Integrand size = 25, antiderivative size = 318 \[ \int \frac {x^2 \sqrt {7+x}}{\sqrt {3+2 x+5 x^2}} \, dx=-\frac {52}{125} \sqrt {7+x} \sqrt {3+2 x+5 x^2}+\frac {2}{25} (7+x)^{3/2} \sqrt {3+2 x+5 x^2}-\frac {1842 \sqrt {7+x} \sqrt {3+2 x+5 x^2}}{125 \left (3 \sqrt {130}+5 (7+x)\right )}+\frac {1842 \sqrt {3} \sqrt [4]{26} \sqrt {\frac {3+2 x+5 x^2}{\left (78+\sqrt {130} (7+x)\right )^2}} \left (78+\sqrt {130} (7+x)\right ) E\left (2 \arctan \left (\frac {\sqrt [4]{\frac {5}{26}} \sqrt {7+x}}{\sqrt {3}}\right )|\frac {1}{390} \left (195+17 \sqrt {130}\right )\right )}{125\ 5^{3/4} \sqrt {3+2 x+5 x^2}}-\frac {3 \sqrt {3} \sqrt [4]{26} \left (307-26 \sqrt {130}\right ) \sqrt {\frac {3+2 x+5 x^2}{\left (78+\sqrt {130} (7+x)\right )^2}} \left (78+\sqrt {130} (7+x)\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{\frac {5}{26}} \sqrt {7+x}}{\sqrt {3}}\right ),\frac {1}{390} \left (195+17 \sqrt {130}\right )\right )}{125\ 5^{3/4} \sqrt {3+2 x+5 x^2}} \] Output:
-52/125*(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)-1842*(7+x)^(1/2)*(5*x^2+2*x+3)^(1/2)/(375*130^(1/2)+4375+625*x)+1842/62 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))*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)-3/625* 3^(1/2)*26^(1/4)*(307-26*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))*5^(1/4)/(5*x^2+2 *x+3)^(1/2)
Result contains complex when optimal does not.
Time = 33.95 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.34 \[ \int \frac {x^2 \sqrt {7+x}}{\sqrt {3+2 x+5 x^2}} \, dx=\frac {2}{125} \sqrt {7+x} (9+5 x) \sqrt {3+2 x+5 x^2}+\frac {2 (7+x)^{3/2} \left (-\frac {11973 \sqrt {-\frac {i}{34 i+\sqrt {14}}} \left (3+2 x+5 x^2\right )}{(7+x)^2}+\frac {307 i \sqrt {13} \left (17 \sqrt {2}+i \sqrt {7}\right ) \sqrt {\frac {34 i+\sqrt {14}-\frac {234 i}{7+x}}{34 i+\sqrt {14}}} \sqrt {\frac {-34 i+\sqrt {14}+\frac {234 i}{7+x}}{-34 i+\sqrt {14}}} E\left (i \text {arcsinh}\left (\frac {3 \sqrt {-\frac {26 i}{34 i+\sqrt {14}}}}{\sqrt {7+x}}\right )|\frac {34 i+\sqrt {14}}{34 i-\sqrt {14}}\right )}{\sqrt {7+x}}+\frac {\sqrt {13} \left (-149 i \sqrt {2}+307 \sqrt {7}\right ) \sqrt {\frac {34 i+\sqrt {14}-\frac {234 i}{7+x}}{34 i+\sqrt {14}}} \sqrt {\frac {-34 i+\sqrt {14}+\frac {234 i}{7+x}}{-34 i+\sqrt {14}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {3 \sqrt {-\frac {26 i}{34 i+\sqrt {14}}}}{\sqrt {7+x}}\right ),\frac {34 i+\sqrt {14}}{34 i-\sqrt {14}}\right )}{\sqrt {7+x}}\right )}{8125 \sqrt {-\frac {i}{34 i+\sqrt {14}}} \sqrt {3+2 x+5 x^2}} \] Input:
Integrate[(x^2*Sqrt[7 + x])/Sqrt[3 + 2*x + 5*x^2],x]
Output:
(2*Sqrt[7 + x]*(9 + 5*x)*Sqrt[3 + 2*x + 5*x^2])/125 + (2*(7 + x)^(3/2)*((- 11973*Sqrt[(-I)/(34*I + Sqrt[14])]*(3 + 2*x + 5*x^2))/(7 + x)^2 + ((307*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 + Sq rt[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]*((-149* I)*Sqrt[2] + 307*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]))/(8125*Sqrt[(-I)/(34*I + Sq rt[14])]*Sqrt[3 + 2*x + 5*x^2])
Result contains complex when optimal does not.
Time = 0.46 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.74, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {1283, 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.
| \(\displaystyle \int \frac {x^2 \sqrt {x+7}}{\sqrt {5 x^2+2 x+3}} \, dx\) |
| \(\Big \downarrow \) 1283 |
| \(\displaystyle \frac {2}{25} x \sqrt {x+7} \sqrt {5 x^2+2 x+3}-\frac {1}{25} \int \frac {3 \left (-9 x^2+17 x+14\right )}{\sqrt {x+7} \sqrt {5 x^2+2 x+3}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{25} x \sqrt {x+7} \sqrt {5 x^2+2 x+3}-\frac {3}{25} \int \frac {-9 x^2+17 x+14}{\sqrt {x+7} \sqrt {5 x^2+2 x+3}}dx\) |
| \(\Big \downarrow \) 2184 |
| \(\displaystyle \frac {2}{25} x \sqrt {x+7} \sqrt {5 x^2+2 x+3}-\frac {3}{25} \left (\frac {2}{15} \int \frac {3 (307 x+121)}{2 \sqrt {x+7} \sqrt {5 x^2+2 x+3}}dx-\frac {6}{5} \sqrt {x+7} \sqrt {5 x^2+2 x+3}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{25} x \sqrt {x+7} \sqrt {5 x^2+2 x+3}-\frac {3}{25} \left (\frac {1}{5} \int \frac {307 x+121}{\sqrt {x+7} \sqrt {5 x^2+2 x+3}}dx-\frac {6}{5} \sqrt {x+7} \sqrt {5 x^2+2 x+3}\right )\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {2}{25} x \sqrt {x+7} \sqrt {5 x^2+2 x+3}-\frac {3}{25} \left (\frac {1}{5} \left (307 \int \frac {\sqrt {x+7}}{\sqrt {5 x^2+2 x+3}}dx-2028 \int \frac {1}{\sqrt {x+7} \sqrt {5 x^2+2 x+3}}dx\right )-\frac {6}{5} \sqrt {x+7} \sqrt {5 x^2+2 x+3}\right )\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {2}{25} x \sqrt {x+7} \sqrt {5 x^2+2 x+3}-\frac {3}{25} \left (-\frac {6}{5} \sqrt {x+7} \sqrt {5 x^2+2 x+3}+\frac {1}{5} \left (\frac {614 i \sqrt {x+7} \int \frac {\sqrt {\frac {i \left (5 x+i \sqrt {14}+1\right )}{34 i+\sqrt {14}}+1}}{\sqrt {\frac {i \left (5 x+i \sqrt {14}+1\right )}{2 \sqrt {14}}+1}}d\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}}{5 \sqrt {\frac {x+7}{34-i \sqrt {14}}}}-\frac {4056 i \sqrt {\frac {x+7}{34-i \sqrt {14}}} \int \frac {1}{\sqrt {\frac {i \left (5 x+i \sqrt {14}+1\right )}{2 \sqrt {14}}+1} \sqrt {\frac {i \left (5 x+i \sqrt {14}+1\right )}{34 i+\sqrt {14}}+1}}d\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}}{\sqrt {x+7}}\right )\right )\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {2}{25} x \sqrt {x+7} \sqrt {5 x^2+2 x+3}-\frac {3}{25} \left (-\frac {6}{5} \sqrt {x+7} \sqrt {5 x^2+2 x+3}+\frac {1}{5} \left (\frac {614 i \sqrt {x+7} \int \frac {\sqrt {\frac {i \left (5 x+i \sqrt {14}+1\right )}{34 i+\sqrt {14}}+1}}{\sqrt {\frac {i \left (5 x+i \sqrt {14}+1\right )}{2 \sqrt {14}}+1}}d\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}}{5 \sqrt {\frac {x+7}{34-i \sqrt {14}}}}-\frac {4056 i \sqrt {\frac {x+7}{34-i \sqrt {14}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}\right ),\frac {2 \sqrt {14}}{34 i+\sqrt {14}}\right )}{\sqrt {x+7}}\right )\right )\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {2}{25} x \sqrt {x+7} \sqrt {5 x^2+2 x+3}-\frac {3}{25} \left (-\frac {6}{5} \sqrt {x+7} \sqrt {5 x^2+2 x+3}+\frac {1}{5} \left (\frac {614 i \sqrt {x+7} E\left (\arcsin \left (\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}\right )|\frac {2 \sqrt {14}}{34 i+\sqrt {14}}\right )}{5 \sqrt {\frac {x+7}{34-i \sqrt {14}}}}-\frac {4056 i \sqrt {\frac {x+7}{34-i \sqrt {14}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}\right ),\frac {2 \sqrt {14}}{34 i+\sqrt {14}}\right )}{\sqrt {x+7}}\right )\right )\) |
Input:
Int[(x^2*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 - (3*((-6*Sqrt[7 + x]*Sqrt[3 + 2*x + 5*x^2])/5 + ((((614*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])] - ((4056*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
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 = 2.67 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.16
| method | result | size |
| risch | \(\frac {2 \left (9+5 x \right ) \sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}}{125}+\frac {\left (-\frac {726 \left (\frac {34}{5}-\frac {i \sqrt {14}}{5}\right ) \sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}-\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}+\frac {i \sqrt {14}}{5}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}, \sqrt {\frac {-\frac {34}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\right )}{125 \sqrt {5 x^{3}+37 x^{2}+17 x +21}}-\frac {1842 \left (\frac {34}{5}-\frac {i \sqrt {14}}{5}\right ) \sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}-\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}+\frac {i \sqrt {14}}{5}}}\, \left (\left (-\frac {34}{5}-\frac {i \sqrt {14}}{5}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}, \sqrt {\frac {-\frac {34}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\right )+\left (-\frac {1}{5}+\frac {i \sqrt {14}}{5}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}, \sqrt {\frac {-\frac {34}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\right )\right )}{125 \sqrt {5 x^{3}+37 x^{2}+17 x +21}}\right ) \sqrt {\left (x +7\right ) \left (5 x^{2}+2 x +3\right )}}{\sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}}\) | \(368\) |
| default | \(-\frac {2 \sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}\, \left (6084 i \sqrt {14}\, \sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}\, \sqrt {\frac {i \sqrt {14}-5 x -1}{i \sqrt {14}+34}}\, \sqrt {\frac {i \sqrt {14}+5 x +1}{-34+i \sqrt {14}}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}, \sqrt {-\frac {-34+i \sqrt {14}}{i \sqrt {14}+34}}\right )+8658 \sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}\, \sqrt {\frac {i \sqrt {14}-5 x -1}{i \sqrt {14}+34}}\, \sqrt {\frac {i \sqrt {14}+5 x +1}{-34+i \sqrt {14}}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}, \sqrt {-\frac {-34+i \sqrt {14}}{i \sqrt {14}+34}}\right )-215514 \sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}\, \sqrt {\frac {i \sqrt {14}-5 x -1}{i \sqrt {14}+34}}\, \sqrt {\frac {i \sqrt {14}+5 x +1}{-34+i \sqrt {14}}}\, \operatorname {EllipticE}\left (\sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}, \sqrt {-\frac {-34+i \sqrt {14}}{i \sqrt {14}+34}}\right )-125 x^{4}-1150 x^{3}-2090 x^{2}-1290 x -945\right )}{625 \left (5 x^{3}+37 x^{2}+17 x +21\right )}\) | \(382\) |
| elliptic | \(\frac {\sqrt {\left (x +7\right ) \left (5 x^{2}+2 x +3\right )}\, \left (\frac {2 x \sqrt {5 x^{3}+37 x^{2}+17 x +21}}{25}+\frac {18 \sqrt {5 x^{3}+37 x^{2}+17 x +21}}{125}-\frac {726 \left (\frac {34}{5}-\frac {i \sqrt {14}}{5}\right ) \sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}-\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}+\frac {i \sqrt {14}}{5}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}, \sqrt {\frac {-\frac {34}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\right )}{125 \sqrt {5 x^{3}+37 x^{2}+17 x +21}}-\frac {1842 \left (\frac {34}{5}-\frac {i \sqrt {14}}{5}\right ) \sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}-\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}+\frac {i \sqrt {14}}{5}}}\, \left (\left (-\frac {34}{5}-\frac {i \sqrt {14}}{5}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}, \sqrt {\frac {-\frac {34}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\right )+\left (-\frac {1}{5}+\frac {i \sqrt {14}}{5}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}, \sqrt {\frac {-\frac {34}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\right )\right )}{125 \sqrt {5 x^{3}+37 x^{2}+17 x +21}}\right )}{\sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}}\) | \(382\) |
Input:
int(x^2*(x+7)^(1/2)/(5*x^2+2*x+3)^(1/2),x,method=_RETURNVERBOSE)
Output:
2/125*(9+5*x)*(x+7)^(1/2)*(5*x^2+2*x+3)^(1/2)+(-726/125*(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))-1842/125*(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 = 0.08 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.16 \[ \int \frac {x^2 \sqrt {7+x}}{\sqrt {3+2 x+5 x^2}} \, dx=\frac {2}{125} \, \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (5 \, x + 9\right )} \sqrt {x + 7} + \frac {19088}{3125} \, \sqrt {5} {\rm weierstrassPInverse}\left (\frac {4456}{75}, -\frac {348704}{3375}, x + \frac {37}{15}\right ) + \frac {1842}{625} \, \sqrt {5} {\rm weierstrassZeta}\left (\frac {4456}{75}, -\frac {348704}{3375}, {\rm weierstrassPInverse}\left (\frac {4456}{75}, -\frac {348704}{3375}, x + \frac {37}{15}\right )\right ) \] Input:
integrate(x^2*(7+x)^(1/2)/(5*x^2+2*x+3)^(1/2),x, algorithm="fricas")
Output:
2/125*sqrt(5*x^2 + 2*x + 3)*(5*x + 9)*sqrt(x + 7) + 19088/3125*sqrt(5)*wei erstrassPInverse(4456/75, -348704/3375, x + 37/15) + 1842/625*sqrt(5)*weie rstrassZeta(4456/75, -348704/3375, weierstrassPInverse(4456/75, -348704/33 75, x + 37/15))
\[ \int \frac {x^2 \sqrt {7+x}}{\sqrt {3+2 x+5 x^2}} \, dx=\int \frac {x^{2} \sqrt {x + 7}}{\sqrt {5 x^{2} + 2 x + 3}}\, dx \] Input:
integrate(x**2*(7+x)**(1/2)/(5*x**2+2*x+3)**(1/2),x)
Output:
Integral(x**2*sqrt(x + 7)/sqrt(5*x**2 + 2*x + 3), x)
\[ \int \frac {x^2 \sqrt {7+x}}{\sqrt {3+2 x+5 x^2}} \, dx=\int { \frac {\sqrt {x + 7} x^{2}}{\sqrt {5 \, x^{2} + 2 \, x + 3}} \,d x } \] Input:
integrate(x^2*(7+x)^(1/2)/(5*x^2+2*x+3)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(x + 7)*x^2/sqrt(5*x^2 + 2*x + 3), x)
\[ \int \frac {x^2 \sqrt {7+x}}{\sqrt {3+2 x+5 x^2}} \, dx=\int { \frac {\sqrt {x + 7} x^{2}}{\sqrt {5 \, x^{2} + 2 \, x + 3}} \,d x } \] Input:
integrate(x^2*(7+x)^(1/2)/(5*x^2+2*x+3)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(x + 7)*x^2/sqrt(5*x^2 + 2*x + 3), x)
Timed out. \[ \int \frac {x^2 \sqrt {7+x}}{\sqrt {3+2 x+5 x^2}} \, dx=\int \frac {x^2\,\sqrt {x+7}}{\sqrt {5\,x^2+2\,x+3}} \,d x \] Input:
int((x^2*(x + 7)^(1/2))/(2*x + 5*x^2 + 3)^(1/2),x)
Output:
int((x^2*(x + 7)^(1/2))/(2*x + 5*x^2 + 3)^(1/2), x)
\[ \int \frac {x^2 \sqrt {7+x}}{\sqrt {3+2 x+5 x^2}} \, dx=\frac {2 \sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}\, x}{25}-\frac {51 \sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}}{925}+\frac {2763 \left (\int \frac {\sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}\, x^{2}}{5 x^{3}+37 x^{2}+17 x +21}d x \right )}{1850}-\frac {2241 \left (\int \frac {\sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}}{5 x^{3}+37 x^{2}+17 x +21}d x \right )}{1850} \] Input:
int(x^2*(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) + 2763*int((sqrt(x + 7)*sqrt(5*x**2 + 2*x + 3)*x**2)/(5*x**3 + 37 *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