\(\int \frac {\sqrt {7+x}}{x \sqrt {3+2 x+5 x^2}} \, dx\) [106]

Optimal result

Integrand size = 25, antiderivative size = 287 \[ \int \frac {\sqrt {7+x}}{x \sqrt {3+2 x+5 x^2}} \, dx=-\sqrt {\frac {7}{3}} \text {arctanh}\left (\frac {\sqrt {\frac {3}{7}} \sqrt {7+x}}{\sqrt {3+2 x+5 x^2}}\right )+\frac {\sqrt {3} \sqrt [4]{\frac {26}{5}} \left (7 \sqrt {5}-3 \sqrt {26}\right ) \sqrt {\frac {3+2 x+5 x^2}{\left (78+\sqrt {130} (7+x)\right )^2}} \left (78+\sqrt {130} (7+x)\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{\frac {5}{26}} \sqrt {7+x}}{\sqrt {3}}\right ),\frac {1}{390} \left (195+17 \sqrt {130}\right )\right )}{11 \sqrt {3+2 x+5 x^2}}+\frac {\left (479-42 \sqrt {130}\right ) \sqrt {\frac {3+2 x+5 x^2}{\left (78+\sqrt {130} (7+x)\right )^2}} \left (78+\sqrt {130} (7+x)\right ) \operatorname {EllipticPi}\left (\frac {5460+479 \sqrt {130}}{10920},2 \arctan \left (\frac {\sqrt [4]{\frac {5}{26}} \sqrt {7+x}}{\sqrt {3}}\right ),\frac {1}{390} \left (195+17 \sqrt {130}\right )\right )}{22 \sqrt {3} \sqrt [4]{130} \sqrt {3+2 x+5 x^2}} \] Output:

-1/3*21^(1/2)*arctanh(1/7*21^(1/2)*(7+x)^(1/2)/(5*x^2+2*x+3)^(1/2))+1/55*3 
^(1/2)*26^(1/4)*5^(3/4)*(7*5^(1/2)-3*26^(1/2))*((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))/(5*x^ 
2+2*x+3)^(1/2)+1/8580*(479-42*130^(1/2))*((5*x^2+2*x+3)/(78+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)*130^(3/4)/(5*x^2+2*x+3)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 20.57 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt {7+x}}{x \sqrt {3+2 x+5 x^2}} \, dx=-\frac {2 i \sqrt {\frac {-i+\sqrt {14}-5 i x}{34 i+\sqrt {14}}} \sqrt {\frac {i+\sqrt {14}+5 i x}{-34 i+\sqrt {14}}} \sqrt {7+x} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {5} \sqrt {-\frac {i (7+x)}{34 i+\sqrt {14}}}\right ),\frac {34 i+\sqrt {14}}{34 i-\sqrt {14}}\right )-\operatorname {EllipticPi}\left (\frac {1}{35} \left (34-i \sqrt {14}\right ),i \text {arcsinh}\left (\sqrt {5} \sqrt {-\frac {i (7+x)}{34 i+\sqrt {14}}}\right ),\frac {34 i+\sqrt {14}}{34 i-\sqrt {14}}\right )\right )}{\sqrt {5} \sqrt {-\frac {i (7+x)}{34 i+\sqrt {14}}} \sqrt {3+2 x+5 x^2}} \] Input:

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

Output:

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

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 0.64 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.09, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {1284, 1172, 321, 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.

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

\(\Big \downarrow \) 1284

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

\(\Big \downarrow \) 1172

\(\displaystyle 7 \int \frac {1}{x \sqrt {x+7} \sqrt {5 x^2+2 x+3}}dx+\frac {2 i \sqrt {\frac {x+7}{34-i \sqrt {14}}} \int \frac {1}{\sqrt {\frac {i \left (5 x+i \sqrt {14}+1\right )}{2 \sqrt {14}}+1} \sqrt {\frac {i \left (5 x+i \sqrt {14}+1\right )}{34 i+\sqrt {14}}+1}}d\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}}{\sqrt {x+7}}\)

\(\Big \downarrow \) 321

\(\displaystyle 7 \int \frac {1}{x \sqrt {x+7} \sqrt {5 x^2+2 x+3}}dx+\frac {2 i \sqrt {\frac {x+7}{34-i \sqrt {14}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}\right ),\frac {2 \sqrt {14}}{34 i+\sqrt {14}}\right )}{\sqrt {x+7}}\)

\(\Big \downarrow \) 1279

\(\displaystyle \frac {14 \sqrt {5 x-i \sqrt {14}+1} \sqrt {5 x+i \sqrt {14}+1} \int \frac {1}{2 x \sqrt {x+7} \sqrt {5 x-i \sqrt {14}+1} \sqrt {5 x+i \sqrt {14}+1}}dx}{\sqrt {5 x^2+2 x+3}}+\frac {2 i \sqrt {\frac {x+7}{34-i \sqrt {14}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}\right ),\frac {2 \sqrt {14}}{34 i+\sqrt {14}}\right )}{\sqrt {x+7}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {7 \sqrt {5 x-i \sqrt {14}+1} \sqrt {5 x+i \sqrt {14}+1} \int \frac {1}{x \sqrt {x+7} \sqrt {5 x-i \sqrt {14}+1} \sqrt {5 x+i \sqrt {14}+1}}dx}{\sqrt {5 x^2+2 x+3}}+\frac {2 i \sqrt {\frac {x+7}{34-i \sqrt {14}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}\right ),\frac {2 \sqrt {14}}{34 i+\sqrt {14}}\right )}{\sqrt {x+7}}\)

\(\Big \downarrow \) 187

\(\displaystyle \frac {2 i \sqrt {\frac {x+7}{34-i \sqrt {14}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}\right ),\frac {2 \sqrt {14}}{34 i+\sqrt {14}}\right )}{\sqrt {x+7}}-\frac {14 \sqrt {5 x-i \sqrt {14}+1} \sqrt {5 x+i \sqrt {14}+1} \int -\frac {1}{x \sqrt {5 (x+7)-i \sqrt {14}-34} \sqrt {5 (x+7)+i \sqrt {14}-34}}d\sqrt {x+7}}{\sqrt {5 x^2+2 x+3}}\)

\(\Big \downarrow \) 413

\(\displaystyle \frac {2 i \sqrt {\frac {x+7}{34-i \sqrt {14}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}\right ),\frac {2 \sqrt {14}}{34 i+\sqrt {14}}\right )}{\sqrt {x+7}}-\frac {14 \sqrt {5 x-i \sqrt {14}+1} \sqrt {5 x+i \sqrt {14}+1} \sqrt {1-\frac {5 (x+7)}{34+i \sqrt {14}}} \int -\frac {1}{x \sqrt {5 (x+7)+i \sqrt {14}-34} \sqrt {1-\frac {5 (x+7)}{34+i \sqrt {14}}}}d\sqrt {x+7}}{\sqrt {5 x^2+2 x+3} \sqrt {5 (x+7)-i \sqrt {14}-34}}\)

\(\Big \downarrow \) 413

\(\displaystyle \frac {2 i \sqrt {\frac {x+7}{34-i \sqrt {14}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}\right ),\frac {2 \sqrt {14}}{34 i+\sqrt {14}}\right )}{\sqrt {x+7}}-\frac {14 \sqrt {5 x-i \sqrt {14}+1} \sqrt {5 x+i \sqrt {14}+1} \sqrt {1-\frac {5 (x+7)}{34-i \sqrt {14}}} \sqrt {1-\frac {5 (x+7)}{34+i \sqrt {14}}} \int -\frac {1}{x \sqrt {1-\frac {5 (x+7)}{34-i \sqrt {14}}} \sqrt {1-\frac {5 (x+7)}{34+i \sqrt {14}}}}d\sqrt {x+7}}{\sqrt {5 x^2+2 x+3} \sqrt {5 (x+7)-i \sqrt {14}-34} \sqrt {5 (x+7)+i \sqrt {14}-34}}\)

\(\Big \downarrow \) 412

\(\displaystyle \frac {2 i \sqrt {\frac {x+7}{34-i \sqrt {14}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}\right ),\frac {2 \sqrt {14}}{34 i+\sqrt {14}}\right )}{\sqrt {x+7}}-\frac {2 \sqrt {\frac {1}{5} \left (34-i \sqrt {14}\right )} \sqrt {5 x-i \sqrt {14}+1} \sqrt {5 x+i \sqrt {14}+1} \sqrt {1-\frac {5 (x+7)}{34-i \sqrt {14}}} \sqrt {1-\frac {5 (x+7)}{34+i \sqrt {14}}} \operatorname {EllipticPi}\left (\frac {1}{35} \left (34-i \sqrt {14}\right ),\arcsin \left (\frac {\sqrt {5} \sqrt {x+7}}{\sqrt {34-i \sqrt {14}}}\right ),\frac {34 i+\sqrt {14}}{34 i-\sqrt {14}}\right )}{\sqrt {5 x^2+2 x+3} \sqrt {5 (x+7)-i \sqrt {14}-34} \sqrt {5 (x+7)+i \sqrt {14}-34}}\)

Input:

Int[Sqrt[7 + x]/(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] - (2*Sqrt[(34 - I*Sqrt[14])/5]*Sqrt[1 - I*Sqrt[14] + 5*x]*Sqrt[1 + 
I*Sqrt[14] + 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]*S 
qrt[7 + x])/Sqrt[34 - I*Sqrt[14]]], (34*I + Sqrt[14])/(34*I - Sqrt[14])])/ 
(Sqrt[3 + 2*x + 5*x^2]*Sqrt[-34 - I*Sqrt[14] + 5*(7 + x)]*Sqrt[-34 + I*Sqr 
t[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 321

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])
 

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 1172

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]
 

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]
 

rule 1284

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 2.61 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.70

method	result	size
default	\(-\frac {2 \sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}\, \left (-34+i \sqrt {14}\right ) \sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}\, \sqrt {\frac {i \sqrt {14}-5 x -1}{i \sqrt {14}+34}}\, \sqrt {\frac {i \sqrt {14}+5 x +1}{-34+i \sqrt {14}}}\, \left (\operatorname {EllipticF}\left (\sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}, \sqrt {-\frac {-34+i \sqrt {14}}{i \sqrt {14}+34}}\right )-\operatorname {EllipticPi}\left (\sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}, \frac {34}{35}-\frac {i \sqrt {14}}{35}, \sqrt {-\frac {-34+i \sqrt {14}}{i \sqrt {14}+34}}\right )\right )}{5 \left (5 x^{3}+37 x^{2}+17 x +21\right )}\)	\(201\)
elliptic	\(\frac {\sqrt {\left (x +7\right ) \left (5 x^{2}+2 x +3\right )}\, \left (\frac {2 \left (\frac {34}{5}-\frac {i \sqrt {14}}{5}\right ) \sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}-\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}+\frac {i \sqrt {14}}{5}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}, \sqrt {\frac {-\frac {34}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\right )}{\sqrt {5 x^{3}+37 x^{2}+17 x +21}}-\frac {2 \left (\frac {34}{5}-\frac {i \sqrt {14}}{5}\right ) \sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}-\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}+\frac {i \sqrt {14}}{5}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}, \frac {34}{35}-\frac {i \sqrt {14}}{35}, \sqrt {\frac {-\frac {34}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\right )}{\sqrt {5 x^{3}+37 x^{2}+17 x +21}}\right )}{\sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}}\)	\(294\)

Input:

int((x+7)^(1/2)/x/(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))-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]

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

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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