Optimal. Leaf size=56 \[ -\frac{3 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{b^{5/2}}+\frac{1}{b \sqrt{x} (a x+b)}-\frac{3}{b^2 \sqrt{x}} \]
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Rubi [A] time = 0.0191806, antiderivative size = 56, 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 = {263, 51, 63, 205} \[ -\frac{3 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{b^{5/2}}+\frac{1}{b \sqrt{x} (a x+b)}-\frac{3}{b^2 \sqrt{x}} \]
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 )^2 x^{7/2}} \, dx &=\int \frac{1}{x^{3/2} (b+a x)^2} \, dx\\ &=\frac{1}{b \sqrt{x} (b+a x)}+\frac{3 \int \frac{1}{x^{3/2} (b+a x)} \, dx}{2 b}\\ &=-\frac{3}{b^2 \sqrt{x}}+\frac{1}{b \sqrt{x} (b+a x)}-\frac{(3 a) \int \frac{1}{\sqrt{x} (b+a x)} \, dx}{2 b^2}\\ &=-\frac{3}{b^2 \sqrt{x}}+\frac{1}{b \sqrt{x} (b+a x)}-\frac{(3 a) \operatorname{Subst}\left (\int \frac{1}{b+a x^2} \, dx,x,\sqrt{x}\right )}{b^2}\\ &=-\frac{3}{b^2 \sqrt{x}}+\frac{1}{b \sqrt{x} (b+a x)}-\frac{3 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{b^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0044137, size = 25, normalized size = 0.45 \[ -\frac{2 \, _2F_1\left (-\frac{1}{2},2;\frac{1}{2};-\frac{a x}{b}\right )}{b^2 \sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 48, normalized size = 0.9 \begin{align*} -{\frac{a}{{b}^{2} \left ( ax+b \right ) }\sqrt{x}}-3\,{\frac{a}{{b}^{2}\sqrt{ab}}\arctan \left ({\frac{a\sqrt{x}}{\sqrt{ab}}} \right ) }-2\,{\frac{1}{{b}^{2}\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 = 1.74837, size = 323, normalized size = 5.77 \begin{align*} \left [\frac{3 \,{\left (a x^{2} + b x\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 (3 \, a x + 2 \, b\right )} \sqrt{x}}{2 \,{\left (a b^{2} x^{2} + b^{3} x\right )}}, \frac{3 \,{\left (a x^{2} + b x\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{\frac{a}{b}}}{a \sqrt{x}}\right ) -{\left (3 \, a x + 2 \, b\right )} \sqrt{x}}{a b^{2} x^{2} + b^{3} x}\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 = 1.12187, size = 66, normalized size = 1.18 \begin{align*} -\frac{3 \, a \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{2}} - \frac{3 \, a x + 2 \, b}{{\left (a x^{\frac{3}{2}} + b \sqrt{x}\right )} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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