Integrand size = 20, antiderivative size = 81 \[ \int \frac {x^3 (A+B x)}{\left (a+b x^2\right )^{3/2}} \, dx=-\frac {x^2 (A+B x)}{b \sqrt {a+b x^2}}+\frac {(4 A+3 B x) \sqrt {a+b x^2}}{2 b^2}-\frac {3 a B \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{5/2}} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {833, 794, 223, 212} \[ \int \frac {x^3 (A+B x)}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {\sqrt {a+b x^2} (4 A+3 B x)}{2 b^2}-\frac {x^2 (A+B x)}{b \sqrt {a+b x^2}}-\frac {3 a B \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{5/2}} \]
[In]
[Out]
Rule 212
Rule 223
Rule 794
Rule 833
Rubi steps \begin{align*} \text {integral}& = -\frac {x^2 (A+B x)}{b \sqrt {a+b x^2}}+\frac {\int \frac {x (2 a A+3 a B x)}{\sqrt {a+b x^2}} \, dx}{a b} \\ & = -\frac {x^2 (A+B x)}{b \sqrt {a+b x^2}}+\frac {(4 A+3 B x) \sqrt {a+b x^2}}{2 b^2}-\frac {(3 a B) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{2 b^2} \\ & = -\frac {x^2 (A+B x)}{b \sqrt {a+b x^2}}+\frac {(4 A+3 B x) \sqrt {a+b x^2}}{2 b^2}-\frac {(3 a B) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{2 b^2} \\ & = -\frac {x^2 (A+B x)}{b \sqrt {a+b x^2}}+\frac {(4 A+3 B x) \sqrt {a+b x^2}}{2 b^2}-\frac {3 a B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{5/2}} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.91 \[ \int \frac {x^3 (A+B x)}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {4 a A+3 a B x+2 A b x^2+b B x^3}{2 b^2 \sqrt {a+b x^2}}+\frac {3 a B \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{2 b^{5/2}} \]
[In]
[Out]
Time = 3.43 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.95
method | result | size |
risch | \(\frac {\left (B x +2 A \right ) \sqrt {b \,x^{2}+a}}{2 b^{2}}+\frac {a B x}{b^{2} \sqrt {b \,x^{2}+a}}-\frac {3 a B \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {5}{2}}}+\frac {a A}{b^{2} \sqrt {b \,x^{2}+a}}\) | \(77\) |
default | \(B \left (\frac {x^{3}}{2 b \sqrt {b \,x^{2}+a}}-\frac {3 a \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )}{2 b}\right )+A \left (\frac {x^{2}}{\sqrt {b \,x^{2}+a}\, b}+\frac {2 a}{b^{2} \sqrt {b \,x^{2}+a}}\right )\) | \(98\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 197, normalized size of antiderivative = 2.43 \[ \int \frac {x^3 (A+B x)}{\left (a+b x^2\right )^{3/2}} \, dx=\left [\frac {3 \, {\left (B a b x^{2} + B a^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (B b^{2} x^{3} + 2 \, A b^{2} x^{2} + 3 \, B a b x + 4 \, A a b\right )} \sqrt {b x^{2} + a}}{4 \, {\left (b^{4} x^{2} + a b^{3}\right )}}, \frac {3 \, {\left (B a b x^{2} + B a^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (B b^{2} x^{3} + 2 \, A b^{2} x^{2} + 3 \, B a b x + 4 \, A a b\right )} \sqrt {b x^{2} + a}}{2 \, {\left (b^{4} x^{2} + a b^{3}\right )}}\right ] \]
[In]
[Out]
Time = 3.63 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.44 \[ \int \frac {x^3 (A+B x)}{\left (a+b x^2\right )^{3/2}} \, dx=A \left (\begin {cases} \frac {2 a}{b^{2} \sqrt {a + b x^{2}}} + \frac {x^{2}}{b \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{4}}{4 a^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) + B \left (\frac {3 \sqrt {a} x}{2 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {3 a \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{2 b^{\frac {5}{2}}} + \frac {x^{3}}{2 \sqrt {a} b \sqrt {1 + \frac {b x^{2}}{a}}}\right ) \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.05 \[ \int \frac {x^3 (A+B x)}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {B x^{3}}{2 \, \sqrt {b x^{2} + a} b} + \frac {A x^{2}}{\sqrt {b x^{2} + a} b} + \frac {3 \, B a x}{2 \, \sqrt {b x^{2} + a} b^{2}} - \frac {3 \, B a \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {5}{2}}} + \frac {2 \, A a}{\sqrt {b x^{2} + a} b^{2}} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.86 \[ \int \frac {x^3 (A+B x)}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {{\left ({\left (\frac {B x}{b} + \frac {2 \, A}{b}\right )} x + \frac {3 \, B a}{b^{2}}\right )} x + \frac {4 \, A a}{b^{2}}}{2 \, \sqrt {b x^{2} + a}} + \frac {3 \, B a \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{2 \, b^{\frac {5}{2}}} \]
[In]
[Out]
Timed out. \[ \int \frac {x^3 (A+B x)}{\left (a+b x^2\right )^{3/2}} \, dx=\int \frac {x^3\,\left (A+B\,x\right )}{{\left (b\,x^2+a\right )}^{3/2}} \,d x \]
[In]
[Out]