Integrand size = 25, antiderivative size = 343 \[ \int \frac {x^3 \sqrt {7+x}}{\sqrt {3+2 x+5 x^2}} \, dx=\frac {6954 \sqrt {7+x} \sqrt {3+2 x+5 x^2}}{4375}-\frac {654}{875} (7+x)^{3/2} \sqrt {3+2 x+5 x^2}+\frac {2}{35} (7+x)^{5/2} \sqrt {3+2 x+5 x^2}+\frac {240034 \sqrt {7+x} \sqrt {3+2 x+5 x^2}}{4375 \left (3 \sqrt {130}+5 (7+x)\right )}-\frac {240034 \sqrt {3} \sqrt [4]{26} \sqrt {\frac {3+2 x+5 x^2}{\left (78+\sqrt {130} (7+x)\right )^2}} \left (78+\sqrt {130} (7+x)\right ) E\left (2 \arctan \left (\frac {\sqrt [4]{\frac {5}{26}} \sqrt {7+x}}{\sqrt {3}}\right )|\frac {1}{390} \left (195+17 \sqrt {130}\right )\right )}{4375\ 5^{3/4} \sqrt {3+2 x+5 x^2}}+\frac {\sqrt {3} \sqrt [4]{26} \left (120017-10431 \sqrt {130}\right ) \sqrt {\frac {3+2 x+5 x^2}{\left (78+\sqrt {130} (7+x)\right )^2}} \left (78+\sqrt {130} (7+x)\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{\frac {5}{26}} \sqrt {7+x}}{\sqrt {3}}\right ),\frac {1}{390} \left (195+17 \sqrt {130}\right )\right )}{4375\ 5^{3/4} \sqrt {3+2 x+5 x^2}} \] Output:
6954/4375*(7+x)^(1/2)*(5*x^2+2*x+3)^(1/2)-654/875*(7+x)^(3/2)*(5*x^2+2*x+3 )^(1/2)+2/35*(7+x)^(5/2)*(5*x^2+2*x+3)^(1/2)+240034*(7+x)^(1/2)*(5*x^2+2*x +3)^(1/2)/(13125*130^(1/2)+153125+21875*x)-240034/21875*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+663 0*130^(1/2))^(1/2))*5^(1/4)/(5*x^2+2*x+3)^(1/2)+1/21875*3^(1/2)*26^(1/4)*( 120017-10431*130^(1/2))*((5*x^2+2*x+3)/(78+130^(1/2)*(7+x))^2)^(1/2)*(78+1 30^(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^(1/4)/(5*x^2+2*x+3)^(1/2)
Result contains complex when optimal does not.
Time = 34.25 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.25 \[ \int \frac {x^3 \sqrt {7+x}}{\sqrt {3+2 x+5 x^2}} \, dx=\frac {2 \sqrt {7+x} \sqrt {3+2 x+5 x^2} \left (-1843+115 x+125 x^2\right )}{4375}-\frac {2 (7+x)^{3/2} \left (-\frac {4680663 \sqrt {-\frac {i}{34 i+\sqrt {14}}} \left (3+2 x+5 x^2\right )}{(7+x)^2}+\frac {120017 i \sqrt {13} \left (17 \sqrt {2}+i \sqrt {7}\right ) \sqrt {\frac {34 i+\sqrt {14}-\frac {234 i}{7+x}}{34 i+\sqrt {14}}} \sqrt {\frac {-34 i+\sqrt {14}+\frac {234 i}{7+x}}{-34 i+\sqrt {14}}} E\left (i \text {arcsinh}\left (\frac {3 \sqrt {-\frac {26 i}{34 i+\sqrt {14}}}}{\sqrt {7+x}}\right )|\frac {34 i+\sqrt {14}}{34 i-\sqrt {14}}\right )}{\sqrt {7+x}}+\frac {\sqrt {13} \left (-6244 i \sqrt {2}+120017 \sqrt {7}\right ) \sqrt {\frac {34 i+\sqrt {14}-\frac {234 i}{7+x}}{34 i+\sqrt {14}}} \sqrt {\frac {-34 i+\sqrt {14}+\frac {234 i}{7+x}}{-34 i+\sqrt {14}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {3 \sqrt {-\frac {26 i}{34 i+\sqrt {14}}}}{\sqrt {7+x}}\right ),\frac {34 i+\sqrt {14}}{34 i-\sqrt {14}}\right )}{\sqrt {7+x}}\right )}{853125 \sqrt {-\frac {i}{34 i+\sqrt {14}}} \sqrt {3+2 x+5 x^2}} \] Input:
Integrate[(x^3*Sqrt[7 + x])/Sqrt[3 + 2*x + 5*x^2],x]
Output:
(2*Sqrt[7 + x]*Sqrt[3 + 2*x + 5*x^2]*(-1843 + 115*x + 125*x^2))/4375 - (2* (7 + x)^(3/2)*((-4680663*Sqrt[(-I)/(34*I + Sqrt[14])]*(3 + 2*x + 5*x^2))/( 7 + x)^2 + ((120017*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]*((-6244*I)*Sqrt[2] + 120017*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]))/ (853125*Sqrt[(-I)/(34*I + Sqrt[14])]*Sqrt[3 + 2*x + 5*x^2])
Result contains complex when optimal does not.
Time = 0.57 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.78, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {1283, 2184, 27, 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.
| \(\displaystyle \int \frac {x^3 \sqrt {x+7}}{\sqrt {5 x^2+2 x+3}} \, dx\) |
| \(\Big \downarrow \) 1283 |
| \(\displaystyle \frac {2}{35} x^2 \sqrt {x+7} \sqrt {5 x^2+2 x+3}-\frac {1}{35} \int \frac {x \left (-23 x^2+85 x+84\right )}{\sqrt {x+7} \sqrt {5 x^2+2 x+3}}dx\) |
| \(\Big \downarrow \) 2184 |
| \(\displaystyle \frac {1}{35} \left (\frac {46}{25} (x+7)^{3/2} \sqrt {5 x^2+2 x+3}-\frac {2}{25} \int \frac {7944 x^2+15187 x+3703}{2 \sqrt {x+7} \sqrt {5 x^2+2 x+3}}dx\right )+\frac {2}{35} \sqrt {x+7} \sqrt {5 x^2+2 x+3} x^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{35} \left (\frac {46}{25} (x+7)^{3/2} \sqrt {5 x^2+2 x+3}-\frac {1}{25} \int \frac {7944 x^2+15187 x+3703}{\sqrt {x+7} \sqrt {5 x^2+2 x+3}}dx\right )+\frac {2}{35} \sqrt {x+7} \sqrt {5 x^2+2 x+3} x^2\) |
| \(\Big \downarrow \) 2184 |
| \(\displaystyle \frac {1}{35} \left (\frac {1}{25} \left (-\frac {2}{15} \int -\frac {3 (120017 x+26501)}{2 \sqrt {x+7} \sqrt {5 x^2+2 x+3}}dx-\frac {5296}{5} \sqrt {x+7} \sqrt {5 x^2+2 x+3}\right )+\frac {46}{25} \sqrt {5 x^2+2 x+3} (x+7)^{3/2}\right )+\frac {2}{35} \sqrt {x+7} \sqrt {5 x^2+2 x+3} x^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{35} \left (\frac {1}{25} \left (\frac {1}{5} \int \frac {120017 x+26501}{\sqrt {x+7} \sqrt {5 x^2+2 x+3}}dx-\frac {5296}{5} \sqrt {x+7} \sqrt {5 x^2+2 x+3}\right )+\frac {46}{25} \sqrt {5 x^2+2 x+3} (x+7)^{3/2}\right )+\frac {2}{35} \sqrt {x+7} \sqrt {5 x^2+2 x+3} x^2\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {1}{35} \left (\frac {1}{25} \left (\frac {1}{5} \left (120017 \int \frac {\sqrt {x+7}}{\sqrt {5 x^2+2 x+3}}dx-813618 \int \frac {1}{\sqrt {x+7} \sqrt {5 x^2+2 x+3}}dx\right )-\frac {5296}{5} \sqrt {x+7} \sqrt {5 x^2+2 x+3}\right )+\frac {46}{25} \sqrt {5 x^2+2 x+3} (x+7)^{3/2}\right )+\frac {2}{35} \sqrt {x+7} \sqrt {5 x^2+2 x+3} x^2\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {2}{35} \sqrt {x+7} \sqrt {5 x^2+2 x+3} x^2+\frac {1}{35} \left (\frac {46}{25} \sqrt {5 x^2+2 x+3} (x+7)^{3/2}+\frac {1}{25} \left (-\frac {5296}{5} \sqrt {x+7} \sqrt {5 x^2+2 x+3}+\frac {1}{5} \left (\frac {240034 i \sqrt {x+7} \int \frac {\sqrt {\frac {i \left (5 x+i \sqrt {14}+1\right )}{34 i+\sqrt {14}}+1}}{\sqrt {\frac {i \left (5 x+i \sqrt {14}+1\right )}{2 \sqrt {14}}+1}}d\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}}{5 \sqrt {\frac {x+7}{34-i \sqrt {14}}}}-\frac {1627236 i \sqrt {\frac {x+7}{34-i \sqrt {14}}} \int \frac {1}{\sqrt {\frac {i \left (5 x+i \sqrt {14}+1\right )}{2 \sqrt {14}}+1} \sqrt {\frac {i \left (5 x+i \sqrt {14}+1\right )}{34 i+\sqrt {14}}+1}}d\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}}{\sqrt {x+7}}\right )\right )\right )\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {2}{35} \sqrt {x+7} \sqrt {5 x^2+2 x+3} x^2+\frac {1}{35} \left (\frac {46}{25} \sqrt {5 x^2+2 x+3} (x+7)^{3/2}+\frac {1}{25} \left (-\frac {5296}{5} \sqrt {x+7} \sqrt {5 x^2+2 x+3}+\frac {1}{5} \left (\frac {240034 i \sqrt {x+7} \int \frac {\sqrt {\frac {i \left (5 x+i \sqrt {14}+1\right )}{34 i+\sqrt {14}}+1}}{\sqrt {\frac {i \left (5 x+i \sqrt {14}+1\right )}{2 \sqrt {14}}+1}}d\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}}{5 \sqrt {\frac {x+7}{34-i \sqrt {14}}}}-\frac {1627236 i \sqrt {\frac {x+7}{34-i \sqrt {14}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}\right ),\frac {2 \sqrt {14}}{34 i+\sqrt {14}}\right )}{\sqrt {x+7}}\right )\right )\right )\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {2}{35} \sqrt {x+7} \sqrt {5 x^2+2 x+3} x^2+\frac {1}{35} \left (\frac {46}{25} \sqrt {5 x^2+2 x+3} (x+7)^{3/2}+\frac {1}{25} \left (-\frac {5296}{5} \sqrt {x+7} \sqrt {5 x^2+2 x+3}+\frac {1}{5} \left (\frac {240034 i \sqrt {x+7} E\left (\arcsin \left (\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}\right )|\frac {2 \sqrt {14}}{34 i+\sqrt {14}}\right )}{5 \sqrt {\frac {x+7}{34-i \sqrt {14}}}}-\frac {1627236 i \sqrt {\frac {x+7}{34-i \sqrt {14}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}\right ),\frac {2 \sqrt {14}}{34 i+\sqrt {14}}\right )}{\sqrt {x+7}}\right )\right )\right )\) |
Input:
Int[(x^3*Sqrt[7 + x])/Sqrt[3 + 2*x + 5*x^2],x]
Output:
(2*x^2*Sqrt[7 + x]*Sqrt[3 + 2*x + 5*x^2])/35 + ((46*(7 + x)^(3/2)*Sqrt[3 + 2*x + 5*x^2])/25 + ((-5296*Sqrt[7 + x]*Sqrt[3 + 2*x + 5*x^2])/5 + ((((240 034*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*S qrt[14])] - ((1627236*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])/5)/25)/35
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*(d + e*x)^(m - 1)*Sqrt[f + g*x ]*(Sqrt[a + b*x + c*x^2]/(c*(2*m + 1))), x] - Simp[1/(c*(2*m + 1)) Int[(( d + e*x)^(m - 2)/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]))*Simp[e*(b*d*f + a*( d*g + 2*e*f*(m - 1))) - c*d^2*f*(2*m + 1) + (a*e^2*g*(2*m - 1) - c*d*(4*e*f *m + d*g*(2*m + 1)) + b*e*(2*d*g + e*f*(2*m - 1)))*x + e*(2*b*e*g*m - c*(e* f + d*g*(4*m - 1)))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && I ntegerQ[2*m] && GtQ[m, 1]
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 = 4.68 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.09
| method | result | size |
| risch | \(\frac {2 \left (125 x^{2}+115 x -1843\right ) \sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}}{4375}+\frac {\left (\frac {53002 \left (\frac {34}{5}-\frac {i \sqrt {14}}{5}\right ) \sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}-\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}+\frac {i \sqrt {14}}{5}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}, \sqrt {\frac {-\frac {34}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\right )}{4375 \sqrt {5 x^{3}+37 x^{2}+17 x +21}}+\frac {240034 \left (\frac {34}{5}-\frac {i \sqrt {14}}{5}\right ) \sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}-\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}+\frac {i \sqrt {14}}{5}}}\, \left (\left (-\frac {34}{5}-\frac {i \sqrt {14}}{5}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}, \sqrt {\frac {-\frac {34}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\right )+\left (-\frac {1}{5}+\frac {i \sqrt {14}}{5}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}, \sqrt {\frac {-\frac {34}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\right )\right )}{4375 \sqrt {5 x^{3}+37 x^{2}+17 x +21}}\right ) \sqrt {\left (x +7\right ) \left (5 x^{2}+2 x +3\right )}}{\sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}}\) | \(373\) |
| default | \(\frac {2 \sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}\, \left (813618 i \sqrt {14}\, \sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}\, \sqrt {\frac {i \sqrt {14}-5 x -1}{i \sqrt {14}+34}}\, \sqrt {\frac {i \sqrt {14}+5 x +1}{-34+i \sqrt {14}}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}, \sqrt {-\frac {-34+i \sqrt {14}}{i \sqrt {14}+34}}\right )+3125 x^{5}+420966 \sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}\, \sqrt {\frac {i \sqrt {14}-5 x -1}{i \sqrt {14}+34}}\, \sqrt {\frac {i \sqrt {14}+5 x +1}{-34+i \sqrt {14}}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}, \sqrt {-\frac {-34+i \sqrt {14}}{i \sqrt {14}+34}}\right )-28083978 \sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}\, \sqrt {\frac {i \sqrt {14}-5 x -1}{i \sqrt {14}+34}}\, \sqrt {\frac {i \sqrt {14}+5 x +1}{-34+i \sqrt {14}}}\, \operatorname {EllipticE}\left (\sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}, \sqrt {-\frac {-34+i \sqrt {14}}{i \sqrt {14}+34}}\right )+26000 x^{4}-14175 x^{3}-318055 x^{2}-144580 x -193515\right )}{21875 \left (5 x^{3}+37 x^{2}+17 x +21\right )}\) | \(387\) |
| elliptic | \(\frac {\sqrt {\left (x +7\right ) \left (5 x^{2}+2 x +3\right )}\, \left (\frac {46 x \sqrt {5 x^{3}+37 x^{2}+17 x +21}}{875}-\frac {3686 \sqrt {5 x^{3}+37 x^{2}+17 x +21}}{4375}+\frac {53002 \left (\frac {34}{5}-\frac {i \sqrt {14}}{5}\right ) \sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}-\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}+\frac {i \sqrt {14}}{5}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}, \sqrt {\frac {-\frac {34}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\right )}{4375 \sqrt {5 x^{3}+37 x^{2}+17 x +21}}+\frac {240034 \left (\frac {34}{5}-\frac {i \sqrt {14}}{5}\right ) \sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}-\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}+\frac {i \sqrt {14}}{5}}}\, \left (\left (-\frac {34}{5}-\frac {i \sqrt {14}}{5}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}, \sqrt {\frac {-\frac {34}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\right )+\left (-\frac {1}{5}+\frac {i \sqrt {14}}{5}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}, \sqrt {\frac {-\frac {34}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\right )\right )}{4375 \sqrt {5 x^{3}+37 x^{2}+17 x +21}}+\frac {2 x^{2} \sqrt {5 x^{3}+37 x^{2}+17 x +21}}{35}\right )}{\sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}}\) | \(404\) |
Input:
int(x^3*(x+7)^(1/2)/(5*x^2+2*x+3)^(1/2),x,method=_RETURNVERBOSE)
Output:
2/4375*(125*x^2+115*x-1843)*(x+7)^(1/2)*(5*x^2+2*x+3)^(1/2)+(53002/4375*(3 4/5-1/5*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*1 4^(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))+24 0034/4375*(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))))*((x+7)*(5*x^2+2*x+3))^(1/2)/(x+7)^(1/2)/(5*x^2+2*x+3)^(1/2)
Time = 0.07 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.16 \[ \int \frac {x^3 \sqrt {7+x}}{\sqrt {3+2 x+5 x^2}} \, dx=\frac {2}{4375} \, {\left (125 \, x^{2} + 115 \, x - 1843\right )} \sqrt {5 \, x^{2} + 2 \, x + 3} \sqrt {x + 7} - \frac {8086228}{328125} \, \sqrt {5} {\rm weierstrassPInverse}\left (\frac {4456}{75}, -\frac {348704}{3375}, x + \frac {37}{15}\right ) - \frac {240034}{21875} \, \sqrt {5} {\rm weierstrassZeta}\left (\frac {4456}{75}, -\frac {348704}{3375}, {\rm weierstrassPInverse}\left (\frac {4456}{75}, -\frac {348704}{3375}, x + \frac {37}{15}\right )\right ) \] Input:
integrate(x^3*(7+x)^(1/2)/(5*x^2+2*x+3)^(1/2),x, algorithm="fricas")
Output:
2/4375*(125*x^2 + 115*x - 1843)*sqrt(5*x^2 + 2*x + 3)*sqrt(x + 7) - 808622 8/328125*sqrt(5)*weierstrassPInverse(4456/75, -348704/3375, x + 37/15) - 2 40034/21875*sqrt(5)*weierstrassZeta(4456/75, -348704/3375, weierstrassPInv erse(4456/75, -348704/3375, 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)
\[ \int \frac {x^3 \sqrt {7+x}}{\sqrt {3+2 x+5 x^2}} \, dx=\int { \frac {\sqrt {x + 7} x^{3}}{\sqrt {5 \, x^{2} + 2 \, x + 3}} \,d x } \] Input:
integrate(x^3*(7+x)^(1/2)/(5*x^2+2*x+3)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(x + 7)*x^3/sqrt(5*x^2 + 2*x + 3), x)
\[ \int \frac {x^3 \sqrt {7+x}}{\sqrt {3+2 x+5 x^2}} \, dx=\int { \frac {\sqrt {x + 7} x^{3}}{\sqrt {5 \, x^{2} + 2 \, x + 3}} \,d x } \] Input:
integrate(x^3*(7+x)^(1/2)/(5*x^2+2*x+3)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(x + 7)*x^3/sqrt(5*x^2 + 2*x + 3), 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)
\[ \int \frac {x^3 \sqrt {7+x}}{\sqrt {3+2 x+5 x^2}} \, dx=\frac {2 \sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}\, x^{2}}{35}+\frac {46 \sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}\, x}{875}-\frac {3273 \sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}}{32375}-\frac {360051 \left (\int \frac {\sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}\, x^{2}}{5 x^{3}+37 x^{2}+17 x +21}d x \right )}{64750}-\frac {15843 \left (\int \frac {\sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}}{5 x^{3}+37 x^{2}+17 x +21}d x \right )}{64750} \] Input:
int(x^3*(7+x)^(1/2)/(5*x^2+2*x+3)^(1/2),x)
Output:
(3700*sqrt(x + 7)*sqrt(5*x**2 + 2*x + 3)*x**2 + 3404*sqrt(x + 7)*sqrt(5*x* *2 + 2*x + 3)*x - 6546*sqrt(x + 7)*sqrt(5*x**2 + 2*x + 3) - 360051*int((sq rt(x + 7)*sqrt(5*x**2 + 2*x + 3)*x**2)/(5*x**3 + 37*x**2 + 17*x + 21),x) - 15843*int((sqrt(x + 7)*sqrt(5*x**2 + 2*x + 3))/(5*x**3 + 37*x**2 + 17*x + 21),x))/64750