Integrand size = 22, antiderivative size = 252 \[ \int \frac {\sqrt {7+x}}{\sqrt {3+2 x+5 x^2}} \, dx=\frac {2 \sqrt {7+x} \sqrt {3+2 x+5 x^2}}{3 \sqrt {130}+5 (7+x)}-\frac {2 \sqrt {3} \sqrt [4]{26} \sqrt {\frac {3+2 x+5 x^2}{\left (78+\sqrt {130} (7+x)\right )^2}} \left (78+\sqrt {130} (7+x)\right ) E\left (2 \arctan \left (\frac {\sqrt [4]{\frac {5}{26}} \sqrt {7+x}}{\sqrt {3}}\right )|\frac {1}{390} \left (195+17 \sqrt {130}\right )\right )}{5^{3/4} \sqrt {3+2 x+5 x^2}}+\frac {\sqrt {3} \sqrt [4]{26} \sqrt {\frac {3+2 x+5 x^2}{\left (78+\sqrt {130} (7+x)\right )^2}} \left (78+\sqrt {130} (7+x)\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{\frac {5}{26}} \sqrt {7+x}}{\sqrt {3}}\right ),\frac {1}{390} \left (195+17 \sqrt {130}\right )\right )}{5^{3/4} \sqrt {3+2 x+5 x^2}} \] Output:
2*(7+x)^(1/2)*(5*x^2+2*x+3)^(1/2)/(5*x+35+3*130^(1/2))-2/5*3^(1/2)*26^(1/4 )*((5*x^2+2*x+3)/(78+130^(1/2)*(7+x))^2)^(1/2)*(78+130^(1/2)*(7+x))*Ellipt icE(sin(2*arctan(1/78*5^(1/4)*26^(3/4)*(7+x)^(1/2)*3^(1/2))),1/390*(76050+ 6630*130^(1/2))^(1/2))*5^(1/4)/(5*x^2+2*x+3)^(1/2)+1/5*3^(1/2)*26^(1/4)*(( 5*x^2+2*x+3)/(78+130^(1/2)*(7+x))^2)^(1/2)*(78+130^(1/2)*(7+x))*InverseJac obiAM(2*arctan(1/78*5^(1/4)*26^(3/4)*(7+x)^(1/2)*3^(1/2)),1/390*(76050+663 0*130^(1/2))^(1/2))*5^(1/4)/(5*x^2+2*x+3)^(1/2)
Result contains complex when optimal does not.
Time = 30.34 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {7+x}}{\sqrt {3+2 x+5 x^2}} \, dx=-\frac {2 \left (-34 i+\sqrt {14}\right ) \sqrt {\frac {-i+\sqrt {14}-5 i x}{34 i+\sqrt {14}}} \sqrt {\frac {i+\sqrt {14}+5 i x}{-34 i+\sqrt {14}}} \sqrt {7+x} \left (E\left (i \text {arcsinh}\left (\sqrt {5} \sqrt {-\frac {i (7+x)}{34 i+\sqrt {14}}}\right )|\frac {34 i+\sqrt {14}}{34 i-\sqrt {14}}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {5} \sqrt {-\frac {i (7+x)}{34 i+\sqrt {14}}}\right ),\frac {34 i+\sqrt {14}}{34 i-\sqrt {14}}\right )\right )}{5 \sqrt {5} \sqrt {-\frac {i (7+x)}{34 i+\sqrt {14}}} \sqrt {3+2 x+5 x^2}} \] Input:
Integrate[Sqrt[7 + x]/Sqrt[3 + 2*x + 5*x^2],x]
Output:
(-2*(-34*I + Sqrt[14])*Sqrt[(-I + Sqrt[14] - (5*I)*x)/(34*I + Sqrt[14])]*S qrt[(I + Sqrt[14] + (5*I)*x)/(-34*I + Sqrt[14])]*Sqrt[7 + x]*(EllipticE[I* ArcSinh[Sqrt[5]*Sqrt[((-I)*(7 + x))/(34*I + Sqrt[14])]], (34*I + Sqrt[14]) /(34*I - Sqrt[14])] - EllipticF[I*ArcSinh[Sqrt[5]*Sqrt[((-I)*(7 + x))/(34* I + Sqrt[14])]], (34*I + Sqrt[14])/(34*I - Sqrt[14])]))/(5*Sqrt[5]*Sqrt[(( -I)*(7 + x))/(34*I + Sqrt[14])]*Sqrt[3 + 2*x + 5*x^2])
Result contains complex when optimal does not.
Time = 0.22 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.35, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1172, 327}
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.
| \(\displaystyle \int \frac {\sqrt {x+7}}{\sqrt {5 x^2+2 x+3}} \, dx\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {2 i \sqrt {x+7} \int \frac {\sqrt {\frac {i \left (5 x+i \sqrt {14}+1\right )}{34 i+\sqrt {14}}+1}}{\sqrt {\frac {i \left (5 x+i \sqrt {14}+1\right )}{2 \sqrt {14}}+1}}d\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}}{5 \sqrt {\frac {x+7}{34-i \sqrt {14}}}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {2 i \sqrt {x+7} E\left (\arcsin \left (\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}\right )|\frac {2 \sqrt {14}}{34 i+\sqrt {14}}\right )}{5 \sqrt {\frac {x+7}{34-i \sqrt {14}}}}\) |
Input:
Int[Sqrt[7 + x]/Sqrt[3 + 2*x + 5*x^2],x]
Output:
(((2*I)/5)*Sqrt[7 + x]*EllipticE[ArcSin[Sqrt[(-I)*(1 + I*Sqrt[14] + 5*x)]/ (2^(3/4)*7^(1/4))], (2*Sqrt[14])/(34*I + Sqrt[14])])/Sqrt[(7 + x)/(34 - I* Sqrt[14])]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[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]
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 = 2.84 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.14
| method | result | size |
| default | \(-\frac {2 \sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}\, \left (-34+i \sqrt {14}\right ) \sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}\, \sqrt {\frac {i \sqrt {14}-5 x -1}{i \sqrt {14}+34}}\, \sqrt {\frac {i \sqrt {14}+5 x +1}{-34+i \sqrt {14}}}\, \left (i \operatorname {EllipticF}\left (\sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}, \sqrt {-\frac {-34+i \sqrt {14}}{i \sqrt {14}+34}}\right ) \sqrt {14}-i \operatorname {EllipticE}\left (\sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}, \sqrt {-\frac {-34+i \sqrt {14}}{i \sqrt {14}+34}}\right ) \sqrt {14}+34 \operatorname {EllipticF}\left (\sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}, \sqrt {-\frac {-34+i \sqrt {14}}{i \sqrt {14}+34}}\right )-34 \operatorname {EllipticE}\left (\sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}, \sqrt {-\frac {-34+i \sqrt {14}}{i \sqrt {14}+34}}\right )\right )}{25 \left (5 x^{3}+37 x^{2}+17 x +21\right )}\) | \(287\) |
| elliptic | \(\frac {\sqrt {\left (x +7\right ) \left (5 x^{2}+2 x +3\right )}\, \left (\frac {14 \left (\frac {34}{5}-\frac {i \sqrt {14}}{5}\right ) \sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}-\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}+\frac {i \sqrt {14}}{5}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}, \sqrt {\frac {-\frac {34}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\right )}{\sqrt {5 x^{3}+37 x^{2}+17 x +21}}+\frac {2 \left (\frac {34}{5}-\frac {i \sqrt {14}}{5}\right ) \sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}-\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}+\frac {i \sqrt {14}}{5}}}\, \left (\left (-\frac {34}{5}-\frac {i \sqrt {14}}{5}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}, \sqrt {\frac {-\frac {34}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\right )+\left (-\frac {1}{5}+\frac {i \sqrt {14}}{5}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}, \sqrt {\frac {-\frac {34}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\right )\right )}{\sqrt {5 x^{3}+37 x^{2}+17 x +21}}\right )}{\sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}}\) | \(343\) |
Input:
int((x+7)^(1/2)/(5*x^2+2*x+3)^(1/2),x,method=_RETURNVERBOSE)
Output:
-2/25*(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)*(I*EllipticF((-5*(x+7)/(-34+I*14^(1/2)))^(1/2) ,(-(-34+I*14^(1/2))/(I*14^(1/2)+34))^(1/2))*14^(1/2)-I*EllipticE((-5*(x+7) /(-34+I*14^(1/2)))^(1/2),(-(-34+I*14^(1/2))/(I*14^(1/2)+34))^(1/2))*14^(1/ 2)+34*EllipticF((-5*(x+7)/(-34+I*14^(1/2)))^(1/2),(-(-34+I*14^(1/2))/(I*14 ^(1/2)+34))^(1/2))-34*EllipticE((-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 = 26, normalized size of antiderivative = 0.10 \[ \int \frac {\sqrt {7+x}}{\sqrt {3+2 x+5 x^2}} \, dx=\frac {136}{75} \, \sqrt {5} {\rm weierstrassPInverse}\left (\frac {4456}{75}, -\frac {348704}{3375}, x + \frac {37}{15}\right ) - \frac {2}{5} \, \sqrt {5} {\rm weierstrassZeta}\left (\frac {4456}{75}, -\frac {348704}{3375}, {\rm weierstrassPInverse}\left (\frac {4456}{75}, -\frac {348704}{3375}, x + \frac {37}{15}\right )\right ) \] Input:
integrate((7+x)^(1/2)/(5*x^2+2*x+3)^(1/2),x, algorithm="fricas")
Output:
136/75*sqrt(5)*weierstrassPInverse(4456/75, -348704/3375, x + 37/15) - 2/5 *sqrt(5)*weierstrassZeta(4456/75, -348704/3375, weierstrassPInverse(4456/7 5, -348704/3375, x + 37/15))
\[ \int \frac {\sqrt {7+x}}{\sqrt {3+2 x+5 x^2}} \, dx=\int \frac {\sqrt {x + 7}}{\sqrt {5 x^{2} + 2 x + 3}}\, dx \] Input:
integrate((7+x)**(1/2)/(5*x**2+2*x+3)**(1/2),x)
Output:
Integral(sqrt(x + 7)/sqrt(5*x**2 + 2*x + 3), x)
\[ \int \frac {\sqrt {7+x}}{\sqrt {3+2 x+5 x^2}} \, dx=\int { \frac {\sqrt {x + 7}}{\sqrt {5 \, x^{2} + 2 \, x + 3}} \,d x } \] Input:
integrate((7+x)^(1/2)/(5*x^2+2*x+3)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(x + 7)/sqrt(5*x^2 + 2*x + 3), x)
\[ \int \frac {\sqrt {7+x}}{\sqrt {3+2 x+5 x^2}} \, dx=\int { \frac {\sqrt {x + 7}}{\sqrt {5 \, x^{2} + 2 \, x + 3}} \,d x } \] Input:
integrate((7+x)^(1/2)/(5*x^2+2*x+3)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(x + 7)/sqrt(5*x^2 + 2*x + 3), x)
Timed out. \[ \int \frac {\sqrt {7+x}}{\sqrt {3+2 x+5 x^2}} \, dx=\int \frac {\sqrt {x+7}}{\sqrt {5\,x^2+2\,x+3}} \,d x \] Input:
int((x + 7)^(1/2)/(2*x + 5*x^2 + 3)^(1/2),x)
Output:
int((x + 7)^(1/2)/(2*x + 5*x^2 + 3)^(1/2), x)
\[ \int \frac {\sqrt {7+x}}{\sqrt {3+2 x+5 x^2}} \, dx=\int \frac {\sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}}{5 x^{2}+2 x +3}d x \] Input:
int((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**2 + 2*x + 3),x)