Optimal. Leaf size=69 \[ -\frac{2 \sqrt{x}}{b^2 \sqrt{a+b x}}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{b^{5/2}}-\frac{2 x^{3/2}}{3 b (a+b x)^{3/2}} \]
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Rubi [A] time = 0.021607, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {47, 63, 217, 206} \[ -\frac{2 \sqrt{x}}{b^2 \sqrt{a+b x}}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{b^{5/2}}-\frac{2 x^{3/2}}{3 b (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{x^{3/2}}{(a+b x)^{5/2}} \, dx &=-\frac{2 x^{3/2}}{3 b (a+b x)^{3/2}}+\frac{\int \frac{\sqrt{x}}{(a+b x)^{3/2}} \, dx}{b}\\ &=-\frac{2 x^{3/2}}{3 b (a+b x)^{3/2}}-\frac{2 \sqrt{x}}{b^2 \sqrt{a+b x}}+\frac{\int \frac{1}{\sqrt{x} \sqrt{a+b x}} \, dx}{b^2}\\ &=-\frac{2 x^{3/2}}{3 b (a+b x)^{3/2}}-\frac{2 \sqrt{x}}{b^2 \sqrt{a+b x}}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\sqrt{x}\right )}{b^2}\\ &=-\frac{2 x^{3/2}}{3 b (a+b x)^{3/2}}-\frac{2 \sqrt{x}}{b^2 \sqrt{a+b x}}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{a+b x}}\right )}{b^2}\\ &=-\frac{2 x^{3/2}}{3 b (a+b x)^{3/2}}-\frac{2 \sqrt{x}}{b^2 \sqrt{a+b x}}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{b^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.130184, size = 80, normalized size = 1.16 \[ \frac{6 \sqrt{a} (a+b x) \sqrt{\frac{b x}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )-2 \sqrt{b} \sqrt{x} (3 a+4 b x)}{3 b^{5/2} (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.025, size = 0, normalized size = 0. \begin{align*} \int{{x}^{{\frac{3}{2}}} \left ( bx+a \right ) ^{-{\frac{5}{2}}}}\, dx \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 = 2.08751, size = 451, normalized size = 6.54 \begin{align*} \left [\frac{3 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt{b} \log \left (2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x} + a\right ) - 2 \,{\left (4 \, b^{2} x + 3 \, a b\right )} \sqrt{b x + a} \sqrt{x}}{3 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}}, -\frac{2 \,{\left (3 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) +{\left (4 \, b^{2} x + 3 \, a b\right )} \sqrt{b x + a} \sqrt{x}\right )}}{3 \,{\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 [B] time = 6.02493, size = 328, normalized size = 4.75 \begin{align*} \frac{6 a^{\frac{39}{2}} b^{11} x^{\frac{27}{2}} \sqrt{1 + \frac{b x}{a}} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{3 a^{\frac{39}{2}} b^{\frac{27}{2}} x^{\frac{27}{2}} \sqrt{1 + \frac{b x}{a}} + 3 a^{\frac{37}{2}} b^{\frac{29}{2}} x^{\frac{29}{2}} \sqrt{1 + \frac{b x}{a}}} + \frac{6 a^{\frac{37}{2}} b^{12} x^{\frac{29}{2}} \sqrt{1 + \frac{b x}{a}} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{3 a^{\frac{39}{2}} b^{\frac{27}{2}} x^{\frac{27}{2}} \sqrt{1 + \frac{b x}{a}} + 3 a^{\frac{37}{2}} b^{\frac{29}{2}} x^{\frac{29}{2}} \sqrt{1 + \frac{b x}{a}}} - \frac{6 a^{19} b^{\frac{23}{2}} x^{14}}{3 a^{\frac{39}{2}} b^{\frac{27}{2}} x^{\frac{27}{2}} \sqrt{1 + \frac{b x}{a}} + 3 a^{\frac{37}{2}} b^{\frac{29}{2}} x^{\frac{29}{2}} \sqrt{1 + \frac{b x}{a}}} - \frac{8 a^{18} b^{\frac{25}{2}} x^{15}}{3 a^{\frac{39}{2}} b^{\frac{27}{2}} x^{\frac{27}{2}} \sqrt{1 + \frac{b x}{a}} + 3 a^{\frac{37}{2}} b^{\frac{29}{2}} x^{\frac{29}{2}} \sqrt{1 + \frac{b x}{a}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 58.8763, size = 223, normalized size = 3.23 \begin{align*} -\frac{{\left (\frac{3 \, \log \left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2}\right )}{\sqrt{b}} + \frac{8 \,{\left (3 \, a{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{4} \sqrt{b} + 3 \, a^{2}{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac{3}{2}} + 2 \, a^{3} b^{\frac{5}{2}}\right )}}{{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3}}\right )}{\left | b \right |}}{3 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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