Integrand size = 25, antiderivative size = 324 \[ \int \frac {x^3}{\sqrt {7+x} \sqrt {3+2 x+5 x^2}} \, dx=-\frac {506}{375} \sqrt {7+x} \sqrt {3+2 x+5 x^2}+\frac {2}{25} (7+x)^{3/2} \sqrt {3+2 x+5 x^2}+\frac {20374 \sqrt {7+x} \sqrt {3+2 x+5 x^2}}{375 \left (3 \sqrt {130}+5 (7+x)\right )}-\frac {20374 \sqrt [4]{26} \sqrt {\frac {3+2 x+5 x^2}{\left (78+\sqrt {130} (7+x)\right )^2}} \left (78+\sqrt {130} (7+x)\right ) E\left (2 \arctan \left (\frac {\sqrt [4]{\frac {5}{26}} \sqrt {7+x}}{\sqrt {3}}\right )|\frac {1}{390} \left (195+17 \sqrt {130}\right )\right )}{125 \sqrt {3} 5^{3/4} \sqrt {3+2 x+5 x^2}}-\frac {\left (23141 \sqrt {5}-10187 \sqrt {26}\right ) \sqrt {\frac {3+2 x+5 x^2}{\left (78+\sqrt {130} (7+x)\right )^2}} \left (78+\sqrt {130} (7+x)\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{\frac {5}{26}} \sqrt {7+x}}{\sqrt {3}}\right ),\frac {1}{390} \left (195+17 \sqrt {130}\right )\right )}{125 \sqrt {3} 5^{3/4} \sqrt [4]{26} \sqrt {3+2 x+5 x^2}} \] Output:
-506/375*(7+x)^(1/2)*(5*x^2+2*x+3)^(1/2)+2/25*(7+x)^(3/2)*(5*x^2+2*x+3)^(1 /2)+20374*(7+x)^(1/2)*(5*x^2+2*x+3)^(1/2)/(1125*130^(1/2)+13125+1875*x)-20 374/1875*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)*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) -1/48750*(23141*5^(1/2)-10187*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))*3^(1/2)*5^(1 /4)*26^(3/4)/(5*x^2+2*x+3)^(1/2)
Result contains complex when optimal does not.
Time = 34.17 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.31 \[ \int \frac {x^3}{\sqrt {7+x} \sqrt {3+2 x+5 x^2}} \, dx=\frac {2}{375} \sqrt {7+x} (-148+15 x) \sqrt {3+2 x+5 x^2}-\frac {(7+x)^{3/2} \left (-\frac {794586 \sqrt {-\frac {i}{34 i+\sqrt {14}}} \left (3+2 x+5 x^2\right )}{(7+x)^2}+\frac {20374 i \sqrt {13} \left (17 \sqrt {2}+i \sqrt {7}\right ) \sqrt {\frac {34 i+\sqrt {14}-\frac {234 i}{7+x}}{34 i+\sqrt {14}}} \sqrt {\frac {-34 i+\sqrt {14}+\frac {234 i}{7+x}}{-34 i+\sqrt {14}}} E\left (i \text {arcsinh}\left (\frac {3 \sqrt {-\frac {26 i}{34 i+\sqrt {14}}}}{\sqrt {7+x}}\right )|\frac {34 i+\sqrt {14}}{34 i-\sqrt {14}}\right )}{\sqrt {7+x}}+\frac {\sqrt {13} \left (757 i \sqrt {2}+20374 \sqrt {7}\right ) \sqrt {\frac {34 i+\sqrt {14}-\frac {234 i}{7+x}}{34 i+\sqrt {14}}} \sqrt {\frac {-34 i+\sqrt {14}+\frac {234 i}{7+x}}{-34 i+\sqrt {14}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {3 \sqrt {-\frac {26 i}{34 i+\sqrt {14}}}}{\sqrt {7+x}}\right ),\frac {34 i+\sqrt {14}}{34 i-\sqrt {14}}\right )}{\sqrt {7+x}}\right )}{73125 \sqrt {-\frac {i}{34 i+\sqrt {14}}} \sqrt {3+2 x+5 x^2}} \] Input:
Integrate[x^3/(Sqrt[7 + x]*Sqrt[3 + 2*x + 5*x^2]),x]
Output:
(2*Sqrt[7 + x]*(-148 + 15*x)*Sqrt[3 + 2*x + 5*x^2])/375 - ((7 + x)^(3/2)*( (-794586*Sqrt[(-I)/(34*I + Sqrt[14])]*(3 + 2*x + 5*x^2))/(7 + x)^2 + ((203 74*I)*Sqrt[13]*(17*Sqrt[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])]*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] + (Sqrt[13]*(( 757*I)*Sqrt[2] + 20374*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]))/(73125*Sqrt[(-I)/(34 *I + Sqrt[14])]*Sqrt[3 + 2*x + 5*x^2])
Result contains complex when optimal does not.
Time = 0.48 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.72, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {1278, 2184, 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.
| \(\displaystyle \int \frac {x^3}{\sqrt {x+7} \sqrt {5 x^2+2 x+3}} \, dx\) |
| \(\Big \downarrow \) 1278 |
| \(\displaystyle \frac {2}{25} x \sqrt {x+7} \sqrt {5 x^2+2 x+3}-\frac {1}{25} \int \frac {148 x^2+51 x+42}{\sqrt {x+7} \sqrt {5 x^2+2 x+3}}dx\) |
| \(\Big \downarrow \) 2184 |
| \(\displaystyle \frac {1}{25} \left (-\frac {2}{15} \int -\frac {10187 x+1886}{2 \sqrt {x+7} \sqrt {5 x^2+2 x+3}}dx-\frac {296}{15} \sqrt {x+7} \sqrt {5 x^2+2 x+3}\right )+\frac {2}{25} \sqrt {x+7} \sqrt {5 x^2+2 x+3} x\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{25} \left (\frac {1}{15} \int \frac {10187 x+1886}{\sqrt {x+7} \sqrt {5 x^2+2 x+3}}dx-\frac {296}{15} \sqrt {x+7} \sqrt {5 x^2+2 x+3}\right )+\frac {2}{25} \sqrt {x+7} \sqrt {5 x^2+2 x+3} x\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {1}{25} \left (\frac {1}{15} \left (10187 \int \frac {\sqrt {x+7}}{\sqrt {5 x^2+2 x+3}}dx-69423 \int \frac {1}{\sqrt {x+7} \sqrt {5 x^2+2 x+3}}dx\right )-\frac {296}{15} \sqrt {x+7} \sqrt {5 x^2+2 x+3}\right )+\frac {2}{25} \sqrt {x+7} \sqrt {5 x^2+2 x+3} x\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {2}{25} \sqrt {x+7} \sqrt {5 x^2+2 x+3} x+\frac {1}{25} \left (-\frac {296}{15} \sqrt {x+7} \sqrt {5 x^2+2 x+3}+\frac {1}{15} \left (\frac {20374 i \sqrt {x+7} \int \frac {\sqrt {\frac {i \left (5 x+i \sqrt {14}+1\right )}{34 i+\sqrt {14}}+1}}{\sqrt {\frac {i \left (5 x+i \sqrt {14}+1\right )}{2 \sqrt {14}}+1}}d\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}}{5 \sqrt {\frac {x+7}{34-i \sqrt {14}}}}-\frac {138846 i \sqrt {\frac {x+7}{34-i \sqrt {14}}} \int \frac {1}{\sqrt {\frac {i \left (5 x+i \sqrt {14}+1\right )}{2 \sqrt {14}}+1} \sqrt {\frac {i \left (5 x+i \sqrt {14}+1\right )}{34 i+\sqrt {14}}+1}}d\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}}{\sqrt {x+7}}\right )\right )\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {2}{25} \sqrt {x+7} \sqrt {5 x^2+2 x+3} x+\frac {1}{25} \left (-\frac {296}{15} \sqrt {x+7} \sqrt {5 x^2+2 x+3}+\frac {1}{15} \left (\frac {20374 i \sqrt {x+7} \int \frac {\sqrt {\frac {i \left (5 x+i \sqrt {14}+1\right )}{34 i+\sqrt {14}}+1}}{\sqrt {\frac {i \left (5 x+i \sqrt {14}+1\right )}{2 \sqrt {14}}+1}}d\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}}{5 \sqrt {\frac {x+7}{34-i \sqrt {14}}}}-\frac {138846 i \sqrt {\frac {x+7}{34-i \sqrt {14}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}\right ),\frac {2 \sqrt {14}}{34 i+\sqrt {14}}\right )}{\sqrt {x+7}}\right )\right )\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {2}{25} \sqrt {x+7} \sqrt {5 x^2+2 x+3} x+\frac {1}{25} \left (-\frac {296}{15} \sqrt {x+7} \sqrt {5 x^2+2 x+3}+\frac {1}{15} \left (\frac {20374 i \sqrt {x+7} E\left (\arcsin \left (\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}\right )|\frac {2 \sqrt {14}}{34 i+\sqrt {14}}\right )}{5 \sqrt {\frac {x+7}{34-i \sqrt {14}}}}-\frac {138846 i \sqrt {\frac {x+7}{34-i \sqrt {14}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}\right ),\frac {2 \sqrt {14}}{34 i+\sqrt {14}}\right )}{\sqrt {x+7}}\right )\right )\) |
Input:
Int[x^3/(Sqrt[7 + x]*Sqrt[3 + 2*x + 5*x^2]),x]
Output:
(2*x*Sqrt[7 + x]*Sqrt[3 + 2*x + 5*x^2])/25 + ((-296*Sqrt[7 + x]*Sqrt[3 + 2 *x + 5*x^2])/15 + ((((20374*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])] - ((138846*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)/25
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/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]
Int[((d_.) + (e_.)*(x_))^(m_)/(Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)* (x_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[2*e^2*(d + e*x)^(m - 2)*Sqrt[f + g *x]*(Sqrt[a + b*x + c*x^2]/(c*g*(2*m - 1))), x] - Simp[1/(c*g*(2*m - 1)) Int[((d + e*x)^(m - 3)/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]))*Simp[b*d*e^2* f + a*e^2*(d*g + 2*e*f*(m - 2)) - c*d^3*g*(2*m - 1) + e*(e*(2*b*d*g + e*(b* f + a*g)*(2*m - 3)) + c*d*(2*e*f - 3*d*g*(2*m - 1)))*x + 2*e^2*(c*e*f - 3*c *d*g + b*e*g)*(m - 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IntegerQ[2*m] && GeQ[m, 2]
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, S imp[f*(d + e*x)^(m + q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(c*e^q*(m + q + 2*p + 1)) Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q - 1) - c *d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[ q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && Pol yQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && !(IGt Q[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Result contains complex when optimal does not.
Time = 2.87 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.14
| method | result | size |
| risch | \(\frac {2 \left (-148+15 x \right ) \sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}}{375}+\frac {\left (\frac {3772 \left (\frac {34}{5}-\frac {i \sqrt {14}}{5}\right ) \sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}-\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}+\frac {i \sqrt {14}}{5}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}, \sqrt {\frac {-\frac {34}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\right )}{375 \sqrt {5 x^{3}+37 x^{2}+17 x +21}}+\frac {20374 \left (\frac {34}{5}-\frac {i \sqrt {14}}{5}\right ) \sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}-\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}+\frac {i \sqrt {14}}{5}}}\, \left (\left (-\frac {34}{5}-\frac {i \sqrt {14}}{5}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}, \sqrt {\frac {-\frac {34}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\right )+\left (-\frac {1}{5}+\frac {i \sqrt {14}}{5}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}, \sqrt {\frac {-\frac {34}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\right )\right )}{375 \sqrt {5 x^{3}+37 x^{2}+17 x +21}}\right ) \sqrt {\left (x +7\right ) \left (5 x^{2}+2 x +3\right )}}{\sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}}\) | \(368\) |
| default | \(\frac {2 \sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}\, \left (69423 i \sqrt {14}\, \sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}\, \sqrt {\frac {i \sqrt {14}-5 x -1}{i \sqrt {14}+34}}\, \sqrt {\frac {i \sqrt {14}+5 x +1}{-34+i \sqrt {14}}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}, \sqrt {-\frac {-34+i \sqrt {14}}{i \sqrt {14}+34}}\right )+23376 \sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}\, \sqrt {\frac {i \sqrt {14}-5 x -1}{i \sqrt {14}+34}}\, \sqrt {\frac {i \sqrt {14}+5 x +1}{-34+i \sqrt {14}}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}, \sqrt {-\frac {-34+i \sqrt {14}}{i \sqrt {14}+34}}\right )-2383758 \sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}\, \sqrt {\frac {i \sqrt {14}-5 x -1}{i \sqrt {14}+34}}\, \sqrt {\frac {i \sqrt {14}+5 x +1}{-34+i \sqrt {14}}}\, \operatorname {EllipticE}\left (\sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}, \sqrt {-\frac {-34+i \sqrt {14}}{i \sqrt {14}+34}}\right )+375 x^{4}-925 x^{3}-26105 x^{2}-11005 x -15540\right )}{1875 \left (5 x^{3}+37 x^{2}+17 x +21\right )}\) | \(382\) |
| elliptic | \(\frac {\sqrt {\left (x +7\right ) \left (5 x^{2}+2 x +3\right )}\, \left (\frac {2 x \sqrt {5 x^{3}+37 x^{2}+17 x +21}}{25}-\frac {296 \sqrt {5 x^{3}+37 x^{2}+17 x +21}}{375}+\frac {3772 \left (\frac {34}{5}-\frac {i \sqrt {14}}{5}\right ) \sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}-\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}+\frac {i \sqrt {14}}{5}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}, \sqrt {\frac {-\frac {34}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\right )}{375 \sqrt {5 x^{3}+37 x^{2}+17 x +21}}+\frac {20374 \left (\frac {34}{5}-\frac {i \sqrt {14}}{5}\right ) \sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}-\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}+\frac {i \sqrt {14}}{5}}}\, \left (\left (-\frac {34}{5}-\frac {i \sqrt {14}}{5}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}, \sqrt {\frac {-\frac {34}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\right )+\left (-\frac {1}{5}+\frac {i \sqrt {14}}{5}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}, \sqrt {\frac {-\frac {34}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\right )\right )}{375 \sqrt {5 x^{3}+37 x^{2}+17 x +21}}\right )}{\sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}}\) | \(382\) |
Input:
int(x^3/(x+7)^(1/2)/(5*x^2+2*x+3)^(1/2),x,method=_RETURNVERBOSE)
Output:
2/375*(-148+15*x)*(x+7)^(1/2)*(5*x^2+2*x+3)^(1/2)+(3772/375*(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))+20374/375*(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))*Ellipti cE(((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))))*((x +7)*(5*x^2+2*x+3))^(1/2)/(x+7)^(1/2)/(5*x^2+2*x+3)^(1/2)
Time = 0.08 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.15 \[ \int \frac {x^3}{\sqrt {7+x} \sqrt {3+2 x+5 x^2}} \, dx=\frac {2}{375} \, \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (15 \, x - 148\right )} \sqrt {x + 7} - \frac {697258}{28125} \, \sqrt {5} {\rm weierstrassPInverse}\left (\frac {4456}{75}, -\frac {348704}{3375}, x + \frac {37}{15}\right ) - \frac {20374}{1875} \, \sqrt {5} {\rm weierstrassZeta}\left (\frac {4456}{75}, -\frac {348704}{3375}, {\rm weierstrassPInverse}\left (\frac {4456}{75}, -\frac {348704}{3375}, x + \frac {37}{15}\right )\right ) \] Input:
integrate(x^3/(7+x)^(1/2)/(5*x^2+2*x+3)^(1/2),x, algorithm="fricas")
Output:
2/375*sqrt(5*x^2 + 2*x + 3)*(15*x - 148)*sqrt(x + 7) - 697258/28125*sqrt(5 )*weierstrassPInverse(4456/75, -348704/3375, x + 37/15) - 20374/1875*sqrt( 5)*weierstrassZeta(4456/75, -348704/3375, weierstrassPInverse(4456/75, -34 8704/3375, x + 37/15))
\[ \int \frac {x^3}{\sqrt {7+x} \sqrt {3+2 x+5 x^2}} \, dx=\int \frac {x^{3}}{\sqrt {x + 7} \sqrt {5 x^{2} + 2 x + 3}}\, dx \] Input:
integrate(x**3/(7+x)**(1/2)/(5*x**2+2*x+3)**(1/2),x)
Output:
Integral(x**3/(sqrt(x + 7)*sqrt(5*x**2 + 2*x + 3)), x)
\[ \int \frac {x^3}{\sqrt {7+x} \sqrt {3+2 x+5 x^2}} \, dx=\int { \frac {x^{3}}{\sqrt {5 \, x^{2} + 2 \, x + 3} \sqrt {x + 7}} \,d x } \] Input:
integrate(x^3/(7+x)^(1/2)/(5*x^2+2*x+3)^(1/2),x, algorithm="maxima")
Output:
integrate(x^3/(sqrt(5*x^2 + 2*x + 3)*sqrt(x + 7)), x)
\[ \int \frac {x^3}{\sqrt {7+x} \sqrt {3+2 x+5 x^2}} \, dx=\int { \frac {x^{3}}{\sqrt {5 \, x^{2} + 2 \, x + 3} \sqrt {x + 7}} \,d x } \] Input:
integrate(x^3/(7+x)^(1/2)/(5*x^2+2*x+3)^(1/2),x, algorithm="giac")
Output:
integrate(x^3/(sqrt(5*x^2 + 2*x + 3)*sqrt(x + 7)), x)
Timed out. \[ \int \frac {x^3}{\sqrt {7+x} \sqrt {3+2 x+5 x^2}} \, dx=\int \frac {x^3}{\sqrt {x+7}\,\sqrt {5\,x^2+2\,x+3}} \,d x \] Input:
int(x^3/((x + 7)^(1/2)*(2*x + 5*x^2 + 3)^(1/2)),x)
Output:
int(x^3/((x + 7)^(1/2)*(2*x + 5*x^2 + 3)^(1/2)), x)
\[ \int \frac {x^3}{\sqrt {7+x} \sqrt {3+2 x+5 x^2}} \, dx=\frac {2 \sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}\, x}{25}-\frac {51 \sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}}{925}-\frac {10187 \left (\int \frac {\sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}\, x^{2}}{5 x^{3}+37 x^{2}+17 x +21}d x \right )}{1850}-\frac {2241 \left (\int \frac {\sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}}{5 x^{3}+37 x^{2}+17 x +21}d x \right )}{1850} \] Input:
int(x^3/(7+x)^(1/2)/(5*x^2+2*x+3)^(1/2),x)
Output:
(148*sqrt(x + 7)*sqrt(5*x**2 + 2*x + 3)*x - 102*sqrt(x + 7)*sqrt(5*x**2 + 2*x + 3) - 10187*int((sqrt(x + 7)*sqrt(5*x**2 + 2*x + 3)*x**2)/(5*x**3 + 3 7*x**2 + 17*x + 21),x) - 2241*int((sqrt(x + 7)*sqrt(5*x**2 + 2*x + 3))/(5* x**3 + 37*x**2 + 17*x + 21),x))/1850