Integrand size = 25, antiderivative size = 182 \[ \int (2-5 x) \sqrt {x} \sqrt {2+5 x+3 x^2} \, dx=-\frac {2476 \sqrt {x} (2+3 x)}{2835 \sqrt {2+5 x+3 x^2}}+\frac {4}{945} \sqrt {x} (430+639 x) \sqrt {2+5 x+3 x^2}-\frac {10}{21} \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2}+\frac {2476 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{2835 \sqrt {2+5 x+3 x^2}}-\frac {164 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{189 \sqrt {2+5 x+3 x^2}} \]
-10/21*(3*x^2+5*x+2)^(3/2)*x^(1/2)-2476/2835*(2+3*x)*x^(1/2)/(3*x^2+5*x+2) ^(1/2)+2476/2835*(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)-164/189* (1+x)^(3/2)*(1/(1+x))^(1/2)*EllipticF(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)+4/945*(430+639*x)*x^(1/2) *(3*x^2+5*x+2)^(1/2)
Result contains complex when optimal does not.
Time = 21.15 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.90 \[ \int (2-5 x) \sqrt {x} \sqrt {2+5 x+3 x^2} \, dx=\frac {-2 \left (2476+3730 x-3354 x^2-1935 x^3+8748 x^4+6075 x^5\right )-2476 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )+16 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right ),\frac {3}{2}\right )}{2835 \sqrt {x} \sqrt {2+5 x+3 x^2}} \]
(-2*(2476 + 3730*x - 3354*x^2 - 1935*x^3 + 8748*x^4 + 6075*x^5) - (2476*I) *Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[2 /3]/Sqrt[x]], 3/2] + (16*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(3/2) *EllipticF[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2])/(2835*Sqrt[x]*Sqrt[2 + 5*x + 3*x^2])
Time = 0.36 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {1236, 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.
| \(\displaystyle \int (2-5 x) \sqrt {x} \sqrt {3 x^2+5 x+2} \, dx\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {2}{21} \int \frac {(71 x+5) \sqrt {3 x^2+5 x+2}}{\sqrt {x}}dx-\frac {10}{21} \sqrt {x} \left (3 x^2+5 x+2\right )^{3/2}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {2}{21} \left (\frac {2}{45} \sqrt {x} (639 x+430) \sqrt {3 x^2+5 x+2}-\frac {2}{45} \int \frac {619 x+410}{2 \sqrt {x} \sqrt {3 x^2+5 x+2}}dx\right )-\frac {10}{21} \sqrt {x} \left (3 x^2+5 x+2\right )^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{21} \left (\frac {2}{45} \sqrt {x} (639 x+430) \sqrt {3 x^2+5 x+2}-\frac {1}{45} \int \frac {619 x+410}{\sqrt {x} \sqrt {3 x^2+5 x+2}}dx\right )-\frac {10}{21} \sqrt {x} \left (3 x^2+5 x+2\right )^{3/2}\) |
| \(\Big \downarrow \) 1240 |
| \(\displaystyle \frac {2}{21} \left (\frac {2}{45} \sqrt {x} (639 x+430) \sqrt {3 x^2+5 x+2}-\frac {2}{45} \int \frac {619 x+410}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}\right )-\frac {10}{21} \sqrt {x} \left (3 x^2+5 x+2\right )^{3/2}\) |
| \(\Big \downarrow \) 1503 |
| \(\displaystyle \frac {2}{21} \left (\frac {2}{45} \sqrt {x} (639 x+430) \sqrt {3 x^2+5 x+2}-\frac {2}{45} \left (410 \int \frac {1}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+619 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}\right )\right )-\frac {10}{21} \sqrt {x} \left (3 x^2+5 x+2\right )^{3/2}\) |
| \(\Big \downarrow \) 1413 |
| \(\displaystyle \frac {2}{21} \left (\frac {2}{45} \sqrt {x} (639 x+430) \sqrt {3 x^2+5 x+2}-\frac {2}{45} \left (619 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+\frac {205 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {3 x^2+5 x+2}}\right )\right )-\frac {10}{21} \sqrt {x} \left (3 x^2+5 x+2\right )^{3/2}\) |
| \(\Big \downarrow \) 1456 |
| \(\displaystyle \frac {2}{21} \left (\frac {2}{45} \sqrt {x} (639 x+430) \sqrt {3 x^2+5 x+2}-\frac {2}{45} \left (\frac {205 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {3 x^2+5 x+2}}+619 \left (\frac {\sqrt {x} (3 x+2)}{3 \sqrt {3 x^2+5 x+2}}-\frac {\sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{3 \sqrt {3 x^2+5 x+2}}\right )\right )\right )-\frac {10}{21} \sqrt {x} \left (3 x^2+5 x+2\right )^{3/2}\) |
(-10*Sqrt[x]*(2 + 5*x + 3*x^2)^(3/2))/21 + (2*((2*Sqrt[x]*(430 + 639*x)*Sq rt[2 + 5*x + 3*x^2])/45 - (2*(619*((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])) + (205*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x) /(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/Sqrt[2 + 5*x + 3*x^2]))/45))/2 1
3.11.37.3.1 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 = 122, normalized size of antiderivative = 0.67
| method | result | size |
| default | \(\frac {-\frac {30 x^{5}}{7}+\frac {418 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{2835}-\frac {1238 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{8505}-\frac {216 x^{4}}{35}+\frac {86 x^{3}}{63}+\frac {4712 x^{2}}{945}+\frac {328 x}{189}}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(122\) |
| risch | \(-\frac {2 \left (675 x^{2}-153 x -410\right ) \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}{945}-\frac {\left (\frac {164 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{567 \sqrt {3 x^{3}+5 x^{2}+2 x}}+\frac {1238 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{2835 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right ) \sqrt {x \left (3 x^{2}+5 x +2\right )}}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(188\) |
| elliptic | \(\frac {\sqrt {x \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {10 x^{2} \sqrt {3 x^{3}+5 x^{2}+2 x}}{7}+\frac {34 x \sqrt {3 x^{3}+5 x^{2}+2 x}}{105}+\frac {164 \sqrt {3 x^{3}+5 x^{2}+2 x}}{189}-\frac {164 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{567 \sqrt {3 x^{3}+5 x^{2}+2 x}}-\frac {1238 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{2835 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right )}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(217\) |
2/8505/x^(1/2)/(3*x^2+5*x+2)^(1/2)*(-18225*x^5+627*(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))-619*(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))-26244*x^4+5805*x^3+21204*x^2+7380*x)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.29 \[ \int (2-5 x) \sqrt {x} \sqrt {2+5 x+3 x^2} \, dx=-\frac {2}{945} \, {\left (675 \, x^{2} - 153 \, x - 410\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x} - \frac {68}{729} \, \sqrt {3} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) + \frac {2476}{2835} \, \sqrt {3} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) \]
-2/945*(675*x^2 - 153*x - 410)*sqrt(3*x^2 + 5*x + 2)*sqrt(x) - 68/729*sqrt (3)*weierstrassPInverse(28/27, 80/729, x + 5/9) + 2476/2835*sqrt(3)*weiers trassZeta(28/27, 80/729, weierstrassPInverse(28/27, 80/729, x + 5/9))
\[ \int (2-5 x) \sqrt {x} \sqrt {2+5 x+3 x^2} \, dx=- \int \left (- 2 \sqrt {x} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 5 x^{\frac {3}{2}} \sqrt {3 x^{2} + 5 x + 2}\, dx \]
-Integral(-2*sqrt(x)*sqrt(3*x**2 + 5*x + 2), x) - Integral(5*x**(3/2)*sqrt (3*x**2 + 5*x + 2), x)
\[ \int (2-5 x) \sqrt {x} \sqrt {2+5 x+3 x^2} \, dx=\int { -\sqrt {3 \, x^{2} + 5 \, x + 2} {\left (5 \, x - 2\right )} \sqrt {x} \,d x } \]
\[ \int (2-5 x) \sqrt {x} \sqrt {2+5 x+3 x^2} \, dx=\int { -\sqrt {3 \, x^{2} + 5 \, x + 2} {\left (5 \, x - 2\right )} \sqrt {x} \,d x } \]
Timed out. \[ \int (2-5 x) \sqrt {x} \sqrt {2+5 x+3 x^2} \, dx=-\int \sqrt {x}\,\left (5\,x-2\right )\,\sqrt {3\,x^2+5\,x+2} \,d x \]