Integrand size = 20, antiderivative size = 67 \[ \int \frac {x^2 \left (c+d x^2\right )}{\left (a+b x^2\right )^2} \, dx=\frac {d x}{b^2}-\frac {(b c-a d) x}{2 b^2 \left (a+b x^2\right )}+\frac {(b c-3 a d) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} b^{5/2}} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {466, 396, 211} \[ \int \frac {x^2 \left (c+d x^2\right )}{\left (a+b x^2\right )^2} \, dx=\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (b c-3 a d)}{2 \sqrt {a} b^{5/2}}-\frac {x (b c-a d)}{2 b^2 \left (a+b x^2\right )}+\frac {d x}{b^2} \]
[In]
[Out]
Rule 211
Rule 396
Rule 466
Rubi steps \begin{align*} \text {integral}& = -\frac {(b c-a d) x}{2 b^2 \left (a+b x^2\right )}-\frac {\int \frac {-b c+a d-2 b d x^2}{a+b x^2} \, dx}{2 b^2} \\ & = \frac {d x}{b^2}-\frac {(b c-a d) x}{2 b^2 \left (a+b x^2\right )}+\frac {(b c-3 a d) \int \frac {1}{a+b x^2} \, dx}{2 b^2} \\ & = \frac {d x}{b^2}-\frac {(b c-a d) x}{2 b^2 \left (a+b x^2\right )}+\frac {(b c-3 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} b^{5/2}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.01 \[ \int \frac {x^2 \left (c+d x^2\right )}{\left (a+b x^2\right )^2} \, dx=\frac {d x}{b^2}-\frac {(b c-a d) x}{2 b^2 \left (a+b x^2\right )}-\frac {(-b c+3 a d) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} b^{5/2}} \]
[In]
[Out]
Time = 2.66 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.88
method | result | size |
default | \(\frac {d x}{b^{2}}-\frac {\frac {\left (-\frac {a d}{2}+\frac {b c}{2}\right ) x}{b \,x^{2}+a}+\frac {\left (3 a d -b c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}}{b^{2}}\) | \(59\) |
risch | \(\frac {d x}{b^{2}}+\frac {\left (\frac {a d}{2}-\frac {b c}{2}\right ) x}{b^{2} \left (b \,x^{2}+a \right )}-\frac {3 \ln \left (b x -\sqrt {-a b}\right ) a d}{4 b^{2} \sqrt {-a b}}+\frac {\ln \left (b x -\sqrt {-a b}\right ) c}{4 b \sqrt {-a b}}+\frac {3 \ln \left (-b x -\sqrt {-a b}\right ) a d}{4 b^{2} \sqrt {-a b}}-\frac {\ln \left (-b x -\sqrt {-a b}\right ) c}{4 b \sqrt {-a b}}\) | \(135\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 202, normalized size of antiderivative = 3.01 \[ \int \frac {x^2 \left (c+d x^2\right )}{\left (a+b x^2\right )^2} \, dx=\left [\frac {4 \, a b^{2} d x^{3} + {\left (a b c - 3 \, a^{2} d + {\left (b^{2} c - 3 \, a b d\right )} x^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} + 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) - 2 \, {\left (a b^{2} c - 3 \, a^{2} b d\right )} x}{4 \, {\left (a b^{4} x^{2} + a^{2} b^{3}\right )}}, \frac {2 \, a b^{2} d x^{3} + {\left (a b c - 3 \, a^{2} d + {\left (b^{2} c - 3 \, a b d\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) - {\left (a b^{2} c - 3 \, a^{2} b d\right )} x}{2 \, {\left (a b^{4} x^{2} + a^{2} b^{3}\right )}}\right ] \]
[In]
[Out]
Time = 0.29 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.70 \[ \int \frac {x^2 \left (c+d x^2\right )}{\left (a+b x^2\right )^2} \, dx=\frac {x \left (a d - b c\right )}{2 a b^{2} + 2 b^{3} x^{2}} + \frac {\sqrt {- \frac {1}{a b^{5}}} \cdot \left (3 a d - b c\right ) \log {\left (- a b^{2} \sqrt {- \frac {1}{a b^{5}}} + x \right )}}{4} - \frac {\sqrt {- \frac {1}{a b^{5}}} \cdot \left (3 a d - b c\right ) \log {\left (a b^{2} \sqrt {- \frac {1}{a b^{5}}} + x \right )}}{4} + \frac {d x}{b^{2}} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.90 \[ \int \frac {x^2 \left (c+d x^2\right )}{\left (a+b x^2\right )^2} \, dx=-\frac {{\left (b c - a d\right )} x}{2 \, {\left (b^{3} x^{2} + a b^{2}\right )}} + \frac {d x}{b^{2}} + \frac {{\left (b c - 3 \, a d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{2}} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.87 \[ \int \frac {x^2 \left (c+d x^2\right )}{\left (a+b x^2\right )^2} \, dx=\frac {d x}{b^{2}} + \frac {{\left (b c - 3 \, a d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{2}} - \frac {b c x - a d x}{2 \, {\left (b x^{2} + a\right )} b^{2}} \]
[In]
[Out]
Time = 5.24 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.88 \[ \int \frac {x^2 \left (c+d x^2\right )}{\left (a+b x^2\right )^2} \, dx=\frac {x\,\left (\frac {a\,d}{2}-\frac {b\,c}{2}\right )}{b^3\,x^2+a\,b^2}+\frac {d\,x}{b^2}-\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (3\,a\,d-b\,c\right )}{2\,\sqrt {a}\,b^{5/2}} \]
[In]
[Out]