Optimal. Leaf size=107 \[ -\frac{5 A b-3 a B}{3 a^2 b x^{3/2}}+\frac{5 A b-3 a B}{a^3 \sqrt{x}}+\frac{\sqrt{b} (5 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{7/2}}+\frac{A b-a B}{a b x^{3/2} (a+b x)} \]
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Rubi [A] time = 0.0464709, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {78, 51, 63, 205} \[ -\frac{5 A b-3 a B}{3 a^2 b x^{3/2}}+\frac{5 A b-3 a B}{a^3 \sqrt{x}}+\frac{\sqrt{b} (5 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{7/2}}+\frac{A b-a B}{a b x^{3/2} (a+b x)} \]
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)^2} \, dx &=\frac{A b-a B}{a b x^{3/2} (a+b x)}-\frac{\left (-\frac{5 A b}{2}+\frac{3 a B}{2}\right ) \int \frac{1}{x^{5/2} (a+b x)} \, dx}{a b}\\ &=-\frac{5 A b-3 a B}{3 a^2 b x^{3/2}}+\frac{A b-a B}{a b x^{3/2} (a+b x)}-\frac{(5 A b-3 a B) \int \frac{1}{x^{3/2} (a+b x)} \, dx}{2 a^2}\\ &=-\frac{5 A b-3 a B}{3 a^2 b x^{3/2}}+\frac{5 A b-3 a B}{a^3 \sqrt{x}}+\frac{A b-a B}{a b x^{3/2} (a+b x)}+\frac{(b (5 A b-3 a B)) \int \frac{1}{\sqrt{x} (a+b x)} \, dx}{2 a^3}\\ &=-\frac{5 A b-3 a B}{3 a^2 b x^{3/2}}+\frac{5 A b-3 a B}{a^3 \sqrt{x}}+\frac{A b-a B}{a b x^{3/2} (a+b x)}+\frac{(b (5 A b-3 a B)) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sqrt{x}\right )}{a^3}\\ &=-\frac{5 A b-3 a B}{3 a^2 b x^{3/2}}+\frac{5 A b-3 a B}{a^3 \sqrt{x}}+\frac{A b-a B}{a b x^{3/2} (a+b x)}+\frac{\sqrt{b} (5 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0177684, size = 64, normalized size = 0.6 \[ \frac{(a+b x) (3 a B-5 A b) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};-\frac{b x}{a}\right )+3 a (A b-a B)}{3 a^2 b x^{3/2} (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 113, normalized size = 1.1 \begin{align*} -{\frac{2\,A}{3\,{a}^{2}}{x}^{-{\frac{3}{2}}}}+4\,{\frac{Ab}{{a}^{3}\sqrt{x}}}-2\,{\frac{B}{{a}^{2}\sqrt{x}}}+{\frac{{b}^{2}A}{{a}^{3} \left ( bx+a \right ) }\sqrt{x}}-{\frac{Bb}{{a}^{2} \left ( bx+a \right ) }\sqrt{x}}+5\,{\frac{{b}^{2}A}{{a}^{3}\sqrt{ab}}\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) }-3\,{\frac{Bb}{{a}^{2}\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 = 2.36935, size = 572, normalized size = 5.35 \begin{align*} \left [-\frac{3 \,{\left ({\left (3 \, B a b - 5 \, A b^{2}\right )} x^{3} +{\left (3 \, B a^{2} - 5 \, A a 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 (2 \, A a^{2} + 3 \,{\left (3 \, B a b - 5 \, A b^{2}\right )} x^{2} + 2 \,{\left (3 \, B a^{2} - 5 \, A a b\right )} x\right )} \sqrt{x}}{6 \,{\left (a^{3} b x^{3} + a^{4} x^{2}\right )}}, \frac{3 \,{\left ({\left (3 \, B a b - 5 \, A b^{2}\right )} x^{3} +{\left (3 \, B a^{2} - 5 \, A a b\right )} x^{2}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b \sqrt{x}}\right ) -{\left (2 \, A a^{2} + 3 \,{\left (3 \, B a b - 5 \, A b^{2}\right )} x^{2} + 2 \,{\left (3 \, B a^{2} - 5 \, A a b\right )} x\right )} \sqrt{x}}{3 \,{\left (a^{3} b x^{3} + a^{4} x^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 112.651, size = 983, normalized size = 9.19 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18567, size = 115, normalized size = 1.07 \begin{align*} -\frac{{\left (3 \, B a b - 5 \, A b^{2}\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{3}} - \frac{B a b \sqrt{x} - A b^{2} \sqrt{x}}{{\left (b x + a\right )} a^{3}} - \frac{2 \,{\left (3 \, B a x - 6 \, A b x + A a\right )}}{3 \, a^{3} x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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