Integrand size = 29, antiderivative size = 173 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{3/2}} \, dx=-\frac {1}{210} (136-2493 x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}-\frac {(47+x) \left (2+5 x+3 x^2\right )^{3/2}}{7 \sqrt {3+2 x}}+\frac {2411 \sqrt {-2-5 x-3 x^2} E\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{60 \sqrt {3} \sqrt {2+5 x+3 x^2}}-\frac {4427 \sqrt {-2-5 x-3 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right ),-\frac {2}{3}\right )}{84 \sqrt {3} \sqrt {2+5 x+3 x^2}} \]
-1/7*(47+x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^(1/2)+2411/180*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) -4427/252*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)-1/210*(136-2493*x)*(3+2*x)^(1/2)*(3*x^2+5*x+ 2)^(1/2)
Time = 27.92 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.11 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{3/2}} \, dx=\frac {28772+73094 x+53340 x^2+18846 x^3+8208 x^4-1620 x^5+16877 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{3/2} \sqrt {\frac {2+3 x}{3+2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )-3596 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{3/2} \sqrt {\frac {2+3 x}{3+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right ),\frac {3}{5}\right )}{1260 \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \]
(28772 + 73094*x + 53340*x^2 + 18846*x^3 + 8208*x^4 - 1620*x^5 + 16877*Sqr t[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^(3/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*Ell ipticE[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5] - 3596*Sqrt[5]*Sqrt[(1 + x)/( 3 + 2*x)]*(3 + 2*x)^(3/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticF[ArcSin[Sqrt[ 5/3]/Sqrt[3 + 2*x]], 3/5])/(1260*Sqrt[3 + 2*x]*Sqrt[2 + 5*x + 3*x^2])
Time = 0.33 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {1230, 25, 1231, 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.
| \(\displaystyle \int \frac {(5-x) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle -\frac {3}{14} \int -\frac {(277 x+231) \sqrt {3 x^2+5 x+2}}{\sqrt {2 x+3}}dx-\frac {(x+47) \left (3 x^2+5 x+2\right )^{3/2}}{7 \sqrt {2 x+3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3}{14} \int \frac {(277 x+231) \sqrt {3 x^2+5 x+2}}{\sqrt {2 x+3}}dx-\frac {(x+47) \left (3 x^2+5 x+2\right )^{3/2}}{7 \sqrt {2 x+3}}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {3}{14} \left (-\frac {1}{90} \int -\frac {16877 x+14248}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx-\frac {1}{45} \sqrt {2 x+3} \sqrt {3 x^2+5 x+2} (136-2493 x)\right )-\frac {(x+47) \left (3 x^2+5 x+2\right )^{3/2}}{7 \sqrt {2 x+3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3}{14} \left (\frac {1}{90} \int \frac {16877 x+14248}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx-\frac {1}{45} (136-2493 x) \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}\right )-\frac {(x+47) \left (3 x^2+5 x+2\right )^{3/2}}{7 \sqrt {2 x+3}}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {3}{14} \left (\frac {1}{90} \left (\frac {16877}{2} \int \frac {\sqrt {2 x+3}}{\sqrt {3 x^2+5 x+2}}dx-\frac {22135}{2} \int \frac {1}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx\right )-\frac {1}{45} (136-2493 x) \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}\right )-\frac {(x+47) \left (3 x^2+5 x+2\right )^{3/2}}{7 \sqrt {2 x+3}}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {3}{14} \left (\frac {1}{90} \left (\frac {16877 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {3} \sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {22135 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {3}}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {1}{45} (136-2493 x) \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}\right )-\frac {(x+47) \left (3 x^2+5 x+2\right )^{3/2}}{7 \sqrt {2 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{14} \left (\frac {1}{90} \left (\frac {16877 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}-\frac {22135 \sqrt {-3 x^2-5 x-2} \int \frac {1}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3 x^2+5 x+2}}\right )-\frac {1}{45} (136-2493 x) \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}\right )-\frac {(x+47) \left (3 x^2+5 x+2\right )^{3/2}}{7 \sqrt {2 x+3}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {3}{14} \left (\frac {1}{90} \left (\frac {16877 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}-\frac {22135 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {1}{45} (136-2493 x) \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}\right )-\frac {(x+47) \left (3 x^2+5 x+2\right )^{3/2}}{7 \sqrt {2 x+3}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {3}{14} \left (\frac {1}{90} \left (\frac {16877 \sqrt {-3 x^2-5 x-2} E\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {22135 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {1}{45} (136-2493 x) \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}\right )-\frac {(x+47) \left (3 x^2+5 x+2\right )^{3/2}}{7 \sqrt {2 x+3}}\) |
-1/7*((47 + x)*(2 + 5*x + 3*x^2)^(3/2))/Sqrt[3 + 2*x] + (3*(-1/45*((136 - 2493*x)*Sqrt[3 + 2*x]*Sqrt[2 + 5*x + 3*x^2]) + ((16877*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]) - (22135*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))/14
3.26.88.3.1 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)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Simp[p/(e^2*(m + 1)*(m + 2*p + 2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1)*Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m + 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, - 1] || EqQ[p, 1] || (IntegerQ[p] && !RationalQ[m])) && NeQ[m, -1] && !ILtQ [m + 2*p + 1, 0] && (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[(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 = 0.34 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.84
| method | result | size |
| default | \(\frac {\sqrt {3 x^{2}+5 x +2}\, \sqrt {3+2 x}\, \left (7887 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )-16877 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )-24300 x^{5}+123120 x^{4}+282690 x^{3}-718830 x^{2}-1435140 x -581040\right )}{113400 x^{3}+359100 x^{2}+359100 x +113400}\) | \(146\) |
| elliptic | \(\frac {\sqrt {\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {65 \left (6 x^{2}+10 x +4\right )}{16 \sqrt {\left (x +\frac {3}{2}\right ) \left (6 x^{2}+10 x +4\right )}}-\frac {3 x^{2} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{14}+\frac {247 x \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{140}-\frac {2029 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{840}-\frac {3562 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{1575 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}-\frac {2411 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, \left (\frac {E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{3}-F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )\right )}{900 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{\sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(259\) |
1/18900*(3*x^2+5*x+2)^(1/2)*(3+2*x)^(1/2)*(7887*(-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))-1 6877*(-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))-24300*x^5+123120*x^4+282690*x^3-718830*x^2-1 435140*x-581040)/(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 = 81, normalized size of antiderivative = 0.47 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{3/2}} \, dx=-\frac {64199 \, \sqrt {6} {\left (2 \, x + 3\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + 303786 \, \sqrt {6} {\left (2 \, x + 3\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) + 108 \, {\left (90 \, x^{3} - 606 \, x^{2} - 97 \, x + 3228\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{22680 \, {\left (2 \, x + 3\right )}} \]
-1/22680*(64199*sqrt(6)*(2*x + 3)*weierstrassPInverse(19/27, -28/729, x + 19/18) + 303786*sqrt(6)*(2*x + 3)*weierstrassZeta(19/27, -28/729, weierstr assPInverse(19/27, -28/729, x + 19/18)) + 108*(90*x^3 - 606*x^2 - 97*x + 3 228)*sqrt(3*x^2 + 5*x + 2)*sqrt(2*x + 3))/(2*x + 3)
\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{3/2}} \, dx=- \int \left (- \frac {10 \sqrt {3 x^{2} + 5 x + 2}}{2 x \sqrt {2 x + 3} + 3 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {23 x \sqrt {3 x^{2} + 5 x + 2}}{2 x \sqrt {2 x + 3} + 3 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {10 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{2 x \sqrt {2 x + 3} + 3 \sqrt {2 x + 3}}\right )\, dx - \int \frac {3 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{2 x \sqrt {2 x + 3} + 3 \sqrt {2 x + 3}}\, dx \]
-Integral(-10*sqrt(3*x**2 + 5*x + 2)/(2*x*sqrt(2*x + 3) + 3*sqrt(2*x + 3)) , x) - Integral(-23*x*sqrt(3*x**2 + 5*x + 2)/(2*x*sqrt(2*x + 3) + 3*sqrt(2 *x + 3)), x) - Integral(-10*x**2*sqrt(3*x**2 + 5*x + 2)/(2*x*sqrt(2*x + 3) + 3*sqrt(2*x + 3)), x) - Integral(3*x**3*sqrt(3*x**2 + 5*x + 2)/(2*x*sqrt (2*x + 3) + 3*sqrt(2*x + 3)), x)
\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{3/2}} \, dx=\int { -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{3/2}} \, dx=\int { -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{3/2}} \, dx=-\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{{\left (2\,x+3\right )}^{3/2}} \,d x \]