Integrand size = 20, antiderivative size = 131 \[ \int \frac {5+x}{\sqrt {2 x+3 x^2+x^3}} \, dx=\frac {2 x (2+x)}{\sqrt {2 x+3 x^2+x^3}}-\frac {2 \sqrt {2} \sqrt {x} (1+x) \sqrt {\frac {2+x}{1+x}} E\left (\arctan \left (\sqrt {x}\right )|\frac {1}{2}\right )}{\sqrt {2 x+3 x^2+x^3}}+\frac {5 \sqrt {2} \sqrt {x} (1+x) \sqrt {\frac {2+x}{1+x}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),\frac {1}{2}\right )}{\sqrt {2 x+3 x^2+x^3}} \] Output:
2*x*(2+x)/(x^3+3*x^2+2*x)^(1/2)-2*2^(1/2)*x^(1/2)*(1+x)*((2+x)/(1+x))^(1/2 )*EllipticE(x^(1/2)/(1+x)^(1/2),1/2*2^(1/2))/(x^3+3*x^2+2*x)^(1/2)+5*2^(1/ 2)*x^(1/2)*(1+x)*((2+x)/(1+x))^(1/2)*InverseJacobiAM(arctan(x^(1/2)),1/2*2 ^(1/2))/(x^3+3*x^2+2*x)^(1/2)
Result contains complex when optimal does not.
Time = 22.25 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.02 \[ \int \frac {5+x}{\sqrt {2 x+3 x^2+x^3}} \, dx=\frac {4+6 x+2 x^2+2 i \sqrt {2} \sqrt {1+\frac {1}{x}} x^{3/2} \sqrt {\frac {2+x}{x}} E\left (i \text {arcsinh}\left (\frac {\sqrt {2}}{\sqrt {x}}\right )|\frac {1}{2}\right )+3 i \sqrt {2} \sqrt {1+\frac {1}{x}} x^{3/2} \sqrt {\frac {2+x}{x}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2}}{\sqrt {x}}\right ),\frac {1}{2}\right )}{\sqrt {x \left (2+3 x+x^2\right )}} \] Input:
Integrate[(5 + x)/Sqrt[2*x + 3*x^2 + x^3],x]
Output:
(4 + 6*x + 2*x^2 + (2*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*x^(3/2)*Sqrt[(2 + x)/x]* EllipticE[I*ArcSinh[Sqrt[2]/Sqrt[x]], 1/2] + (3*I)*Sqrt[2]*Sqrt[1 + x^(-1) ]*x^(3/2)*Sqrt[(2 + x)/x]*EllipticF[I*ArcSinh[Sqrt[2]/Sqrt[x]], 1/2])/Sqrt [x*(2 + 3*x + x^2)]
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.
\(\displaystyle \int \frac {x+5}{\sqrt {x^3+3 x^2+2 x}} \, dx\) |
\(\Big \downarrow \) 1985 |
\(\displaystyle \int \frac {x+5}{\sqrt {x^3+3 x^2+2 x}}dx\) |
Input:
Int[(5 + x)/Sqrt[2*x + 3*x^2 + x^3],x]
Output:
$Aborted
Int[((A_) + (B_.)*(x_)^(j_.))*((b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.) + (c_.)* (x_)^(r_.))^(p_.), x_Symbol] :> Unintegrable[(A + B*x^(n - q))*(b*x^n + c*x ^(2*n - q) + a*x^q)^p, x] /; FreeQ[{a, b, c, A, B, n, p, q}, x] && EqQ[j, n - q] && EqQ[r, 2*n - q]
Time = 0.19 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.88
method | result | size |
default | \(\frac {\sqrt {2 x +4}\, \sqrt {-1-x}\, \sqrt {-2 x}\, \left (-\operatorname {EllipticE}\left (\frac {\sqrt {2 x +4}}{2}, \sqrt {2}\right )-\operatorname {EllipticF}\left (\frac {\sqrt {2 x +4}}{2}, \sqrt {2}\right )\right )}{\sqrt {x^{3}+3 x^{2}+2 x}}+\frac {5 \sqrt {2 x +4}\, \sqrt {-1-x}\, \sqrt {-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {2 x +4}}{2}, \sqrt {2}\right )}{\sqrt {x^{3}+3 x^{2}+2 x}}\) | \(115\) |
elliptic | \(\frac {\sqrt {2 x +4}\, \sqrt {-1-x}\, \sqrt {-2 x}\, \left (-\operatorname {EllipticE}\left (\frac {\sqrt {2 x +4}}{2}, \sqrt {2}\right )-\operatorname {EllipticF}\left (\frac {\sqrt {2 x +4}}{2}, \sqrt {2}\right )\right )}{\sqrt {x^{3}+3 x^{2}+2 x}}+\frac {5 \sqrt {2 x +4}\, \sqrt {-1-x}\, \sqrt {-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {2 x +4}}{2}, \sqrt {2}\right )}{\sqrt {x^{3}+3 x^{2}+2 x}}\) | \(115\) |
Input:
int((5+x)/(x^3+3*x^2+2*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
(2*x+4)^(1/2)*(-1-x)^(1/2)*(-2*x)^(1/2)/(x^3+3*x^2+2*x)^(1/2)*(-EllipticE( 1/2*(2*x+4)^(1/2),2^(1/2))-EllipticF(1/2*(2*x+4)^(1/2),2^(1/2)))+5*(2*x+4) ^(1/2)*(-1-x)^(1/2)*(-2*x)^(1/2)/(x^3+3*x^2+2*x)^(1/2)*EllipticF(1/2*(2*x+ 4)^(1/2),2^(1/2))
Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.15 \[ \int \frac {5+x}{\sqrt {2 x+3 x^2+x^3}} \, dx=8 \, {\rm weierstrassPInverse}\left (4, 0, x + 1\right ) - 2 \, {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, x + 1\right )\right ) \] Input:
integrate((5+x)/(x^3+3*x^2+2*x)^(1/2),x, algorithm="fricas")
Output:
8*weierstrassPInverse(4, 0, x + 1) - 2*weierstrassZeta(4, 0, weierstrassPI nverse(4, 0, x + 1))
\[ \int \frac {5+x}{\sqrt {2 x+3 x^2+x^3}} \, dx=\int \frac {x + 5}{\sqrt {x \left (x + 1\right ) \left (x + 2\right )}}\, dx \] Input:
integrate((5+x)/(x**3+3*x**2+2*x)**(1/2),x)
Output:
Integral((x + 5)/sqrt(x*(x + 1)*(x + 2)), x)
\[ \int \frac {5+x}{\sqrt {2 x+3 x^2+x^3}} \, dx=\int { \frac {x + 5}{\sqrt {x^{3} + 3 \, x^{2} + 2 \, x}} \,d x } \] Input:
integrate((5+x)/(x^3+3*x^2+2*x)^(1/2),x, algorithm="maxima")
Output:
integrate((x + 5)/sqrt(x^3 + 3*x^2 + 2*x), x)
\[ \int \frac {5+x}{\sqrt {2 x+3 x^2+x^3}} \, dx=\int { \frac {x + 5}{\sqrt {x^{3} + 3 \, x^{2} + 2 \, x}} \,d x } \] Input:
integrate((5+x)/(x^3+3*x^2+2*x)^(1/2),x, algorithm="giac")
Output:
integrate((x + 5)/sqrt(x^3 + 3*x^2 + 2*x), x)
Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.43 \[ \int \frac {5+x}{\sqrt {2 x+3 x^2+x^3}} \, dx=\frac {2\,\sqrt {-x}\,\left (\mathrm {E}\left (\mathrm {asin}\left (\sqrt {-x-1}\right )\middle |-1\right )-5\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-x-1}\right )\middle |-1\right )\right )\,\sqrt {-x-1}\,\sqrt {x+2}}{\sqrt {x^3+3\,x^2+2\,x}} \] Input:
int((x + 5)/(2*x + 3*x^2 + x^3)^(1/2),x)
Output:
(2*(-x)^(1/2)*(ellipticE(asin((- x - 1)^(1/2)), -1) - 5*ellipticF(asin((- x - 1)^(1/2)), -1))*(- x - 1)^(1/2)*(x + 2)^(1/2))/(2*x + 3*x^2 + x^3)^(1/ 2)
\[ \int \frac {5+x}{\sqrt {2 x+3 x^2+x^3}} \, dx=5 \left (\int \frac {\sqrt {x}\, \sqrt {x^{2}+3 x +2}}{x^{3}+3 x^{2}+2 x}d x \right )+\int \frac {\sqrt {x}\, \sqrt {x^{2}+3 x +2}}{x^{2}+3 x +2}d x \] Input:
int((5+x)/(x^3+3*x^2+2*x)^(1/2),x)
Output:
5*int((sqrt(x)*sqrt(x**2 + 3*x + 2))/(x**3 + 3*x**2 + 2*x),x) + int((sqrt( x)*sqrt(x**2 + 3*x + 2))/(x**2 + 3*x + 2),x)