Integrand size = 29, antiderivative size = 136 \[ \int \frac {5-x}{(3+2 x)^{3/2} \sqrt {2+5 x+3 x^2}} \, dx=-\frac {26 \sqrt {2+5 x+3 x^2}}{5 \sqrt {3+2 x}}+\frac {13 \sqrt {3} \sqrt {-2-5 x-3 x^2} E\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{5 \sqrt {2+5 x+3 x^2}}-\frac {\sqrt {-2-5 x-3 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {2+5 x+3 x^2}} \]
-1/3*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)+13/5*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)-26/5*(3*x^2+5*x+2)^(1/2) /(3+2*x)^(1/2)
Time = 31.27 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.76 \[ \int \frac {5-x}{(3+2 x)^{3/2} \sqrt {2+5 x+3 x^2}} \, dx=\frac {(1+x) \sqrt {\frac {2+3 x}{3+2 x}} \left (13 E\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )-12 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right ),\frac {3}{5}\right )\right )}{\sqrt {5} \sqrt {\frac {1+x}{3+2 x}} \sqrt {2+5 x+3 x^2}} \]
((1 + x)*Sqrt[(2 + 3*x)/(3 + 2*x)]*(13*EllipticE[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5] - 12*EllipticF[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5]))/(Sqrt[ 5]*Sqrt[(1 + x)/(3 + 2*x)]*Sqrt[2 + 5*x + 3*x^2])
Time = 0.29 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {1237, 27, 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.
| \(\displaystyle \int \frac {5-x}{(2 x+3)^{3/2} \sqrt {3 x^2+5 x+2}} \, dx\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -\frac {2}{5} \int -\frac {39 x+56}{2 \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx-\frac {26 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \int \frac {39 x+56}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx-\frac {26 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {1}{5} \left (\frac {39}{2} \int \frac {\sqrt {2 x+3}}{\sqrt {3 x^2+5 x+2}}dx-\frac {5}{2} \int \frac {1}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx\right )-\frac {26 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {1}{5} \left (\frac {13 \sqrt {3} \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {3} \sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3 x^2+5 x+2}}-\frac {5 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {3}}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {26 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \left (\frac {13 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3 x^2+5 x+2}}-\frac {5 \sqrt {-3 x^2-5 x-2} \int \frac {1}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3 x^2+5 x+2}}\right )-\frac {26 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {1}{5} \left (\frac {13 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3 x^2+5 x+2}}-\frac {5 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {26 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {1}{5} \left (\frac {13 \sqrt {3} \sqrt {-3 x^2-5 x-2} E\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{\sqrt {3 x^2+5 x+2}}-\frac {5 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {26 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}\) |
(-26*Sqrt[2 + 5*x + 3*x^2])/(5*Sqrt[3 + 2*x]) + ((13*Sqrt[3]*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/Sqrt[2 + 5*x + 3*x ^2] - (5*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]))/5
3.27.10.3.1 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[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b* x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1) *(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[ (c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 ] && (IntegerQ[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 = 0.34 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.96
| method | result | size |
| default | \(-\frac {\sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}\, \left (17 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )+13 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )+1170 x^{2}+1950 x +780\right )}{75 \left (6 x^{3}+19 x^{2}+19 x +6\right )}\) | \(131\) |
| elliptic | \(\frac {\sqrt {\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {13 \left (6 x^{2}+10 x +4\right )}{5 \sqrt {\left (x +\frac {3}{2}\right ) \left (6 x^{2}+10 x +4\right )}}-\frac {56 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{75 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}-\frac {13 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, \left (\frac {E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{3}-F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )\right )}{25 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{\sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(198\) |
-1/75*(3+2*x)^(1/2)*(3*x^2+5*x+2)^(1/2)*(17*(-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))+13*(- 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))+1170*x^2+1950*x+780)/(6*x^3+19*x^2+19*x+6)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.49 \[ \int \frac {5-x}{(3+2 x)^{3/2} \sqrt {2+5 x+3 x^2}} \, dx=\frac {89 \, \sqrt {6} {\left (2 \, x + 3\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) - 234 \, \sqrt {6} {\left (2 \, x + 3\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) - 468 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{90 \, {\left (2 \, x + 3\right )}} \]
1/90*(89*sqrt(6)*(2*x + 3)*weierstrassPInverse(19/27, -28/729, x + 19/18) - 234*sqrt(6)*(2*x + 3)*weierstrassZeta(19/27, -28/729, weierstrassPInvers e(19/27, -28/729, x + 19/18)) - 468*sqrt(3*x^2 + 5*x + 2)*sqrt(2*x + 3))/( 2*x + 3)
\[ \int \frac {5-x}{(3+2 x)^{3/2} \sqrt {2+5 x+3 x^2}} \, dx=- \int \frac {x}{2 x \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 3 \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {5}{2 x \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 3 \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \]
-Integral(x/(2*x*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 3*sqrt(2*x + 3)*sq rt(3*x**2 + 5*x + 2)), x) - Integral(-5/(2*x*sqrt(2*x + 3)*sqrt(3*x**2 + 5 *x + 2) + 3*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)), x)
\[ \int \frac {5-x}{(3+2 x)^{3/2} \sqrt {2+5 x+3 x^2}} \, dx=\int { -\frac {x - 5}{\sqrt {3 \, x^{2} + 5 \, x + 2} {\left (2 \, x + 3\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {5-x}{(3+2 x)^{3/2} \sqrt {2+5 x+3 x^2}} \, dx=\int { -\frac {x - 5}{\sqrt {3 \, x^{2} + 5 \, x + 2} {\left (2 \, x + 3\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {5-x}{(3+2 x)^{3/2} \sqrt {2+5 x+3 x^2}} \, dx=-\int \frac {x-5}{{\left (2\,x+3\right )}^{3/2}\,\sqrt {3\,x^2+5\,x+2}} \,d x \]