\(\int \frac {(3+4 x) \sqrt {x+2 x^2-2 x^4}}{(1+2 x) (1+2 x+x^3)} \, dx\) [1180]

Optimal result

Integrand size = 39, antiderivative size = 87 \[ \int \frac {(3+4 x) \sqrt {x+2 x^2-2 x^4}}{(1+2 x) \left (1+2 x+x^3\right )} \, dx=2 \sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt {x+2 x^2-2 x^4}}{-1-2 x+2 x^3}\right )-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x \sqrt {x+2 x^2-2 x^4}}{-1-2 x+2 x^3}\right ) \] Output:

2*2^(1/2)*arctan(2^(1/2)*x*(-2*x^4+2*x^2+x)^(1/2)/(2*x^3-2*x-1))-2*3^(1/2) 
*arctan(3^(1/2)*x*(-2*x^4+2*x^2+x)^(1/2)/(2*x^3-2*x-1))
 

Mathematica [A] (verified)

Time = 2.25 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.13 \[ \int \frac {(3+4 x) \sqrt {x+2 x^2-2 x^4}}{(1+2 x) \left (1+2 x+x^3\right )} \, dx=\frac {2 \sqrt {x+2 x^2-2 x^4} \left (\sqrt {2} \text {arctanh}\left (\frac {x^{3/2}}{\sqrt {-\frac {1}{2}-x+x^3}}\right )-\sqrt {3} \text {arctanh}\left (\frac {\sqrt {3} x^{3/2}}{\sqrt {-1-2 x+2 x^3}}\right )\right )}{\sqrt {x} \sqrt {-1-2 x+2 x^3}} \] Input:

Integrate[((3 + 4*x)*Sqrt[x + 2*x^2 - 2*x^4])/((1 + 2*x)*(1 + 2*x + x^3)), 
x]
 

Output:

(2*Sqrt[x + 2*x^2 - 2*x^4]*(Sqrt[2]*ArcTanh[x^(3/2)/Sqrt[-1/2 - x + x^3]] 
- Sqrt[3]*ArcTanh[(Sqrt[3]*x^(3/2))/Sqrt[-1 - 2*x + 2*x^3]]))/(Sqrt[x]*Sqr 
t[-1 - 2*x + 2*x^3])
 

Rubi [F]

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.

\(\displaystyle \int \frac {(4 x+3) \sqrt {-2 x^4+2 x^2+x}}{(2 x+1) \left (x^3+2 x+1\right )} \, dx\)

\(\Big \downarrow \) 2467

\(\displaystyle \frac {\sqrt {-2 x^4+2 x^2+x} \int \frac {\sqrt {x} (4 x+3) \sqrt {-2 x^3+2 x+1}}{(2 x+1) \left (x^3+2 x+1\right )}dx}{\sqrt {x} \sqrt {-2 x^3+2 x+1}}\)

\(\Big \downarrow \) 2035

\(\displaystyle \frac {2 \sqrt {-2 x^4+2 x^2+x} \int \frac {x (4 x+3) \sqrt {-2 x^3+2 x+1}}{(2 x+1) \left (x^3+2 x+1\right )}d\sqrt {x}}{\sqrt {x} \sqrt {-2 x^3+2 x+1}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {2 \sqrt {-2 x^4+2 x^2+x} \int \left (\frac {\sqrt {-2 x^3+2 x+1} \left (-2 x^2+3 x-4\right )}{x^3+2 x+1}+\frac {4 \sqrt {-2 x^3+2 x+1}}{2 x+1}\right )d\sqrt {x}}{\sqrt {x} \sqrt {-2 x^3+2 x+1}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \sqrt {-2 x^4+2 x^2+x} \left (2 i \int \frac {\sqrt {-2 x^3+2 x+1}}{i-\sqrt {2} \sqrt {x}}d\sqrt {x}+2 i \int \frac {\sqrt {-2 x^3+2 x+1}}{\sqrt {2} \sqrt {x}+i}d\sqrt {x}-4 \int \frac {\sqrt {-2 x^3+2 x+1}}{x^3+2 x+1}d\sqrt {x}+3 \int \frac {x \sqrt {-2 x^3+2 x+1}}{x^3+2 x+1}d\sqrt {x}-2 \int \frac {x^2 \sqrt {-2 x^3+2 x+1}}{x^3+2 x+1}d\sqrt {x}\right )}{\sqrt {x} \sqrt {-2 x^3+2 x+1}}\)

Input:

Int[((3 + 4*x)*Sqrt[x + 2*x^2 - 2*x^4])/((1 + 2*x)*(1 + 2*x + x^3)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 6.07 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.67

method	result	size
default	\(-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {-2 x^{4}+2 x^{2}+x}\, \sqrt {3}}{3 x^{2}}\right )+2 \sqrt {2}\, \arctan \left (\frac {\sqrt {-2 x^{4}+2 x^{2}+x}\, \sqrt {2}}{2 x^{2}}\right )\)	\(58\)
pseudoelliptic	\(-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {-2 x^{4}+2 x^{2}+x}\, \sqrt {3}}{3 x^{2}}\right )+2 \sqrt {2}\, \arctan \left (\frac {\sqrt {-2 x^{4}+2 x^{2}+x}\, \sqrt {2}}{2 x^{2}}\right )\)	\(58\)
trager	\(\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (-\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{3}+4 \sqrt {-2 x^{4}+2 x^{2}+x}\, x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right )}{1+2 x}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{3}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x -6 \sqrt {-2 x^{4}+2 x^{2}+x}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right )}{x^{3}+2 x +1}\right )\)	\(130\)
elliptic	\(\text {Expression too large to display}\)	\(413672\)

Input:

int((3+4*x)*(-2*x^4+2*x^2+x)^(1/2)/(1+2*x)/(x^3+2*x+1),x,method=_RETURNVER 
BOSE)
 

Output:

-2*3^(1/2)*arctan(1/3*(-2*x^4+2*x^2+x)^(1/2)/x^2*3^(1/2))+2*2^(1/2)*arctan 
(1/2*(-2*x^4+2*x^2+x)^(1/2)/x^2*2^(1/2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (75) = 150\).

Time = 0.29 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.93 \[ \int \frac {(3+4 x) \sqrt {x+2 x^2-2 x^4}}{(1+2 x) \left (1+2 x+x^3\right )} \, dx=\frac {2}{5} \, \sqrt {2} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {-2 \, x^{4} + 2 \, x^{2} + x} {\left (4 \, x^{3} - 4 \, x^{2} - x + 1\right )}}{16 \, x^{5} - 16 \, x^{4} - 12 \, x^{3} + 8 \, x^{2} + 4 \, x - 1}\right ) - \frac {1}{5} \, \sqrt {2} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {-2 \, x^{4} + 2 \, x^{2} + x} {\left (4 \, x^{2} + 5 \, x + 2\right )}}{32 \, x^{5} + 80 \, x^{4} + 84 \, x^{3} + 40 \, x^{2} + 6 \, x - 1}\right ) - \sqrt {3} \arctan \left (\frac {2 \, \sqrt {3} \sqrt {-2 \, x^{4} + 2 \, x^{2} + x} x}{5 \, x^{3} - 2 \, x - 1}\right ) \] Input:

integrate((3+4*x)*(-2*x^4+2*x^2+x)^(1/2)/(1+2*x)/(x^3+2*x+1),x, algorithm= 
"fricas")
 

Output:

2/5*sqrt(2)*arctan(2*sqrt(2)*sqrt(-2*x^4 + 2*x^2 + x)*(4*x^3 - 4*x^2 - x + 
 1)/(16*x^5 - 16*x^4 - 12*x^3 + 8*x^2 + 4*x - 1)) - 1/5*sqrt(2)*arctan(2*s 
qrt(2)*sqrt(-2*x^4 + 2*x^2 + x)*(4*x^2 + 5*x + 2)/(32*x^5 + 80*x^4 + 84*x^ 
3 + 40*x^2 + 6*x - 1)) - sqrt(3)*arctan(2*sqrt(3)*sqrt(-2*x^4 + 2*x^2 + x) 
*x/(5*x^3 - 2*x - 1))
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(3+4 x) \sqrt {x+2 x^2-2 x^4}}{(1+2 x) \left (1+2 x+x^3\right )} \, dx=\text {Timed out} \] Input:

integrate((3+4*x)*(-2*x**4+2*x**2+x)**(1/2)/(1+2*x)/(x**3+2*x+1),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(3+4 x) \sqrt {x+2 x^2-2 x^4}}{(1+2 x) \left (1+2 x+x^3\right )} \, dx=\int { \frac {\sqrt {-2 \, x^{4} + 2 \, x^{2} + x} {\left (4 \, x + 3\right )}}{{\left (x^{3} + 2 \, x + 1\right )} {\left (2 \, x + 1\right )}} \,d x } \] Input:

integrate((3+4*x)*(-2*x^4+2*x^2+x)^(1/2)/(1+2*x)/(x^3+2*x+1),x, algorithm= 
"maxima")
 

Output:

integrate(sqrt(-2*x^4 + 2*x^2 + x)*(4*x + 3)/((x^3 + 2*x + 1)*(2*x + 1)), 
x)
 

Giac [F]

\[ \int \frac {(3+4 x) \sqrt {x+2 x^2-2 x^4}}{(1+2 x) \left (1+2 x+x^3\right )} \, dx=\int { \frac {\sqrt {-2 \, x^{4} + 2 \, x^{2} + x} {\left (4 \, x + 3\right )}}{{\left (x^{3} + 2 \, x + 1\right )} {\left (2 \, x + 1\right )}} \,d x } \] Input:

integrate((3+4*x)*(-2*x^4+2*x^2+x)^(1/2)/(1+2*x)/(x^3+2*x+1),x, algorithm= 
"giac")
 

Output:

integrate(sqrt(-2*x^4 + 2*x^2 + x)*(4*x + 3)/((x^3 + 2*x + 1)*(2*x + 1)), 
x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(3+4 x) \sqrt {x+2 x^2-2 x^4}}{(1+2 x) \left (1+2 x+x^3\right )} \, dx=\int \frac {\left (4\,x+3\right )\,\sqrt {-2\,x^4+2\,x^2+x}}{\left (2\,x+1\right )\,\left (x^3+2\,x+1\right )} \,d x \] Input:

int(((4*x + 3)*(x + 2*x^2 - 2*x^4)^(1/2))/((2*x + 1)*(2*x + x^3 + 1)),x)
 

Output:

int(((4*x + 3)*(x + 2*x^2 - 2*x^4)^(1/2))/((2*x + 1)*(2*x + x^3 + 1)), x)
 

Reduce [F]

\[ \int \frac {(3+4 x) \sqrt {x+2 x^2-2 x^4}}{(1+2 x) \left (1+2 x+x^3\right )} \, dx=\sqrt {2}\, \mathit {atan} \left (\frac {4 \sqrt {-2 x^{3}+2 x +1}\, \sqrt {2}\, x^{3}-2 \sqrt {-2 x^{3}+2 x +1}\, \sqrt {2}\, x -\sqrt {-2 x^{3}+2 x +1}\, \sqrt {2}}{8 \sqrt {x}\, x^{4}-8 \sqrt {x}\, x^{2}-4 \sqrt {x}\, x}\right )-\frac {18 \sqrt {x}\, \sqrt {-2 x^{3}+2 x +1}}{13}-\frac {168 \left (\int \frac {\sqrt {x}\, \sqrt {-2 x^{3}+2 x +1}\, x^{6}}{4 x^{7}+2 x^{6}+4 x^{5}+4 x^{4}-7 x^{3}-12 x^{2}-6 x -1}d x \right )}{13}-\frac {84 \left (\int \frac {\sqrt {x}\, \sqrt {-2 x^{3}+2 x +1}\, x^{5}}{4 x^{7}+2 x^{6}+4 x^{5}+4 x^{4}-7 x^{3}-12 x^{2}-6 x -1}d x \right )}{13}+12 \left (\int \frac {\sqrt {x}\, \sqrt {-2 x^{3}+2 x +1}\, x^{5}}{2 x^{6}+2 x^{4}+x^{3}-4 x^{2}-4 x -1}d x \right )-\frac {252 \left (\int \frac {\sqrt {x}\, \sqrt {-2 x^{3}+2 x +1}\, x^{4}}{4 x^{7}+2 x^{6}+4 x^{5}+4 x^{4}-7 x^{3}-12 x^{2}-6 x -1}d x \right )}{13}+18 \left (\int \frac {\sqrt {x}\, \sqrt {-2 x^{3}+2 x +1}\, x^{3}}{2 x^{6}+2 x^{4}+x^{3}-4 x^{2}-4 x -1}d x \right )-\frac {81 \left (\int \frac {\sqrt {x}\, \sqrt {-2 x^{3}+2 x +1}\, x^{2}}{4 x^{7}+2 x^{6}+4 x^{5}+4 x^{4}-7 x^{3}-12 x^{2}-6 x -1}d x \right )}{13}-\frac {180 \left (\int \frac {\sqrt {x}\, \sqrt {-2 x^{3}+2 x +1}\, x}{4 x^{7}+2 x^{6}+4 x^{5}+4 x^{4}-7 x^{3}-12 x^{2}-6 x -1}d x \right )}{13}-12 \left (\int \frac {\sqrt {x}\, \sqrt {-2 x^{3}+2 x +1}\, x}{2 x^{6}+2 x^{4}+x^{3}-4 x^{2}-4 x -1}d x \right )-\frac {9 \left (\int \frac {\sqrt {x}\, \sqrt {-2 x^{3}+2 x +1}}{4 x^{8}+2 x^{7}+4 x^{6}+4 x^{5}-7 x^{4}-12 x^{3}-6 x^{2}-x}d x \right )}{13}-\frac {72 \left (\int \frac {\sqrt {x}\, \sqrt {-2 x^{3}+2 x +1}}{4 x^{7}+2 x^{6}+4 x^{5}+4 x^{4}-7 x^{3}-12 x^{2}-6 x -1}d x \right )}{13}-9 \left (\int \frac {\sqrt {x}\, \sqrt {-2 x^{3}+2 x +1}}{2 x^{6}+2 x^{4}+x^{3}-4 x^{2}-4 x -1}d x \right ) \] Input:

int((3+4*x)*(-2*x^4+2*x^2+x)^(1/2)/(1+2*x)/(x^3+2*x+1),x)
 

Output:

(13*sqrt(2)*atan((4*sqrt( - 2*x**3 + 2*x + 1)*sqrt(2)*x**3 - 2*sqrt( - 2*x 
**3 + 2*x + 1)*sqrt(2)*x - sqrt( - 2*x**3 + 2*x + 1)*sqrt(2))/(8*sqrt(x)*x 
**4 - 8*sqrt(x)*x**2 - 4*sqrt(x)*x)) - 18*sqrt(x)*sqrt( - 2*x**3 + 2*x + 1 
) - 168*int((sqrt(x)*sqrt( - 2*x**3 + 2*x + 1)*x**6)/(4*x**7 + 2*x**6 + 4* 
x**5 + 4*x**4 - 7*x**3 - 12*x**2 - 6*x - 1),x) - 84*int((sqrt(x)*sqrt( - 2 
*x**3 + 2*x + 1)*x**5)/(4*x**7 + 2*x**6 + 4*x**5 + 4*x**4 - 7*x**3 - 12*x* 
*2 - 6*x - 1),x) + 156*int((sqrt(x)*sqrt( - 2*x**3 + 2*x + 1)*x**5)/(2*x** 
6 + 2*x**4 + x**3 - 4*x**2 - 4*x - 1),x) - 252*int((sqrt(x)*sqrt( - 2*x**3 
 + 2*x + 1)*x**4)/(4*x**7 + 2*x**6 + 4*x**5 + 4*x**4 - 7*x**3 - 12*x**2 - 
6*x - 1),x) + 234*int((sqrt(x)*sqrt( - 2*x**3 + 2*x + 1)*x**3)/(2*x**6 + 2 
*x**4 + x**3 - 4*x**2 - 4*x - 1),x) - 81*int((sqrt(x)*sqrt( - 2*x**3 + 2*x 
 + 1)*x**2)/(4*x**7 + 2*x**6 + 4*x**5 + 4*x**4 - 7*x**3 - 12*x**2 - 6*x - 
1),x) - 180*int((sqrt(x)*sqrt( - 2*x**3 + 2*x + 1)*x)/(4*x**7 + 2*x**6 + 4 
*x**5 + 4*x**4 - 7*x**3 - 12*x**2 - 6*x - 1),x) - 156*int((sqrt(x)*sqrt( - 
 2*x**3 + 2*x + 1)*x)/(2*x**6 + 2*x**4 + x**3 - 4*x**2 - 4*x - 1),x) - 9*i 
nt((sqrt(x)*sqrt( - 2*x**3 + 2*x + 1))/(4*x**8 + 2*x**7 + 4*x**6 + 4*x**5 
- 7*x**4 - 12*x**3 - 6*x**2 - x),x) - 72*int((sqrt(x)*sqrt( - 2*x**3 + 2*x 
 + 1))/(4*x**7 + 2*x**6 + 4*x**5 + 4*x**4 - 7*x**3 - 12*x**2 - 6*x - 1),x) 
 - 117*int((sqrt(x)*sqrt( - 2*x**3 + 2*x + 1))/(2*x**6 + 2*x**4 + x**3 - 4 
*x**2 - 4*x - 1),x))/13