Optimal. Leaf size=65 \[ \frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{3/2}}-\frac{\sqrt{a+b x}}{2 x^2}-\frac{b \sqrt{a+b x}}{4 a x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0183526, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {47, 51, 63, 208} \[ \frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{3/2}}-\frac{\sqrt{a+b x}}{2 x^2}-\frac{b \sqrt{a+b x}}{4 a x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 47
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x}}{x^3} \, dx &=-\frac{\sqrt{a+b x}}{2 x^2}+\frac{1}{4} b \int \frac{1}{x^2 \sqrt{a+b x}} \, dx\\ &=-\frac{\sqrt{a+b x}}{2 x^2}-\frac{b \sqrt{a+b x}}{4 a x}-\frac{b^2 \int \frac{1}{x \sqrt{a+b x}} \, dx}{8 a}\\ &=-\frac{\sqrt{a+b x}}{2 x^2}-\frac{b \sqrt{a+b x}}{4 a x}-\frac{b \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x}\right )}{4 a}\\ &=-\frac{\sqrt{a+b x}}{2 x^2}-\frac{b \sqrt{a+b x}}{4 a x}+\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0243131, size = 35, normalized size = 0.54 \[ -\frac{2 b^2 (a+b x)^{3/2} \, _2F_1\left (\frac{3}{2},3;\frac{5}{2};\frac{b x}{a}+1\right )}{3 a^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.008, size = 53, normalized size = 0.8 \begin{align*} 2\,{b}^{2} \left ({\frac{1}{{b}^{2}{x}^{2}} \left ( -1/8\,{\frac{ \left ( bx+a \right ) ^{3/2}}{a}}-1/8\,\sqrt{bx+a} \right ) }+1/8\,{\frac{1}{{a}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.5508, size = 292, normalized size = 4.49 \begin{align*} \left [\frac{\sqrt{a} b^{2} x^{2} \log \left (\frac{b x + 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) - 2 \,{\left (a b x + 2 \, a^{2}\right )} \sqrt{b x + a}}{8 \, a^{2} x^{2}}, -\frac{\sqrt{-a} b^{2} x^{2} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) +{\left (a b x + 2 \, a^{2}\right )} \sqrt{b x + a}}{4 \, a^{2} x^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 4.86688, size = 97, normalized size = 1.49 \begin{align*} - \frac{a}{2 \sqrt{b} x^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{3 \sqrt{b}}{4 x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{b^{\frac{3}{2}}}{4 a \sqrt{x} \sqrt{\frac{a}{b x} + 1}} + \frac{b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{4 a^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.1688, size = 89, normalized size = 1.37 \begin{align*} -\frac{\frac{b^{3} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} + \frac{{\left (b x + a\right )}^{\frac{3}{2}} b^{3} + \sqrt{b x + a} a b^{3}}{a b^{2} x^{2}}}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]