Integrand size = 25, antiderivative size = 210 \[ \int (2-5 x) \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2} \, dx=-\frac {424 \sqrt {x} (2+3 x)}{1155 \sqrt {2+5 x+3 x^2}}-\frac {4}{385} \sqrt {x} (55+39 x) \sqrt {2+5 x+3 x^2}+\frac {4}{231} \sqrt {x} (65+84 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {10}{33} \sqrt {x} \left (2+5 x+3 x^2\right )^{5/2}+\frac {424 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{1155 \sqrt {2+5 x+3 x^2}}-\frac {36 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{77 \sqrt {2+5 x+3 x^2}} \]
4/231*(65+84*x)*(3*x^2+5*x+2)^(3/2)*x^(1/2)-10/33*(3*x^2+5*x+2)^(5/2)*x^(1 /2)-424/1155*(2+3*x)*x^(1/2)/(3*x^2+5*x+2)^(1/2)+424/1155*(1+x)^(3/2)*(1/( 1+x))^(1/2)*EllipticE(x^(1/2)/(1+x)^(1/2),1/2*I*2^(1/2))*2^(1/2)*((2+3*x)/ (1+x))^(1/2)/(3*x^2+5*x+2)^(1/2)-36/77*(1+x)^(3/2)*(1/(1+x))^(1/2)*Ellipti cF(x^(1/2)/(1+x)^(1/2),1/2*I*2^(1/2))*2^(1/2)*((2+3*x)/(1+x))^(1/2)/(3*x^2 +5*x+2)^(1/2)-4/385*(55+39*x)*x^(1/2)*(3*x^2+5*x+2)^(1/2)
Result contains complex when optimal does not.
Time = 21.21 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.82 \[ \int (2-5 x) \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2} \, dx=\frac {-424 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )-2 \left (424+520 x-3106 x^2-6140 x^3+3497 x^4+17775 x^5+16065 x^6+4725 x^7+58 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right ),\frac {3}{2}\right )\right )}{1155 \sqrt {x} \sqrt {2+5 x+3 x^2}} \]
((-424*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(3/2)*EllipticE[I*ArcSi nh[Sqrt[2/3]/Sqrt[x]], 3/2] - 2*(424 + 520*x - 3106*x^2 - 6140*x^3 + 3497* x^4 + 17775*x^5 + 16065*x^6 + 4725*x^7 + (58*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*S qrt[3 + 2/x]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2]))/(1155* Sqrt[x]*Sqrt[2 + 5*x + 3*x^2])
Time = 0.41 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {1236, 1231, 27, 1231, 27, 1240, 1503, 1413, 1456}
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.
| \(\displaystyle \int (2-5 x) \sqrt {x} \left (3 x^2+5 x+2\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {2}{33} \int \frac {(108 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{\sqrt {x}}dx-\frac {10}{33} \sqrt {x} \left (3 x^2+5 x+2\right )^{5/2}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {2}{33} \left (\frac {2}{7} \sqrt {x} (84 x+65) \left (3 x^2+5 x+2\right )^{3/2}-\frac {2}{63} \int \frac {27 (39 x+20) \sqrt {3 x^2+5 x+2}}{2 \sqrt {x}}dx\right )-\frac {10}{33} \sqrt {x} \left (3 x^2+5 x+2\right )^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{33} \left (\frac {2}{7} \sqrt {x} (84 x+65) \left (3 x^2+5 x+2\right )^{3/2}-\frac {3}{7} \int \frac {(39 x+20) \sqrt {3 x^2+5 x+2}}{\sqrt {x}}dx\right )-\frac {10}{33} \sqrt {x} \left (3 x^2+5 x+2\right )^{5/2}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {2}{33} \left (\frac {2}{7} \sqrt {x} (84 x+65) \left (3 x^2+5 x+2\right )^{3/2}-\frac {3}{7} \left (\frac {2}{5} \sqrt {x} (39 x+55) \sqrt {3 x^2+5 x+2}-\frac {2}{45} \int -\frac {9 (53 x+45)}{\sqrt {x} \sqrt {3 x^2+5 x+2}}dx\right )\right )-\frac {10}{33} \sqrt {x} \left (3 x^2+5 x+2\right )^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{33} \left (\frac {2}{7} \sqrt {x} (84 x+65) \left (3 x^2+5 x+2\right )^{3/2}-\frac {3}{7} \left (\frac {2}{5} \int \frac {53 x+45}{\sqrt {x} \sqrt {3 x^2+5 x+2}}dx+\frac {2}{5} \sqrt {x} \sqrt {3 x^2+5 x+2} (39 x+55)\right )\right )-\frac {10}{33} \sqrt {x} \left (3 x^2+5 x+2\right )^{5/2}\) |
| \(\Big \downarrow \) 1240 |
| \(\displaystyle \frac {2}{33} \left (\frac {2}{7} \sqrt {x} (84 x+65) \left (3 x^2+5 x+2\right )^{3/2}-\frac {3}{7} \left (\frac {4}{5} \int \frac {53 x+45}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+\frac {2}{5} \sqrt {x} \sqrt {3 x^2+5 x+2} (39 x+55)\right )\right )-\frac {10}{33} \sqrt {x} \left (3 x^2+5 x+2\right )^{5/2}\) |
| \(\Big \downarrow \) 1503 |
| \(\displaystyle \frac {2}{33} \left (\frac {2}{7} \sqrt {x} (84 x+65) \left (3 x^2+5 x+2\right )^{3/2}-\frac {3}{7} \left (\frac {4}{5} \left (45 \int \frac {1}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+53 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}\right )+\frac {2}{5} \sqrt {x} \sqrt {3 x^2+5 x+2} (39 x+55)\right )\right )-\frac {10}{33} \sqrt {x} \left (3 x^2+5 x+2\right )^{5/2}\) |
| \(\Big \downarrow \) 1413 |
| \(\displaystyle \frac {2}{33} \left (\frac {2}{7} \sqrt {x} (84 x+65) \left (3 x^2+5 x+2\right )^{3/2}-\frac {3}{7} \left (\frac {4}{5} \left (53 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+\frac {45 (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {2} \sqrt {3 x^2+5 x+2}}\right )+\frac {2}{5} \sqrt {x} \sqrt {3 x^2+5 x+2} (39 x+55)\right )\right )-\frac {10}{33} \sqrt {x} \left (3 x^2+5 x+2\right )^{5/2}\) |
| \(\Big \downarrow \) 1456 |
| \(\displaystyle \frac {2}{33} \left (\frac {2}{7} \sqrt {x} (84 x+65) \left (3 x^2+5 x+2\right )^{3/2}-\frac {3}{7} \left (\frac {4}{5} \left (\frac {45 (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {2} \sqrt {3 x^2+5 x+2}}+53 \left (\frac {\sqrt {x} (3 x+2)}{3 \sqrt {3 x^2+5 x+2}}-\frac {\sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{3 \sqrt {3 x^2+5 x+2}}\right )\right )+\frac {2}{5} \sqrt {x} \sqrt {3 x^2+5 x+2} (39 x+55)\right )\right )-\frac {10}{33} \sqrt {x} \left (3 x^2+5 x+2\right )^{5/2}\) |
(-10*Sqrt[x]*(2 + 5*x + 3*x^2)^(5/2))/33 + (2*((2*Sqrt[x]*(65 + 84*x)*(2 + 5*x + 3*x^2)^(3/2))/7 - (3*((2*Sqrt[x]*(55 + 39*x)*Sqrt[2 + 5*x + 3*x^2]) /5 + (4*(53*((Sqrt[x]*(2 + 3*x))/(3*Sqrt[2 + 5*x + 3*x^2]) - (Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/(3*Sqrt[2 + 5*x + 3*x^2])) + (45*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt [x]], -1/2])/(Sqrt[2]*Sqrt[2 + 5*x + 3*x^2])))/5))/7))/33
3.11.46.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ (c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)) Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c ^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x ] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || !R ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Integer Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
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[((f_) + (g_.)*(x_))/(Sqrt[x_]*Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[2 Subst[Int[(f + g*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x, Sqrt[x]], x] /; FreeQ[{a, b, c, f, g}, x]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b ^2 - 4*a*c, 2]}, Simp[(2*a + (b - q)*x^2)*(Sqrt[(2*a + (b + q)*x^2)/(2*a + (b - q)*x^2)]/(2*a*Rt[(b - q)/(2*a), 2]*Sqrt[a + b*x^2 + c*x^4]))*EllipticF [ArcTan[Rt[(b - q)/(2*a), 2]*x], -2*(q/(b - q))], x] /; PosQ[(b - q)/a]] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]
Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[x*((b - q + 2*c*x^2)/(2*c*Sqrt[a + b*x^2 + c*x^4 ])), x] - Simp[Rt[(b - q)/(2*a), 2]*(2*a + (b - q)*x^2)*(Sqrt[(2*a + (b + q )*x^2)/(2*a + (b - q)*x^2)]/(2*c*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[ArcTan [Rt[(b - q)/(2*a), 2]*x], -2*(q/(b - q))], x] /; PosQ[(b - q)/a]] /; FreeQ[ {a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[d Int[1/Sqrt[a + b*x^2 + c*x^4] , x], x] + Simp[e Int[x^2/Sqrt[a + b*x^2 + c*x^4], x], x] /; PosQ[(b + q) /a] || PosQ[(b - q)/a]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0]
Time = 0.23 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.63
| method | result | size |
| default | \(\frac {-\frac {90 x^{7}}{11}-\frac {306 x^{6}}{11}-\frac {2370 x^{5}}{77}+\frac {32 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{1155}-\frac {212 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3465}-\frac {6994 x^{4}}{1155}+\frac {2456 x^{3}}{231}+\frac {7484 x^{2}}{1155}+\frac {72 x}{77}}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(132\) |
| risch | \(-\frac {2 \left (1575 x^{4}+2730 x^{3}+325 x^{2}-1196 x -270\right ) \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}{1155}-\frac {\left (\frac {12 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{77 \sqrt {3 x^{3}+5 x^{2}+2 x}}+\frac {212 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{1155 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right ) \sqrt {x \left (3 x^{2}+5 x +2\right )}}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(198\) |
| elliptic | \(\frac {\sqrt {x \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {30 x^{4} \sqrt {3 x^{3}+5 x^{2}+2 x}}{11}-\frac {52 x^{3} \sqrt {3 x^{3}+5 x^{2}+2 x}}{11}-\frac {130 x^{2} \sqrt {3 x^{3}+5 x^{2}+2 x}}{231}+\frac {2392 x \sqrt {3 x^{3}+5 x^{2}+2 x}}{1155}+\frac {36 \sqrt {3 x^{3}+5 x^{2}+2 x}}{77}-\frac {12 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{77 \sqrt {3 x^{3}+5 x^{2}+2 x}}-\frac {212 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{1155 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right )}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(259\) |
2/3465/(3*x^2+5*x+2)^(1/2)/x^(1/2)*(-14175*x^7-48195*x^6-53325*x^5+48*(6*x +4)^(1/2)*(3+3*x)^(1/2)*6^(1/2)*(-x)^(1/2)*EllipticF(1/2*(6*x+4)^(1/2),I*2 ^(1/2))-106*(6*x+4)^(1/2)*(3+3*x)^(1/2)*6^(1/2)*(-x)^(1/2)*EllipticE(1/2*( 6*x+4)^(1/2),I*2^(1/2))-10491*x^4+18420*x^3+11226*x^2+1620*x)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.30 \[ \int (2-5 x) \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2} \, dx=-\frac {2}{1155} \, {\left (1575 \, x^{4} + 2730 \, x^{3} + 325 \, x^{2} - 1196 \, x - 270\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x} - \frac {32}{297} \, \sqrt {3} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) + \frac {424}{1155} \, \sqrt {3} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) \]
-2/1155*(1575*x^4 + 2730*x^3 + 325*x^2 - 1196*x - 270)*sqrt(3*x^2 + 5*x + 2)*sqrt(x) - 32/297*sqrt(3)*weierstrassPInverse(28/27, 80/729, x + 5/9) + 424/1155*sqrt(3)*weierstrassZeta(28/27, 80/729, weierstrassPInverse(28/27, 80/729, x + 5/9))
\[ \int (2-5 x) \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2} \, dx=- \int \left (- 4 \sqrt {x} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 19 x^{\frac {5}{2}} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int 15 x^{\frac {7}{2}} \sqrt {3 x^{2} + 5 x + 2}\, dx \]
-Integral(-4*sqrt(x)*sqrt(3*x**2 + 5*x + 2), x) - Integral(19*x**(5/2)*sqr t(3*x**2 + 5*x + 2), x) - Integral(15*x**(7/2)*sqrt(3*x**2 + 5*x + 2), x)
\[ \int (2-5 x) \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2} \, dx=\int { -{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (5 \, x - 2\right )} \sqrt {x} \,d x } \]
\[ \int (2-5 x) \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2} \, dx=\int { -{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (5 \, x - 2\right )} \sqrt {x} \,d x } \]
Timed out. \[ \int (2-5 x) \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2} \, dx=-\int \sqrt {x}\,\left (5\,x-2\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2} \,d x \]