Integrand size = 25, antiderivative size = 183 \[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{3/2}} \, dx=\frac {5848 \sqrt {x} (2+3 x)}{315 \sqrt {2+5 x+3 x^2}}+\frac {2}{105} \sqrt {x} (1045+531 x) \sqrt {2+5 x+3 x^2}-\frac {2 (14+5 x) \left (2+5 x+3 x^2\right )^{3/2}}{7 \sqrt {x}}-\frac {5848 \sqrt {2} \sqrt {2+5 x+3 x^2} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{315 \sqrt {1+x} \sqrt {2+3 x}}+\frac {482 \sqrt {2} \sqrt {1+x} \sqrt {2+3 x} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{21 \sqrt {2+5 x+3 x^2}} \] Output:
5848/315*x^(1/2)*(2+3*x)/(3*x^2+5*x+2)^(1/2)+2/105*x^(1/2)*(1045+531*x)*(3 *x^2+5*x+2)^(1/2)-2/7*(14+5*x)*(3*x^2+5*x+2)^(3/2)/x^(1/2)-5848/315*2^(1/2 )*(3*x^2+5*x+2)^(1/2)*EllipticE(x^(1/2)/(1+x)^(1/2),1/2*I*2^(1/2))/(1+x)^( 1/2)/(2+3*x)^(1/2)+482/21*2^(1/2)*(1+x)^(1/2)*(2+3*x)^(1/2)*InverseJacobiA M(arctan(x^(1/2)),1/2*I*2^(1/2))/(3*x^2+5*x+2)^(1/2)
Result contains complex when optimal does not.
Time = 21.22 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.89 \[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{3/2}} \, dx=\frac {-2 \left (-3328-7390 x+177 x^2+9855 x^3+7641 x^4+2025 x^5\right )+5848 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )+1382 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right ),\frac {3}{2}\right )}{315 \sqrt {x} \sqrt {2+5 x+3 x^2}} \] Input:
Integrate[((2 - 5*x)*(2 + 5*x + 3*x^2)^(3/2))/x^(3/2),x]
Output:
(-2*(-3328 - 7390*x + 177*x^2 + 9855*x^3 + 7641*x^4 + 2025*x^5) + (5848*I) *Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[2 /3]/Sqrt[x]], 3/2] + (1382*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(3/ 2)*EllipticF[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2])/(315*Sqrt[x]*Sqrt[2 + 5*x + 3*x^2])
Time = 0.38 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {1230, 27, 1231, 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.
| \(\displaystyle \int \frac {(2-5 x) \left (3 x^2+5 x+2\right )^{3/2}}{x^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle -\frac {6}{7} \int -\frac {(59 x+50) \sqrt {3 x^2+5 x+2}}{2 \sqrt {x}}dx-\frac {2 (5 x+14) \left (3 x^2+5 x+2\right )^{3/2}}{7 \sqrt {x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{7} \int \frac {(59 x+50) \sqrt {3 x^2+5 x+2}}{\sqrt {x}}dx-\frac {2 (5 x+14) \left (3 x^2+5 x+2\right )^{3/2}}{7 \sqrt {x}}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {3}{7} \left (\frac {2}{45} \sqrt {x} (531 x+1045) \sqrt {3 x^2+5 x+2}-\frac {2}{45} \int -\frac {1462 x+1205}{\sqrt {x} \sqrt {3 x^2+5 x+2}}dx\right )-\frac {2 (5 x+14) \left (3 x^2+5 x+2\right )^{3/2}}{7 \sqrt {x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3}{7} \left (\frac {2}{45} \int \frac {1462 x+1205}{\sqrt {x} \sqrt {3 x^2+5 x+2}}dx+\frac {2}{45} \sqrt {x} \sqrt {3 x^2+5 x+2} (531 x+1045)\right )-\frac {2 (5 x+14) \left (3 x^2+5 x+2\right )^{3/2}}{7 \sqrt {x}}\) |
| \(\Big \downarrow \) 1240 |
| \(\displaystyle \frac {3}{7} \left (\frac {4}{45} \int \frac {1462 x+1205}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+\frac {2}{45} \sqrt {x} \sqrt {3 x^2+5 x+2} (531 x+1045)\right )-\frac {2 (5 x+14) \left (3 x^2+5 x+2\right )^{3/2}}{7 \sqrt {x}}\) |
| \(\Big \downarrow \) 1503 |
| \(\displaystyle \frac {3}{7} \left (\frac {4}{45} \left (1205 \int \frac {1}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+1462 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}\right )+\frac {2}{45} \sqrt {x} \sqrt {3 x^2+5 x+2} (531 x+1045)\right )-\frac {2 (5 x+14) \left (3 x^2+5 x+2\right )^{3/2}}{7 \sqrt {x}}\) |
| \(\Big \downarrow \) 1413 |
| \(\displaystyle \frac {3}{7} \left (\frac {4}{45} \left (1462 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+\frac {1205 (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {2} \sqrt {3 x^2+5 x+2}}\right )+\frac {2}{45} \sqrt {x} \sqrt {3 x^2+5 x+2} (531 x+1045)\right )-\frac {2 (5 x+14) \left (3 x^2+5 x+2\right )^{3/2}}{7 \sqrt {x}}\) |
| \(\Big \downarrow \) 1456 |
| \(\displaystyle \frac {3}{7} \left (\frac {4}{45} \left (\frac {1205 (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {2} \sqrt {3 x^2+5 x+2}}+1462 \left (\frac {\sqrt {x} (3 x+2)}{3 \sqrt {3 x^2+5 x+2}}-\frac {\sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{3 \sqrt {3 x^2+5 x+2}}\right )\right )+\frac {2}{45} \sqrt {x} \sqrt {3 x^2+5 x+2} (531 x+1045)\right )-\frac {2 (5 x+14) \left (3 x^2+5 x+2\right )^{3/2}}{7 \sqrt {x}}\) |
Input:
Int[((2 - 5*x)*(2 + 5*x + 3*x^2)^(3/2))/x^(3/2),x]
Output:
(-2*(14 + 5*x)*(2 + 5*x + 3*x^2)^(3/2))/(7*Sqrt[x]) + (3*((2*Sqrt[x]*(1045 + 531*x)*Sqrt[2 + 5*x + 3*x^2])/45 + (4*(1462*((Sqrt[x]*(2 + 3*x))/(3*Sqr t[2 + 5*x + 3*x^2]) - (Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[A rcTan[Sqrt[x]], -1/2])/(3*Sqrt[2 + 5*x + 3*x^2])) + (1205*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/(Sqrt[2]*Sqrt[2 + 5*x + 3*x^2])))/45))/7
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)*(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[((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 = 1.06 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.67
| method | result | size |
| default | \(-\frac {2 \left (6075 x^{5}+771 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, \operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )-1462 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, \operatorname {EllipticE}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )+22923 x^{4}+29565 x^{3}+26847 x^{2}+21690 x +7560\right )}{945 \sqrt {3 x^{2}+5 x +2}\, \sqrt {x}}\) | \(123\) |
| elliptic | \(\frac {\sqrt {\left (3 x^{2}+5 x +2\right ) x}\, \left (-\frac {8 \left (3 x^{2}+5 x +2\right )}{\sqrt {\left (3 x^{2}+5 x +2\right ) x}}-\frac {30 x^{2} \sqrt {3 x^{3}+5 x^{2}+2 x}}{7}-\frac {316 x \sqrt {3 x^{3}+5 x^{2}+2 x}}{35}-\frac {62 \sqrt {3 x^{3}+5 x^{2}+2 x}}{21}+\frac {482 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{63 \sqrt {3 x^{3}+5 x^{2}+2 x}}+\frac {2924 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {\operatorname {EllipticE}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-\operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{315 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right )}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(243\) |
Input:
int((2-5*x)*(3*x^2+5*x+2)^(3/2)/x^(3/2),x,method=_RETURNVERBOSE)
Output:
-2/945*(6075*x^5+771*(6*x+4)^(1/2)*(3+3*x)^(1/2)*6^(1/2)*(-x)^(1/2)*Ellipt icF(1/2*(6*x+4)^(1/2),I*2^(1/2))-1462*(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))+22923*x^4+29565*x^3+2684 7*x^2+21690*x+7560)/(3*x^2+5*x+2)^(1/2)/x^(1/2)
Time = 0.09 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.36 \[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{3/2}} \, dx=\frac {2 \, {\left (7070 \, \sqrt {3} x {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) - 26316 \, \sqrt {3} x {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) - 27 \, {\left (225 \, x^{3} + 474 \, x^{2} + 155 \, x + 420\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}\right )}}{2835 \, x} \] Input:
integrate((2-5*x)*(3*x^2+5*x+2)^(3/2)/x^(3/2),x, algorithm="fricas")
Output:
2/2835*(7070*sqrt(3)*x*weierstrassPInverse(28/27, 80/729, x + 5/9) - 26316 *sqrt(3)*x*weierstrassZeta(28/27, 80/729, weierstrassPInverse(28/27, 80/72 9, x + 5/9)) - 27*(225*x^3 + 474*x^2 + 155*x + 420)*sqrt(3*x^2 + 5*x + 2)* sqrt(x))/x
\[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{3/2}} \, dx=- \int \left (- \frac {4 \sqrt {3 x^{2} + 5 x + 2}}{x^{\frac {3}{2}}}\right )\, dx - \int 19 \sqrt {x} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int 15 x^{\frac {3}{2}} \sqrt {3 x^{2} + 5 x + 2}\, dx \] Input:
integrate((2-5*x)*(3*x**2+5*x+2)**(3/2)/x**(3/2),x)
Output:
-Integral(-4*sqrt(3*x**2 + 5*x + 2)/x**(3/2), x) - Integral(19*sqrt(x)*sqr t(3*x**2 + 5*x + 2), x) - Integral(15*x**(3/2)*sqrt(3*x**2 + 5*x + 2), x)
\[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{3/2}} \, dx=\int { -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (5 \, x - 2\right )}}{x^{\frac {3}{2}}} \,d x } \] Input:
integrate((2-5*x)*(3*x^2+5*x+2)^(3/2)/x^(3/2),x, algorithm="maxima")
Output:
-integrate((3*x^2 + 5*x + 2)^(3/2)*(5*x - 2)/x^(3/2), x)
\[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{3/2}} \, dx=\int { -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (5 \, x - 2\right )}}{x^{\frac {3}{2}}} \,d x } \] Input:
integrate((2-5*x)*(3*x^2+5*x+2)^(3/2)/x^(3/2),x, algorithm="giac")
Output:
integrate(-(3*x^2 + 5*x + 2)^(3/2)*(5*x - 2)/x^(3/2), x)
Timed out. \[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{3/2}} \, dx=\int -\frac {\left (5\,x-2\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{x^{3/2}} \,d x \] Input:
int(-((5*x - 2)*(5*x + 3*x^2 + 2)^(3/2))/x^(3/2),x)
Output:
int(-((5*x - 2)*(5*x + 3*x^2 + 2)^(3/2))/x^(3/2), x)
\[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{3/2}} \, dx=\frac {-\frac {30 \sqrt {3 x^{2}+5 x +2}\, x^{3}}{7}-\frac {316 \sqrt {3 x^{2}+5 x +2}\, x^{2}}{35}-\frac {62 \sqrt {3 x^{2}+5 x +2}\, x}{21}-8 \sqrt {3 x^{2}+5 x +2}+\frac {482 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}{3 x^{3}+5 x^{2}+2 x}d x \right )}{21}+\frac {2924 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}{3 x^{2}+5 x +2}d x \right )}{105}}{\sqrt {x}} \] Input:
int((2-5*x)*(3*x^2+5*x+2)^(3/2)/x^(3/2),x)
Output:
(2*( - 225*sqrt(3*x**2 + 5*x + 2)*x**3 - 474*sqrt(3*x**2 + 5*x + 2)*x**2 - 155*sqrt(3*x**2 + 5*x + 2)*x - 420*sqrt(3*x**2 + 5*x + 2) + 1205*sqrt(x)* int((sqrt(x)*sqrt(3*x**2 + 5*x + 2))/(3*x**3 + 5*x**2 + 2*x),x) + 1462*sqr t(x)*int((sqrt(x)*sqrt(3*x**2 + 5*x + 2))/(3*x**2 + 5*x + 2),x)))/(105*sqr t(x))