Integrand size = 20, antiderivative size = 54 \[ \int \frac {x^3}{a+b+2 a x^2+a x^4} \, dx=-\frac {\arctan \left (\frac {\sqrt {a} \left (1+x^2\right )}{\sqrt {b}}\right )}{2 \sqrt {a} \sqrt {b}}+\frac {\log \left (a+b+2 a x^2+a x^4\right )}{4 a} \] Output:
-1/2*arctan(a^(1/2)*(x^2+1)/b^(1/2))/a^(1/2)/b^(1/2)+1/4*ln(a*x^4+2*a*x^2+ a+b)/a
Time = 0.01 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.91 \[ \int \frac {x^3}{a+b+2 a x^2+a x^4} \, dx=\frac {-\frac {2 \sqrt {a} \arctan \left (\frac {\sqrt {a} \left (1+x^2\right )}{\sqrt {b}}\right )}{\sqrt {b}}+\log \left (b+a \left (1+x^2\right )^2\right )}{4 a} \] Input:
Integrate[x^3/(a + b + 2*a*x^2 + a*x^4),x]
Output:
((-2*Sqrt[a]*ArcTan[(Sqrt[a]*(1 + x^2))/Sqrt[b]])/Sqrt[b] + Log[b + a*(1 + x^2)^2])/(4*a)
Time = 0.39 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.19, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1434, 1142, 27, 1083, 217, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x^3}{a x^4+2 a x^2+a+b} \, dx\) |
\(\Big \downarrow \) 1434 |
\(\displaystyle \frac {1}{2} \int \frac {x^2}{a x^4+2 a x^2+a+b}dx^2\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle \frac {1}{2} \left (\frac {\int \frac {2 a \left (x^2+1\right )}{a x^4+2 a x^2+a+b}dx^2}{2 a}-\int \frac {1}{a x^4+2 a x^2+a+b}dx^2\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \left (\int \frac {x^2+1}{a x^4+2 a x^2+a+b}dx^2-\int \frac {1}{a x^4+2 a x^2+a+b}dx^2\right )\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {1}{2} \left (2 \int \frac {1}{-x^4-4 a b}d\left (2 a x^2+2 a\right )+\int \frac {x^2+1}{a x^4+2 a x^2+a+b}dx^2\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {1}{2} \left (\int \frac {x^2+1}{a x^4+2 a x^2+a+b}dx^2-\frac {\arctan \left (\frac {2 a x^2+2 a}{2 \sqrt {a} \sqrt {b}}\right )}{\sqrt {a} \sqrt {b}}\right )\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {1}{2} \left (\frac {\log \left (a x^4+2 a x^2+a+b\right )}{2 a}-\frac {\arctan \left (\frac {2 a x^2+2 a}{2 \sqrt {a} \sqrt {b}}\right )}{\sqrt {a} \sqrt {b}}\right )\) |
Input:
Int[x^3/(a + b + 2*a*x^2 + a*x^4),x]
Output:
(-(ArcTan[(2*a + 2*a*x^2)/(2*Sqrt[a]*Sqrt[b])]/(Sqrt[a]*Sqrt[b])) + Log[a + b + 2*a*x^2 + a*x^4]/(2*a))/2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp [1/2 Subst[Int[x^((m - 1)/2)*(a + b*x + c*x^2)^p, x], x, x^2], x] /; Free Q[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]
Time = 0.07 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.87
method | result | size |
default | \(\frac {\ln \left (a \,x^{4}+2 a \,x^{2}+a +b \right )}{4 a}-\frac {\arctan \left (\frac {2 a \,x^{2}+2 a}{2 \sqrt {a b}}\right )}{2 \sqrt {a b}}\) | \(47\) |
risch | \(\frac {\ln \left (\left (a \sqrt {-a b}-a b \right ) x^{2}+a \sqrt {-a b}+\sqrt {-a b}\, b \right )}{4 a}+\frac {\ln \left (\left (a \sqrt {-a b}-a b \right ) x^{2}+a \sqrt {-a b}+\sqrt {-a b}\, b \right ) \sqrt {-a b}}{4 b a}+\frac {\ln \left (\left (-a \sqrt {-a b}-a b \right ) x^{2}-a \sqrt {-a b}-\sqrt {-a b}\, b \right )}{4 a}-\frac {\ln \left (\left (-a \sqrt {-a b}-a b \right ) x^{2}-a \sqrt {-a b}-\sqrt {-a b}\, b \right ) \sqrt {-a b}}{4 b a}\) | \(186\) |
Input:
int(x^3/(a*x^4+2*a*x^2+a+b),x,method=_RETURNVERBOSE)
Output:
1/4*ln(a*x^4+2*a*x^2+a+b)/a-1/2/(a*b)^(1/2)*arctan(1/2*(2*a*x^2+2*a)/(a*b) ^(1/2))
Time = 0.08 (sec) , antiderivative size = 131, normalized size of antiderivative = 2.43 \[ \int \frac {x^3}{a+b+2 a x^2+a x^4} \, dx=\left [\frac {b \log \left (a x^{4} + 2 \, a x^{2} + a + b\right ) - \sqrt {-a b} \log \left (\frac {a x^{4} + 2 \, a x^{2} + 2 \, \sqrt {-a b} {\left (x^{2} + 1\right )} + a - b}{a x^{4} + 2 \, a x^{2} + a + b}\right )}{4 \, a b}, \frac {b \log \left (a x^{4} + 2 \, a x^{2} + a + b\right ) + 2 \, \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{a x^{2} + a}\right )}{4 \, a b}\right ] \] Input:
integrate(x^3/(a*x^4+2*a*x^2+a+b),x, algorithm="fricas")
Output:
[1/4*(b*log(a*x^4 + 2*a*x^2 + a + b) - sqrt(-a*b)*log((a*x^4 + 2*a*x^2 + 2 *sqrt(-a*b)*(x^2 + 1) + a - b)/(a*x^4 + 2*a*x^2 + a + b)))/(a*b), 1/4*(b*l og(a*x^4 + 2*a*x^2 + a + b) + 2*sqrt(a*b)*arctan(sqrt(a*b)/(a*x^2 + a)))/( a*b)]
Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (48) = 96\).
Time = 0.31 (sec) , antiderivative size = 117, normalized size of antiderivative = 2.17 \[ \int \frac {x^3}{a+b+2 a x^2+a x^4} \, dx=\left (\frac {1}{4 a} - \frac {\sqrt {- a^{3} b}}{4 a^{2} b}\right ) \log {\left (x^{2} + \frac {- 4 a b \left (\frac {1}{4 a} - \frac {\sqrt {- a^{3} b}}{4 a^{2} b}\right ) + a + b}{a} \right )} + \left (\frac {1}{4 a} + \frac {\sqrt {- a^{3} b}}{4 a^{2} b}\right ) \log {\left (x^{2} + \frac {- 4 a b \left (\frac {1}{4 a} + \frac {\sqrt {- a^{3} b}}{4 a^{2} b}\right ) + a + b}{a} \right )} \] Input:
integrate(x**3/(a*x**4+2*a*x**2+a+b),x)
Output:
(1/(4*a) - sqrt(-a**3*b)/(4*a**2*b))*log(x**2 + (-4*a*b*(1/(4*a) - sqrt(-a **3*b)/(4*a**2*b)) + a + b)/a) + (1/(4*a) + sqrt(-a**3*b)/(4*a**2*b))*log( x**2 + (-4*a*b*(1/(4*a) + sqrt(-a**3*b)/(4*a**2*b)) + a + b)/a)
Time = 0.11 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.78 \[ \int \frac {x^3}{a+b+2 a x^2+a x^4} \, dx=-\frac {\arctan \left (\frac {a x^{2} + a}{\sqrt {a b}}\right )}{2 \, \sqrt {a b}} + \frac {\log \left (a x^{4} + 2 \, a x^{2} + a + b\right )}{4 \, a} \] Input:
integrate(x^3/(a*x^4+2*a*x^2+a+b),x, algorithm="maxima")
Output:
-1/2*arctan((a*x^2 + a)/sqrt(a*b))/sqrt(a*b) + 1/4*log(a*x^4 + 2*a*x^2 + a + b)/a
Time = 0.14 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.78 \[ \int \frac {x^3}{a+b+2 a x^2+a x^4} \, dx=-\frac {\arctan \left (\frac {a x^{2} + a}{\sqrt {a b}}\right )}{2 \, \sqrt {a b}} + \frac {\log \left (a x^{4} + 2 \, a x^{2} + a + b\right )}{4 \, a} \] Input:
integrate(x^3/(a*x^4+2*a*x^2+a+b),x, algorithm="giac")
Output:
-1/2*arctan((a*x^2 + a)/sqrt(a*b))/sqrt(a*b) + 1/4*log(a*x^4 + 2*a*x^2 + a + b)/a
Time = 0.09 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.57 \[ \int \frac {x^3}{a+b+2 a x^2+a x^4} \, dx=\frac {\ln \left (a\,x^4+2\,a\,x^2+a+b\right )}{4\,a}-\frac {\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {b}}{a+b}+\frac {a^{3/2}}{\sqrt {b}\,\left (a+b\right )}+\frac {\sqrt {a}\,\sqrt {b}\,x^2}{a+b}+\frac {a^{3/2}\,x^2}{\sqrt {b}\,\left (a+b\right )}\right )}{2\,\sqrt {a}\,\sqrt {b}} \] Input:
int(x^3/(a + b + 2*a*x^2 + a*x^4),x)
Output:
log(a + b + 2*a*x^2 + a*x^4)/(4*a) - atan((a^(1/2)*b^(1/2))/(a + b) + a^(3 /2)/(b^(1/2)*(a + b)) + (a^(1/2)*b^(1/2)*x^2)/(a + b) + (a^(3/2)*x^2)/(b^( 1/2)*(a + b)))/(2*a^(1/2)*b^(1/2))
Time = 0.15 (sec) , antiderivative size = 196, normalized size of antiderivative = 3.63 \[ \int \frac {x^3}{a+b+2 a x^2+a x^4} \, dx=\frac {2 \sqrt {\sqrt {a}\, \sqrt {a +b}+a}\, \sqrt {\sqrt {a}\, \sqrt {a +b}-a}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {a}\, \sqrt {a +b}-a}\, \sqrt {2}-2 \sqrt {a}\, x}{\sqrt {\sqrt {a}\, \sqrt {a +b}+a}\, \sqrt {2}}\right )+2 \sqrt {\sqrt {a}\, \sqrt {a +b}+a}\, \sqrt {\sqrt {a}\, \sqrt {a +b}-a}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {a}\, \sqrt {a +b}-a}\, \sqrt {2}+2 \sqrt {a}\, x}{\sqrt {\sqrt {a}\, \sqrt {a +b}+a}\, \sqrt {2}}\right )+\mathrm {log}\left (-\sqrt {\sqrt {a}\, \sqrt {a +b}-a}\, \sqrt {2}\, x +\sqrt {a +b}+\sqrt {a}\, x^{2}\right ) b +\mathrm {log}\left (\sqrt {\sqrt {a}\, \sqrt {a +b}-a}\, \sqrt {2}\, x +\sqrt {a +b}+\sqrt {a}\, x^{2}\right ) b}{4 a b} \] Input:
int(x^3/(a*x^4+2*a*x^2+a+b),x)
Output:
(2*sqrt(sqrt(a)*sqrt(a + b) + a)*sqrt(sqrt(a)*sqrt(a + b) - a)*atan((sqrt( sqrt(a)*sqrt(a + b) - a)*sqrt(2) - 2*sqrt(a)*x)/(sqrt(sqrt(a)*sqrt(a + b) + a)*sqrt(2))) + 2*sqrt(sqrt(a)*sqrt(a + b) + a)*sqrt(sqrt(a)*sqrt(a + b) - a)*atan((sqrt(sqrt(a)*sqrt(a + b) - a)*sqrt(2) + 2*sqrt(a)*x)/(sqrt(sqrt (a)*sqrt(a + b) + a)*sqrt(2))) + log( - sqrt(sqrt(a)*sqrt(a + b) - a)*sqrt (2)*x + sqrt(a + b) + sqrt(a)*x**2)*b + log(sqrt(sqrt(a)*sqrt(a + b) - a)* sqrt(2)*x + sqrt(a + b) + sqrt(a)*x**2)*b)/(4*a*b)