Integrand size = 25, antiderivative size = 210 \[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{11/2}} \, dx=-\frac {5438 \sqrt {x} (2+3 x)}{315 \sqrt {2+5 x+3 x^2}}+\frac {5438 \sqrt {2+5 x+3 x^2}}{315 \sqrt {x}}+\frac {(1446+4055 x) \sqrt {2+5 x+3 x^2}}{315 x^{5/2}}-\frac {4 (7-15 x) \left (2+5 x+3 x^2\right )^{3/2}}{63 x^{9/2}}+\frac {5438 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{315 \sqrt {2+5 x+3 x^2}}-\frac {899 (1+x) \sqrt {\frac {2+3 x}{1+x}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{21 \sqrt {2} \sqrt {2+5 x+3 x^2}} \]
-4/63*(7-15*x)*(3*x^2+5*x+2)^(3/2)/x^(9/2)-5438/315*(2+3*x)*x^(1/2)/(3*x^2 +5*x+2)^(1/2)-899/42*(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)+5438 /315*(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)+1/315*(1446+4055*x)* (3*x^2+5*x+2)^(1/2)/x^(5/2)+5438/315*(3*x^2+5*x+2)^(1/2)/x^(1/2)
Result contains complex when optimal does not.
Time = 21.16 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.76 \[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{11/2}} \, dx=\frac {-1120-3200 x+7424 x^2+44480 x^3+64706 x^4+29730 x^5-10876 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{11/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )-2609 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{11/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right ),\frac {3}{2}\right )}{630 x^{9/2} \sqrt {2+5 x+3 x^2}} \]
(-1120 - 3200*x + 7424*x^2 + 44480*x^3 + 64706*x^4 + 29730*x^5 - (10876*I) *Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(11/2)*EllipticE[I*ArcSinh[Sqrt[ 2/3]/Sqrt[x]], 3/2] - (2609*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(1 1/2)*EllipticF[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2])/(630*x^(9/2)*Sqrt[2 + 5 *x + 3*x^2])
Time = 0.41 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.06, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {1229, 1229, 27, 1237, 27, 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.
| \(\displaystyle \int \frac {(2-5 x) \left (3 x^2+5 x+2\right )^{3/2}}{x^{11/2}} \, dx\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle -\frac {1}{21} \int \frac {(285 x+241) \sqrt {3 x^2+5 x+2}}{x^{7/2}}dx-\frac {4 (7-15 x) \left (3 x^2+5 x+2\right )^{3/2}}{63 x^{9/2}}\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle \frac {1}{21} \left (\frac {1}{15} \int -\frac {13485 x+10876}{2 x^{3/2} \sqrt {3 x^2+5 x+2}}dx+\frac {\sqrt {3 x^2+5 x+2} (4055 x+1446)}{15 x^{5/2}}\right )-\frac {4 (7-15 x) \left (3 x^2+5 x+2\right )^{3/2}}{63 x^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{21} \left (\frac {(4055 x+1446) \sqrt {3 x^2+5 x+2}}{15 x^{5/2}}-\frac {1}{30} \int \frac {13485 x+10876}{x^{3/2} \sqrt {3 x^2+5 x+2}}dx\right )-\frac {4 (7-15 x) \left (3 x^2+5 x+2\right )^{3/2}}{63 x^{9/2}}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {1}{21} \left (\frac {1}{30} \left (\int -\frac {3 (5438 x+4495)}{\sqrt {x} \sqrt {3 x^2+5 x+2}}dx+\frac {10876 \sqrt {3 x^2+5 x+2}}{\sqrt {x}}\right )+\frac {\sqrt {3 x^2+5 x+2} (4055 x+1446)}{15 x^{5/2}}\right )-\frac {4 (7-15 x) \left (3 x^2+5 x+2\right )^{3/2}}{63 x^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{21} \left (\frac {1}{30} \left (\frac {10876 \sqrt {3 x^2+5 x+2}}{\sqrt {x}}-3 \int \frac {5438 x+4495}{\sqrt {x} \sqrt {3 x^2+5 x+2}}dx\right )+\frac {\sqrt {3 x^2+5 x+2} (4055 x+1446)}{15 x^{5/2}}\right )-\frac {4 (7-15 x) \left (3 x^2+5 x+2\right )^{3/2}}{63 x^{9/2}}\) |
| \(\Big \downarrow \) 1240 |
| \(\displaystyle \frac {1}{21} \left (\frac {1}{30} \left (\frac {10876 \sqrt {3 x^2+5 x+2}}{\sqrt {x}}-6 \int \frac {5438 x+4495}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}\right )+\frac {\sqrt {3 x^2+5 x+2} (4055 x+1446)}{15 x^{5/2}}\right )-\frac {4 (7-15 x) \left (3 x^2+5 x+2\right )^{3/2}}{63 x^{9/2}}\) |
| \(\Big \downarrow \) 1503 |
| \(\displaystyle \frac {1}{21} \left (\frac {1}{30} \left (\frac {10876 \sqrt {3 x^2+5 x+2}}{\sqrt {x}}-6 \left (4495 \int \frac {1}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+5438 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}\right )\right )+\frac {\sqrt {3 x^2+5 x+2} (4055 x+1446)}{15 x^{5/2}}\right )-\frac {4 (7-15 x) \left (3 x^2+5 x+2\right )^{3/2}}{63 x^{9/2}}\) |
| \(\Big \downarrow \) 1413 |
| \(\displaystyle \frac {1}{21} \left (\frac {1}{30} \left (\frac {10876 \sqrt {3 x^2+5 x+2}}{\sqrt {x}}-6 \left (5438 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+\frac {4495 (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {2} \sqrt {3 x^2+5 x+2}}\right )\right )+\frac {\sqrt {3 x^2+5 x+2} (4055 x+1446)}{15 x^{5/2}}\right )-\frac {4 (7-15 x) \left (3 x^2+5 x+2\right )^{3/2}}{63 x^{9/2}}\) |
| \(\Big \downarrow \) 1456 |
| \(\displaystyle \frac {1}{21} \left (\frac {1}{30} \left (\frac {10876 \sqrt {3 x^2+5 x+2}}{\sqrt {x}}-6 \left (\frac {4495 (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {2} \sqrt {3 x^2+5 x+2}}+5438 \left (\frac {\sqrt {x} (3 x+2)}{3 \sqrt {3 x^2+5 x+2}}-\frac {\sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{3 \sqrt {3 x^2+5 x+2}}\right )\right )\right )+\frac {\sqrt {3 x^2+5 x+2} (4055 x+1446)}{15 x^{5/2}}\right )-\frac {4 (7-15 x) \left (3 x^2+5 x+2\right )^{3/2}}{63 x^{9/2}}\) |
(-4*(7 - 15*x)*(2 + 5*x + 3*x^2)^(3/2))/(63*x^(9/2)) + (((1446 + 4055*x)*S qrt[2 + 5*x + 3*x^2])/(15*x^(5/2)) + ((10876*Sqrt[2 + 5*x + 3*x^2])/Sqrt[x ] - 6*(5438*((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])) + (4495*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sq rt[x]], -1/2])/(Sqrt[2]*Sqrt[2 + 5*x + 3*x^2])))/30)/21
3.11.52.3.1 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/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2))*(c* d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x), x] - Simp[p/(e^2*(m + 1 )*(m + 2)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2 )^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m + 1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c *(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1) - b*(d*g*( m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g }, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] && !ILtQ[m + 2*p + 3, 0]
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[((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 = 0.27 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.64
| method | result | size |
| default | \(\frac {2829 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x^{4}-5438 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x^{4}+97884 x^{6}+252330 x^{5}+259374 x^{4}+133440 x^{3}+22272 x^{2}-9600 x -3360}{1890 \sqrt {3 x^{2}+5 x +2}\, x^{\frac {9}{2}}}\) | \(134\) |
| risch | \(\frac {16314 x^{6}+42055 x^{5}+43229 x^{4}+22240 x^{3}+3712 x^{2}-1600 x -560}{315 x^{\frac {9}{2}} \sqrt {3 x^{2}+5 x +2}}-\frac {\left (\frac {899 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{126 \sqrt {3 x^{3}+5 x^{2}+2 x}}+\frac {2719 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{315 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right ) \sqrt {x \left (3 x^{2}+5 x +2\right )}}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(208\) |
| elliptic | \(\frac {\sqrt {x \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {8 \sqrt {3 x^{3}+5 x^{2}+2 x}}{9 x^{5}}-\frac {20 \sqrt {3 x^{3}+5 x^{2}+2 x}}{63 x^{4}}+\frac {842 \sqrt {3 x^{3}+5 x^{2}+2 x}}{105 x^{3}}+\frac {991 \sqrt {3 x^{3}+5 x^{2}+2 x}}{63 x^{2}}+\frac {\frac {5438}{105} x^{2}+\frac {5438}{63} x +\frac {10876}{315}}{\sqrt {x \left (3 x^{2}+5 x +2\right )}}-\frac {899 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{126 \sqrt {3 x^{3}+5 x^{2}+2 x}}-\frac {2719 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{315 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right )}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(269\) |
1/1890*(2829*(6*x+4)^(1/2)*(3+3*x)^(1/2)*6^(1/2)*(-x)^(1/2)*EllipticF(1/2* (6*x+4)^(1/2),I*2^(1/2))*x^4-5438*(6*x+4)^(1/2)*(3+3*x)^(1/2)*6^(1/2)*(-x) ^(1/2)*EllipticE(1/2*(6*x+4)^(1/2),I*2^(1/2))*x^4+97884*x^6+252330*x^5+259 374*x^4+133440*x^3+22272*x^2-9600*x-3360)/(3*x^2+5*x+2)^(1/2)/x^(9/2)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.35 \[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{11/2}} \, dx=-\frac {13265 \, \sqrt {3} x^{5} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) - 48942 \, \sqrt {3} x^{5} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) - 9 \, {\left (5438 \, x^{4} + 4955 \, x^{3} + 2526 \, x^{2} - 100 \, x - 280\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}}{2835 \, x^{5}} \]
-1/2835*(13265*sqrt(3)*x^5*weierstrassPInverse(28/27, 80/729, x + 5/9) - 4 8942*sqrt(3)*x^5*weierstrassZeta(28/27, 80/729, weierstrassPInverse(28/27, 80/729, x + 5/9)) - 9*(5438*x^4 + 4955*x^3 + 2526*x^2 - 100*x - 280)*sqrt (3*x^2 + 5*x + 2)*sqrt(x))/x^5
\[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{11/2}} \, dx=- \int \left (- \frac {4 \sqrt {3 x^{2} + 5 x + 2}}{x^{\frac {11}{2}}}\right )\, dx - \int \frac {19 \sqrt {3 x^{2} + 5 x + 2}}{x^{\frac {7}{2}}}\, dx - \int \frac {15 \sqrt {3 x^{2} + 5 x + 2}}{x^{\frac {5}{2}}}\, dx \]
-Integral(-4*sqrt(3*x**2 + 5*x + 2)/x**(11/2), x) - Integral(19*sqrt(3*x** 2 + 5*x + 2)/x**(7/2), x) - Integral(15*sqrt(3*x**2 + 5*x + 2)/x**(5/2), x )
\[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{11/2}} \, dx=\int { -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (5 \, x - 2\right )}}{x^{\frac {11}{2}}} \,d x } \]
\[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{11/2}} \, dx=\int { -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (5 \, x - 2\right )}}{x^{\frac {11}{2}}} \,d x } \]
Timed out. \[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{11/2}} \, dx=\int -\frac {\left (5\,x-2\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{x^{11/2}} \,d x \]