Optimal. Leaf size=82 \[ -\frac{5 x^{3/2}}{4 b^2 (a+b x)}-\frac{15 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 b^{7/2}}-\frac{x^{5/2}}{2 b (a+b x)^2}+\frac{15 \sqrt{x}}{4 b^3} \]
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Rubi [A] time = 0.0234385, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {47, 50, 63, 205} \[ -\frac{5 x^{3/2}}{4 b^2 (a+b x)}-\frac{15 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 b^{7/2}}-\frac{x^{5/2}}{2 b (a+b x)^2}+\frac{15 \sqrt{x}}{4 b^3} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{x^{5/2}}{(a+b x)^3} \, dx &=-\frac{x^{5/2}}{2 b (a+b x)^2}+\frac{5 \int \frac{x^{3/2}}{(a+b x)^2} \, dx}{4 b}\\ &=-\frac{x^{5/2}}{2 b (a+b x)^2}-\frac{5 x^{3/2}}{4 b^2 (a+b x)}+\frac{15 \int \frac{\sqrt{x}}{a+b x} \, dx}{8 b^2}\\ &=\frac{15 \sqrt{x}}{4 b^3}-\frac{x^{5/2}}{2 b (a+b x)^2}-\frac{5 x^{3/2}}{4 b^2 (a+b x)}-\frac{(15 a) \int \frac{1}{\sqrt{x} (a+b x)} \, dx}{8 b^3}\\ &=\frac{15 \sqrt{x}}{4 b^3}-\frac{x^{5/2}}{2 b (a+b x)^2}-\frac{5 x^{3/2}}{4 b^2 (a+b x)}-\frac{(15 a) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sqrt{x}\right )}{4 b^3}\\ &=\frac{15 \sqrt{x}}{4 b^3}-\frac{x^{5/2}}{2 b (a+b x)^2}-\frac{5 x^{3/2}}{4 b^2 (a+b x)}-\frac{15 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 b^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0047717, size = 27, normalized size = 0.33 \[ \frac{2 x^{7/2} \, _2F_1\left (3,\frac{7}{2};\frac{9}{2};-\frac{b x}{a}\right )}{7 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 66, normalized size = 0.8 \begin{align*} 2\,{\frac{\sqrt{x}}{{b}^{3}}}+{\frac{9\,a}{4\,{b}^{2} \left ( bx+a \right ) ^{2}}{x}^{{\frac{3}{2}}}}+{\frac{7\,{a}^{2}}{4\,{b}^{3} \left ( bx+a \right ) ^{2}}\sqrt{x}}-{\frac{15\,a}{4\,{b}^{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 = 1.32697, size = 443, normalized size = 5.4 \begin{align*} \left [\frac{15 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x - 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) + 2 \,{\left (8 \, b^{2} x^{2} + 25 \, a b x + 15 \, a^{2}\right )} \sqrt{x}}{8 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}}, -\frac{15 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{x} \sqrt{\frac{a}{b}}}{a}\right ) -{\left (8 \, b^{2} x^{2} + 25 \, a b x + 15 \, a^{2}\right )} \sqrt{x}}{4 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\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 = 1.25916, size = 80, normalized size = 0.98 \begin{align*} -\frac{15 \, a \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} b^{3}} + \frac{2 \, \sqrt{x}}{b^{3}} + \frac{9 \, a b x^{\frac{3}{2}} + 7 \, a^{2} \sqrt{x}}{4 \,{\left (b x + a\right )}^{2} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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