Integrand size = 29, antiderivative size = 137 \[ \int \frac {(5-x) \sqrt {3+2 x}}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {3+2 x} (29+35 x)}{\sqrt {2+5 x+3 x^2}}+\frac {70 \sqrt {-2-5 x-3 x^2} E\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{\sqrt {3} \sqrt {2+5 x+3 x^2}}-\frac {94 \sqrt {-2-5 x-3 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {2+5 x+3 x^2}} \]
-2*(29+35*x)*(3+2*x)^(1/2)/(3*x^2+5*x+2)^(1/2)+70/3*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)-94 /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)
Time = 31.27 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.30 \[ \int \frac {(5-x) \sqrt {3+2 x}}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx=-\frac {10 \sqrt {3+2 x} (121+139 x)-350 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^2 \sqrt {\frac {2+3 x}{3+2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )+68 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^2 \sqrt {\frac {2+3 x}{3+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right ),\frac {3}{5}\right )}{15 (3+2 x) \sqrt {2+5 x+3 x^2}} \]
-1/15*(10*Sqrt[3 + 2*x]*(121 + 139*x) - 350*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] + 68*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 = 140, 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 = {1234, 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.
\(\displaystyle \int \frac {(5-x) \sqrt {2 x+3}}{\left (3 x^2+5 x+2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 1234 |
\(\displaystyle -2 \int -\frac {35 x+29}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx-\frac {2 \sqrt {2 x+3} (35 x+29)}{\sqrt {3 x^2+5 x+2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle 2 \int \frac {35 x+29}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx-\frac {2 \sqrt {2 x+3} (35 x+29)}{\sqrt {3 x^2+5 x+2}}\) |
\(\Big \downarrow \) 1269 |
\(\displaystyle 2 \left (\frac {35}{2} \int \frac {\sqrt {2 x+3}}{\sqrt {3 x^2+5 x+2}}dx-\frac {47}{2} \int \frac {1}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx\right )-\frac {2 \sqrt {2 x+3} (35 x+29)}{\sqrt {3 x^2+5 x+2}}\) |
\(\Big \downarrow \) 1172 |
\(\displaystyle 2 \left (\frac {35 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {3} \sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {47 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {3}}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {2 \sqrt {2 x+3} (35 x+29)}{\sqrt {3 x^2+5 x+2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \left (\frac {35 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}-\frac {47 \sqrt {-3 x^2-5 x-2} \int \frac {1}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3 x^2+5 x+2}}\right )-\frac {2 \sqrt {2 x+3} (35 x+29)}{\sqrt {3 x^2+5 x+2}}\) |
\(\Big \downarrow \) 321 |
\(\displaystyle 2 \left (\frac {35 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}-\frac {47 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {2 \sqrt {2 x+3} (35 x+29)}{\sqrt {3 x^2+5 x+2}}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle 2 \left (\frac {35 \sqrt {-3 x^2-5 x-2} E\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {47 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {2 \sqrt {2 x+3} (35 x+29)}{\sqrt {3 x^2+5 x+2}}\) |
(-2*Sqrt[3 + 2*x]*(29 + 35*x))/Sqrt[2 + 5*x + 3*x^2] + 2*((35*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]) - (47*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]))
3.27.16.3.1 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*(a + b*x + c*x^2)^(p + 1)*( (f*b - 2*a*g + (2*c*f - b*g)*x)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*Simp[g *(2*a*e*m + b*d*(2*p + 3)) - f*(b*e*m + 2*c*d*(2*p + 3)) - e*(2*c*f - b*g)* (m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1 ] && GtQ[m, 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[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 | \(\frac {2 \sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}\, \left (18 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )-35 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )-3150 x^{2}-7335 x -3915\right )}{45 \left (6 x^{3}+19 x^{2}+19 x +6\right )}\) | \(131\) |
elliptic | \(\frac {\sqrt {\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {2 \left (9+6 x \right ) \left (\frac {29}{3}+\frac {35 x}{3}\right )}{\sqrt {\left (x^{2}+\frac {5}{3} x +\frac {2}{3}\right ) \left (9+6 x \right )}}-\frac {58 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{15 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}-\frac {14 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, \left (\frac {E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{3}-F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )\right )}{3 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{\sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(198\) |
2/45*(3+2*x)^(1/2)*(3*x^2+5*x+2)^(1/2)*(18*(-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))-35*(-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))-3150*x^2-7335*x-3915)/(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 = 0.63 \[ \int \frac {(5-x) \sqrt {3+2 x}}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx=-\frac {143 \, \sqrt {6} {\left (3 \, x^{2} + 5 \, x + 2\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + 630 \, \sqrt {6} {\left (3 \, x^{2} + 5 \, x + 2\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) + 54 \, \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (35 \, x + 29\right )} \sqrt {2 \, x + 3}}{27 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}} \]
-1/27*(143*sqrt(6)*(3*x^2 + 5*x + 2)*weierstrassPInverse(19/27, -28/729, x + 19/18) + 630*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) *(35*x + 29)*sqrt(2*x + 3))/(3*x^2 + 5*x + 2)
\[ \int \frac {(5-x) \sqrt {3+2 x}}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx=- \int \left (- \frac {5 \sqrt {2 x + 3}}{3 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 5 x \sqrt {3 x^{2} + 5 x + 2} + 2 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac {x \sqrt {2 x + 3}}{3 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 5 x \sqrt {3 x^{2} + 5 x + 2} + 2 \sqrt {3 x^{2} + 5 x + 2}}\, dx \]
-Integral(-5*sqrt(2*x + 3)/(3*x**2*sqrt(3*x**2 + 5*x + 2) + 5*x*sqrt(3*x** 2 + 5*x + 2) + 2*sqrt(3*x**2 + 5*x + 2)), x) - Integral(x*sqrt(2*x + 3)/(3 *x**2*sqrt(3*x**2 + 5*x + 2) + 5*x*sqrt(3*x**2 + 5*x + 2) + 2*sqrt(3*x**2 + 5*x + 2)), x)
\[ \int \frac {(5-x) \sqrt {3+2 x}}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx=\int { -\frac {\sqrt {2 \, x + 3} {\left (x - 5\right )}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(5-x) \sqrt {3+2 x}}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx=\int { -\frac {\sqrt {2 \, x + 3} {\left (x - 5\right )}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(5-x) \sqrt {3+2 x}}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx=-\int \frac {\sqrt {2\,x+3}\,\left (x-5\right )}{{\left (3\,x^2+5\,x+2\right )}^{3/2}} \,d x \]