Optimal. Leaf size=83 \[ -\frac{5 x^3}{24 b^2 \left (a+b x^2\right )^2}-\frac{5 x}{16 b^3 \left (a+b x^2\right )}+\frac{5 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{16 \sqrt{a} b^{7/2}}-\frac{x^5}{6 b \left (a+b x^2\right )^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.039702, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {28, 288, 205} \[ -\frac{5 x^3}{24 b^2 \left (a+b x^2\right )^2}-\frac{5 x}{16 b^3 \left (a+b x^2\right )}+\frac{5 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{16 \sqrt{a} b^{7/2}}-\frac{x^5}{6 b \left (a+b x^2\right )^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 28
Rule 288
Rule 205
Rubi steps
\begin{align*} \int \frac{x^6}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx &=b^4 \int \frac{x^6}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac{x^5}{6 b \left (a+b x^2\right )^3}+\frac{1}{6} \left (5 b^2\right ) \int \frac{x^4}{\left (a b+b^2 x^2\right )^3} \, dx\\ &=-\frac{x^5}{6 b \left (a+b x^2\right )^3}-\frac{5 x^3}{24 b^2 \left (a+b x^2\right )^2}+\frac{5}{8} \int \frac{x^2}{\left (a b+b^2 x^2\right )^2} \, dx\\ &=-\frac{x^5}{6 b \left (a+b x^2\right )^3}-\frac{5 x^3}{24 b^2 \left (a+b x^2\right )^2}-\frac{5 x}{16 b^3 \left (a+b x^2\right )}+\frac{5 \int \frac{1}{a b+b^2 x^2} \, dx}{16 b^2}\\ &=-\frac{x^5}{6 b \left (a+b x^2\right )^3}-\frac{5 x^3}{24 b^2 \left (a+b x^2\right )^2}-\frac{5 x}{16 b^3 \left (a+b x^2\right )}+\frac{5 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{16 \sqrt{a} b^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.037152, size = 66, normalized size = 0.8 \[ \frac{5 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{16 \sqrt{a} b^{7/2}}-\frac{x \left (15 a^2+40 a b x^2+33 b^2 x^4\right )}{48 b^3 \left (a+b x^2\right )^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.049, size = 58, normalized size = 0.7 \begin{align*}{\frac{1}{ \left ( b{x}^{2}+a \right ) ^{3}} \left ( -{\frac{11\,{x}^{5}}{16\,b}}-{\frac{5\,a{x}^{3}}{6\,{b}^{2}}}-{\frac{5\,{a}^{2}x}{16\,{b}^{3}}} \right ) }+{\frac{5}{16\,{b}^{3}}\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 = 1.66429, size = 544, normalized size = 6.55 \begin{align*} \left [-\frac{66 \, a b^{3} x^{5} + 80 \, a^{2} b^{2} x^{3} + 30 \, a^{3} b x + 15 \,{\left (b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}\right )} \sqrt{-a b} \log \left (\frac{b x^{2} - 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right )}{96 \,{\left (a b^{7} x^{6} + 3 \, a^{2} b^{6} x^{4} + 3 \, a^{3} b^{5} x^{2} + a^{4} b^{4}\right )}}, -\frac{33 \, a b^{3} x^{5} + 40 \, a^{2} b^{2} x^{3} + 15 \, a^{3} b x - 15 \,{\left (b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{a}\right )}{48 \,{\left (a b^{7} x^{6} + 3 \, a^{2} b^{6} x^{4} + 3 \, a^{3} b^{5} x^{2} + a^{4} b^{4}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.657521, size = 133, normalized size = 1.6 \begin{align*} - \frac{5 \sqrt{- \frac{1}{a b^{7}}} \log{\left (- a b^{3} \sqrt{- \frac{1}{a b^{7}}} + x \right )}}{32} + \frac{5 \sqrt{- \frac{1}{a b^{7}}} \log{\left (a b^{3} \sqrt{- \frac{1}{a b^{7}}} + x \right )}}{32} - \frac{15 a^{2} x + 40 a b x^{3} + 33 b^{2} x^{5}}{48 a^{3} b^{3} + 144 a^{2} b^{4} x^{2} + 144 a b^{5} x^{4} + 48 b^{6} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.13428, size = 76, normalized size = 0.92 \begin{align*} \frac{5 \, \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{16 \, \sqrt{a b} b^{3}} - \frac{33 \, b^{2} x^{5} + 40 \, a b x^{3} + 15 \, a^{2} x}{48 \,{\left (b x^{2} + a\right )}^{3} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]