Integrand size = 25, antiderivative size = 51 \[ \int \frac {1+x}{\sqrt {5-2 x} \sqrt {2+3 x+x^2}} \, dx=-\frac {\sqrt {7} \sqrt {1+x} \sqrt {2+x} E\left (\arcsin \left (\frac {1}{3} \sqrt {5-2 x}\right )|\frac {9}{7}\right )}{\sqrt {2+3 x+x^2}} \] Output:
-7^(1/2)*(1+x)^(1/2)*(2+x)^(1/2)*EllipticE(1/3*(5-2*x)^(1/2),3/7*7^(1/2))/ (x^2+3*x+2)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(149\) vs. \(2(51)=102\).
Time = 31.54 (sec) , antiderivative size = 149, normalized size of antiderivative = 2.92 \[ \int \frac {1+x}{\sqrt {5-2 x} \sqrt {2+3 x+x^2}} \, dx=\frac {12+18 x+6 x^2+9 (5-2 x)^{3/2} \sqrt {\frac {1+x}{-5+2 x}} \sqrt {\frac {2+x}{-5+2 x}} E\left (\arcsin \left (\frac {3}{\sqrt {5-2 x}}\right )|\frac {7}{9}\right )-2 (5-2 x)^{3/2} \sqrt {\frac {1+x}{-5+2 x}} \sqrt {\frac {2+x}{-5+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {3}{\sqrt {5-2 x}}\right ),\frac {7}{9}\right )}{3 \sqrt {5-2 x} \sqrt {2+3 x+x^2}} \] Input:
Integrate[(1 + x)/(Sqrt[5 - 2*x]*Sqrt[2 + 3*x + x^2]),x]
Output:
(12 + 18*x + 6*x^2 + 9*(5 - 2*x)^(3/2)*Sqrt[(1 + x)/(-5 + 2*x)]*Sqrt[(2 + x)/(-5 + 2*x)]*EllipticE[ArcSin[3/Sqrt[5 - 2*x]], 7/9] - 2*(5 - 2*x)^(3/2) *Sqrt[(1 + x)/(-5 + 2*x)]*Sqrt[(2 + x)/(-5 + 2*x)]*EllipticF[ArcSin[3/Sqrt [5 - 2*x]], 7/9])/(3*Sqrt[5 - 2*x]*Sqrt[2 + 3*x + x^2])
Time = 0.29 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1268, 123}
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.
\(\displaystyle \int \frac {x+1}{\sqrt {5-2 x} \sqrt {x^2+3 x+2}} \, dx\) |
\(\Big \downarrow \) 1268 |
\(\displaystyle \frac {\sqrt {x+1} \sqrt {x+2} \int \frac {\sqrt {x+1}}{\sqrt {5-2 x} \sqrt {x+2}}dx}{\sqrt {x^2+3 x+2}}\) |
\(\Big \downarrow \) 123 |
\(\displaystyle -\frac {\sqrt {7} \sqrt {x+1} \sqrt {x+2} E\left (\arcsin \left (\frac {1}{3} \sqrt {5-2 x}\right )|\frac {9}{7}\right )}{\sqrt {x^2+3 x+2}}\) |
Input:
Int[(1 + x)/(Sqrt[5 - 2*x]*Sqrt[2 + 3*x + x^2]),x]
Output:
-((Sqrt[7]*Sqrt[1 + x]*Sqrt[2 + x]*EllipticE[ArcSin[Sqrt[5 - 2*x]/3], 9/7] )/Sqrt[2 + 3*x + x^2])
Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ )]), x_] :> Simp[(2/b)*Rt[-(b*e - a*f)/d, 2]*EllipticE[ArcSin[Sqrt[a + b*x] /Rt[-(b*c - a*d)/d, 2]], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] && !L tQ[-(b*c - a*d)/d, 0] && !(SimplerQ[c + d*x, a + b*x] && GtQ[-d/(b*c - a*d ), 0] && GtQ[d/(d*e - c*f), 0] && !LtQ[(b*c - a*d)/b, 0])
Int[((d_) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/((d + e*x)^FracPart[p]*(a/d + (c*x)/e)^FracPart[p]) Int[(d + e*x)^(m + p)*(f + g*x)^n*(a/d + (c/e)*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && EqQ[c*d^2 - b*d*e + a*e^2, 0]
Time = 1.12 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.82
method | result | size |
default | \(-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right ) \sqrt {2 x +4}\, \sqrt {14 x +14}}{2 \sqrt {x^{2}+3 x +2}}\) | \(42\) |
elliptic | \(\frac {\sqrt {-\left (-5+2 x \right ) \left (x^{2}+3 x +2\right )}\, \left (-\frac {\sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \operatorname {EllipticF}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )}{7 \sqrt {-2 x^{3}-x^{2}+11 x +10}}-\frac {\sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \left (\frac {7 \operatorname {EllipticE}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )}{2}-\operatorname {EllipticF}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )\right )}{7 \sqrt {-2 x^{3}-x^{2}+11 x +10}}\right )}{\sqrt {5-2 x}\, \sqrt {x^{2}+3 x +2}}\) | \(167\) |
Input:
int((x+1)/(5-2*x)^(1/2)/(x^2+3*x+2)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/2*EllipticE(1/3*(5-2*x)^(1/2),3/7*7^(1/2))*(2*x+4)^(1/2)*(14*x+14)^(1/2 )/(x^2+3*x+2)^(1/2)
Time = 0.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.49 \[ \int \frac {1+x}{\sqrt {5-2 x} \sqrt {2+3 x+x^2}} \, dx=-\frac {5}{6} \, \sqrt {-2} {\rm weierstrassPInverse}\left (\frac {67}{3}, \frac {440}{27}, x + \frac {1}{6}\right ) + \sqrt {-2} {\rm weierstrassZeta}\left (\frac {67}{3}, \frac {440}{27}, {\rm weierstrassPInverse}\left (\frac {67}{3}, \frac {440}{27}, x + \frac {1}{6}\right )\right ) \] Input:
integrate((1+x)/(5-2*x)^(1/2)/(x^2+3*x+2)^(1/2),x, algorithm="fricas")
Output:
-5/6*sqrt(-2)*weierstrassPInverse(67/3, 440/27, x + 1/6) + sqrt(-2)*weiers trassZeta(67/3, 440/27, weierstrassPInverse(67/3, 440/27, x + 1/6))
\[ \int \frac {1+x}{\sqrt {5-2 x} \sqrt {2+3 x+x^2}} \, dx=\int \frac {x + 1}{\sqrt {\left (x + 1\right ) \left (x + 2\right )} \sqrt {5 - 2 x}}\, dx \] Input:
integrate((1+x)/(5-2*x)**(1/2)/(x**2+3*x+2)**(1/2),x)
Output:
Integral((x + 1)/(sqrt((x + 1)*(x + 2))*sqrt(5 - 2*x)), x)
\[ \int \frac {1+x}{\sqrt {5-2 x} \sqrt {2+3 x+x^2}} \, dx=\int { \frac {x + 1}{\sqrt {x^{2} + 3 \, x + 2} \sqrt {-2 \, x + 5}} \,d x } \] Input:
integrate((1+x)/(5-2*x)^(1/2)/(x^2+3*x+2)^(1/2),x, algorithm="maxima")
Output:
integrate((x + 1)/(sqrt(x^2 + 3*x + 2)*sqrt(-2*x + 5)), x)
\[ \int \frac {1+x}{\sqrt {5-2 x} \sqrt {2+3 x+x^2}} \, dx=\int { \frac {x + 1}{\sqrt {x^{2} + 3 \, x + 2} \sqrt {-2 \, x + 5}} \,d x } \] Input:
integrate((1+x)/(5-2*x)^(1/2)/(x^2+3*x+2)^(1/2),x, algorithm="giac")
Output:
integrate((x + 1)/(sqrt(x^2 + 3*x + 2)*sqrt(-2*x + 5)), x)
Timed out. \[ \int \frac {1+x}{\sqrt {5-2 x} \sqrt {2+3 x+x^2}} \, dx=\int \frac {x+1}{\sqrt {5-2\,x}\,\sqrt {x^2+3\,x+2}} \,d x \] Input:
int((x + 1)/((5 - 2*x)^(1/2)*(3*x + x^2 + 2)^(1/2)),x)
Output:
int((x + 1)/((5 - 2*x)^(1/2)*(3*x + x^2 + 2)^(1/2)), x)
\[ \int \frac {1+x}{\sqrt {5-2 x} \sqrt {2+3 x+x^2}} \, dx=-\left (\int \frac {\sqrt {-2 x +5}\, \sqrt {x^{2}+3 x +2}}{2 x^{2}-x -10}d x \right ) \] Input:
int((1+x)/(5-2*x)^(1/2)/(x^2+3*x+2)^(1/2),x)
Output:
- int((sqrt( - 2*x + 5)*sqrt(x**2 + 3*x + 2))/(2*x**2 - x - 10),x)