\(\int \frac {x}{\sqrt {7+x} \sqrt {3+2 x+5 x^2}} \, dx\) [112]

Optimal result

Integrand size = 23, antiderivative size = 268 \[ \int \frac {x}{\sqrt {7+x} \sqrt {3+2 x+5 x^2}} \, dx=\frac {2 \sqrt {7+x} \sqrt {3+2 x+5 x^2}}{3 \sqrt {130}+5 (7+x)}-\frac {2 \sqrt {3} \sqrt [4]{26} \sqrt {\frac {3+2 x+5 x^2}{\left (78+\sqrt {130} (7+x)\right )^2}} \left (78+\sqrt {130} (7+x)\right ) E\left (2 \arctan \left (\frac {\sqrt [4]{\frac {5}{26}} \sqrt {7+x}}{\sqrt {3}}\right )|\frac {1}{390} \left (195+17 \sqrt {130}\right )\right )}{5^{3/4} \sqrt {3+2 x+5 x^2}}-\frac {\left (7 \sqrt {5}-3 \sqrt {26}\right ) \sqrt {\frac {3+2 x+5 x^2}{\left (78+\sqrt {130} (7+x)\right )^2}} \left (78+\sqrt {130} (7+x)\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{\frac {5}{26}} \sqrt {7+x}}{\sqrt {3}}\right ),\frac {1}{390} \left (195+17 \sqrt {130}\right )\right )}{\sqrt {3} 5^{3/4} \sqrt [4]{26} \sqrt {3+2 x+5 x^2}} \] 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)-1/390*(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))*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)*5^(1/4)*26^(3/4)/(5*x^2+2*x+3)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 22.54 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.46 \[ \int \frac {x}{\sqrt {7+x} \sqrt {3+2 x+5 x^2}} \, dx=\frac {(7+x)^{3/2} \left (\frac {78 \sqrt {-\frac {i}{34 i+\sqrt {14}}} \left (3+2 x+5 x^2\right )}{(7+x)^2}+\frac {2 \sqrt {13} \left (-17 i \sqrt {2}+\sqrt {7}\right ) \sqrt {\frac {34 i+\sqrt {14}-\frac {234 i}{7+x}}{34 i+\sqrt {14}}} \sqrt {\frac {-34 i+\sqrt {14}+\frac {234 i}{7+x}}{-34 i+\sqrt {14}}} E\left (i \text {arcsinh}\left (\frac {3 \sqrt {-\frac {26 i}{34 i+\sqrt {14}}}}{\sqrt {7+x}}\right )|\frac {34 i+\sqrt {14}}{34 i-\sqrt {14}}\right )}{\sqrt {7+x}}-\frac {i \sqrt {13} \left (\sqrt {2}-2 i \sqrt {7}\right ) \sqrt {\frac {34 i+\sqrt {14}-\frac {234 i}{7+x}}{34 i+\sqrt {14}}} \sqrt {\frac {-34 i+\sqrt {14}+\frac {234 i}{7+x}}{-34 i+\sqrt {14}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {3 \sqrt {-\frac {26 i}{34 i+\sqrt {14}}}}{\sqrt {7+x}}\right ),\frac {34 i+\sqrt {14}}{34 i-\sqrt {14}}\right )}{\sqrt {7+x}}\right )}{195 \sqrt {-\frac {i}{34 i+\sqrt {14}}} \sqrt {3+2 x+5 x^2}} \] Input:

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

Output:

((7 + x)^(3/2)*((78*Sqrt[(-I)/(34*I + Sqrt[14])]*(3 + 2*x + 5*x^2))/(7 + x 
)^2 + (2*Sqrt[13]*((-17*I)*Sqrt[2] + Sqrt[7])*Sqrt[(34*I + Sqrt[14] - (234 
*I)/(7 + x))/(34*I + Sqrt[14])]*Sqrt[(-34*I + Sqrt[14] + (234*I)/(7 + x))/ 
(-34*I + Sqrt[14])]*EllipticE[I*ArcSinh[(3*Sqrt[(-26*I)/(34*I + Sqrt[14])] 
)/Sqrt[7 + x]], (34*I + Sqrt[14])/(34*I - Sqrt[14])])/Sqrt[7 + x] - (I*Sqr 
t[13]*(Sqrt[2] - (2*I)*Sqrt[7])*Sqrt[(34*I + Sqrt[14] - (234*I)/(7 + x))/( 
34*I + Sqrt[14])]*Sqrt[(-34*I + Sqrt[14] + (234*I)/(7 + x))/(-34*I + Sqrt[ 
14])]*EllipticF[I*ArcSinh[(3*Sqrt[(-26*I)/(34*I + Sqrt[14])])/Sqrt[7 + x]] 
, (34*I + Sqrt[14])/(34*I - Sqrt[14])])/Sqrt[7 + x]))/(195*Sqrt[(-I)/(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 = 0.34 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.65, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {1269, 1172, 321, 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.

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

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1172

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 327

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

Input:

Int[x/(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])] - ((14*I)*Sqrt[(7 + x)/(34 - I*Sqrt[14])]*EllipticF[ArcSin[Sqrt 
[(-I)*(1 + I*Sqrt[14] + 5*x)]/(2^(3/4)*7^(1/4))], (2*Sqrt[14])/(34*I + Sqr 
t[14])])/Sqrt[7 + x]
 

Defintions of rubi rules used

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 327

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]
 

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 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 2.95 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.81

method	result	size
elliptic	\(\frac {2 \sqrt {\left (x +7\right ) \left (5 x^{2}+2 x +3\right )}\, \left (\frac {34}{5}-\frac {i \sqrt {14}}{5}\right ) \sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}-\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}+\frac {i \sqrt {14}}{5}}}\, \left (\left (-\frac {34}{5}-\frac {i \sqrt {14}}{5}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}, \sqrt {\frac {-\frac {34}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\right )+\left (-\frac {1}{5}+\frac {i \sqrt {14}}{5}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}, \sqrt {\frac {-\frac {34}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\right )\right )}{\sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}\, \sqrt {5 x^{3}+37 x^{2}+17 x +21}}\)	\(216\)
default	\(\frac {2 \sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}\, \left (-34+i \sqrt {14}\right ) \sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}\, \sqrt {\frac {i \sqrt {14}-5 x -1}{i \sqrt {14}+34}}\, \sqrt {\frac {i \sqrt {14}+5 x +1}{-34+i \sqrt {14}}}\, \left (i \operatorname {EllipticE}\left (\sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}, \sqrt {-\frac {-34+i \sqrt {14}}{i \sqrt {14}+34}}\right ) \sqrt {14}-i \operatorname {EllipticF}\left (\sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}, \sqrt {-\frac {-34+i \sqrt {14}}{i \sqrt {14}+34}}\right ) \sqrt {14}+34 \operatorname {EllipticE}\left (\sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}, \sqrt {-\frac {-34+i \sqrt {14}}{i \sqrt {14}+34}}\right )+\operatorname {EllipticF}\left (\sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}, \sqrt {-\frac {-34+i \sqrt {14}}{i \sqrt {14}+34}}\right )\right )}{25 \left (5 x^{3}+37 x^{2}+17 x +21\right )}\)	\(285\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.10 \[ \int \frac {x}{\sqrt {7+x} \sqrt {3+2 x+5 x^2}} \, dx=-\frac {74}{75} \, \sqrt {5} {\rm weierstrassPInverse}\left (\frac {4456}{75}, -\frac {348704}{3375}, x + \frac {37}{15}\right ) - \frac {2}{5} \, \sqrt {5} {\rm weierstrassZeta}\left (\frac {4456}{75}, -\frac {348704}{3375}, {\rm weierstrassPInverse}\left (\frac {4456}{75}, -\frac {348704}{3375}, x + \frac {37}{15}\right )\right ) \] Input:

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

Output:

-74/75*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))
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x}{\sqrt {7+x} \sqrt {3+2 x+5 x^2}} \, dx=\int \frac {\sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}\, x}{5 x^{3}+37 x^{2}+17 x +21}d x \] Input:

int(x/(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)*x)/(5*x**3 + 37*x**2 + 17*x + 21), 
x)