Optimal. Leaf size=95 \[ \frac{35 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 b^{9/2}}+\frac{7}{4 b^2 x^{3/2} (a x+b)}+\frac{35 a}{4 b^4 \sqrt{x}}+\frac{1}{2 b x^{3/2} (a x+b)^2}-\frac{35}{12 b^3 x^{3/2}} \]
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Rubi [A] time = 0.0324622, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {263, 51, 63, 205} \[ \frac{35 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 b^{9/2}}+\frac{7}{4 b^2 x^{3/2} (a x+b)}+\frac{35 a}{4 b^4 \sqrt{x}}+\frac{1}{2 b x^{3/2} (a x+b)^2}-\frac{35}{12 b^3 x^{3/2}} \]
Antiderivative was successfully verified.
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Rule 263
Rule 51
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x}\right )^3 x^{11/2}} \, dx &=\int \frac{1}{x^{5/2} (b+a x)^3} \, dx\\ &=\frac{1}{2 b x^{3/2} (b+a x)^2}+\frac{7 \int \frac{1}{x^{5/2} (b+a x)^2} \, dx}{4 b}\\ &=\frac{1}{2 b x^{3/2} (b+a x)^2}+\frac{7}{4 b^2 x^{3/2} (b+a x)}+\frac{35 \int \frac{1}{x^{5/2} (b+a x)} \, dx}{8 b^2}\\ &=-\frac{35}{12 b^3 x^{3/2}}+\frac{1}{2 b x^{3/2} (b+a x)^2}+\frac{7}{4 b^2 x^{3/2} (b+a x)}-\frac{(35 a) \int \frac{1}{x^{3/2} (b+a x)} \, dx}{8 b^3}\\ &=-\frac{35}{12 b^3 x^{3/2}}+\frac{35 a}{4 b^4 \sqrt{x}}+\frac{1}{2 b x^{3/2} (b+a x)^2}+\frac{7}{4 b^2 x^{3/2} (b+a x)}+\frac{\left (35 a^2\right ) \int \frac{1}{\sqrt{x} (b+a x)} \, dx}{8 b^4}\\ &=-\frac{35}{12 b^3 x^{3/2}}+\frac{35 a}{4 b^4 \sqrt{x}}+\frac{1}{2 b x^{3/2} (b+a x)^2}+\frac{7}{4 b^2 x^{3/2} (b+a x)}+\frac{\left (35 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{b+a x^2} \, dx,x,\sqrt{x}\right )}{4 b^4}\\ &=-\frac{35}{12 b^3 x^{3/2}}+\frac{35 a}{4 b^4 \sqrt{x}}+\frac{1}{2 b x^{3/2} (b+a x)^2}+\frac{7}{4 b^2 x^{3/2} (b+a x)}+\frac{35 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 b^{9/2}}\\ \end{align*}
Mathematica [C] time = 0.0054574, size = 27, normalized size = 0.28 \[ -\frac{2 \, _2F_1\left (-\frac{3}{2},3;-\frac{1}{2};-\frac{a x}{b}\right )}{3 b^3 x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 79, normalized size = 0.8 \begin{align*}{\frac{11\,{a}^{3}}{4\,{b}^{4} \left ( ax+b \right ) ^{2}}{x}^{{\frac{3}{2}}}}+{\frac{13\,{a}^{2}}{4\,{b}^{3} \left ( ax+b \right ) ^{2}}\sqrt{x}}+{\frac{35\,{a}^{2}}{4\,{b}^{4}}\arctan \left ({a\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{2}{3\,{b}^{3}}{x}^{-{\frac{3}{2}}}}+6\,{\frac{a}{{b}^{4}\sqrt{x}}} \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 = 1.9296, size = 545, normalized size = 5.74 \begin{align*} \left [\frac{105 \,{\left (a^{3} x^{4} + 2 \, a^{2} b x^{3} + a b^{2} x^{2}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{a x + 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - b}{a x + b}\right ) + 2 \,{\left (105 \, a^{3} x^{3} + 175 \, a^{2} b x^{2} + 56 \, a b^{2} x - 8 \, b^{3}\right )} \sqrt{x}}{24 \,{\left (a^{2} b^{4} x^{4} + 2 \, a b^{5} x^{3} + b^{6} x^{2}\right )}}, -\frac{105 \,{\left (a^{3} x^{4} + 2 \, a^{2} b x^{3} + a b^{2} x^{2}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{\frac{a}{b}}}{a \sqrt{x}}\right ) -{\left (105 \, a^{3} x^{3} + 175 \, a^{2} b x^{2} + 56 \, a b^{2} x - 8 \, b^{3}\right )} \sqrt{x}}{12 \,{\left (a^{2} b^{4} x^{4} + 2 \, a b^{5} x^{3} + b^{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 = 1.11787, size = 96, normalized size = 1.01 \begin{align*} \frac{35 \, a^{2} \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} b^{4}} + \frac{2 \,{\left (9 \, a x - b\right )}}{3 \, b^{4} x^{\frac{3}{2}}} + \frac{11 \, a^{3} x^{\frac{3}{2}} + 13 \, a^{2} b \sqrt{x}}{4 \,{\left (a x + b\right )}^{2} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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