Optimal. Leaf size=153 \[ \frac{x (b d-5 a e)}{8 a b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{x (b d-a e)}{4 b^2 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (a+b x^2\right ) (3 a e+b d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{3/2} b^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.119201, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121, Rules used = {1250, 455, 385, 205} \[ \frac{x (b d-5 a e)}{8 a b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{x (b d-a e)}{4 b^2 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (a+b x^2\right ) (3 a e+b d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{3/2} b^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1250
Rule 455
Rule 385
Rule 205
Rubi steps
\begin{align*} \int \frac{x^2 \left (d+e x^2\right )}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx &=\frac{\left (b^2 \left (a b+b^2 x^2\right )\right ) \int \frac{x^2 \left (d+e x^2\right )}{\left (a b+b^2 x^2\right )^3} \, dx}{\sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{(b d-a e) x}{4 b^2 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (a b+b^2 x^2\right ) \int \frac{-b (b d-a e)-4 b^2 e x^2}{\left (a b+b^2 x^2\right )^2} \, dx}{4 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{(b d-5 a e) x}{8 a b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{(b d-a e) x}{4 b^2 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left ((b d+3 a e) \left (a b+b^2 x^2\right )\right ) \int \frac{1}{a b+b^2 x^2} \, dx}{8 a b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{(b d-5 a e) x}{8 a b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{(b d-a e) x}{4 b^2 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{(b d+3 a e) \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{3/2} b^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ \end{align*}
Mathematica [A] time = 0.0630064, size = 108, normalized size = 0.71 \[ \frac{\left (a+b x^2\right )^2 (3 a e+b d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )-\sqrt{a} \sqrt{b} x \left (3 a^2 e+a b \left (d+5 e x^2\right )-b^2 d x^2\right )}{8 a^{3/2} b^{5/2} \left (a+b x^2\right ) \sqrt{\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.016, size = 188, normalized size = 1.2 \begin{align*} -{\frac{b{x}^{2}+a}{8\,{b}^{2}a} \left ( -3\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{4}a{b}^{2}e-\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){x}^{4}{b}^{3}d+5\,\sqrt{ab}{x}^{3}abe-\sqrt{ab}{x}^{3}{b}^{2}d-6\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{2}{a}^{2}be-2\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{2}a{b}^{2}d+3\,\sqrt{ab}x{a}^{2}e+\sqrt{ab}xabd-3\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){a}^{3}e-\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){a}^{2}bd \right ){\frac{1}{\sqrt{ab}}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{-{\frac{3}{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 = 1.56592, size = 621, normalized size = 4.06 \begin{align*} \left [\frac{2 \,{\left (a b^{3} d - 5 \, a^{2} b^{2} e\right )} x^{3} -{\left ({\left (b^{3} d + 3 \, a b^{2} e\right )} x^{4} + a^{2} b d + 3 \, a^{3} e + 2 \,{\left (a b^{2} d + 3 \, a^{2} b e\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^{2} b^{2} d + 3 \, a^{3} b e\right )} x}{16 \,{\left (a^{2} b^{5} x^{4} + 2 \, a^{3} b^{4} x^{2} + a^{4} b^{3}\right )}}, \frac{{\left (a b^{3} d - 5 \, a^{2} b^{2} e\right )} x^{3} +{\left ({\left (b^{3} d + 3 \, a b^{2} e\right )} x^{4} + a^{2} b d + 3 \, a^{3} e + 2 \,{\left (a b^{2} d + 3 \, a^{2} b e\right )} x^{2}\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) -{\left (a^{2} b^{2} d + 3 \, a^{3} b e\right )} x}{8 \,{\left (a^{2} b^{5} x^{4} + 2 \, a^{3} b^{4} x^{2} + a^{4} b^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \left (d + e x^{2}\right )}{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]