Optimal. Leaf size=86 \[ -\frac{3 A b-2 a B}{2 a^2 \sqrt{a+b x^2}}+\frac{(3 A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{A}{2 a x^2 \sqrt{a+b x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0671464, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {446, 78, 51, 63, 208} \[ -\frac{3 A b-2 a B}{2 a^2 \sqrt{a+b x^2}}+\frac{(3 A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{A}{2 a x^2 \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 446
Rule 78
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{A+B x^2}{x^3 \left (a+b x^2\right )^{3/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{A+B x}{x^2 (a+b x)^{3/2}} \, dx,x,x^2\right )\\ &=-\frac{A}{2 a x^2 \sqrt{a+b x^2}}+\frac{\left (-\frac{3 A b}{2}+a B\right ) \operatorname{Subst}\left (\int \frac{1}{x (a+b x)^{3/2}} \, dx,x,x^2\right )}{2 a}\\ &=-\frac{3 A b-2 a B}{2 a^2 \sqrt{a+b x^2}}-\frac{A}{2 a x^2 \sqrt{a+b x^2}}-\frac{(3 A b-2 a B) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^2\right )}{4 a^2}\\ &=-\frac{3 A b-2 a B}{2 a^2 \sqrt{a+b x^2}}-\frac{A}{2 a x^2 \sqrt{a+b x^2}}-\frac{(3 A b-2 a B) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^2}\right )}{2 a^2 b}\\ &=-\frac{3 A b-2 a B}{2 a^2 \sqrt{a+b x^2}}-\frac{A}{2 a x^2 \sqrt{a+b x^2}}+\frac{(3 A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0184403, size = 57, normalized size = 0.66 \[ \frac{x^2 (2 a B-3 A b) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{b x^2}{a}+1\right )-a A}{2 a^2 x^2 \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.009, size = 109, normalized size = 1.3 \begin{align*}{\frac{B}{a}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{B\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{A}{2\,a{x}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{3\,Ab}{2\,{a}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{3\,Ab}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}} \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 = 1.7149, size = 506, normalized size = 5.88 \begin{align*} \left [-\frac{{\left ({\left (2 \, B a b - 3 \, A b^{2}\right )} x^{4} +{\left (2 \, B a^{2} - 3 \, A a b\right )} x^{2}\right )} \sqrt{a} \log \left (-\frac{b x^{2} + 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (A a^{2} -{\left (2 \, B a^{2} - 3 \, A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{4 \,{\left (a^{3} b x^{4} + a^{4} x^{2}\right )}}, \frac{{\left ({\left (2 \, B a b - 3 \, A b^{2}\right )} x^{4} +{\left (2 \, B a^{2} - 3 \, A a b\right )} x^{2}\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) -{\left (A a^{2} -{\left (2 \, B a^{2} - 3 \, A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{2 \,{\left (a^{3} b x^{4} + a^{4} x^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 20.9669, size = 262, normalized size = 3.05 \begin{align*} A \left (- \frac{1}{2 a \sqrt{b} x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{3 \sqrt{b}}{2 a^{2} x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{3 b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2 a^{\frac{5}{2}}}\right ) + B \left (\frac{2 a^{3} \sqrt{1 + \frac{b x^{2}}{a}}}{2 a^{\frac{9}{2}} + 2 a^{\frac{7}{2}} b x^{2}} + \frac{a^{3} \log{\left (\frac{b x^{2}}{a} \right )}}{2 a^{\frac{9}{2}} + 2 a^{\frac{7}{2}} b x^{2}} - \frac{2 a^{3} \log{\left (\sqrt{1 + \frac{b x^{2}}{a}} + 1 \right )}}{2 a^{\frac{9}{2}} + 2 a^{\frac{7}{2}} b x^{2}} + \frac{a^{2} b x^{2} \log{\left (\frac{b x^{2}}{a} \right )}}{2 a^{\frac{9}{2}} + 2 a^{\frac{7}{2}} b x^{2}} - \frac{2 a^{2} b x^{2} \log{\left (\sqrt{1 + \frac{b x^{2}}{a}} + 1 \right )}}{2 a^{\frac{9}{2}} + 2 a^{\frac{7}{2}} b x^{2}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.13625, size = 134, normalized size = 1.56 \begin{align*} \frac{{\left (2 \, B a - 3 \, A b\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{2 \, \sqrt{-a} a^{2}} + \frac{2 \,{\left (b x^{2} + a\right )} B a - 2 \, B a^{2} - 3 \,{\left (b x^{2} + a\right )} A b + 2 \, A a b}{2 \,{\left ({\left (b x^{2} + a\right )}^{\frac{3}{2}} - \sqrt{b x^{2} + a} a\right )} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]