Integrand size = 19, antiderivative size = 58 \[ \int \frac {A+B x^2}{\sqrt {a+b x^2}} \, dx=\frac {B x \sqrt {a+b x^2}}{2 b}+\frac {(2 A b-a B) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{3/2}} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {396, 223, 212} \[ \int \frac {A+B x^2}{\sqrt {a+b x^2}} \, dx=\frac {(2 A b-a B) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{3/2}}+\frac {B x \sqrt {a+b x^2}}{2 b} \]
[In]
[Out]
Rule 212
Rule 223
Rule 396
Rubi steps \begin{align*} \text {integral}& = \frac {B x \sqrt {a+b x^2}}{2 b}-\frac {(-2 A b+a B) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{2 b} \\ & = \frac {B x \sqrt {a+b x^2}}{2 b}-\frac {(-2 A b+a B) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{2 b} \\ & = \frac {B x \sqrt {a+b x^2}}{2 b}+\frac {(2 A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{3/2}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.12 \[ \int \frac {A+B x^2}{\sqrt {a+b x^2}} \, dx=\frac {B x \sqrt {a+b x^2}}{2 b}+\frac {(2 A b-a B) \text {arctanh}\left (\frac {\sqrt {b} x}{-\sqrt {a}+\sqrt {a+b x^2}}\right )}{b^{3/2}} \]
[In]
[Out]
Time = 2.82 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.83
method | result | size |
risch | \(\frac {B x \sqrt {b \,x^{2}+a}}{2 b}+\frac {\left (2 A b -B a \right ) \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\) | \(48\) |
default | \(\frac {A \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}+B \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )\) | \(63\) |
pseudoelliptic | \(\frac {B x \sqrt {b}\, \sqrt {b \,x^{2}+a}+2 A \,\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right ) b -B \,\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right ) a}{2 b^{\frac {3}{2}}}\) | \(64\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.90 \[ \int \frac {A+B x^2}{\sqrt {a+b x^2}} \, dx=\left [\frac {2 \, \sqrt {b x^{2} + a} B b x - {\left (B a - 2 \, A b\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right )}{4 \, b^{2}}, \frac {\sqrt {b x^{2} + a} B b x + {\left (B a - 2 \, A b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right )}{2 \, b^{2}}\right ] \]
[In]
[Out]
Time = 0.26 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.41 \[ \int \frac {A+B x^2}{\sqrt {a+b x^2}} \, dx=\begin {cases} \frac {B x \sqrt {a + b x^{2}}}{2 b} + \left (A - \frac {B a}{2 b}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: b \neq 0 \\\frac {A x + \frac {B x^{3}}{3}}{\sqrt {a}} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.81 \[ \int \frac {A+B x^2}{\sqrt {a+b x^2}} \, dx=\frac {\sqrt {b x^{2} + a} B x}{2 \, b} - \frac {B a \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {3}{2}}} + \frac {A \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {b}} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.83 \[ \int \frac {A+B x^2}{\sqrt {a+b x^2}} \, dx=\frac {\sqrt {b x^{2} + a} B x}{2 \, b} + \frac {{\left (B a - 2 \, A b\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{2 \, b^{\frac {3}{2}}} \]
[In]
[Out]
Time = 5.41 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.48 \[ \int \frac {A+B x^2}{\sqrt {a+b x^2}} \, dx=\left \{\begin {array}{cl} \frac {B\,x^3+3\,A\,x}{3\,\sqrt {a}} & \text {\ if\ \ }b=0\\ \frac {A\,\ln \left (\sqrt {b}\,x+\sqrt {b\,x^2+a}\right )}{\sqrt {b}}-\frac {B\,a\,\ln \left (2\,\sqrt {b}\,x+2\,\sqrt {b\,x^2+a}\right )}{2\,b^{3/2}}+\frac {B\,x\,\sqrt {b\,x^2+a}}{2\,b} & \text {\ if\ \ }b\neq 0 \end {array}\right . \]
[In]
[Out]