Integrand size = 22, antiderivative size = 101 \[ \int \frac {1}{\sqrt {7+x} \sqrt {3+2 x+5 x^2}} \, dx=\frac {\sqrt {\frac {3+2 x+5 x^2}{\left (78+\sqrt {130} (7+x)\right )^2}} \left (78+\sqrt {130} (7+x)\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{\frac {5}{26}} \sqrt {7+x}}{\sqrt {3}}\right ),\frac {1}{390} \left (195+17 \sqrt {130}\right )\right )}{\sqrt {3} \sqrt [4]{130} \sqrt {3+2 x+5 x^2}} \] Output:
1/390*((5*x^2+2*x+3)/(78+130^(1/2)*(7+x))^2)^(1/2)*(78+130^(1/2)*(7+x))*In verseJacobiAM(2*arctan(1/78*5^(1/4)*26^(3/4)*(7+x)^(1/2)*3^(1/2)),1/390*(7 6050+6630*130^(1/2))^(1/2))*3^(1/2)*130^(3/4)/(5*x^2+2*x+3)^(1/2)
Result contains complex when optimal does not.
Time = 31.36 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.51 \[ \int \frac {1}{\sqrt {7+x} \sqrt {3+2 x+5 x^2}} \, dx=\frac {i (7+x) \sqrt {1+\frac {234 i}{\left (-34 i+\sqrt {14}\right ) (7+x)}} \sqrt {\frac {2}{13}-\frac {36 i}{\left (34 i+\sqrt {14}\right ) (7+x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {3 \sqrt {-\frac {26 i}{34 i+\sqrt {14}}}}{\sqrt {7+x}}\right ),\frac {34 i+\sqrt {14}}{34 i-\sqrt {14}}\right )}{3 \sqrt {-\frac {i}{34 i+\sqrt {14}}} \sqrt {3+2 x+5 x^2}} \] Input:
Integrate[1/(Sqrt[7 + x]*Sqrt[3 + 2*x + 5*x^2]),x]
Output:
((I/3)*(7 + x)*Sqrt[1 + (234*I)/((-34*I + Sqrt[14])*(7 + x))]*Sqrt[2/13 - (36*I)/((34*I + Sqrt[14])*(7 + x))]*EllipticF[I*ArcSinh[(3*Sqrt[(-26*I)/(3 4*I + Sqrt[14])])/Sqrt[7 + x]], (34*I + Sqrt[14])/(34*I - Sqrt[14])])/(Sqr t[(-I)/(34*I + Sqrt[14])]*Sqrt[3 + 2*x + 5*x^2])
Result contains complex when optimal does not.
Time = 0.23 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.84, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1172, 321}
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.
\(\displaystyle \int \frac {1}{\sqrt {x+7} \sqrt {5 x^2+2 x+3}} \, dx\) |
\(\Big \downarrow \) 1172 |
\(\displaystyle \frac {2 i \sqrt {\frac {x+7}{34-i \sqrt {14}}} \int \frac {1}{\sqrt {\frac {i \left (5 x+i \sqrt {14}+1\right )}{2 \sqrt {14}}+1} \sqrt {\frac {i \left (5 x+i \sqrt {14}+1\right )}{34 i+\sqrt {14}}+1}}d\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}}{\sqrt {x+7}}\) |
\(\Big \downarrow \) 321 |
\(\displaystyle \frac {2 i \sqrt {\frac {x+7}{34-i \sqrt {14}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}\right ),\frac {2 \sqrt {14}}{34 i+\sqrt {14}}\right )}{\sqrt {x+7}}\) |
Input:
Int[1/(Sqrt[7 + x]*Sqrt[3 + 2*x + 5*x^2]),x]
Output:
((2*I)*Sqrt[(7 + x)/(34 - I*Sqrt[14])]*EllipticF[ArcSin[Sqrt[(-I)*(1 + I*S qrt[14] + 5*x)]/(2^(3/4)*7^(1/4))], (2*Sqrt[14])/(34*I + Sqrt[14])])/Sqrt[ 7 + x]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sy mbol] :> Simp[2*Rt[b^2 - 4*a*c, 2]*(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2 )/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*e - e *Rt[b^2 - 4*a*c, 2])))^m)) Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2* c*d - b*e - e*Rt[b^2 - 4*a*c, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^ 2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b, c, d, e }, x] && EqQ[m^2, 1/4]
Result contains complex when optimal does not.
Time = 2.31 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.49
method | result | size |
default | \(-\frac {2 \sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}\, \left (-34+i \sqrt {14}\right ) \sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}\, \sqrt {\frac {i \sqrt {14}-5 x -1}{i \sqrt {14}+34}}\, \sqrt {\frac {i \sqrt {14}+5 x +1}{-34+i \sqrt {14}}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}, \sqrt {-\frac {-34+i \sqrt {14}}{i \sqrt {14}+34}}\right )}{5 \left (5 x^{3}+37 x^{2}+17 x +21\right )}\) | \(150\) |
elliptic | \(\frac {2 \sqrt {\left (x +7\right ) \left (5 x^{2}+2 x +3\right )}\, \left (\frac {34}{5}-\frac {i \sqrt {14}}{5}\right ) \sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}-\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}+\frac {i \sqrt {14}}{5}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}, \sqrt {\frac {-\frac {34}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\right )}{\sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}\, \sqrt {5 x^{3}+37 x^{2}+17 x +21}}\) | \(159\) |
Input:
int(1/(x+7)^(1/2)/(5*x^2+2*x+3)^(1/2),x,method=_RETURNVERBOSE)
Output:
-2/5*(x+7)^(1/2)*(5*x^2+2*x+3)^(1/2)*(-34+I*14^(1/2))*(-5*(x+7)/(-34+I*14^ (1/2)))^(1/2)*((I*14^(1/2)-5*x-1)/(I*14^(1/2)+34))^(1/2)*((I*14^(1/2)+5*x+ 1)/(-34+I*14^(1/2)))^(1/2)*EllipticF((-5*(x+7)/(-34+I*14^(1/2)))^(1/2),(-( -34+I*14^(1/2))/(I*14^(1/2)+34))^(1/2))/(5*x^3+37*x^2+17*x+21)
Time = 0.07 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.11 \[ \int \frac {1}{\sqrt {7+x} \sqrt {3+2 x+5 x^2}} \, dx=\frac {2}{5} \, \sqrt {5} {\rm weierstrassPInverse}\left (\frac {4456}{75}, -\frac {348704}{3375}, x + \frac {37}{15}\right ) \] Input:
integrate(1/(7+x)^(1/2)/(5*x^2+2*x+3)^(1/2),x, algorithm="fricas")
Output:
2/5*sqrt(5)*weierstrassPInverse(4456/75, -348704/3375, x + 37/15)
\[ \int \frac {1}{\sqrt {7+x} \sqrt {3+2 x+5 x^2}} \, dx=\int \frac {1}{\sqrt {x + 7} \sqrt {5 x^{2} + 2 x + 3}}\, dx \] Input:
integrate(1/(7+x)**(1/2)/(5*x**2+2*x+3)**(1/2),x)
Output:
Integral(1/(sqrt(x + 7)*sqrt(5*x**2 + 2*x + 3)), x)
\[ \int \frac {1}{\sqrt {7+x} \sqrt {3+2 x+5 x^2}} \, dx=\int { \frac {1}{\sqrt {5 \, x^{2} + 2 \, x + 3} \sqrt {x + 7}} \,d x } \] Input:
integrate(1/(7+x)^(1/2)/(5*x^2+2*x+3)^(1/2),x, algorithm="maxima")
Output:
integrate(1/(sqrt(5*x^2 + 2*x + 3)*sqrt(x + 7)), x)
\[ \int \frac {1}{\sqrt {7+x} \sqrt {3+2 x+5 x^2}} \, dx=\int { \frac {1}{\sqrt {5 \, x^{2} + 2 \, x + 3} \sqrt {x + 7}} \,d x } \] Input:
integrate(1/(7+x)^(1/2)/(5*x^2+2*x+3)^(1/2),x, algorithm="giac")
Output:
integrate(1/(sqrt(5*x^2 + 2*x + 3)*sqrt(x + 7)), x)
Timed out. \[ \int \frac {1}{\sqrt {7+x} \sqrt {3+2 x+5 x^2}} \, dx=\int \frac {1}{\sqrt {x+7}\,\sqrt {5\,x^2+2\,x+3}} \,d x \] Input:
int(1/((x + 7)^(1/2)*(2*x + 5*x^2 + 3)^(1/2)),x)
Output:
int(1/((x + 7)^(1/2)*(2*x + 5*x^2 + 3)^(1/2)), x)
\[ \int \frac {1}{\sqrt {7+x} \sqrt {3+2 x+5 x^2}} \, dx=\int \frac {\sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}}{5 x^{3}+37 x^{2}+17 x +21}d x \] Input:
int(1/(7+x)^(1/2)/(5*x^2+2*x+3)^(1/2),x)
Output:
int((sqrt(x + 7)*sqrt(5*x**2 + 2*x + 3))/(5*x**3 + 37*x**2 + 17*x + 21),x)