Integrand size = 29, antiderivative size = 141 \[ \int \frac {5-x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}} \, dx=-\frac {6 \sqrt {3+2 x} (37+47 x)}{5 \sqrt {2+5 x+3 x^2}}+\frac {94 \sqrt {3} \sqrt {-2-5 x-3 x^2} E\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{5 \sqrt {2+5 x+3 x^2}}-\frac {70 \sqrt {-2-5 x-3 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {2+5 x+3 x^2}} \]
-6/5*(37+47*x)*(3+2*x)^(1/2)/(3*x^2+5*x+2)^(1/2)-70/3*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)+ 94/5*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)
Time = 31.28 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.26 \[ \int \frac {5-x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}} \, dx=\frac {-10 \sqrt {3+2 x} (29+35 x)+94 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^2 \sqrt {\frac {2+3 x}{3+2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )-24 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^2 \sqrt {\frac {2+3 x}{3+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right ),\frac {3}{5}\right )}{5 (3+2 x) \sqrt {2+5 x+3 x^2}} \]
(-10*Sqrt[3 + 2*x]*(29 + 35*x) + 94*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2 *x)^2*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticE[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5] - 24*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])/(5*(3 + 2*x)*Sqrt[ 2 + 5*x + 3*x^2])
Time = 0.29 (sec) , antiderivative size = 144, 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 = {1235, 25, 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.
| \(\displaystyle \int \frac {5-x}{\sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle -\frac {2}{5} \int -\frac {141 x+124}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx-\frac {6 \sqrt {2 x+3} (47 x+37)}{5 \sqrt {3 x^2+5 x+2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2}{5} \int \frac {141 x+124}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx-\frac {6 \sqrt {2 x+3} (47 x+37)}{5 \sqrt {3 x^2+5 x+2}}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {2}{5} \left (\frac {141}{2} \int \frac {\sqrt {2 x+3}}{\sqrt {3 x^2+5 x+2}}dx-\frac {175}{2} \int \frac {1}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx\right )-\frac {6 \sqrt {2 x+3} (47 x+37)}{5 \sqrt {3 x^2+5 x+2}}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {2}{5} \left (\frac {47 \sqrt {3} \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {3} \sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3 x^2+5 x+2}}-\frac {175 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {3}}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {6 \sqrt {2 x+3} (47 x+37)}{5 \sqrt {3 x^2+5 x+2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{5} \left (\frac {47 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3 x^2+5 x+2}}-\frac {175 \sqrt {-3 x^2-5 x-2} \int \frac {1}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3 x^2+5 x+2}}\right )-\frac {6 \sqrt {2 x+3} (47 x+37)}{5 \sqrt {3 x^2+5 x+2}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {2}{5} \left (\frac {47 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3 x^2+5 x+2}}-\frac {175 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {6 \sqrt {2 x+3} (47 x+37)}{5 \sqrt {3 x^2+5 x+2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {2}{5} \left (\frac {47 \sqrt {3} \sqrt {-3 x^2-5 x-2} E\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{\sqrt {3 x^2+5 x+2}}-\frac {175 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {6 \sqrt {2 x+3} (47 x+37)}{5 \sqrt {3 x^2+5 x+2}}\) |
(-6*Sqrt[3 + 2*x]*(37 + 47*x))/(5*Sqrt[2 + 5*x + 3*x^2]) + (2*((47*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] - (175*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*S qrt[1 + x]], -2/3])/(Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])))/5
3.27.17.3.1 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)*(f*(b*c*d - b^2*e + 2 *a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m *(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && LtQ[p, -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 = 0.33 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.93
| method | result | size |
| default | \(\frac {2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3+2 x}\, \left (17 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )-47 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )-4230 x^{2}-9675 x -4995\right )}{75 \left (6 x^{3}+19 x^{2}+19 x +6\right )}\) | \(131\) |
| elliptic | \(\frac {\sqrt {\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {2 \left (9+6 x \right ) \left (\frac {37}{5}+\frac {47 x}{5}\right )}{\sqrt {\left (x^{2}+\frac {5}{3} x +\frac {2}{3}\right ) \left (9+6 x \right )}}-\frac {248 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{75 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}-\frac {94 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, \left (\frac {E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{3}-F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )\right )}{25 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{\sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(198\) |
2/75*(3*x^2+5*x+2)^(1/2)*(3+2*x)^(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))-47*(-2 0-30*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))-4230*x^2-9675*x-4995)/(6*x^3+19*x^2+19*x+6)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.61 \[ \int \frac {5-x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}} \, dx=-\frac {149 \, \sqrt {6} {\left (3 \, x^{2} + 5 \, x + 2\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + 846 \, \sqrt {6} {\left (3 \, x^{2} + 5 \, x + 2\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) + 54 \, \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (47 \, x + 37\right )} \sqrt {2 \, x + 3}}{45 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}} \]
-1/45*(149*sqrt(6)*(3*x^2 + 5*x + 2)*weierstrassPInverse(19/27, -28/729, x + 19/18) + 846*sqrt(6)*(3*x^2 + 5*x + 2)*weierstrassZeta(19/27, -28/729, weierstrassPInverse(19/27, -28/729, x + 19/18)) + 54*sqrt(3*x^2 + 5*x + 2) *(47*x + 37)*sqrt(2*x + 3))/(3*x^2 + 5*x + 2)
\[ \int \frac {5-x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}} \, dx=- \int \frac {x}{3 x^{2} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 5 x \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 2 \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {5}{3 x^{2} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 5 x \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 2 \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \]
-Integral(x/(3*x**2*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 5*x*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 2*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)), x) - Integral(-5/(3*x**2*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 5*x*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 2*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)), x)
\[ \int \frac {5-x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}} \, dx=\int { -\frac {x - 5}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} \sqrt {2 \, x + 3}} \,d x } \]
\[ \int \frac {5-x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}} \, dx=\int { -\frac {x - 5}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} \sqrt {2 \, x + 3}} \,d x } \]
Timed out. \[ \int \frac {5-x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}} \, dx=-\int \frac {x-5}{\sqrt {2\,x+3}\,{\left (3\,x^2+5\,x+2\right )}^{3/2}} \,d x \]