Optimal. Leaf size=68 \[ -\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{5/2}}+\frac{3 b \sqrt{a+b x}}{4 a^2 x}-\frac{\sqrt{a+b x}}{2 a x^2} \]
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Rubi [A] time = 0.0165941, antiderivative size = 68, normalized size of antiderivative = 1., 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} \[ -\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{5/2}}+\frac{3 b \sqrt{a+b x}}{4 a^2 x}-\frac{\sqrt{a+b x}}{2 a x^2} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x^3 \sqrt{a+b x}} \, dx &=-\frac{\sqrt{a+b x}}{2 a x^2}-\frac{(3 b) \int \frac{1}{x^2 \sqrt{a+b x}} \, dx}{4 a}\\ &=-\frac{\sqrt{a+b x}}{2 a x^2}+\frac{3 b \sqrt{a+b x}}{4 a^2 x}+\frac{\left (3 b^2\right ) \int \frac{1}{x \sqrt{a+b x}} \, dx}{8 a^2}\\ &=-\frac{\sqrt{a+b x}}{2 a x^2}+\frac{3 b \sqrt{a+b x}}{4 a^2 x}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x}\right )}{4 a^2}\\ &=-\frac{\sqrt{a+b x}}{2 a x^2}+\frac{3 b \sqrt{a+b x}}{4 a^2 x}-\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.010191, size = 33, normalized size = 0.49 \[ -\frac{2 b^2 \sqrt{a+b x} \, _2F_1\left (\frac{1}{2},3;\frac{3}{2};\frac{b x}{a}+1\right )}{a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 66, normalized size = 1. \begin{align*} 2\,{b}^{2} \left ( -1/4\,{\frac{\sqrt{bx+a}}{a{b}^{2}{x}^{2}}}-3/4\,{\frac{1}{a} \left ( -1/2\,{\frac{\sqrt{bx+a}}{abx}}+1/2\,{\frac{1}{{a}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) } \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 = 1.55904, size = 301, normalized size = 4.43 \begin{align*} \left [\frac{3 \, \sqrt{a} b^{2} x^{2} \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \,{\left (3 \, a b x - 2 \, a^{2}\right )} \sqrt{b x + a}}{8 \, a^{3} x^{2}}, \frac{3 \, \sqrt{-a} b^{2} x^{2} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) +{\left (3 \, a b x - 2 \, a^{2}\right )} \sqrt{b x + a}}{4 \, a^{3} x^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.00912, size = 102, normalized size = 1.5 \begin{align*} - \frac{1}{2 \sqrt{b} x^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{\sqrt{b}}{4 a x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{3 b^{\frac{3}{2}}}{4 a^{2} \sqrt{x} \sqrt{\frac{a}{b x} + 1}} - \frac{3 b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{4 a^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2173, size = 93, normalized size = 1.37 \begin{align*} \frac{\frac{3 \, b^{3} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{3 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{3} - 5 \, \sqrt{b x + a} a b^{3}}{a^{2} b^{2} x^{2}}}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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