Integrand size = 29, antiderivative size = 143 \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{\sqrt {3+2 x}} \, dx=\frac {1}{45} (88-9 x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}-\frac {761 \sqrt {-2-5 x-3 x^2} E\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{90 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {191 \sqrt {-2-5 x-3 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right ),-\frac {2}{3}\right )}{18 \sqrt {3} \sqrt {2+5 x+3 x^2}} \]
-761/270*EllipticE(3^(1/2)*(1+x)^(1/2),1/3*I*6^(1/2))*(-3*x^2-5*x-2)^(1/2) *3^(1/2)/(3*x^2+5*x+2)^(1/2)+191/54*EllipticF(3^(1/2)*(1+x)^(1/2),1/3*I*6^ (1/2))*(-3*x^2-5*x-2)^(1/2)*3^(1/2)/(3*x^2+5*x+2)^(1/2)+1/45*(88-9*x)*(3+2 *x)^(1/2)*(3*x^2+5*x+2)^(1/2)
Time = 25.77 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.35 \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{\sqrt {3+2 x}} \, dx=-\frac {2 \sqrt {3+2 x} \left (-62-1049 x-2220 x^2-1071 x^3+162 x^4\right )+761 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^2 \sqrt {\frac {2+3 x}{3+2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )-188 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^2 \sqrt {\frac {2+3 x}{3+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right ),\frac {3}{5}\right )}{270 (3+2 x) \sqrt {2+5 x+3 x^2}} \]
-1/270*(2*Sqrt[3 + 2*x]*(-62 - 1049*x - 2220*x^2 - 1071*x^3 + 162*x^4) + 7 61*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^2*Sqrt[(2 + 3*x)/(3 + 2*x)]*E llipticE[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5] - 188*Sqrt[5]*Sqrt[(1 + x)/ (3 + 2*x)]*(3 + 2*x)^2*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticF[ArcSin[Sqrt[5/3 ]/Sqrt[3 + 2*x]], 3/5])/((3 + 2*x)*Sqrt[2 + 5*x + 3*x^2])
Time = 0.29 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {1231, 1269, 1172, 27, 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.
| \(\displaystyle \int \frac {(5-x) \sqrt {3 x^2+5 x+2}}{\sqrt {2 x+3}} \, dx\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {1}{45} (88-9 x) \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}-\frac {1}{90} \int \frac {761 x+664}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {1}{90} \left (\frac {955}{2} \int \frac {1}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx-\frac {761}{2} \int \frac {\sqrt {2 x+3}}{\sqrt {3 x^2+5 x+2}}dx\right )+\frac {1}{45} \sqrt {2 x+3} \sqrt {3 x^2+5 x+2} (88-9 x)\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {1}{90} \left (\frac {955 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {3}}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {761 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {3} \sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )+\frac {1}{45} \sqrt {2 x+3} \sqrt {3 x^2+5 x+2} (88-9 x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{90} \left (\frac {955 \sqrt {-3 x^2-5 x-2} \int \frac {1}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3 x^2+5 x+2}}-\frac {761 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}\right )+\frac {1}{45} \sqrt {2 x+3} \sqrt {3 x^2+5 x+2} (88-9 x)\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {1}{90} \left (\frac {955 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {761 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}\right )+\frac {1}{45} \sqrt {2 x+3} \sqrt {3 x^2+5 x+2} (88-9 x)\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {1}{90} \left (\frac {955 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {761 \sqrt {-3 x^2-5 x-2} E\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )+\frac {1}{45} \sqrt {2 x+3} \sqrt {3 x^2+5 x+2} (88-9 x)\) |
((88 - 9*x)*Sqrt[3 + 2*x]*Sqrt[2 + 5*x + 3*x^2])/45 + ((-761*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (955*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[ 1 + x]], -2/3])/(Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]))/90
3.26.79.3.1 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[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[(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/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]
Time = 0.42 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.99
| method | result | size |
| default | \(-\frac {\sqrt {3 x^{2}+5 x +2}\, \sqrt {3+2 x}\, \left (291 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )-761 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )+4860 x^{4}-32130 x^{3}-135090 x^{2}-145620 x -47520\right )}{4050 \left (6 x^{3}+19 x^{2}+19 x +6\right )}\) | \(141\) |
| risch | \(-\frac {\left (-88+9 x \right ) \sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}{45}-\frac {\left (\frac {332 \sqrt {45+30 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, F\left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )}{675 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}+\frac {761 \sqrt {45+30 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, \left (-\frac {E\left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )}{2}-F\left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )\right )}{1350 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right ) \sqrt {\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )}}{\sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(198\) |
| elliptic | \(\frac {\sqrt {\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {x \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{5}+\frac {88 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{45}+\frac {332 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{675 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}+\frac {761 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, \left (\frac {E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{3}-F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )\right )}{1350 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{\sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(209\) |
-1/4050*(3*x^2+5*x+2)^(1/2)*(3+2*x)^(1/2)*(291*(-20-30*x)^(1/2)*(3+3*x)^(1 /2)*15^(1/2)*(3+2*x)^(1/2)*EllipticF(1/5*(-20-30*x)^(1/2),1/2*10^(1/2))-76 1*(-20-30*x)^(1/2)*(3+3*x)^(1/2)*15^(1/2)*(3+2*x)^(1/2)*EllipticE(1/5*(-20 -30*x)^(1/2),1/2*10^(1/2))+4860*x^4-32130*x^3-135090*x^2-145620*x-47520)/( 6*x^3+19*x^2+19*x+6)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.36 \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{\sqrt {3+2 x}} \, dx=-\frac {1}{45} \, \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (9 \, x - 88\right )} \sqrt {2 \, x + 3} + \frac {2507}{4860} \, \sqrt {6} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + \frac {761}{270} \, \sqrt {6} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) \]
-1/45*sqrt(3*x^2 + 5*x + 2)*(9*x - 88)*sqrt(2*x + 3) + 2507/4860*sqrt(6)*w eierstrassPInverse(19/27, -28/729, x + 19/18) + 761/270*sqrt(6)*weierstras sZeta(19/27, -28/729, weierstrassPInverse(19/27, -28/729, x + 19/18))
\[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{\sqrt {3+2 x}} \, dx=- \int \left (- \frac {5 \sqrt {3 x^{2} + 5 x + 2}}{\sqrt {2 x + 3}}\right )\, dx - \int \frac {x \sqrt {3 x^{2} + 5 x + 2}}{\sqrt {2 x + 3}}\, dx \]
-Integral(-5*sqrt(3*x**2 + 5*x + 2)/sqrt(2*x + 3), x) - Integral(x*sqrt(3* x**2 + 5*x + 2)/sqrt(2*x + 3), x)
\[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{\sqrt {3+2 x}} \, dx=\int { -\frac {\sqrt {3 \, x^{2} + 5 \, x + 2} {\left (x - 5\right )}}{\sqrt {2 \, x + 3}} \,d x } \]
\[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{\sqrt {3+2 x}} \, dx=\int { -\frac {\sqrt {3 \, x^{2} + 5 \, x + 2} {\left (x - 5\right )}}{\sqrt {2 \, x + 3}} \,d x } \]
Timed out. \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{\sqrt {3+2 x}} \, dx=-\int \frac {\left (x-5\right )\,\sqrt {3\,x^2+5\,x+2}}{\sqrt {2\,x+3}} \,d x \]