Integrand size = 25, antiderivative size = 210 \[ \int \frac {(2-5 x) x^{9/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2 x^{7/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {17512 \sqrt {x} (2+3 x)}{243 \sqrt {2+5 x+3 x^2}}-\frac {4 x^{3/2} (536+645 x)}{9 \sqrt {2+5 x+3 x^2}}+\frac {7540}{81} \sqrt {x} \sqrt {2+5 x+3 x^2}+\frac {17512 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{243 \sqrt {2+5 x+3 x^2}}-\frac {7540 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{81 \sqrt {2+5 x+3 x^2}} \]
2/9*x^(7/2)*(74+95*x)/(3*x^2+5*x+2)^(3/2)-4/9*x^(3/2)*(536+645*x)/(3*x^2+5 *x+2)^(1/2)-17512/243*(2+3*x)*x^(1/2)/(3*x^2+5*x+2)^(1/2)+17512/243*(1+x)^ (3/2)*(1/(1+x))^(1/2)*EllipticE(x^(1/2)/(1+x)^(1/2),1/2*I*2^(1/2))*2^(1/2) *((2+3*x)/(1+x))^(1/2)/(3*x^2+5*x+2)^(1/2)-7540/81*(1+x)^(3/2)*(1/(1+x))^( 1/2)*EllipticF(x^(1/2)/(1+x)^(1/2),1/2*I*2^(1/2))*2^(1/2)*((2+3*x)/(1+x))^ (1/2)/(3*x^2+5*x+2)^(1/2)+7540/81*x^(1/2)*(3*x^2+5*x+2)^(1/2)
Result contains complex when optimal does not.
Time = 21.97 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.84 \[ \int \frac {(2-5 x) x^{9/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {-2 \left (35024+129880 x+155660 x^2+58590 x^3-1512 x^4+135 x^5\right )-17512 i \sqrt {2+\frac {2}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} \left (2+5 x+3 x^2\right ) E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )-5108 i \sqrt {2+\frac {2}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} \left (2+5 x+3 x^2\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right ),\frac {3}{2}\right )}{243 \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2}} \]
(-2*(35024 + 129880*x + 155660*x^2 + 58590*x^3 - 1512*x^4 + 135*x^5) - (17 512*I)*Sqrt[2 + 2/x]*Sqrt[3 + 2/x]*x^(3/2)*(2 + 5*x + 3*x^2)*EllipticE[I*A rcSinh[Sqrt[2/3]/Sqrt[x]], 3/2] - (5108*I)*Sqrt[2 + 2/x]*Sqrt[3 + 2/x]*x^( 3/2)*(2 + 5*x + 3*x^2)*EllipticF[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2])/(243* Sqrt[x]*(2 + 5*x + 3*x^2)^(3/2))
Time = 0.41 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.04, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {1233, 25, 1233, 27, 1236, 25, 1240, 1503, 1413, 1456}
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.
| \(\displaystyle \int \frac {(2-5 x) x^{9/2}}{\left (3 x^2+5 x+2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1233 |
| \(\displaystyle \frac {2}{9} \int -\frac {x^{5/2} (150 x+259)}{\left (3 x^2+5 x+2\right )^{3/2}}dx+\frac {2 (95 x+74) x^{7/2}}{9 \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 x^{7/2} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \int \frac {x^{5/2} (150 x+259)}{\left (3 x^2+5 x+2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 1233 |
| \(\displaystyle \frac {2 x^{7/2} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \left (\frac {2}{3} \int -\frac {3 \sqrt {x} (1885 x+1608)}{2 \sqrt {3 x^2+5 x+2}}dx+\frac {2 (645 x+536) x^{3/2}}{\sqrt {3 x^2+5 x+2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 x^{7/2} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \left (\frac {2 x^{3/2} (645 x+536)}{\sqrt {3 x^2+5 x+2}}-\int \frac {\sqrt {x} (1885 x+1608)}{\sqrt {3 x^2+5 x+2}}dx\right )\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {2 x^{7/2} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \left (-\frac {2}{9} \int -\frac {2189 x+1885}{\sqrt {x} \sqrt {3 x^2+5 x+2}}dx-\frac {3770}{9} \sqrt {3 x^2+5 x+2} \sqrt {x}+\frac {2 (645 x+536) x^{3/2}}{\sqrt {3 x^2+5 x+2}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 x^{7/2} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \left (\frac {2}{9} \int \frac {2189 x+1885}{\sqrt {x} \sqrt {3 x^2+5 x+2}}dx-\frac {3770}{9} \sqrt {3 x^2+5 x+2} \sqrt {x}+\frac {2 (645 x+536) x^{3/2}}{\sqrt {3 x^2+5 x+2}}\right )\) |
| \(\Big \downarrow \) 1240 |
| \(\displaystyle \frac {2 x^{7/2} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \left (\frac {4}{9} \int \frac {2189 x+1885}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}-\frac {3770}{9} \sqrt {3 x^2+5 x+2} \sqrt {x}+\frac {2 (645 x+536) x^{3/2}}{\sqrt {3 x^2+5 x+2}}\right )\) |
| \(\Big \downarrow \) 1503 |
| \(\displaystyle \frac {2 x^{7/2} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \left (\frac {4}{9} \left (1885 \int \frac {1}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+2189 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}\right )-\frac {3770}{9} \sqrt {3 x^2+5 x+2} \sqrt {x}+\frac {2 (645 x+536) x^{3/2}}{\sqrt {3 x^2+5 x+2}}\right )\) |
| \(\Big \downarrow \) 1413 |
| \(\displaystyle \frac {2 x^{7/2} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \left (\frac {4}{9} \left (2189 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+\frac {1885 (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {2} \sqrt {3 x^2+5 x+2}}\right )-\frac {3770}{9} \sqrt {3 x^2+5 x+2} \sqrt {x}+\frac {2 (645 x+536) x^{3/2}}{\sqrt {3 x^2+5 x+2}}\right )\) |
| \(\Big \downarrow \) 1456 |
| \(\displaystyle \frac {2 x^{7/2} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \left (\frac {4}{9} \left (\frac {1885 (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {2} \sqrt {3 x^2+5 x+2}}+2189 \left (\frac {\sqrt {x} (3 x+2)}{3 \sqrt {3 x^2+5 x+2}}-\frac {\sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{3 \sqrt {3 x^2+5 x+2}}\right )\right )-\frac {3770}{9} \sqrt {3 x^2+5 x+2} \sqrt {x}+\frac {2 (645 x+536) x^{3/2}}{\sqrt {3 x^2+5 x+2}}\right )\) |
(2*x^(7/2)*(74 + 95*x))/(9*(2 + 5*x + 3*x^2)^(3/2)) - (2*((2*x^(3/2)*(536 + 645*x))/Sqrt[2 + 5*x + 3*x^2] - (3770*Sqrt[x]*Sqrt[2 + 5*x + 3*x^2])/9 + (4*(2189*((Sqrt[x]*(2 + 3*x))/(3*Sqrt[2 + 5*x + 3*x^2]) - (Sqrt[2]*(1 + x )*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/(3*Sqrt[2 + 5* x + 3*x^2])) + (1885*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt [x]], -1/2])/(Sqrt[2]*Sqrt[2 + 5*x + 3*x^2])))/9))/9
3.11.74.3.1 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[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m - 1))*(a + b*x + c*x^2) ^(p + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g - c *(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Simp[1/(c*( p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Sim p[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a*e*(e*f *(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*( m + p + 1) + 2*c^2*d*f*(m + 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2* p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, b, c, d, e, f, g]) | | !ILtQ[m + 2*p + 3, 0])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Simp[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1 )*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m *(c*e*f + c*d*g - b*e*g) + e*(p + 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[ {a, b, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && EqQ[f, 0])
Int[((f_) + (g_.)*(x_))/(Sqrt[x_]*Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[2 Subst[Int[(f + g*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x, Sqrt[x]], x] /; FreeQ[{a, b, c, f, g}, x]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b ^2 - 4*a*c, 2]}, Simp[(2*a + (b - q)*x^2)*(Sqrt[(2*a + (b + q)*x^2)/(2*a + (b - q)*x^2)]/(2*a*Rt[(b - q)/(2*a), 2]*Sqrt[a + b*x^2 + c*x^4]))*EllipticF [ArcTan[Rt[(b - q)/(2*a), 2]*x], -2*(q/(b - q))], x] /; PosQ[(b - q)/a]] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]
Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[x*((b - q + 2*c*x^2)/(2*c*Sqrt[a + b*x^2 + c*x^4 ])), x] - Simp[Rt[(b - q)/(2*a), 2]*(2*a + (b - q)*x^2)*(Sqrt[(2*a + (b + q )*x^2)/(2*a + (b - q)*x^2)]/(2*c*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[ArcTan [Rt[(b - q)/(2*a), 2]*x], -2*(q/(b - q))], x] /; PosQ[(b - q)/a]] /; FreeQ[ {a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[d Int[1/Sqrt[a + b*x^2 + c*x^4] , x], x] + Simp[e Int[x^2/Sqrt[a + b*x^2 + c*x^4], x], x] /; PosQ[(b + q) /a] || PosQ[(b - q)/a]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0]
Time = 0.21 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.10
| method | result | size |
| elliptic | \(\frac {\sqrt {x \left (3 x^{2}+5 x +2\right )}\, \left (\frac {\left (-\frac {2780}{2187}-\frac {3914 x}{2187}\right ) \sqrt {3 x^{3}+5 x^{2}+2 x}}{\left (x^{2}+\frac {5}{3} x +\frac {2}{3}\right )^{2}}-\frac {2 x \left (-\frac {23345}{729}-\frac {8974 x}{243}\right ) \sqrt {3}}{\sqrt {x \left (x^{2}+\frac {5}{3} x +\frac {2}{3}\right )}}-\frac {10 \sqrt {3 x^{3}+5 x^{2}+2 x}}{81}-\frac {7540 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{243 \sqrt {3 x^{3}+5 x^{2}+2 x}}-\frac {8756 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{243 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right )}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(232\) |
| default | \(\frac {2 \sqrt {3 x^{2}+5 x +2}\, \left (5472 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x^{2}-13134 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x^{2}+9120 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x -21890 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x +3648 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )-8756 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )-405 x^{5}+240948 x^{4}+612270 x^{3}+504936 x^{2}+135720 x \right )}{729 \sqrt {x}\, \left (2+3 x \right )^{2} \left (1+x \right )^{2}}\) | \(302\) |
(x*(3*x^2+5*x+2))^(1/2)/x^(1/2)/(3*x^2+5*x+2)^(1/2)*((-2780/2187-3914/2187 *x)*(3*x^3+5*x^2+2*x)^(1/2)/(x^2+5/3*x+2/3)^2-2*x*(-23345/729-8974/243*x)* 3^(1/2)/(x*(x^2+5/3*x+2/3))^(1/2)-10/81*(3*x^3+5*x^2+2*x)^(1/2)-7540/243*( 6*x+4)^(1/2)*(3+3*x)^(1/2)*(-6*x)^(1/2)/(3*x^3+5*x^2+2*x)^(1/2)*EllipticF( 1/2*(6*x+4)^(1/2),I*2^(1/2))-8756/243*(6*x+4)^(1/2)*(3+3*x)^(1/2)*(-6*x)^( 1/2)/(3*x^3+5*x^2+2*x)^(1/2)*(1/3*EllipticE(1/2*(6*x+4)^(1/2),I*2^(1/2))-E llipticF(1/2*(6*x+4)^(1/2),I*2^(1/2))))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.60 \[ \int \frac {(2-5 x) x^{9/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left (24080 \, \sqrt {3} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) - 78804 \, \sqrt {3} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) + 27 \, {\left (45 \, x^{4} - 26772 \, x^{3} - 68030 \, x^{2} - 56104 \, x - 15080\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}\right )}}{2187 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \]
-2/2187*(24080*sqrt(3)*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*weierstrassPIn verse(28/27, 80/729, x + 5/9) - 78804*sqrt(3)*(9*x^4 + 30*x^3 + 37*x^2 + 2 0*x + 4)*weierstrassZeta(28/27, 80/729, weierstrassPInverse(28/27, 80/729, x + 5/9)) + 27*(45*x^4 - 26772*x^3 - 68030*x^2 - 56104*x - 15080)*sqrt(3* x^2 + 5*x + 2)*sqrt(x))/(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)
Timed out. \[ \int \frac {(2-5 x) x^{9/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(2-5 x) x^{9/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\int { -\frac {{\left (5 \, x - 2\right )} x^{\frac {9}{2}}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {(2-5 x) x^{9/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\int { -\frac {{\left (5 \, x - 2\right )} x^{\frac {9}{2}}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {(2-5 x) x^{9/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\int \frac {x^{9/2}\,\left (5\,x-2\right )}{{\left (3\,x^2+5\,x+2\right )}^{5/2}} \,d x \]