Integrand size = 19, antiderivative size = 324 \[ \int \sqrt {2+6 x+3 x^2+9 x^3} \, dx=-\frac {4}{45} \sqrt {2+6 x+3 x^2+9 x^3}+\frac {2}{15} (1+3 x) \sqrt {2+6 x+3 x^2+9 x^3}+\frac {68 \sqrt {2+6 x+3 x^2+9 x^3}}{45 \left (1+\sqrt {7}+3 x\right )}-\frac {68 \sqrt [4]{7} \left (1+\sqrt {7}+3 x\right ) \sqrt {\frac {2+3 x^2}{\left (1+\sqrt {7}+3 x\right )^2}} \sqrt {2+6 x+3 x^2+9 x^3} E\left (2 \arctan \left (\frac {\sqrt {1+3 x}}{\sqrt [4]{7}}\right )|\frac {1}{14} \left (7+\sqrt {7}\right )\right )}{45 \sqrt {3} \sqrt {1+3 x} \left (2+3 x^2\right )}+\frac {2 \sqrt [4]{7} \left (17+\sqrt {7}\right ) \left (1+\sqrt {7}+3 x\right ) \sqrt {\frac {2+3 x^2}{\left (1+\sqrt {7}+3 x\right )^2}} \sqrt {2+6 x+3 x^2+9 x^3} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {1+3 x}}{\sqrt [4]{7}}\right ),\frac {1}{14} \left (7+\sqrt {7}\right )\right )}{45 \sqrt {3} \sqrt {1+3 x} \left (2+3 x^2\right )} \] Output:
-4/45*(9*x^3+3*x^2+6*x+2)^(1/2)+2/15*(1+3*x)*(9*x^3+3*x^2+6*x+2)^(1/2)+68* (9*x^3+3*x^2+6*x+2)^(1/2)/(45+45*7^(1/2)+135*x)-68/135*7^(1/4)*(1+7^(1/2)+ 3*x)*((3*x^2+2)/(1+7^(1/2)+3*x)^2)^(1/2)*(9*x^3+3*x^2+6*x+2)^(1/2)*Ellipti cE(sin(2*arctan(1/7*(1+3*x)^(1/2)*7^(3/4))),1/14*(98+14*7^(1/2))^(1/2))*3^ (1/2)/(1+3*x)^(1/2)/(3*x^2+2)+2/135*7^(1/4)*(17+7^(1/2))*(1+7^(1/2)+3*x)*( (3*x^2+2)/(1+7^(1/2)+3*x)^2)^(1/2)*(9*x^3+3*x^2+6*x+2)^(1/2)*InverseJacobi AM(2*arctan(1/7*(1+3*x)^(1/2)*7^(3/4)),1/14*(98+14*7^(1/2))^(1/2))*3^(1/2) /(1+3*x)^(1/2)/(3*x^2+2)
Result contains complex when optimal does not.
Time = 6.95 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.67 \[ \int \sqrt {2+6 x+3 x^2+9 x^3} \, dx=\frac {2 \left (2+3 x^2\right ) \left (\sqrt {6+9 x^2} \left (1+12 x+27 x^2\right )+\frac {34 i (1+3 x) E\left (\arcsin \left (\frac {\sqrt {\sqrt {6}-3 i x}}{2^{3/4} \sqrt [4]{3}}\right )|\frac {2 \sqrt {6}}{i+\sqrt {6}}\right )}{\sqrt {\frac {i (1+3 x)}{i+\sqrt {6}}}}+14 i \sqrt {\frac {i (1+3 x)}{i+\sqrt {6}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\sqrt {6}-3 i x}}{2^{3/4} \sqrt [4]{3}}\right ),\frac {2 \sqrt {6}}{i+\sqrt {6}}\right )\right )}{45 \sqrt {6+9 x^2} \sqrt {2+6 x+3 x^2+9 x^3}} \] Input:
Integrate[Sqrt[2 + 6*x + 3*x^2 + 9*x^3],x]
Output:
(2*(2 + 3*x^2)*(Sqrt[6 + 9*x^2]*(1 + 12*x + 27*x^2) + ((34*I)*(1 + 3*x)*El lipticE[ArcSin[Sqrt[Sqrt[6] - (3*I)*x]/(2^(3/4)*3^(1/4))], (2*Sqrt[6])/(I + Sqrt[6])])/Sqrt[(I*(1 + 3*x))/(I + Sqrt[6])] + (14*I)*Sqrt[(I*(1 + 3*x)) /(I + Sqrt[6])]*EllipticF[ArcSin[Sqrt[Sqrt[6] - (3*I)*x]/(2^(3/4)*3^(1/4)) ], (2*Sqrt[6])/(I + Sqrt[6])]))/(45*Sqrt[6 + 9*x^2]*Sqrt[2 + 6*x + 3*x^2 + 9*x^3])
Time = 0.46 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.11, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.684, Rules used = {2477, 27, 493, 27, 687, 27, 599, 25, 27, 1511, 27, 1416, 1509}
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.
| \(\displaystyle \int \sqrt {9 x^3+3 x^2+6 x+2} \, dx\) |
| \(\Big \downarrow \) 2477 |
| \(\displaystyle \frac {\sqrt {9 x^3+3 x^2+6 x+2} \int 3 \sqrt {3 x+1} \sqrt {3 x^2+2}dx}{3 \sqrt {3 x+1} \sqrt {3 x^2+2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {9 x^3+3 x^2+6 x+2} \int \sqrt {3 x+1} \sqrt {3 x^2+2}dx}{\sqrt {3 x+1} \sqrt {3 x^2+2}}\) |
| \(\Big \downarrow \) 493 |
| \(\displaystyle \frac {\sqrt {9 x^3+3 x^2+6 x+2} \left (\frac {2}{15} \int \frac {3 (2-x) \sqrt {3 x+1}}{\sqrt {3 x^2+2}}dx+\frac {2}{15} \sqrt {3 x^2+2} (3 x+1)^{3/2}\right )}{\sqrt {3 x+1} \sqrt {3 x^2+2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {9 x^3+3 x^2+6 x+2} \left (\frac {2}{5} \int \frac {(2-x) \sqrt {3 x+1}}{\sqrt {3 x^2+2}}dx+\frac {2}{15} \sqrt {3 x^2+2} (3 x+1)^{3/2}\right )}{\sqrt {3 x+1} \sqrt {3 x^2+2}}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {\sqrt {9 x^3+3 x^2+6 x+2} \left (\frac {2}{5} \left (\frac {2}{9} \int \frac {3 (17 x+8)}{2 \sqrt {3 x+1} \sqrt {3 x^2+2}}dx-\frac {2}{9} \sqrt {3 x+1} \sqrt {3 x^2+2}\right )+\frac {2}{15} \sqrt {3 x^2+2} (3 x+1)^{3/2}\right )}{\sqrt {3 x+1} \sqrt {3 x^2+2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {9 x^3+3 x^2+6 x+2} \left (\frac {2}{5} \left (\frac {1}{3} \int \frac {17 x+8}{\sqrt {3 x+1} \sqrt {3 x^2+2}}dx-\frac {2}{9} \sqrt {3 x+1} \sqrt {3 x^2+2}\right )+\frac {2}{15} \sqrt {3 x^2+2} (3 x+1)^{3/2}\right )}{\sqrt {3 x+1} \sqrt {3 x^2+2}}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle \frac {\sqrt {9 x^3+3 x^2+6 x+2} \left (\frac {2}{5} \left (-\frac {2}{27} \int -\frac {\sqrt {3} (17 (3 x+1)+7)}{\sqrt {(3 x+1)^2-2 (3 x+1)+7}}d\sqrt {3 x+1}-\frac {2}{9} \sqrt {3 x+1} \sqrt {3 x^2+2}\right )+\frac {2}{15} \sqrt {3 x^2+2} (3 x+1)^{3/2}\right )}{\sqrt {3 x+1} \sqrt {3 x^2+2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {9 x^3+3 x^2+6 x+2} \left (\frac {2}{5} \left (\frac {2}{27} \int \frac {\sqrt {3} (17 (3 x+1)+7)}{\sqrt {(3 x+1)^2-2 (3 x+1)+7}}d\sqrt {3 x+1}-\frac {2}{9} \sqrt {3 x+1} \sqrt {3 x^2+2}\right )+\frac {2}{15} \sqrt {3 x^2+2} (3 x+1)^{3/2}\right )}{\sqrt {3 x+1} \sqrt {3 x^2+2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {9 x^3+3 x^2+6 x+2} \left (\frac {2}{5} \left (\frac {2 \int \frac {17 (3 x+1)+7}{\sqrt {(3 x+1)^2-2 (3 x+1)+7}}d\sqrt {3 x+1}}{9 \sqrt {3}}-\frac {2}{9} \sqrt {3 x+1} \sqrt {3 x^2+2}\right )+\frac {2}{15} \sqrt {3 x^2+2} (3 x+1)^{3/2}\right )}{\sqrt {3 x+1} \sqrt {3 x^2+2}}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {\sqrt {9 x^3+3 x^2+6 x+2} \left (\frac {2}{5} \left (\frac {2 \left (\left (7+17 \sqrt {7}\right ) \int \frac {1}{\sqrt {(3 x+1)^2-2 (3 x+1)+7}}d\sqrt {3 x+1}-17 \sqrt {7} \int \frac {-3 x+\sqrt {7}-1}{\sqrt {7} \sqrt {(3 x+1)^2-2 (3 x+1)+7}}d\sqrt {3 x+1}\right )}{9 \sqrt {3}}-\frac {2}{9} \sqrt {3 x+1} \sqrt {3 x^2+2}\right )+\frac {2}{15} \sqrt {3 x^2+2} (3 x+1)^{3/2}\right )}{\sqrt {3 x+1} \sqrt {3 x^2+2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {9 x^3+3 x^2+6 x+2} \left (\frac {2}{5} \left (\frac {2 \left (\left (7+17 \sqrt {7}\right ) \int \frac {1}{\sqrt {(3 x+1)^2-2 (3 x+1)+7}}d\sqrt {3 x+1}-17 \int \frac {-3 x+\sqrt {7}-1}{\sqrt {(3 x+1)^2-2 (3 x+1)+7}}d\sqrt {3 x+1}\right )}{9 \sqrt {3}}-\frac {2}{9} \sqrt {3 x+1} \sqrt {3 x^2+2}\right )+\frac {2}{15} \sqrt {3 x^2+2} (3 x+1)^{3/2}\right )}{\sqrt {3 x+1} \sqrt {3 x^2+2}}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {\sqrt {9 x^3+3 x^2+6 x+2} \left (\frac {2}{5} \left (\frac {2 \left (\frac {\left (7+17 \sqrt {7}\right ) \left (3 x+\sqrt {7}+1\right ) \sqrt {\frac {(3 x+1)^2-2 (3 x+1)+7}{\left (3 x+\sqrt {7}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {3 x+1}}{\sqrt [4]{7}}\right ),\frac {1}{14} \left (7+\sqrt {7}\right )\right )}{2 \sqrt [4]{7} \sqrt {(3 x+1)^2-2 (3 x+1)+7}}-17 \int \frac {-3 x+\sqrt {7}-1}{\sqrt {(3 x+1)^2-2 (3 x+1)+7}}d\sqrt {3 x+1}\right )}{9 \sqrt {3}}-\frac {2}{9} \sqrt {3 x+1} \sqrt {3 x^2+2}\right )+\frac {2}{15} \sqrt {3 x^2+2} (3 x+1)^{3/2}\right )}{\sqrt {3 x+1} \sqrt {3 x^2+2}}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {\sqrt {9 x^3+3 x^2+6 x+2} \left (\frac {2}{5} \left (\frac {2 \left (\frac {\left (7+17 \sqrt {7}\right ) \left (3 x+\sqrt {7}+1\right ) \sqrt {\frac {(3 x+1)^2-2 (3 x+1)+7}{\left (3 x+\sqrt {7}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {3 x+1}}{\sqrt [4]{7}}\right ),\frac {1}{14} \left (7+\sqrt {7}\right )\right )}{2 \sqrt [4]{7} \sqrt {(3 x+1)^2-2 (3 x+1)+7}}-17 \left (\frac {\sqrt [4]{7} \left (3 x+\sqrt {7}+1\right ) \sqrt {\frac {(3 x+1)^2-2 (3 x+1)+7}{\left (3 x+\sqrt {7}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt {3 x+1}}{\sqrt [4]{7}}\right )|\frac {1}{14} \left (7+\sqrt {7}\right )\right )}{\sqrt {(3 x+1)^2-2 (3 x+1)+7}}-\frac {\sqrt {3 x+1} \sqrt {(3 x+1)^2-2 (3 x+1)+7}}{3 x+\sqrt {7}+1}\right )\right )}{9 \sqrt {3}}-\frac {2}{9} \sqrt {3 x+1} \sqrt {3 x^2+2}\right )+\frac {2}{15} \sqrt {3 x^2+2} (3 x+1)^{3/2}\right )}{\sqrt {3 x+1} \sqrt {3 x^2+2}}\) |
Input:
Int[Sqrt[2 + 6*x + 3*x^2 + 9*x^3],x]
Output:
(Sqrt[2 + 6*x + 3*x^2 + 9*x^3]*((2*(1 + 3*x)^(3/2)*Sqrt[2 + 3*x^2])/15 + ( 2*((-2*Sqrt[1 + 3*x]*Sqrt[2 + 3*x^2])/9 + (2*(-17*(-((Sqrt[1 + 3*x]*Sqrt[7 - 2*(1 + 3*x) + (1 + 3*x)^2])/(1 + Sqrt[7] + 3*x)) + (7^(1/4)*(1 + Sqrt[7 ] + 3*x)*Sqrt[(7 - 2*(1 + 3*x) + (1 + 3*x)^2)/(1 + Sqrt[7] + 3*x)^2]*Ellip ticE[2*ArcTan[Sqrt[1 + 3*x]/7^(1/4)], (7 + Sqrt[7])/14])/Sqrt[7 - 2*(1 + 3 *x) + (1 + 3*x)^2]) + ((7 + 17*Sqrt[7])*(1 + Sqrt[7] + 3*x)*Sqrt[(7 - 2*(1 + 3*x) + (1 + 3*x)^2)/(1 + Sqrt[7] + 3*x)^2]*EllipticF[2*ArcTan[Sqrt[1 + 3*x]/7^(1/4)], (7 + Sqrt[7])/14])/(2*7^(1/4)*Sqrt[7 - 2*(1 + 3*x) + (1 + 3 *x)^2])))/(9*Sqrt[3])))/5))/(Sqrt[1 + 3*x]*Sqrt[2 + 3*x^2])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (c + d*x)^(n + 1)*((a + b*x^2)^p/(d*(n + 2*p + 1))), x] + Simp[2*(p/(d*(n + 2*p + 1))) Int[(c + d*x)^n*(a + b*x^2)^(p - 1)*(a*d - b*c*x), x], x] /; FreeQ[{a, b, c, d, n}, x] && GtQ[p, 0] && NeQ[n + 2*p + 1, 0] && ( !Rationa lQ[n] || LtQ[n, 1]) && !ILtQ[n + 2*p, 0] && IntQuadraticQ[a, 0, b, c, d, n , p, x]
Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] ), x_Symbol] :> Simp[-2/d^2 Subst[Int[(B*c - A*d - B*x^2)/Sqrt[(b*c^2 + a *d^2)/d^2 - 2*b*c*(x^2/d^2) + b*(x^4/d^2)], x], x, Sqrt[c + d*x]], x] /; Fr eeQ[{a, b, c, d, A, B}, x] && PosQ[b/a]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + 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 + c*x^2)^p*Simp [c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x ] /; FreeQ[{a, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && Eq Q[f, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c /a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q ^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 /(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + b*x^2 + c*x^ 4], x], x] - Simp[e/q Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos Q[c/a]
Int[(Px_)^(p_), x_Symbol] :> With[{a = Coeff[Px, x, 0], b = Coeff[Px, x, 1] , c = Coeff[Px, x, 2], d = Coeff[Px, x, 3]}, Simp[Px^p/((c + d*x)^p*(b + d* x^2)^p) Int[(c + d*x)^p*(b + d*x^2)^p, x], x] /; EqQ[b*c - a*d, 0]] /; Fr eeQ[p, x] && PolyQ[Px, x, 3] && !IntegerQ[p]
Result contains complex when optimal does not.
Time = 0.39 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.01
| method | result | size |
| risch | \(\frac {2 \left (1+9 x \right ) \sqrt {9 x^{3}+3 x^{2}+6 x +2}}{45}+\frac {32 \left (-\frac {i \sqrt {6}}{3}+\frac {1}{3}\right ) \sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}\, \sqrt {\frac {x -\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\, \sqrt {\frac {x +\frac {i \sqrt {6}}{3}}{-\frac {1}{3}+\frac {i \sqrt {6}}{3}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )}{15 \sqrt {9 x^{3}+3 x^{2}+6 x +2}}+\frac {68 \left (-\frac {i \sqrt {6}}{3}+\frac {1}{3}\right ) \sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}\, \sqrt {\frac {x -\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\, \sqrt {\frac {x +\frac {i \sqrt {6}}{3}}{-\frac {1}{3}+\frac {i \sqrt {6}}{3}}}\, \left (\left (-\frac {1}{3}-\frac {i \sqrt {6}}{3}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )+\frac {i \sqrt {6}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )}{3}\right )}{15 \sqrt {9 x^{3}+3 x^{2}+6 x +2}}\) | \(326\) |
| default | \(\frac {2 x \sqrt {9 x^{3}+3 x^{2}+6 x +2}}{5}+\frac {2 \sqrt {9 x^{3}+3 x^{2}+6 x +2}}{45}+\frac {32 \left (-\frac {i \sqrt {6}}{3}+\frac {1}{3}\right ) \sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}\, \sqrt {\frac {x -\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\, \sqrt {\frac {x +\frac {i \sqrt {6}}{3}}{-\frac {1}{3}+\frac {i \sqrt {6}}{3}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )}{15 \sqrt {9 x^{3}+3 x^{2}+6 x +2}}+\frac {68 \left (-\frac {i \sqrt {6}}{3}+\frac {1}{3}\right ) \sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}\, \sqrt {\frac {x -\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\, \sqrt {\frac {x +\frac {i \sqrt {6}}{3}}{-\frac {1}{3}+\frac {i \sqrt {6}}{3}}}\, \left (\left (-\frac {1}{3}-\frac {i \sqrt {6}}{3}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )+\frac {i \sqrt {6}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )}{3}\right )}{15 \sqrt {9 x^{3}+3 x^{2}+6 x +2}}\) | \(341\) |
| elliptic | \(\frac {2 x \sqrt {9 x^{3}+3 x^{2}+6 x +2}}{5}+\frac {2 \sqrt {9 x^{3}+3 x^{2}+6 x +2}}{45}+\frac {32 \left (-\frac {i \sqrt {6}}{3}+\frac {1}{3}\right ) \sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}\, \sqrt {\frac {x -\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\, \sqrt {\frac {x +\frac {i \sqrt {6}}{3}}{-\frac {1}{3}+\frac {i \sqrt {6}}{3}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )}{15 \sqrt {9 x^{3}+3 x^{2}+6 x +2}}+\frac {68 \left (-\frac {i \sqrt {6}}{3}+\frac {1}{3}\right ) \sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}\, \sqrt {\frac {x -\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\, \sqrt {\frac {x +\frac {i \sqrt {6}}{3}}{-\frac {1}{3}+\frac {i \sqrt {6}}{3}}}\, \left (\left (-\frac {1}{3}-\frac {i \sqrt {6}}{3}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )+\frac {i \sqrt {6}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )}{3}\right )}{15 \sqrt {9 x^{3}+3 x^{2}+6 x +2}}\) | \(341\) |
Input:
int((9*x^3+3*x^2+6*x+2)^(1/2),x,method=_RETURNVERBOSE)
Output:
2/45*(1+9*x)*(9*x^3+3*x^2+6*x+2)^(1/2)+32/15*(-1/3*I*6^(1/2)+1/3)*((x+1/3) /(-1/3*I*6^(1/2)+1/3))^(1/2)*((x-1/3*I*6^(1/2))/(-1/3-1/3*I*6^(1/2)))^(1/2 )*((x+1/3*I*6^(1/2))/(-1/3+1/3*I*6^(1/2)))^(1/2)/(9*x^3+3*x^2+6*x+2)^(1/2) *EllipticF(((x+1/3)/(-1/3*I*6^(1/2)+1/3))^(1/2),((-1/3+1/3*I*6^(1/2))/(-1/ 3-1/3*I*6^(1/2)))^(1/2))+68/15*(-1/3*I*6^(1/2)+1/3)*((x+1/3)/(-1/3*I*6^(1/ 2)+1/3))^(1/2)*((x-1/3*I*6^(1/2))/(-1/3-1/3*I*6^(1/2)))^(1/2)*((x+1/3*I*6^ (1/2))/(-1/3+1/3*I*6^(1/2)))^(1/2)/(9*x^3+3*x^2+6*x+2)^(1/2)*((-1/3-1/3*I* 6^(1/2))*EllipticE(((x+1/3)/(-1/3*I*6^(1/2)+1/3))^(1/2),((-1/3+1/3*I*6^(1/ 2))/(-1/3-1/3*I*6^(1/2)))^(1/2))+1/3*I*6^(1/2)*EllipticF(((x+1/3)/(-1/3*I* 6^(1/2)+1/3))^(1/2),((-1/3+1/3*I*6^(1/2))/(-1/3-1/3*I*6^(1/2)))^(1/2)))
Time = 0.07 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.14 \[ \int \sqrt {2+6 x+3 x^2+9 x^3} \, dx=\frac {2}{45} \, \sqrt {9 \, x^{3} + 3 \, x^{2} + 6 \, x + 2} {\left (9 \, x + 1\right )} + \frac {44}{81} \, {\rm weierstrassPInverse}\left (-\frac {68}{27}, -\frac {440}{729}, x + \frac {1}{9}\right ) - \frac {68}{45} \, {\rm weierstrassZeta}\left (-\frac {68}{27}, -\frac {440}{729}, {\rm weierstrassPInverse}\left (-\frac {68}{27}, -\frac {440}{729}, x + \frac {1}{9}\right )\right ) \] Input:
integrate((9*x^3+3*x^2+6*x+2)^(1/2),x, algorithm="fricas")
Output:
2/45*sqrt(9*x^3 + 3*x^2 + 6*x + 2)*(9*x + 1) + 44/81*weierstrassPInverse(- 68/27, -440/729, x + 1/9) - 68/45*weierstrassZeta(-68/27, -440/729, weiers trassPInverse(-68/27, -440/729, x + 1/9))
\[ \int \sqrt {2+6 x+3 x^2+9 x^3} \, dx=\int \sqrt {9 x^{3} + 3 x^{2} + 6 x + 2}\, dx \] Input:
integrate((9*x**3+3*x**2+6*x+2)**(1/2),x)
Output:
Integral(sqrt(9*x**3 + 3*x**2 + 6*x + 2), x)
\[ \int \sqrt {2+6 x+3 x^2+9 x^3} \, dx=\int { \sqrt {9 \, x^{3} + 3 \, x^{2} + 6 \, x + 2} \,d x } \] Input:
integrate((9*x^3+3*x^2+6*x+2)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(9*x^3 + 3*x^2 + 6*x + 2), x)
\[ \int \sqrt {2+6 x+3 x^2+9 x^3} \, dx=\int { \sqrt {9 \, x^{3} + 3 \, x^{2} + 6 \, x + 2} \,d x } \] Input:
integrate((9*x^3+3*x^2+6*x+2)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(9*x^3 + 3*x^2 + 6*x + 2), x)
Time = 0.03 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.35 \[ \int \sqrt {2+6 x+3 x^2+9 x^3} \, dx =\text {Too large to display} \] Input:
int((6*x + 3*x^2 + 9*x^3 + 2)^(1/2),x)
Output:
(2*((2*x)/3 + x^2/3 + x^3 + 2/9))/(6*x + 3*x^2 + 9*x^3 + 2)^(1/2) + (18*(x - 4/9)*((2*x)/3 + x^2/3 + x^3 + 2/9))/(5*(6*x + 3*x^2 + 9*x^3 + 2)^(1/2)) + (32*((x + 1/3)/((2^(1/2)*3^(1/2)*1i)/3 + 1/3))^(1/2)*((x + (2^(1/2)*3^( 1/2)*1i)/3)/((2^(1/2)*3^(1/2)*1i)/3 - 1/3))^(1/2)*(-(x - (2^(1/2)*3^(1/2)* 1i)/3)/((2^(1/2)*3^(1/2)*1i)/3 + 1/3))^(1/2)*((2^(1/2)*3^(1/2)*1i)/3 + 1/3 )*ellipticF(asin(((x + 1/3)/((2^(1/2)*3^(1/2)*1i)/3 + 1/3))^(1/2)), -((2^( 1/2)*3^(1/2)*1i)/3 + 1/3)/((2^(1/2)*3^(1/2)*1i)/3 - 1/3)))/(15*(6*x + 3*x^ 2 + 9*x^3 + 2)^(1/2)) + (68*((x + 1/3)/((2^(1/2)*3^(1/2)*1i)/3 + 1/3))^(1/ 2)*((x + (2^(1/2)*3^(1/2)*1i)/3)/((2^(1/2)*3^(1/2)*1i)/3 - 1/3))^(1/2)*(-( x - (2^(1/2)*3^(1/2)*1i)/3)/((2^(1/2)*3^(1/2)*1i)/3 + 1/3))^(1/2)*(((2^(1/ 2)*3^(1/2)*1i)/3 - 1/3)*ellipticE(asin(((x + 1/3)/((2^(1/2)*3^(1/2)*1i)/3 + 1/3))^(1/2)), -((2^(1/2)*3^(1/2)*1i)/3 + 1/3)/((2^(1/2)*3^(1/2)*1i)/3 - 1/3)) - (2^(1/2)*3^(1/2)*ellipticF(asin(((x + 1/3)/((2^(1/2)*3^(1/2)*1i)/3 + 1/3))^(1/2)), -((2^(1/2)*3^(1/2)*1i)/3 + 1/3)/((2^(1/2)*3^(1/2)*1i)/3 - 1/3))*1i)/3)*((2^(1/2)*3^(1/2)*1i)/3 + 1/3))/(15*(6*x + 3*x^2 + 9*x^3 + 2 )^(1/2))
\[ \int \sqrt {2+6 x+3 x^2+9 x^3} \, dx=\frac {2 \sqrt {9 x^{3}+3 x^{2}+6 x +2}\, x}{5}+\frac {4 \sqrt {9 x^{3}+3 x^{2}+6 x +2}}{5}-\frac {6 \left (\int \frac {\sqrt {9 x^{3}+3 x^{2}+6 x +2}}{9 x^{3}+3 x^{2}+6 x +2}d x \right )}{5}-\frac {51 \left (\int \frac {\sqrt {9 x^{3}+3 x^{2}+6 x +2}\, x^{2}}{9 x^{3}+3 x^{2}+6 x +2}d x \right )}{5} \] Input:
int((9*x^3+3*x^2+6*x+2)^(1/2),x)
Output:
(2*sqrt(9*x**3 + 3*x**2 + 6*x + 2)*x + 4*sqrt(9*x**3 + 3*x**2 + 6*x + 2) - 6*int(sqrt(9*x**3 + 3*x**2 + 6*x + 2)/(9*x**3 + 3*x**2 + 6*x + 2),x) - 51 *int((sqrt(9*x**3 + 3*x**2 + 6*x + 2)*x**2)/(9*x**3 + 3*x**2 + 6*x + 2),x) )/5