Integrand size = 23, antiderivative size = 293 \[ \int \frac {x \sqrt {7+x}}{\sqrt {3+2 x+5 x^2}} \, dx=\frac {2}{15} \sqrt {7+x} \sqrt {3+2 x+5 x^2}+\frac {62 \sqrt {7+x} \sqrt {3+2 x+5 x^2}}{15 \left (3 \sqrt {130}+5 (7+x)\right )}-\frac {62 \sqrt [4]{26} \sqrt {\frac {3+2 x+5 x^2}{\left (78+\sqrt {130} (7+x)\right )^2}} \left (78+\sqrt {130} (7+x)\right ) E\left (2 \arctan \left (\frac {\sqrt [4]{\frac {5}{26}} \sqrt {7+x}}{\sqrt {3}}\right )|\frac {1}{390} \left (195+17 \sqrt {130}\right )\right )}{5 \sqrt {3} 5^{3/4} \sqrt {3+2 x+5 x^2}}+\frac {\sqrt [4]{26} \left (31-3 \sqrt {130}\right ) \sqrt {\frac {3+2 x+5 x^2}{\left (78+\sqrt {130} (7+x)\right )^2}} \left (78+\sqrt {130} (7+x)\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{\frac {5}{26}} \sqrt {7+x}}{\sqrt {3}}\right ),\frac {1}{390} \left (195+17 \sqrt {130}\right )\right )}{5 \sqrt {3} 5^{3/4} \sqrt {3+2 x+5 x^2}} \] Output:
2/15*(7+x)^(1/2)*(5*x^2+2*x+3)^(1/2)+62*(7+x)^(1/2)*(5*x^2+2*x+3)^(1/2)/(4 5*130^(1/2)+525+75*x)-62/75*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))*EllipticE(sin(2*arctan(1/78*5^(1/4)*2 6^(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)+1/75*26^(1/4)*(31-3*130^(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))*3 ^(1/2)*5^(1/4)/(5*x^2+2*x+3)^(1/2)
Result contains complex when optimal does not.
Time = 23.16 (sec) , antiderivative size = 419, normalized size of antiderivative = 1.43 \[ \int \frac {x \sqrt {7+x}}{\sqrt {3+2 x+5 x^2}} \, dx=\frac {2}{15} \sqrt {7+x} \sqrt {3+2 x+5 x^2}+\frac {(7+x)^{3/2} \left (\frac {2418 \sqrt {-\frac {i}{34 i+\sqrt {14}}} \left (3+2 x+5 x^2\right )}{(7+x)^2}+\frac {62 \sqrt {13} \left (-17 i \sqrt {2}+\sqrt {7}\right ) \sqrt {\frac {34 i+\sqrt {14}-\frac {234 i}{7+x}}{34 i+\sqrt {14}}} \sqrt {\frac {-34 i+\sqrt {14}+\frac {234 i}{7+x}}{-34 i+\sqrt {14}}} E\left (i \text {arcsinh}\left (\frac {3 \sqrt {-\frac {26 i}{34 i+\sqrt {14}}}}{\sqrt {7+x}}\right )|\frac {34 i+\sqrt {14}}{34 i-\sqrt {14}}\right )}{\sqrt {7+x}}-\frac {2 i \sqrt {13} \left (58 \sqrt {2}-31 i \sqrt {7}\right ) \sqrt {\frac {34 i+\sqrt {14}-\frac {234 i}{7+x}}{34 i+\sqrt {14}}} \sqrt {\frac {-34 i+\sqrt {14}+\frac {234 i}{7+x}}{-34 i+\sqrt {14}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {3 \sqrt {-\frac {26 i}{34 i+\sqrt {14}}}}{\sqrt {7+x}}\right ),\frac {34 i+\sqrt {14}}{34 i-\sqrt {14}}\right )}{\sqrt {7+x}}\right )}{2925 \sqrt {-\frac {i}{34 i+\sqrt {14}}} \sqrt {3+2 x+5 x^2}} \] Input:
Integrate[(x*Sqrt[7 + x])/Sqrt[3 + 2*x + 5*x^2],x]
Output:
(2*Sqrt[7 + x]*Sqrt[3 + 2*x + 5*x^2])/15 + ((7 + x)^(3/2)*((2418*Sqrt[(-I) /(34*I + Sqrt[14])]*(3 + 2*x + 5*x^2))/(7 + x)^2 + (62*Sqrt[13]*((-17*I)*S qrt[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])]*Elliptic E[I*ArcSinh[(3*Sqrt[(-26*I)/(34*I + Sqrt[14])])/Sqrt[7 + x]], (34*I + Sqrt [14])/(34*I - Sqrt[14])])/Sqrt[7 + x] - ((2*I)*Sqrt[13]*(58*Sqrt[2] - (31* I)*Sqrt[7])*Sqrt[(34*I + Sqrt[14] - (234*I)/(7 + x))/(34*I + Sqrt[14])]*Sq rt[(-34*I + Sqrt[14] + (234*I)/(7 + x))/(-34*I + Sqrt[14])]*EllipticF[I*Ar cSinh[(3*Sqrt[(-26*I)/(34*I + Sqrt[14])])/Sqrt[7 + x]], (34*I + Sqrt[14])/ (34*I - Sqrt[14])])/Sqrt[7 + x]))/(2925*Sqrt[(-I)/(34*I + Sqrt[14])]*Sqrt[ 3 + 2*x + 5*x^2])
Result contains complex when optimal does not.
Time = 0.37 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.69, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {1236, 27, 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.
| \(\displaystyle \int \frac {x \sqrt {x+7}}{\sqrt {5 x^2+2 x+3}} \, dx\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {2}{15} \int -\frac {17-31 x}{2 \sqrt {x+7} \sqrt {5 x^2+2 x+3}}dx+\frac {2}{15} \sqrt {x+7} \sqrt {5 x^2+2 x+3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{15} \sqrt {x+7} \sqrt {5 x^2+2 x+3}-\frac {1}{15} \int \frac {17-31 x}{\sqrt {x+7} \sqrt {5 x^2+2 x+3}}dx\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {1}{15} \left (31 \int \frac {\sqrt {x+7}}{\sqrt {5 x^2+2 x+3}}dx-234 \int \frac {1}{\sqrt {x+7} \sqrt {5 x^2+2 x+3}}dx\right )+\frac {2}{15} \sqrt {x+7} \sqrt {5 x^2+2 x+3}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {2}{15} \sqrt {x+7} \sqrt {5 x^2+2 x+3}+\frac {1}{15} \left (\frac {62 i \sqrt {x+7} \int \frac {\sqrt {\frac {i \left (5 x+i \sqrt {14}+1\right )}{34 i+\sqrt {14}}+1}}{\sqrt {\frac {i \left (5 x+i \sqrt {14}+1\right )}{2 \sqrt {14}}+1}}d\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}}{5 \sqrt {\frac {x+7}{34-i \sqrt {14}}}}-\frac {468 i \sqrt {\frac {x+7}{34-i \sqrt {14}}} \int \frac {1}{\sqrt {\frac {i \left (5 x+i \sqrt {14}+1\right )}{2 \sqrt {14}}+1} \sqrt {\frac {i \left (5 x+i \sqrt {14}+1\right )}{34 i+\sqrt {14}}+1}}d\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}}{\sqrt {x+7}}\right )\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {2}{15} \sqrt {x+7} \sqrt {5 x^2+2 x+3}+\frac {1}{15} \left (\frac {62 i \sqrt {x+7} \int \frac {\sqrt {\frac {i \left (5 x+i \sqrt {14}+1\right )}{34 i+\sqrt {14}}+1}}{\sqrt {\frac {i \left (5 x+i \sqrt {14}+1\right )}{2 \sqrt {14}}+1}}d\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}}{5 \sqrt {\frac {x+7}{34-i \sqrt {14}}}}-\frac {468 i \sqrt {\frac {x+7}{34-i \sqrt {14}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}\right ),\frac {2 \sqrt {14}}{34 i+\sqrt {14}}\right )}{\sqrt {x+7}}\right )\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {2}{15} \sqrt {x+7} \sqrt {5 x^2+2 x+3}+\frac {1}{15} \left (\frac {62 i \sqrt {x+7} E\left (\arcsin \left (\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}\right )|\frac {2 \sqrt {14}}{34 i+\sqrt {14}}\right )}{5 \sqrt {\frac {x+7}{34-i \sqrt {14}}}}-\frac {468 i \sqrt {\frac {x+7}{34-i \sqrt {14}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}\right ),\frac {2 \sqrt {14}}{34 i+\sqrt {14}}\right )}{\sqrt {x+7}}\right )\) |
Input:
Int[(x*Sqrt[7 + x])/Sqrt[3 + 2*x + 5*x^2],x]
Output:
(2*Sqrt[7 + x]*Sqrt[3 + 2*x + 5*x^2])/15 + ((((62*I)/5)*Sqrt[7 + x]*Ellipt icE[ArcSin[Sqrt[(-I)*(1 + I*Sqrt[14] + 5*x)]/(2^(3/4)*7^(1/4))], (2*Sqrt[1 4])/(34*I + Sqrt[14])])/Sqrt[(7 + x)/(34 - I*Sqrt[14])] - ((468*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 + Sqrt[14])])/Sqrt[7 + x])/15
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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[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]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Simp[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1 )*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m *(c*e*f + c*d*g - b*e*g) + e*(p + 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[ {a, b, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && EqQ[f, 0])
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]
Result contains complex when optimal does not.
Time = 2.93 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.24
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (x +7\right ) \left (5 x^{2}+2 x +3\right )}\, \left (\frac {2 \sqrt {5 x^{3}+37 x^{2}+17 x +21}}{15}-\frac {34 \left (\frac {34}{5}-\frac {i \sqrt {14}}{5}\right ) \sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}-\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}+\frac {i \sqrt {14}}{5}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}, \sqrt {\frac {-\frac {34}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\right )}{15 \sqrt {5 x^{3}+37 x^{2}+17 x +21}}+\frac {62 \left (\frac {34}{5}-\frac {i \sqrt {14}}{5}\right ) \sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}-\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}+\frac {i \sqrt {14}}{5}}}\, \left (\left (-\frac {34}{5}-\frac {i \sqrt {14}}{5}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}, \sqrt {\frac {-\frac {34}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\right )+\left (-\frac {1}{5}+\frac {i \sqrt {14}}{5}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}, \sqrt {\frac {-\frac {34}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\right )\right )}{15 \sqrt {5 x^{3}+37 x^{2}+17 x +21}}\right )}{\sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}}\) | \(362\) |
| risch | \(\frac {2 \sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}}{15}+\frac {\left (-\frac {34 \left (\frac {34}{5}-\frac {i \sqrt {14}}{5}\right ) \sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}-\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}+\frac {i \sqrt {14}}{5}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}, \sqrt {\frac {-\frac {34}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\right )}{15 \sqrt {5 x^{3}+37 x^{2}+17 x +21}}+\frac {62 \left (\frac {34}{5}-\frac {i \sqrt {14}}{5}\right ) \sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}-\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}+\frac {i \sqrt {14}}{5}}}\, \left (\left (-\frac {34}{5}-\frac {i \sqrt {14}}{5}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}, \sqrt {\frac {-\frac {34}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\right )+\left (-\frac {1}{5}+\frac {i \sqrt {14}}{5}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}, \sqrt {\frac {-\frac {34}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\right )\right )}{15 \sqrt {5 x^{3}+37 x^{2}+17 x +21}}\right ) \sqrt {\left (x +7\right ) \left (5 x^{2}+2 x +3\right )}}{\sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}}\) | \(363\) |
| default | \(\frac {2 \sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}\, \left (234 i \sqrt {14}\, \sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}\, \sqrt {\frac {i \sqrt {14}-5 x -1}{i \sqrt {14}+34}}\, \sqrt {\frac {i \sqrt {14}+5 x +1}{-34+i \sqrt {14}}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}, \sqrt {-\frac {-34+i \sqrt {14}}{i \sqrt {14}+34}}\right )-702 \sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}\, \sqrt {\frac {i \sqrt {14}-5 x -1}{i \sqrt {14}+34}}\, \sqrt {\frac {i \sqrt {14}+5 x +1}{-34+i \sqrt {14}}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}, \sqrt {-\frac {-34+i \sqrt {14}}{i \sqrt {14}+34}}\right )-7254 \sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}\, \sqrt {\frac {i \sqrt {14}-5 x -1}{i \sqrt {14}+34}}\, \sqrt {\frac {i \sqrt {14}+5 x +1}{-34+i \sqrt {14}}}\, \operatorname {EllipticE}\left (\sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}, \sqrt {-\frac {-34+i \sqrt {14}}{i \sqrt {14}+34}}\right )+25 x^{3}+185 x^{2}+85 x +105\right )}{75 \left (5 x^{3}+37 x^{2}+17 x +21\right )}\) | \(377\) |
Input:
int(x*(x+7)^(1/2)/(5*x^2+2*x+3)^(1/2),x,method=_RETURNVERBOSE)
Output:
((x+7)*(5*x^2+2*x+3))^(1/2)/(x+7)^(1/2)/(5*x^2+2*x+3)^(1/2)*(2/15*(5*x^3+3 7*x^2+17*x+21)^(1/2)-34/15*(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)* 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))+62/15*(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))))
Time = 0.07 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.15 \[ \int \frac {x \sqrt {7+x}}{\sqrt {3+2 x+5 x^2}} \, dx=-\frac {2804}{1125} \, \sqrt {5} {\rm weierstrassPInverse}\left (\frac {4456}{75}, -\frac {348704}{3375}, x + \frac {37}{15}\right ) - \frac {62}{75} \, \sqrt {5} {\rm weierstrassZeta}\left (\frac {4456}{75}, -\frac {348704}{3375}, {\rm weierstrassPInverse}\left (\frac {4456}{75}, -\frac {348704}{3375}, x + \frac {37}{15}\right )\right ) + \frac {2}{15} \, \sqrt {5 \, x^{2} + 2 \, x + 3} \sqrt {x + 7} \] Input:
integrate(x*(7+x)^(1/2)/(5*x^2+2*x+3)^(1/2),x, algorithm="fricas")
Output:
-2804/1125*sqrt(5)*weierstrassPInverse(4456/75, -348704/3375, x + 37/15) - 62/75*sqrt(5)*weierstrassZeta(4456/75, -348704/3375, weierstrassPInverse( 4456/75, -348704/3375, x + 37/15)) + 2/15*sqrt(5*x^2 + 2*x + 3)*sqrt(x + 7 )
\[ \int \frac {x \sqrt {7+x}}{\sqrt {3+2 x+5 x^2}} \, dx=\int \frac {x \sqrt {x + 7}}{\sqrt {5 x^{2} + 2 x + 3}}\, dx \] 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)
\[ \int \frac {x \sqrt {7+x}}{\sqrt {3+2 x+5 x^2}} \, dx=\int { \frac {\sqrt {x + 7} x}{\sqrt {5 \, x^{2} + 2 \, x + 3}} \,d x } \] Input:
integrate(x*(7+x)^(1/2)/(5*x^2+2*x+3)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(x + 7)*x/sqrt(5*x^2 + 2*x + 3), x)
\[ \int \frac {x \sqrt {7+x}}{\sqrt {3+2 x+5 x^2}} \, dx=\int { \frac {\sqrt {x + 7} x}{\sqrt {5 \, x^{2} + 2 \, x + 3}} \,d x } \] Input:
integrate(x*(7+x)^(1/2)/(5*x^2+2*x+3)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(x + 7)*x/sqrt(5*x^2 + 2*x + 3), x)
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)
\[ \int \frac {x \sqrt {7+x}}{\sqrt {3+2 x+5 x^2}} \, dx=\frac {7 \sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}}{37}-\frac {31 \left (\int \frac {\sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}\, x^{2}}{5 x^{3}+37 x^{2}+17 x +21}d x \right )}{74}-\frac {119 \left (\int \frac {\sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}}{5 x^{3}+37 x^{2}+17 x +21}d x \right )}{74} \] Input:
int(x*(7+x)^(1/2)/(5*x^2+2*x+3)^(1/2),x)
Output:
(14*sqrt(x + 7)*sqrt(5*x**2 + 2*x + 3) - 31*int((sqrt(x + 7)*sqrt(5*x**2 + 2*x + 3)*x**2)/(5*x**3 + 37*x**2 + 17*x + 21),x) - 119*int((sqrt(x + 7)*s qrt(5*x**2 + 2*x + 3))/(5*x**3 + 37*x**2 + 17*x + 21),x))/74