Optimal. Leaf size=69 \[ \frac{2 (A b-a B)}{a^2 \sqrt{x}}+\frac{2 \sqrt{b} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{5/2}}-\frac{2 A}{3 a x^{3/2}} \]
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Rubi [A] time = 0.0362944, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {78, 51, 63, 205} \[ \frac{2 (A b-a B)}{a^2 \sqrt{x}}+\frac{2 \sqrt{b} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{5/2}}-\frac{2 A}{3 a x^{3/2}} \]
Antiderivative was successfully verified.
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Rule 78
Rule 51
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{A+B x}{x^{5/2} (a+b x)} \, dx &=-\frac{2 A}{3 a x^{3/2}}+\frac{\left (2 \left (-\frac{3 A b}{2}+\frac{3 a B}{2}\right )\right ) \int \frac{1}{x^{3/2} (a+b x)} \, dx}{3 a}\\ &=-\frac{2 A}{3 a x^{3/2}}+\frac{2 (A b-a B)}{a^2 \sqrt{x}}+\frac{(b (A b-a B)) \int \frac{1}{\sqrt{x} (a+b x)} \, dx}{a^2}\\ &=-\frac{2 A}{3 a x^{3/2}}+\frac{2 (A b-a B)}{a^2 \sqrt{x}}+\frac{(2 b (A b-a B)) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sqrt{x}\right )}{a^2}\\ &=-\frac{2 A}{3 a x^{3/2}}+\frac{2 (A b-a B)}{a^2 \sqrt{x}}+\frac{2 \sqrt{b} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0110088, size = 43, normalized size = 0.62 \[ \frac{6 x (A b-a B) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};-\frac{b x}{a}\right )-2 a A}{3 a^2 x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 78, normalized size = 1.1 \begin{align*} -{\frac{2\,A}{3\,a}{x}^{-{\frac{3}{2}}}}+2\,{\frac{Ab}{{a}^{2}\sqrt{x}}}-2\,{\frac{B}{a\sqrt{x}}}+2\,{\frac{A{b}^{2}}{{a}^{2}\sqrt{ab}}\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) }-2\,{\frac{Bb}{a\sqrt{ab}}\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) } \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 = 2.3339, size = 335, normalized size = 4.86 \begin{align*} \left [-\frac{3 \,{\left (B a - A b\right )} x^{2} \sqrt{-\frac{b}{a}} \log \left (\frac{b x + 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - a}{b x + a}\right ) + 2 \,{\left (A a + 3 \,{\left (B a - A b\right )} x\right )} \sqrt{x}}{3 \, a^{2} x^{2}}, \frac{2 \,{\left (3 \,{\left (B a - A b\right )} x^{2} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b \sqrt{x}}\right ) -{\left (A a + 3 \,{\left (B a - A b\right )} x\right )} \sqrt{x}\right )}}{3 \, a^{2} x^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 19.5106, size = 248, normalized size = 3.59 \begin{align*} \begin{cases} \tilde{\infty } \left (- \frac{2 A}{5 x^{\frac{5}{2}}} - \frac{2 B}{3 x^{\frac{3}{2}}}\right ) & \text{for}\: a = 0 \wedge b = 0 \\\frac{- \frac{2 A}{5 x^{\frac{5}{2}}} - \frac{2 B}{3 x^{\frac{3}{2}}}}{b} & \text{for}\: a = 0 \\\frac{- \frac{2 A}{3 x^{\frac{3}{2}}} - \frac{2 B}{\sqrt{x}}}{a} & \text{for}\: b = 0 \\- \frac{2 A}{3 a x^{\frac{3}{2}}} + \frac{2 A b}{a^{2} \sqrt{x}} - \frac{i A b \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{a^{\frac{5}{2}} \sqrt{\frac{1}{b}}} + \frac{i A b \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{a^{\frac{5}{2}} \sqrt{\frac{1}{b}}} - \frac{2 B}{a \sqrt{x}} + \frac{i B \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{a^{\frac{3}{2}} \sqrt{\frac{1}{b}}} - \frac{i B \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{a^{\frac{3}{2}} \sqrt{\frac{1}{b}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14277, size = 74, normalized size = 1.07 \begin{align*} -\frac{2 \,{\left (B a b - A b^{2}\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{2}} - \frac{2 \,{\left (3 \, B a x - 3 \, A b x + A a\right )}}{3 \, a^{2} x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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