Optimal. Leaf size=147 \[ \frac{7 A b-3 a B}{4 a^2 b x^{3/2} (a+b x)}-\frac{5 (7 A b-3 a B)}{12 a^3 b x^{3/2}}+\frac{5 (7 A b-3 a B)}{4 a^4 \sqrt{x}}+\frac{5 \sqrt{b} (7 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{9/2}}+\frac{A b-a B}{2 a b x^{3/2} (a+b x)^2} \]
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Rubi [A] time = 0.0605477, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {78, 51, 63, 205} \[ \frac{7 A b-3 a B}{4 a^2 b x^{3/2} (a+b x)}-\frac{5 (7 A b-3 a B)}{12 a^3 b x^{3/2}}+\frac{5 (7 A b-3 a B)}{4 a^4 \sqrt{x}}+\frac{5 \sqrt{b} (7 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{9/2}}+\frac{A b-a B}{2 a b x^{3/2} (a+b x)^2} \]
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)^3} \, dx &=\frac{A b-a B}{2 a b x^{3/2} (a+b x)^2}-\frac{\left (-\frac{7 A b}{2}+\frac{3 a B}{2}\right ) \int \frac{1}{x^{5/2} (a+b x)^2} \, dx}{2 a b}\\ &=\frac{A b-a B}{2 a b x^{3/2} (a+b x)^2}+\frac{7 A b-3 a B}{4 a^2 b x^{3/2} (a+b x)}+\frac{(5 (7 A b-3 a B)) \int \frac{1}{x^{5/2} (a+b x)} \, dx}{8 a^2 b}\\ &=-\frac{5 (7 A b-3 a B)}{12 a^3 b x^{3/2}}+\frac{A b-a B}{2 a b x^{3/2} (a+b x)^2}+\frac{7 A b-3 a B}{4 a^2 b x^{3/2} (a+b x)}-\frac{(5 (7 A b-3 a B)) \int \frac{1}{x^{3/2} (a+b x)} \, dx}{8 a^3}\\ &=-\frac{5 (7 A b-3 a B)}{12 a^3 b x^{3/2}}+\frac{5 (7 A b-3 a B)}{4 a^4 \sqrt{x}}+\frac{A b-a B}{2 a b x^{3/2} (a+b x)^2}+\frac{7 A b-3 a B}{4 a^2 b x^{3/2} (a+b x)}+\frac{(5 b (7 A b-3 a B)) \int \frac{1}{\sqrt{x} (a+b x)} \, dx}{8 a^4}\\ &=-\frac{5 (7 A b-3 a B)}{12 a^3 b x^{3/2}}+\frac{5 (7 A b-3 a B)}{4 a^4 \sqrt{x}}+\frac{A b-a B}{2 a b x^{3/2} (a+b x)^2}+\frac{7 A b-3 a B}{4 a^2 b x^{3/2} (a+b x)}+\frac{(5 b (7 A b-3 a B)) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sqrt{x}\right )}{4 a^4}\\ &=-\frac{5 (7 A b-3 a B)}{12 a^3 b x^{3/2}}+\frac{5 (7 A b-3 a B)}{4 a^4 \sqrt{x}}+\frac{A b-a B}{2 a b x^{3/2} (a+b x)^2}+\frac{7 A b-3 a B}{4 a^2 b x^{3/2} (a+b x)}+\frac{5 \sqrt{b} (7 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{9/2}}\\ \end{align*}
Mathematica [C] time = 0.0260262, size = 61, normalized size = 0.41 \[ \frac{\frac{3 a^2 (A b-a B)}{(a+b x)^2}+(3 a B-7 A b) \, _2F_1\left (-\frac{3}{2},2;-\frac{1}{2};-\frac{b x}{a}\right )}{6 a^3 b x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 152, normalized size = 1. \begin{align*} -{\frac{2\,A}{3\,{a}^{3}}{x}^{-{\frac{3}{2}}}}+6\,{\frac{Ab}{{a}^{4}\sqrt{x}}}-2\,{\frac{B}{{a}^{3}\sqrt{x}}}+{\frac{11\,{b}^{3}A}{4\,{a}^{4} \left ( bx+a \right ) ^{2}}{x}^{{\frac{3}{2}}}}-{\frac{7\,{b}^{2}B}{4\,{a}^{3} \left ( bx+a \right ) ^{2}}{x}^{{\frac{3}{2}}}}+{\frac{13\,A{b}^{2}}{4\,{a}^{3} \left ( bx+a \right ) ^{2}}\sqrt{x}}-{\frac{9\,Bb}{4\,{a}^{2} \left ( bx+a \right ) ^{2}}\sqrt{x}}+{\frac{35\,A{b}^{2}}{4\,{a}^{4}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{15\,Bb}{4\,{a}^{3}}\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 = 2.51183, size = 821, normalized size = 5.59 \begin{align*} \left [-\frac{15 \,{\left ({\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{4} + 2 \,{\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{3} +{\left (3 \, B a^{3} - 7 \, A a^{2} 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 (8 \, A a^{3} + 15 \,{\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} + 25 \,{\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} + 8 \,{\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x\right )} \sqrt{x}}{24 \,{\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}}, \frac{15 \,{\left ({\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{4} + 2 \,{\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{3} +{\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x^{2}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b \sqrt{x}}\right ) -{\left (8 \, A a^{3} + 15 \,{\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} + 25 \,{\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} + 8 \,{\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x\right )} \sqrt{x}}{12 \,{\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} 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 = 1.21525, size = 146, normalized size = 0.99 \begin{align*} -\frac{5 \,{\left (3 \, B a b - 7 \, A b^{2}\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} a^{4}} - \frac{2 \,{\left (3 \, B a x - 9 \, A b x + A a\right )}}{3 \, a^{4} x^{\frac{3}{2}}} - \frac{7 \, B a b^{2} x^{\frac{3}{2}} - 11 \, A b^{3} x^{\frac{3}{2}} + 9 \, B a^{2} b \sqrt{x} - 13 \, A a b^{2} \sqrt{x}}{4 \,{\left (b x + a\right )}^{2} a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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