Optimal. Leaf size=57 \[ -\frac{3 b}{a^2 \sqrt{a+b x}}+\frac{3 b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{5/2}}-\frac{1}{a x \sqrt{a+b x}} \]
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Rubi [A] time = 0.0165216, antiderivative size = 59, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {51, 63, 208} \[ -\frac{3 \sqrt{a+b x}}{a^2 x}+\frac{3 b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{5/2}}+\frac{2}{a x \sqrt{a+b x}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x^2 (a+b x)^{3/2}} \, dx &=\frac{2}{a x \sqrt{a+b x}}+\frac{3 \int \frac{1}{x^2 \sqrt{a+b x}} \, dx}{a}\\ &=\frac{2}{a x \sqrt{a+b x}}-\frac{3 \sqrt{a+b x}}{a^2 x}-\frac{(3 b) \int \frac{1}{x \sqrt{a+b x}} \, dx}{2 a^2}\\ &=\frac{2}{a x \sqrt{a+b x}}-\frac{3 \sqrt{a+b x}}{a^2 x}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x}\right )}{a^2}\\ &=\frac{2}{a x \sqrt{a+b x}}-\frac{3 \sqrt{a+b x}}{a^2 x}+\frac{3 b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0156457, size = 31, normalized size = 0.54 \[ -\frac{2 b \, _2F_1\left (-\frac{1}{2},2;\frac{1}{2};\frac{b x}{a}+1\right )}{a^2 \sqrt{a+b x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 55, normalized size = 1. \begin{align*} 2\,b \left ( -{\frac{1}{{a}^{2}} \left ( 1/2\,{\frac{\sqrt{bx+a}}{bx}}-3/2\,{\frac{1}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) }-{\frac{1}{{a}^{2}\sqrt{bx+a}}} \right ) \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 = 1.63446, size = 346, normalized size = 6.07 \begin{align*} \left [\frac{3 \,{\left (b^{2} x^{2} + a b x\right )} \sqrt{a} \log \left (\frac{b x + 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) - 2 \,{\left (3 \, a b x + a^{2}\right )} \sqrt{b x + a}}{2 \,{\left (a^{3} b x^{2} + a^{4} x\right )}}, -\frac{3 \,{\left (b^{2} x^{2} + a b x\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) +{\left (3 \, a b x + a^{2}\right )} \sqrt{b x + a}}{a^{3} b x^{2} + a^{4} x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.14375, size = 73, normalized size = 1.28 \begin{align*} - \frac{1}{a \sqrt{b} x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{3 \sqrt{b}}{a^{2} \sqrt{x} \sqrt{\frac{a}{b x} + 1}} + \frac{3 b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{a^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18841, size = 86, normalized size = 1.51 \begin{align*} -\frac{3 \, b \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} - \frac{3 \,{\left (b x + a\right )} b - 2 \, a b}{{\left ({\left (b x + a\right )}^{\frac{3}{2}} - \sqrt{b x + a} a\right )} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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