\(\int \frac {(1+x^2)^2}{a x^6+b (1+x^2)^3} \, dx\) [393]

Optimal result

Integrand size = 25, antiderivative size = 168 \[ \int \frac {\left (1+x^2\right )^2}{a x^6+b \left (1+x^2\right )^3} \, dx=\frac {\arctan \left (\frac {\sqrt {\sqrt [3]{a}+\sqrt [3]{b}} x}{\sqrt [6]{b}}\right )}{3 \sqrt {\sqrt [3]{a}+\sqrt [3]{b}} b^{5/6}}+\frac {\arctan \left (\frac {\sqrt {-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b}} x}{\sqrt [6]{b}}\right )}{3 \sqrt {-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b}} b^{5/6}}+\frac {\arctan \left (\frac {\sqrt {(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b}} x}{\sqrt [6]{b}}\right )}{3 \sqrt {(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b}} b^{5/6}} \]

[Out]

1/3*arctan(x*(a^(1/3)+b^(1/3))^(1/2)/b^(1/6))/b^(5/6)/(a^(1/3)+b^(1/3))^(1/2)+1/3*arctan(x*(-(-1)^(1/3)*a^(1/3
)+b^(1/3))^(1/2)/b^(1/6))/b^(5/6)/(-(-1)^(1/3)*a^(1/3)+b^(1/3))^(1/2)+1/3*arctan(x*((-1)^(2/3)*a^(1/3)+b^(1/3)
)^(1/2)/b^(1/6))/b^(5/6)/((-1)^(2/3)*a^(1/3)+b^(1/3))^(1/2)

Rubi [F]

\[ \int \frac {\left (1+x^2\right )^2}{a x^6+b \left (1+x^2\right )^3} \, dx=\int \frac {\left (1+x^2\right )^2}{a x^6+b \left (1+x^2\right )^3} \, dx \]

[In]

Int[(1 + x^2)^2/(a*x^6 + b*(1 + x^2)^3),x]

[Out]

Defer[Int][(a*x^6 + b*(1 + x^2)^3)^(-1), x] + 2*Defer[Int][x^2/(a*x^6 + b*(1 + x^2)^3), x] + Defer[Int][x^4/(a
*x^6 + b*(1 + x^2)^3), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{b+3 b x^2+3 b x^4+a \left (1+\frac {b}{a}\right ) x^6}+\frac {2 x^2}{b+3 b x^2+3 b x^4+a \left (1+\frac {b}{a}\right ) x^6}+\frac {x^4}{b+3 b x^2+3 b x^4+a \left (1+\frac {b}{a}\right ) x^6}\right ) \, dx \\ & = 2 \int \frac {x^2}{b+3 b x^2+3 b x^4+a \left (1+\frac {b}{a}\right ) x^6} \, dx+\int \frac {1}{b+3 b x^2+3 b x^4+a \left (1+\frac {b}{a}\right ) x^6} \, dx+\int \frac {x^4}{b+3 b x^2+3 b x^4+a \left (1+\frac {b}{a}\right ) x^6} \, dx \\ & = 2 \int \frac {x^2}{a x^6+b \left (1+x^2\right )^3} \, dx+\int \frac {1}{a x^6+b \left (1+x^2\right )^3} \, dx+\int \frac {x^4}{a x^6+b \left (1+x^2\right )^3} \, dx \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.04 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.57 \[ \int \frac {\left (1+x^2\right )^2}{a x^6+b \left (1+x^2\right )^3} \, dx=\frac {1}{6} \text {RootSum}\left [b+3 b \text {$\#$1}^2+3 b \text {$\#$1}^4+a \text {$\#$1}^6+b \text {$\#$1}^6\&,\frac {\log (x-\text {$\#$1})+2 \log (x-\text {$\#$1}) \text {$\#$1}^2+\log (x-\text {$\#$1}) \text {$\#$1}^4}{b \text {$\#$1}+2 b \text {$\#$1}^3+a \text {$\#$1}^5+b \text {$\#$1}^5}\&\right ] \]

[In]

Integrate[(1 + x^2)^2/(a*x^6 + b*(1 + x^2)^3),x]

[Out]

RootSum[b + 3*b*#1^2 + 3*b*#1^4 + a*#1^6 + b*#1^6 & , (Log[x - #1] + 2*Log[x - #1]*#1^2 + Log[x - #1]*#1^4)/(b
*#1 + 2*b*#1^3 + a*#1^5 + b*#1^5) & ]/6

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.82 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.40

method	result	size
default	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a +b \right ) \textit {\_Z}^{6}+3 \textit {\_Z}^{4} b +3 \textit {\_Z}^{2} b +b \right )}{\sum }\frac {\left (\textit {\_R}^{4}+2 \textit {\_R}^{2}+1\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5} a +\textit {\_R}^{5} b +2 \textit {\_R}^{3} b +b \textit {\_R}}\right )}{6}\)	\(67\)
risch	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a +b \right ) \textit {\_Z}^{6}+3 \textit {\_Z}^{4} b +3 \textit {\_Z}^{2} b +b \right )}{\sum }\frac {\left (\textit {\_R}^{4}+2 \textit {\_R}^{2}+1\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5} a +\textit {\_R}^{5} b +2 \textit {\_R}^{3} b +b \textit {\_R}}\right )}{6}\)	\(67\)

[In]

int((x^2+1)^2/(a*x^6+b*(x^2+1)^3),x,method=_RETURNVERBOSE)

[Out]

1/6*sum((_R^4+2*_R^2+1)/(_R^5*a+_R^5*b+2*_R^3*b+_R*b)*ln(x-_R),_R=RootOf((a+b)*_Z^6+3*_Z^4*b+3*_Z^2*b+b))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.92 (sec) , antiderivative size = 5653, normalized size of antiderivative = 33.65 \[ \int \frac {\left (1+x^2\right )^2}{a x^6+b \left (1+x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate((x^2+1)^2/(a*x^6+b*(x^2+1)^3),x, algorithm="fricas")

[Out]

Too large to include

Sympy [A] (verification not implemented)

Time = 1.04 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.25 \[ \int \frac {\left (1+x^2\right )^2}{a x^6+b \left (1+x^2\right )^3} \, dx=\operatorname {RootSum} {\left (t^{6} \cdot \left (46656 a b^{5} + 46656 b^{6}\right ) + 3888 t^{4} b^{4} + 108 t^{2} b^{2} + 1, \left ( t \mapsto t \log {\left (6 t b + x \right )} \right )\right )} \]

[In]

integrate((x**2+1)**2/(a*x**6+b*(x**2+1)**3),x)

[Out]

RootSum(_t**6*(46656*a*b**5 + 46656*b**6) + 3888*_t**4*b**4 + 108*_t**2*b**2 + 1, Lambda(_t, _t*log(6*_t*b + x
)))

Maxima [F]

\[ \int \frac {\left (1+x^2\right )^2}{a x^6+b \left (1+x^2\right )^3} \, dx=\int { \frac {{\left (x^{2} + 1\right )}^{2}}{a x^{6} + {\left (x^{2} + 1\right )}^{3} b} \,d x } \]

[In]

integrate((x^2+1)^2/(a*x^6+b*(x^2+1)^3),x, algorithm="maxima")

[Out]

integrate((x^2 + 1)^2/(a*x^6 + (x^2 + 1)^3*b), x)

Giac [F]

\[ \int \frac {\left (1+x^2\right )^2}{a x^6+b \left (1+x^2\right )^3} \, dx=\int { \frac {{\left (x^{2} + 1\right )}^{2}}{a x^{6} + {\left (x^{2} + 1\right )}^{3} b} \,d x } \]

[In]

integrate((x^2+1)^2/(a*x^6+b*(x^2+1)^3),x, algorithm="giac")

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 10.48 (sec) , antiderivative size = 504, normalized size of antiderivative = 3.00 \[ \int \frac {\left (1+x^2\right )^2}{a x^6+b \left (1+x^2\right )^3} \, dx=\sum _{k=1}^6\ln \left (-a^3\,\left (a+b\right )\,\left (-{\mathrm {root}\left (46656\,a\,b^5\,z^6+46656\,b^6\,z^6+3888\,b^4\,z^4+108\,b^2\,z^2+1,z,k\right )}^2\,b^2\,60-{\mathrm {root}\left (46656\,a\,b^5\,z^6+46656\,b^6\,z^6+3888\,b^4\,z^4+108\,b^2\,z^2+1,z,k\right )}^4\,b^4\,864-{\mathrm {root}\left (46656\,a\,b^5\,z^6+46656\,b^6\,z^6+3888\,b^4\,z^4+108\,b^2\,z^2+1,z,k\right )}^4\,a\,b^3\,864+{\mathrm {root}\left (46656\,a\,b^5\,z^6+46656\,b^6\,z^6+3888\,b^4\,z^4+108\,b^2\,z^2+1,z,k\right )}^3\,b^3\,x\,504+{\mathrm {root}\left (46656\,a\,b^5\,z^6+46656\,b^6\,z^6+3888\,b^4\,z^4+108\,b^2\,z^2+1,z,k\right )}^5\,b^5\,x\,7776+\mathrm {root}\left (46656\,a\,b^5\,z^6+46656\,b^6\,z^6+3888\,b^4\,z^4+108\,b^2\,z^2+1,z,k\right )\,a\,x\,2+\mathrm {root}\left (46656\,a\,b^5\,z^6+46656\,b^6\,z^6+3888\,b^4\,z^4+108\,b^2\,z^2+1,z,k\right )\,b\,x\,8+{\mathrm {root}\left (46656\,a\,b^5\,z^6+46656\,b^6\,z^6+3888\,b^4\,z^4+108\,b^2\,z^2+1,z,k\right )}^2\,a\,b\,12-{\mathrm {root}\left (46656\,a\,b^5\,z^6+46656\,b^6\,z^6+3888\,b^4\,z^4+108\,b^2\,z^2+1,z,k\right )}^3\,a\,b^2\,x\,144+{\mathrm {root}\left (46656\,a\,b^5\,z^6+46656\,b^6\,z^6+3888\,b^4\,z^4+108\,b^2\,z^2+1,z,k\right )}^5\,a\,b^4\,x\,7776-1\right )\,3\right )\,\mathrm {root}\left (46656\,a\,b^5\,z^6+46656\,b^6\,z^6+3888\,b^4\,z^4+108\,b^2\,z^2+1,z,k\right ) \]

[In]

int((x^2 + 1)^2/(b*(x^2 + 1)^3 + a*x^6),x)

[Out]

symsum(log(-3*a^3*(a + b)*(504*root(46656*a*b^5*z^6 + 46656*b^6*z^6 + 3888*b^4*z^4 + 108*b^2*z^2 + 1, z, k)^3*
b^3*x - 864*root(46656*a*b^5*z^6 + 46656*b^6*z^6 + 3888*b^4*z^4 + 108*b^2*z^2 + 1, z, k)^4*b^4 - 864*root(4665
6*a*b^5*z^6 + 46656*b^6*z^6 + 3888*b^4*z^4 + 108*b^2*z^2 + 1, z, k)^4*a*b^3 - 60*root(46656*a*b^5*z^6 + 46656*
b^6*z^6 + 3888*b^4*z^4 + 108*b^2*z^2 + 1, z, k)^2*b^2 + 7776*root(46656*a*b^5*z^6 + 46656*b^6*z^6 + 3888*b^4*z
^4 + 108*b^2*z^2 + 1, z, k)^5*b^5*x + 2*root(46656*a*b^5*z^6 + 46656*b^6*z^6 + 3888*b^4*z^4 + 108*b^2*z^2 + 1,
z, k)*a*x + 8*root(46656*a*b^5*z^6 + 46656*b^6*z^6 + 3888*b^4*z^4 + 108*b^2*z^2 + 1, z, k)*b*x + 12*root(4665
6*a*b^5*z^6 + 46656*b^6*z^6 + 3888*b^4*z^4 + 108*b^2*z^2 + 1, z, k)^2*a*b - 144*root(46656*a*b^5*z^6 + 46656*b
^6*z^6 + 3888*b^4*z^4 + 108*b^2*z^2 + 1, z, k)^3*a*b^2*x + 7776*root(46656*a*b^5*z^6 + 46656*b^6*z^6 + 3888*b^
4*z^4 + 108*b^2*z^2 + 1, z, k)^5*a*b^4*x - 1))*root(46656*a*b^5*z^6 + 46656*b^6*z^6 + 3888*b^4*z^4 + 108*b^2*z
^2 + 1, z, k), k, 1, 6)