\(\int \frac {1}{x \sqrt {7+x} \sqrt {3+2 x+5 x^2}} \, dx\) [114]

Optimal result

Integrand size = 25, antiderivative size = 301 \[ \int \frac {1}{x \sqrt {7+x} \sqrt {3+2 x+5 x^2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {\frac {3}{7}} \sqrt {7+x}}{\sqrt {3+2 x+5 x^2}}\right )}{\sqrt {21}}-\frac {5^{3/4} \sqrt {\frac {3+2 x+5 x^2}{\left (78+\sqrt {130} (7+x)\right )^2}} \left (78+\sqrt {130} (7+x)\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{\frac {5}{26}} \sqrt {7+x}}{\sqrt {3}}\right ),\frac {1}{390} \left (195+17 \sqrt {130}\right )\right )}{\sqrt {3} \sqrt [4]{26} \left (35+3 \sqrt {130}\right ) \sqrt {3+2 x+5 x^2}}+\frac {\sqrt [4]{\frac {5}{26}} \left (7 \sqrt {5}-3 \sqrt {26}\right ) \sqrt {\frac {3+2 x+5 x^2}{\left (78+\sqrt {130} (7+x)\right )^2}} \left (78+\sqrt {130} (7+x)\right ) \operatorname {EllipticPi}\left (\frac {5460+479 \sqrt {130}}{10920},2 \arctan \left (\frac {\sqrt [4]{\frac {5}{26}} \sqrt {7+x}}{\sqrt {3}}\right ),\frac {1}{390} \left (195+17 \sqrt {130}\right )\right )}{14 \sqrt {3} \left (35+3 \sqrt {130}\right ) \sqrt {3+2 x+5 x^2}} \] Output:

-1/21*21^(1/2)*arctanh(1/7*21^(1/2)*(7+x)^(1/2)/(5*x^2+2*x+3)^(1/2))-1/78* 
5^(3/4)*((5*x^2+2*x+3)/(78+130^(1/2)*(7+x))^2)^(1/2)*(78+130^(1/2)*(7+x))* 
InverseJacobiAM(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))*3^(1/2)*26^(3/4)/(35+3*130^(1/2))/(5*x^2+2*x 
+3)^(1/2)+1/1092*5^(1/4)*26^(3/4)*(7*5^(1/2)-3*26^(1/2))*((5*x^2+2*x+3)/(7 
8+130^(1/2)*(7+x))^2)^(1/2)*(78+130^(1/2)*(7+x))*EllipticPi(sin(2*arctan(1 
/78*5^(1/4)*26^(3/4)*(7+x)^(1/2)*3^(1/2))),1/2+479/10920*130^(1/2),1/390*( 
76050+6630*130^(1/2))^(1/2))*3^(1/2)/(35+3*130^(1/2))/(5*x^2+2*x+3)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 21.65 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x \sqrt {7+x} \sqrt {3+2 x+5 x^2}} \, dx=-\frac {i (7+x) \sqrt {\frac {2}{13}+\frac {36 i}{\left (-34 i+\sqrt {14}\right ) (7+x)}} \sqrt {1-\frac {234 i}{\left (34 i+\sqrt {14}\right ) (7+x)}} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {3 \sqrt {-\frac {26 i}{34 i+\sqrt {14}}}}{\sqrt {7+x}}\right ),\frac {34 i+\sqrt {14}}{34 i-\sqrt {14}}\right )-\operatorname {EllipticPi}\left (\frac {7}{234} \left (34-i \sqrt {14}\right ),i \text {arcsinh}\left (\frac {3 \sqrt {-\frac {26 i}{34 i+\sqrt {14}}}}{\sqrt {7+x}}\right ),\frac {34 i+\sqrt {14}}{34 i-\sqrt {14}}\right )\right )}{21 \sqrt {-\frac {i}{34 i+\sqrt {14}}} \sqrt {3+2 x+5 x^2}} \] Input:

Integrate[1/(x*Sqrt[7 + x]*Sqrt[3 + 2*x + 5*x^2]),x]
 

Output:

((-1/21*I)*(7 + x)*Sqrt[2/13 + (36*I)/((-34*I + Sqrt[14])*(7 + x))]*Sqrt[1 
 - (234*I)/((34*I + Sqrt[14])*(7 + x))]*(EllipticF[I*ArcSinh[(3*Sqrt[(-26* 
I)/(34*I + Sqrt[14])])/Sqrt[7 + x]], (34*I + Sqrt[14])/(34*I - Sqrt[14])] 
- EllipticPi[(7*(34 - I*Sqrt[14]))/234, I*ArcSinh[(3*Sqrt[(-26*I)/(34*I + 
Sqrt[14])])/Sqrt[7 + x]], (34*I + Sqrt[14])/(34*I - Sqrt[14])]))/(Sqrt[(-I 
)/(34*I + Sqrt[14])]*Sqrt[3 + 2*x + 5*x^2])
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.50 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.76, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1279, 27, 187, 413, 413, 412}

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.

\(\displaystyle \int \frac {1}{x \sqrt {x+7} \sqrt {5 x^2+2 x+3}} \, dx\)

\(\Big \downarrow \) 1279

\(\displaystyle \frac {2 \sqrt {5 x-i \sqrt {14}+1} \sqrt {5 x+i \sqrt {14}+1} \int \frac {1}{2 x \sqrt {x+7} \sqrt {5 x-i \sqrt {14}+1} \sqrt {5 x+i \sqrt {14}+1}}dx}{\sqrt {5 x^2+2 x+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {5 x-i \sqrt {14}+1} \sqrt {5 x+i \sqrt {14}+1} \int \frac {1}{x \sqrt {x+7} \sqrt {5 x-i \sqrt {14}+1} \sqrt {5 x+i \sqrt {14}+1}}dx}{\sqrt {5 x^2+2 x+3}}\)

\(\Big \downarrow \) 187

\(\displaystyle -\frac {2 \sqrt {5 x-i \sqrt {14}+1} \sqrt {5 x+i \sqrt {14}+1} \int -\frac {1}{x \sqrt {5 (x+7)-i \sqrt {14}-34} \sqrt {5 (x+7)+i \sqrt {14}-34}}d\sqrt {x+7}}{\sqrt {5 x^2+2 x+3}}\)

\(\Big \downarrow \) 413

\(\displaystyle -\frac {2 \sqrt {5 x-i \sqrt {14}+1} \sqrt {5 x+i \sqrt {14}+1} \sqrt {1-\frac {5 (x+7)}{34+i \sqrt {14}}} \int -\frac {1}{x \sqrt {5 (x+7)+i \sqrt {14}-34} \sqrt {1-\frac {5 (x+7)}{34+i \sqrt {14}}}}d\sqrt {x+7}}{\sqrt {5 x^2+2 x+3} \sqrt {5 (x+7)-i \sqrt {14}-34}}\)

\(\Big \downarrow \) 413

\(\displaystyle -\frac {2 \sqrt {5 x-i \sqrt {14}+1} \sqrt {5 x+i \sqrt {14}+1} \sqrt {1-\frac {5 (x+7)}{34-i \sqrt {14}}} \sqrt {1-\frac {5 (x+7)}{34+i \sqrt {14}}} \int -\frac {1}{x \sqrt {1-\frac {5 (x+7)}{34-i \sqrt {14}}} \sqrt {1-\frac {5 (x+7)}{34+i \sqrt {14}}}}d\sqrt {x+7}}{\sqrt {5 x^2+2 x+3} \sqrt {5 (x+7)-i \sqrt {14}-34} \sqrt {5 (x+7)+i \sqrt {14}-34}}\)

\(\Big \downarrow \) 412

\(\displaystyle -\frac {2 \sqrt {\frac {1}{5} \left (34-i \sqrt {14}\right )} \sqrt {5 x-i \sqrt {14}+1} \sqrt {5 x+i \sqrt {14}+1} \sqrt {1-\frac {5 (x+7)}{34-i \sqrt {14}}} \sqrt {1-\frac {5 (x+7)}{34+i \sqrt {14}}} \operatorname {EllipticPi}\left (\frac {1}{35} \left (34-i \sqrt {14}\right ),\arcsin \left (\frac {\sqrt {5} \sqrt {x+7}}{\sqrt {34-i \sqrt {14}}}\right ),\frac {34 i+\sqrt {14}}{34 i-\sqrt {14}}\right )}{7 \sqrt {5 x^2+2 x+3} \sqrt {5 (x+7)-i \sqrt {14}-34} \sqrt {5 (x+7)+i \sqrt {14}-34}}\)

Input:

Int[1/(x*Sqrt[7 + x]*Sqrt[3 + 2*x + 5*x^2]),x]
 

Output:

(-2*Sqrt[(34 - I*Sqrt[14])/5]*Sqrt[1 - I*Sqrt[14] + 5*x]*Sqrt[1 + I*Sqrt[1 
4] + 5*x]*Sqrt[1 - (5*(7 + x))/(34 - I*Sqrt[14])]*Sqrt[1 - (5*(7 + x))/(34 
 + I*Sqrt[14])]*EllipticPi[(34 - I*Sqrt[14])/35, ArcSin[(Sqrt[5]*Sqrt[7 + 
x])/Sqrt[34 - I*Sqrt[14]]], (34*I + Sqrt[14])/(34*I - Sqrt[14])])/(7*Sqrt[ 
3 + 2*x + 5*x^2]*Sqrt[-34 - I*Sqrt[14] + 5*(7 + x)]*Sqrt[-34 + I*Sqrt[14] 
+ 5*(7 + x)])
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 187

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_ 
)]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-2   Subst[Int[1/(Simp[b*c - a*d 
- b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d*g - c*h)/ 
d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, 
g, h}, x] &&  !SimplerQ[e + f*x, c + d*x] &&  !SimplerQ[g + h*x, c + d*x]
 

rule 412

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* 
(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, 
 f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && S 
implerSqrtQ[-f/e, -d/c])
 

rule 413

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]   Int[1/((a + 
 b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, 
e, f}, x] &&  !GtQ[c, 0]
 

rule 1279

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_ 
) + (c_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[Sqrt[b 
 - q + 2*c*x]*(Sqrt[b + q + 2*c*x]/Sqrt[a + b*x + c*x^2])   Int[1/((d + e*x 
)*Sqrt[f + g*x]*Sqrt[b - q + 2*c*x]*Sqrt[b + q + 2*c*x]), x], x]] /; FreeQ[ 
{a, b, c, d, e, f, g}, x]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 2.76 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.52

method	result	size
default	\(\frac {2 \sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}\, \left (-34+i \sqrt {14}\right ) \sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}\, \sqrt {\frac {i \sqrt {14}-5 x -1}{i \sqrt {14}+34}}\, \sqrt {\frac {i \sqrt {14}+5 x +1}{-34+i \sqrt {14}}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}, \frac {34}{35}-\frac {i \sqrt {14}}{35}, \sqrt {-\frac {-34+i \sqrt {14}}{i \sqrt {14}+34}}\right )}{35 \left (5 x^{3}+37 x^{2}+17 x +21\right )}\)	\(158\)
elliptic	\(-\frac {2 \sqrt {\left (x +7\right ) \left (5 x^{2}+2 x +3\right )}\, \left (\frac {34}{5}-\frac {i \sqrt {14}}{5}\right ) \sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}-\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}+\frac {i \sqrt {14}}{5}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}, \frac {34}{35}-\frac {i \sqrt {14}}{35}, \sqrt {\frac {-\frac {34}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\right )}{7 \sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}\, \sqrt {5 x^{3}+37 x^{2}+17 x +21}}\)	\(167\)

Input:

int(1/x/(x+7)^(1/2)/(5*x^2+2*x+3)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

2/35*(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)*EllipticPi((-5*(x+7)/(-34+I*14^(1/2)))^(1/2),34 
/35-1/35*I*14^(1/2),(-(-34+I*14^(1/2))/(I*14^(1/2)+34))^(1/2))/(5*x^3+37*x 
^2+17*x+21)
 

Fricas [F]

\[ \int \frac {1}{x \sqrt {7+x} \sqrt {3+2 x+5 x^2}} \, dx=\int { \frac {1}{\sqrt {5 \, x^{2} + 2 \, x + 3} \sqrt {x + 7} x} \,d x } \] Input:

integrate(1/x/(7+x)^(1/2)/(5*x^2+2*x+3)^(1/2),x, algorithm="fricas")
 

Output:

integral(sqrt(5*x^2 + 2*x + 3)*sqrt(x + 7)/(5*x^4 + 37*x^3 + 17*x^2 + 21*x 
), x)
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \frac {1}{x \sqrt {7+x} \sqrt {3+2 x+5 x^2}} \, dx=\int \frac {1}{x \sqrt {x + 7} \sqrt {5 x^{2} + 2 x + 3}}\, dx \] Input:

integrate(1/x/(7+x)**(1/2)/(5*x**2+2*x+3)**(1/2),x)
 

Output:

Integral(1/(x*sqrt(x + 7)*sqrt(5*x**2 + 2*x + 3)), x)
 

Maxima [F]

\[ \int \frac {1}{x \sqrt {7+x} \sqrt {3+2 x+5 x^2}} \, dx=\int { \frac {1}{\sqrt {5 \, x^{2} + 2 \, x + 3} \sqrt {x + 7} x} \,d x } \] Input:

integrate(1/x/(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), x)
 

Giac [F]

\[ \int \frac {1}{x \sqrt {7+x} \sqrt {3+2 x+5 x^2}} \, dx=\int { \frac {1}{\sqrt {5 \, x^{2} + 2 \, x + 3} \sqrt {x + 7} x} \,d x } \] Input:

integrate(1/x/(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), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x \sqrt {7+x} \sqrt {3+2 x+5 x^2}} \, dx=\int \frac {1}{x\,\sqrt {x+7}\,\sqrt {5\,x^2+2\,x+3}} \,d x \] Input:

int(1/(x*(x + 7)^(1/2)*(2*x + 5*x^2 + 3)^(1/2)),x)
 

Output:

int(1/(x*(x + 7)^(1/2)*(2*x + 5*x^2 + 3)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {1}{x \sqrt {7+x} \sqrt {3+2 x+5 x^2}} \, dx=\int \frac {1}{x \sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}}d x \] Input:

int(1/x/(7+x)^(1/2)/(5*x^2+2*x+3)^(1/2),x)
 

Output:

int(1/x/(7+x)^(1/2)/(5*x^2+2*x+3)^(1/2),x)