}{8 a^5 \sqrt{x}}+\frac{35 \sqrt{b} (3 A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{8 a^{11/2}}+\frac{A b-a B}{3 a b x^{3/2} (a+b x)^3} \]
Antiderivative was successfully verified.
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Rule 27
Rule 78
Rule 51
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac{A+B x}{x^{5/2} (a+b x)^4} \, dx\\ &=\frac{A b-a B}{3 a b x^{3/2} (a+b x)^3}-\frac{\left (-\frac{9 A b}{2}+\frac{3 a B}{2}\right ) \int \frac{1}{x^{5/2} (a+b x)^3} \, dx}{3 a b}\\ &=\frac{A b-a B}{3 a b x^{3/2} (a+b x)^3}+\frac{3 A b-a B}{4 a^2 b x^{3/2} (a+b x)^2}+\frac{(7 (3 A b-a B)) \int \frac{1}{x^{5/2} (a+b x)^2} \, dx}{8 a^2 b}\\ &=\frac{A b-a B}{3 a b x^{3/2} (a+b x)^3}+\frac{3 A b-a B}{4 a^2 b x^{3/2} (a+b x)^2}+\frac{7 (3 A b-a B)}{8 a^3 b x^{3/2} (a+b x)}+\frac{(35 (3 A b-a B)) \int \frac{1}{x^{5/2} (a+b x)} \, dx}{16 a^3 b}\\ &=-\frac{35 (3 A b-a B)}{24 a^4 b x^{3/2}}+\frac{A b-a B}{3 a b x^{3/2} (a+b x)^3}+\frac{3 A b-a B}{4 a^2 b x^{3/2} (a+b x)^2}+\frac{7 (3 A b-a B)}{8 a^3 b x^{3/2} (a+b x)}-\frac{(35 (3 A b-a B)) \int \frac{1}{x^{3/2} (a+b x)} \, dx}{16 a^4}\\ &=-\frac{35 (3 A b-a B)}{24 a^4 b x^{3/2}}+\frac{35 (3 A b-a B)}{8 a^5 \sqrt{x}}+\frac{A b-a B}{3 a b x^{3/2} (a+b x)^3}+\frac{3 A b-a B}{4 a^2 b x^{3/2} (a+b x)^2}+\frac{7 (3 A b-a B)}{8 a^3 b x^{3/2} (a+b x)}+\frac{(35 b (3 A b-a B)) \int \frac{1}{\sqrt{x} (a+b x)} \, dx}{16 a^5}\\ &=-\frac{35 (3 A b-a B)}{24 a^4 b x^{3/2}}+\frac{35 (3 A b-a B)}{8 a^5 \sqrt{x}}+\frac{A b-a B}{3 a b x^{3/2} (a+b x)^3}+\frac{3 A b-a B}{4 a^2 b x^{3/2} (a+b x)^2}+\frac{7 (3 A b-a B)}{8 a^3 b x^{3/2} (a+b x)}+\frac{(35 b (3 A b-a B)) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sqrt{x}\right )}{8 a^5}\\ &=-\frac{35 (3 A b-a B)}{24 a^4 b x^{3/2}}+\frac{35 (3 A b-a B)}{8 a^5 \sqrt{x}}+\frac{A b-a B}{3 a b x^{3/2} (a+b x)^3}+\frac{3 A b-a B}{4 a^2 b x^{3/2} (a+b x)^2}+\frac{7 (3 A b-a B)}{8 a^3 b x^{3/2} (a+b x)}+\frac{35 \sqrt{b} (3 A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{8 a^{11/2}}\\ \end{align*}
Mathematica [C] time = 0.0273225, size = 61, normalized size = 0.34 \[ \frac{\frac{3 a^3 (A b-a B)}{(a+b x)^3}+(3 a B-9 A b) \, _2F_1\left (-\frac{3}{2},3;-\frac{1}{2};-\frac{b x}{a}\right )}{9 a^4 b x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 190, normalized size = 1.1 \begin{align*} -{\frac{2\,A}{3\,{a}^{4}}{x}^{-{\frac{3}{2}}}}+8\,{\frac{Ab}{{a}^{5}\sqrt{x}}}-2\,{\frac{B}{{a}^{4}\sqrt{x}}}+{\frac{41\,{b}^{4}A}{8\,{a}^{5} \left ( bx+a \right ) ^{3}}{x}^{{\frac{5}{2}}}}-{\frac{19\,{b}^{3}B}{8\,{a}^{4} \left ( bx+a \right ) ^{3}}{x}^{{\frac{5}{2}}}}+{\frac{35\,A{b}^{3}}{3\,{a}^{4} \left ( bx+a \right ) ^{3}}{x}^{{\frac{3}{2}}}}-{\frac{17\,{b}^{2}B}{3\,{a}^{3} \left ( bx+a \right ) ^{3}}{x}^{{\frac{3}{2}}}}+{\frac{55\,A{b}^{2}}{8\,{a}^{3} \left ( bx+a \right ) ^{3}}\sqrt{x}}-{\frac{29\,bB}{8\,{a}^{2} \left ( bx+a \right ) ^{3}}\sqrt{x}}+{\frac{105\,A{b}^{2}}{8\,{a}^{5}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{35\,bB}{8\,{a}^{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 = 1.92305, size = 1034, normalized size = 5.81 \begin{align*} \left [-\frac{105 \,{\left ({\left (B a b^{3} - 3 \, A b^{4}\right )} x^{5} + 3 \,{\left (B a^{2} b^{2} - 3 \, A a b^{3}\right )} x^{4} + 3 \,{\left (B a^{3} b - 3 \, A a^{2} b^{2}\right )} x^{3} +{\left (B a^{4} - 3 \, A a^{3} b\right )} x^{2}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x + 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - a}{b x + a}\right ) + 2 \,{\left (16 \, A a^{4} + 105 \,{\left (B a b^{3} - 3 \, A b^{4}\right )} x^{4} + 280 \,{\left (B a^{2} b^{2} - 3 \, A a b^{3}\right )} x^{3} + 231 \,{\left (B a^{3} b - 3 \, A a^{2} b^{2}\right )} x^{2} + 48 \,{\left (B a^{4} - 3 \, A a^{3} b\right )} x\right )} \sqrt{x}}{48 \,{\left (a^{5} b^{3} x^{5} + 3 \, a^{6} b^{2} x^{4} + 3 \, a^{7} b x^{3} + a^{8} x^{2}\right )}}, \frac{105 \,{\left ({\left (B a b^{3} - 3 \, A b^{4}\right )} x^{5} + 3 \,{\left (B a^{2} b^{2} - 3 \, A a b^{3}\right )} x^{4} + 3 \,{\left (B a^{3} b - 3 \, A a^{2} b^{2}\right )} x^{3} +{\left (B a^{4} - 3 \, A a^{3} b\right )} x^{2}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b \sqrt{x}}\right ) -{\left (16 \, A a^{4} + 105 \,{\left (B a b^{3} - 3 \, A b^{4}\right )} x^{4} + 280 \,{\left (B a^{2} b^{2} - 3 \, A a b^{3}\right )} x^{3} + 231 \,{\left (B a^{3} b - 3 \, A a^{2} b^{2}\right )} x^{2} + 48 \,{\left (B a^{4} - 3 \, A a^{3} b\right )} x\right )} \sqrt{x}}{24 \,{\left (a^{5} b^{3} x^{5} + 3 \, a^{6} b^{2} x^{4} + 3 \, a^{7} b x^{3} + a^{8} x^{2}\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 = 1.24819, size = 184, normalized size = 1.03 \begin{align*} -\frac{35 \,{\left (B a b - 3 \, A b^{2}\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{5}} - \frac{105 \, B a b^{3} x^{4} - 315 \, A b^{4} x^{4} + 280 \, B a^{2} b^{2} x^{3} - 840 \, A a b^{3} x^{3} + 231 \, B a^{3} b x^{2} - 693 \, A a^{2} b^{2} x^{2} + 48 \, B a^{4} x - 144 \, A a^{3} b x + 16 \, A a^{4}}{24 \,{\left (b x^{\frac{3}{2}} + a \sqrt{x}\right )}^{3} a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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