Optimal. Leaf size=67 \[ -\frac{x (b c-a d)}{2 b^2 \left (a+b x^2\right )}+\frac{(b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{5/2}}+\frac{d x}{b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0507307, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {455, 388, 205} \[ -\frac{x (b c-a d)}{2 b^2 \left (a+b x^2\right )}+\frac{(b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{5/2}}+\frac{d x}{b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 455
Rule 388
Rule 205
Rubi steps
\begin{align*} \int \frac{x^2 \left (c+d x^2\right )}{\left (a+b x^2\right )^2} \, dx &=-\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*}
Mathematica [A] time = 0.0682567, size = 68, normalized size = 1.01 \[ -\frac{x (b c-a d)}{2 b^2 \left (a+b x^2\right )}-\frac{(3 a d-b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{5/2}}+\frac{d x}{b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.009, size = 82, normalized size = 1.2 \begin{align*}{\frac{dx}{{b}^{2}}}+{\frac{axd}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{cx}{2\,b \left ( b{x}^{2}+a \right ) }}-{\frac{3\,ad}{2\,{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{c}{2\,b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \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 = 1.51835, size = 433, normalized size = 6.46 \begin{align*} \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 ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.646734, size = 114, normalized size = 1.7 \begin{align*} \frac{x \left (a d - b c\right )}{2 a b^{2} + 2 b^{3} x^{2}} + \frac{\sqrt{- \frac{1}{a b^{5}}} \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}}} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.17406, size = 78, normalized size = 1.16 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]