Integrand size = 25, antiderivative size = 252 \[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{15/2}} \, dx=-\frac {6907 \sqrt {x} (2+3 x)}{10010 \sqrt {2+5 x+3 x^2}}+\frac {204 \sqrt {2+5 x+3 x^2}}{385 x^{5/2}}-\frac {1231 \sqrt {2+5 x+3 x^2}}{2002 x^{3/2}}+\frac {6907 \sqrt {2+5 x+3 x^2}}{10010 \sqrt {x}}+\frac {(1834+3445 x) \sqrt {2+5 x+3 x^2}}{1001 x^{9/2}}-\frac {4 (11-25 x) \left (2+5 x+3 x^2\right )^{3/2}}{143 x^{13/2}}+\frac {6907 \sqrt {2+5 x+3 x^2} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{5005 \sqrt {2} \sqrt {1+x} \sqrt {2+3 x}}-\frac {3693 \sqrt {1+x} \sqrt {2+3 x} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{2002 \sqrt {2} \sqrt {2+5 x+3 x^2}} \] Output:
-6907/10010*x^(1/2)*(2+3*x)/(3*x^2+5*x+2)^(1/2)+204/385*(3*x^2+5*x+2)^(1/2 )/x^(5/2)-1231/2002*(3*x^2+5*x+2)^(1/2)/x^(3/2)+6907/10010*(3*x^2+5*x+2)^( 1/2)/x^(1/2)+1/1001*(1834+3445*x)*(3*x^2+5*x+2)^(1/2)/x^(9/2)-4/143*(11-25 *x)*(3*x^2+5*x+2)^(3/2)/x^(13/2)+6907/10010*2^(1/2)*(3*x^2+5*x+2)^(1/2)*El lipticE(x^(1/2)/(1+x)^(1/2),1/2*I*2^(1/2))/(1+x)^(1/2)/(2+3*x)^(1/2)-3693/ 4004*2^(1/2)*(1+x)^(1/2)*(2+3*x)^(1/2)*InverseJacobiAM(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 = 21.23 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.67 \[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{15/2}} \, dx=\frac {-24640-67200 x+125440 x^2+654400 x^3+840316 x^4+361120 x^5-29726 x^6-36930 x^7-13814 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{15/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )-4651 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{15/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right ),\frac {3}{2}\right )}{20020 x^{13/2} \sqrt {2+5 x+3 x^2}} \] Input:
Integrate[((2 - 5*x)*(2 + 5*x + 3*x^2)^(3/2))/x^(15/2),x]
Output:
(-24640 - 67200*x + 125440*x^2 + 654400*x^3 + 840316*x^4 + 361120*x^5 - 29 726*x^6 - 36930*x^7 - (13814*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^( 15/2)*EllipticE[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2] - (4651*I)*Sqrt[2]*Sqrt [1 + x^(-1)]*Sqrt[3 + 2/x]*x^(15/2)*EllipticF[I*ArcSinh[Sqrt[2/3]/Sqrt[x]] , 3/2])/(20020*x^(13/2)*Sqrt[2 + 5*x + 3*x^2])
Time = 0.51 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.10, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {1229, 27, 1229, 27, 1237, 1237, 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.
| \(\displaystyle \int \frac {(2-5 x) \left (3 x^2+5 x+2\right )^{3/2}}{x^{15/2}} \, dx\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle -\frac {3}{143} \int \frac {3 (155 x+131) \sqrt {3 x^2+5 x+2}}{x^{11/2}}dx-\frac {4 (11-25 x) \left (3 x^2+5 x+2\right )^{3/2}}{143 x^{13/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {9}{143} \int \frac {(155 x+131) \sqrt {3 x^2+5 x+2}}{x^{11/2}}dx-\frac {4 (11-25 x) \left (3 x^2+5 x+2\right )^{3/2}}{143 x^{13/2}}\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle -\frac {9}{143} \left (-\frac {1}{63} \int -\frac {3 (2305 x+1768)}{2 x^{7/2} \sqrt {3 x^2+5 x+2}}dx-\frac {\sqrt {3 x^2+5 x+2} (3445 x+1834)}{63 x^{9/2}}\right )-\frac {4 (11-25 x) \left (3 x^2+5 x+2\right )^{3/2}}{143 x^{13/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {9}{143} \left (\frac {1}{42} \int \frac {2305 x+1768}{x^{7/2} \sqrt {3 x^2+5 x+2}}dx-\frac {(3445 x+1834) \sqrt {3 x^2+5 x+2}}{63 x^{9/2}}\right )-\frac {4 (11-25 x) \left (3 x^2+5 x+2\right )^{3/2}}{143 x^{13/2}}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -\frac {9}{143} \left (\frac {1}{42} \left (-\frac {1}{5} \int \frac {7956 x+6155}{x^{5/2} \sqrt {3 x^2+5 x+2}}dx-\frac {1768 \sqrt {3 x^2+5 x+2}}{5 x^{5/2}}\right )-\frac {(3445 x+1834) \sqrt {3 x^2+5 x+2}}{63 x^{9/2}}\right )-\frac {4 (11-25 x) \left (3 x^2+5 x+2\right )^{3/2}}{143 x^{13/2}}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -\frac {9}{143} \left (\frac {1}{42} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {18465 x+13814}{2 x^{3/2} \sqrt {3 x^2+5 x+2}}dx+\frac {6155 \sqrt {3 x^2+5 x+2}}{3 x^{3/2}}\right )-\frac {1768 \sqrt {3 x^2+5 x+2}}{5 x^{5/2}}\right )-\frac {(3445 x+1834) \sqrt {3 x^2+5 x+2}}{63 x^{9/2}}\right )-\frac {4 (11-25 x) \left (3 x^2+5 x+2\right )^{3/2}}{143 x^{13/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {9}{143} \left (\frac {1}{42} \left (\frac {1}{5} \left (\frac {1}{6} \int \frac {18465 x+13814}{x^{3/2} \sqrt {3 x^2+5 x+2}}dx+\frac {6155 \sqrt {3 x^2+5 x+2}}{3 x^{3/2}}\right )-\frac {1768 \sqrt {3 x^2+5 x+2}}{5 x^{5/2}}\right )-\frac {(3445 x+1834) \sqrt {3 x^2+5 x+2}}{63 x^{9/2}}\right )-\frac {4 (11-25 x) \left (3 x^2+5 x+2\right )^{3/2}}{143 x^{13/2}}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -\frac {9}{143} \left (\frac {1}{42} \left (\frac {1}{5} \left (\frac {1}{6} \left (-\int -\frac {3 (6907 x+6155)}{\sqrt {x} \sqrt {3 x^2+5 x+2}}dx-\frac {13814 \sqrt {3 x^2+5 x+2}}{\sqrt {x}}\right )+\frac {6155 \sqrt {3 x^2+5 x+2}}{3 x^{3/2}}\right )-\frac {1768 \sqrt {3 x^2+5 x+2}}{5 x^{5/2}}\right )-\frac {(3445 x+1834) \sqrt {3 x^2+5 x+2}}{63 x^{9/2}}\right )-\frac {4 (11-25 x) \left (3 x^2+5 x+2\right )^{3/2}}{143 x^{13/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {9}{143} \left (\frac {1}{42} \left (\frac {1}{5} \left (\frac {1}{6} \left (3 \int \frac {6907 x+6155}{\sqrt {x} \sqrt {3 x^2+5 x+2}}dx-\frac {13814 \sqrt {3 x^2+5 x+2}}{\sqrt {x}}\right )+\frac {6155 \sqrt {3 x^2+5 x+2}}{3 x^{3/2}}\right )-\frac {1768 \sqrt {3 x^2+5 x+2}}{5 x^{5/2}}\right )-\frac {(3445 x+1834) \sqrt {3 x^2+5 x+2}}{63 x^{9/2}}\right )-\frac {4 (11-25 x) \left (3 x^2+5 x+2\right )^{3/2}}{143 x^{13/2}}\) |
| \(\Big \downarrow \) 1240 |
| \(\displaystyle -\frac {9}{143} \left (\frac {1}{42} \left (\frac {1}{5} \left (\frac {1}{6} \left (6 \int \frac {6907 x+6155}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}-\frac {13814 \sqrt {3 x^2+5 x+2}}{\sqrt {x}}\right )+\frac {6155 \sqrt {3 x^2+5 x+2}}{3 x^{3/2}}\right )-\frac {1768 \sqrt {3 x^2+5 x+2}}{5 x^{5/2}}\right )-\frac {(3445 x+1834) \sqrt {3 x^2+5 x+2}}{63 x^{9/2}}\right )-\frac {4 (11-25 x) \left (3 x^2+5 x+2\right )^{3/2}}{143 x^{13/2}}\) |
| \(\Big \downarrow \) 1503 |
| \(\displaystyle -\frac {9}{143} \left (\frac {1}{42} \left (\frac {1}{5} \left (\frac {1}{6} \left (6 \left (6155 \int \frac {1}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+6907 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}\right )-\frac {13814 \sqrt {3 x^2+5 x+2}}{\sqrt {x}}\right )+\frac {6155 \sqrt {3 x^2+5 x+2}}{3 x^{3/2}}\right )-\frac {1768 \sqrt {3 x^2+5 x+2}}{5 x^{5/2}}\right )-\frac {(3445 x+1834) \sqrt {3 x^2+5 x+2}}{63 x^{9/2}}\right )-\frac {4 (11-25 x) \left (3 x^2+5 x+2\right )^{3/2}}{143 x^{13/2}}\) |
| \(\Big \downarrow \) 1413 |
| \(\displaystyle -\frac {9}{143} \left (\frac {1}{42} \left (\frac {1}{5} \left (\frac {1}{6} \left (6 \left (6907 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+\frac {6155 (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {2} \sqrt {3 x^2+5 x+2}}\right )-\frac {13814 \sqrt {3 x^2+5 x+2}}{\sqrt {x}}\right )+\frac {6155 \sqrt {3 x^2+5 x+2}}{3 x^{3/2}}\right )-\frac {1768 \sqrt {3 x^2+5 x+2}}{5 x^{5/2}}\right )-\frac {(3445 x+1834) \sqrt {3 x^2+5 x+2}}{63 x^{9/2}}\right )-\frac {4 (11-25 x) \left (3 x^2+5 x+2\right )^{3/2}}{143 x^{13/2}}\) |
| \(\Big \downarrow \) 1456 |
| \(\displaystyle -\frac {9}{143} \left (\frac {1}{42} \left (\frac {1}{5} \left (\frac {1}{6} \left (6 \left (\frac {6155 (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {2} \sqrt {3 x^2+5 x+2}}+6907 \left (\frac {\sqrt {x} (3 x+2)}{3 \sqrt {3 x^2+5 x+2}}-\frac {\sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{3 \sqrt {3 x^2+5 x+2}}\right )\right )-\frac {13814 \sqrt {3 x^2+5 x+2}}{\sqrt {x}}\right )+\frac {6155 \sqrt {3 x^2+5 x+2}}{3 x^{3/2}}\right )-\frac {1768 \sqrt {3 x^2+5 x+2}}{5 x^{5/2}}\right )-\frac {(3445 x+1834) \sqrt {3 x^2+5 x+2}}{63 x^{9/2}}\right )-\frac {4 (11-25 x) \left (3 x^2+5 x+2\right )^{3/2}}{143 x^{13/2}}\) |
Input:
Int[((2 - 5*x)*(2 + 5*x + 3*x^2)^(3/2))/x^(15/2),x]
Output:
(-4*(11 - 25*x)*(2 + 5*x + 3*x^2)^(3/2))/(143*x^(13/2)) - (9*(-1/63*((1834 + 3445*x)*Sqrt[2 + 5*x + 3*x^2])/x^(9/2) + ((-1768*Sqrt[2 + 5*x + 3*x^2]) /(5*x^(5/2)) + ((6155*Sqrt[2 + 5*x + 3*x^2])/(3*x^(3/2)) + ((-13814*Sqrt[2 + 5*x + 3*x^2])/Sqrt[x] + 6*(6907*((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])) + (6155*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/(Sqrt[2]*Sqrt[2 + 5*x + 3*x^2])))/6) /5)/42))/143
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 = 0.96 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.57
| method | result | size |
| default | \(\frac {2256 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, \operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x^{6}-6907 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, \operatorname {EllipticE}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x^{6}+124326 x^{8}+96420 x^{7}-6294 x^{6}+1083360 x^{5}+2520948 x^{4}+1963200 x^{3}+376320 x^{2}-201600 x -73920}{60060 \sqrt {3 x^{2}+5 x +2}\, x^{\frac {13}{2}}}\) | \(144\) |
| elliptic | \(\frac {\sqrt {\left (3 x^{2}+5 x +2\right ) x}\, \left (-\frac {8 \sqrt {3 x^{3}+5 x^{2}+2 x}}{13 x^{7}}-\frac {20 \sqrt {3 x^{3}+5 x^{2}+2 x}}{143 x^{6}}+\frac {630 \sqrt {3 x^{3}+5 x^{2}+2 x}}{143 x^{5}}+\frac {5545 \sqrt {3 x^{3}+5 x^{2}+2 x}}{1001 x^{4}}+\frac {204 \sqrt {3 x^{3}+5 x^{2}+2 x}}{385 x^{3}}-\frac {1231 \sqrt {3 x^{3}+5 x^{2}+2 x}}{2002 x^{2}}+\frac {\frac {20721}{10010} x^{2}+\frac {6907}{2002} x +\frac {6907}{5005}}{\sqrt {\left (3 x^{2}+5 x +2\right ) x}}-\frac {1231 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{4004 \sqrt {3 x^{3}+5 x^{2}+2 x}}-\frac {6907 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {\operatorname {EllipticE}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-\operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{20020 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right )}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(311\) |
Input:
int((2-5*x)*(3*x^2+5*x+2)^(3/2)/x^(15/2),x,method=_RETURNVERBOSE)
Output:
1/60060*(2256*(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^6-6907*(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^6+124326*x^8+96420*x^7-62 94*x^6+1083360*x^5+2520948*x^4+1963200*x^3+376320*x^2-201600*x-73920)/(3*x ^2+5*x+2)^(1/2)/x^(13/2)
Time = 0.08 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.33 \[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{15/2}} \, dx=-\frac {20860 \, \sqrt {3} x^{7} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) - 62163 \, \sqrt {3} x^{7} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) - 9 \, {\left (6907 \, x^{6} - 6155 \, x^{5} + 5304 \, x^{4} + 55450 \, x^{3} + 44100 \, x^{2} - 1400 \, x - 6160\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}}{90090 \, x^{7}} \] Input:
integrate((2-5*x)*(3*x^2+5*x+2)^(3/2)/x^(15/2),x, algorithm="fricas")
Output:
-1/90090*(20860*sqrt(3)*x^7*weierstrassPInverse(28/27, 80/729, x + 5/9) - 62163*sqrt(3)*x^7*weierstrassZeta(28/27, 80/729, weierstrassPInverse(28/27 , 80/729, x + 5/9)) - 9*(6907*x^6 - 6155*x^5 + 5304*x^4 + 55450*x^3 + 4410 0*x^2 - 1400*x - 6160)*sqrt(3*x^2 + 5*x + 2)*sqrt(x))/x^7
Timed out. \[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{15/2}} \, dx=\text {Timed out} \] Input:
integrate((2-5*x)*(3*x**2+5*x+2)**(3/2)/x**(15/2),x)
Output:
Timed out
\[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{15/2}} \, dx=\int { -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (5 \, x - 2\right )}}{x^{\frac {15}{2}}} \,d x } \] Input:
integrate((2-5*x)*(3*x^2+5*x+2)^(3/2)/x^(15/2),x, algorithm="maxima")
Output:
-integrate((3*x^2 + 5*x + 2)^(3/2)*(5*x - 2)/x^(15/2), x)
\[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{15/2}} \, dx=\int { -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (5 \, x - 2\right )}}{x^{\frac {15}{2}}} \,d x } \] Input:
integrate((2-5*x)*(3*x^2+5*x+2)^(3/2)/x^(15/2),x, algorithm="giac")
Output:
integrate(-(3*x^2 + 5*x + 2)^(3/2)*(5*x - 2)/x^(15/2), x)
Timed out. \[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{15/2}} \, dx=\int -\frac {\left (5\,x-2\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{x^{15/2}} \,d x \] Input:
int(-((5*x - 2)*(5*x + 3*x^2 + 2)^(3/2))/x^(15/2),x)
Output:
int(-((5*x - 2)*(5*x + 3*x^2 + 2)^(3/2))/x^(15/2), x)
\[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{15/2}} \, dx=\frac {126360 \sqrt {3 x^{2}+5 x +2}\, x^{3}+76726 \sqrt {3 x^{2}+5 x +2}\, x^{2}+4680 \sqrt {3 x^{2}+5 x +2}\, x -12960 \sqrt {3 x^{2}+5 x +2}+83880 \sqrt {x}\, \left (\int \frac {\sqrt {3 x^{2}+5 x +2}}{3 \sqrt {x}\, x^{8}+5 \sqrt {x}\, x^{7}+2 \sqrt {x}\, x^{6}}d x \right ) x^{6}+46134 \sqrt {x}\, \left (\int \frac {\sqrt {3 x^{2}+5 x +2}}{3 \sqrt {x}\, x^{7}+5 \sqrt {x}\, x^{6}+2 \sqrt {x}\, x^{5}}d x \right ) x^{6}-150280 \sqrt {x}\, \left (\int \frac {\sqrt {3 x^{2}+5 x +2}}{3 \sqrt {x}\, x^{6}+5 \sqrt {x}\, x^{5}+2 \sqrt {x}\, x^{4}}d x \right ) x^{6}-78897 \sqrt {x}\, \left (\int \frac {\sqrt {3 x^{2}+5 x +2}}{3 \sqrt {x}\, x^{5}+5 \sqrt {x}\, x^{4}+2 \sqrt {x}\, x^{3}}d x \right ) x^{6}}{21060 \sqrt {x}\, x^{6}} \] Input:
int((2-5*x)*(3*x^2+5*x+2)^(3/2)/x^(15/2),x)
Output:
(126360*sqrt(3*x**2 + 5*x + 2)*x**3 + 76726*sqrt(3*x**2 + 5*x + 2)*x**2 + 4680*sqrt(3*x**2 + 5*x + 2)*x - 12960*sqrt(3*x**2 + 5*x + 2) + 83880*sqrt( x)*int(sqrt(3*x**2 + 5*x + 2)/(3*sqrt(x)*x**8 + 5*sqrt(x)*x**7 + 2*sqrt(x) *x**6),x)*x**6 + 46134*sqrt(x)*int(sqrt(3*x**2 + 5*x + 2)/(3*sqrt(x)*x**7 + 5*sqrt(x)*x**6 + 2*sqrt(x)*x**5),x)*x**6 - 150280*sqrt(x)*int(sqrt(3*x** 2 + 5*x + 2)/(3*sqrt(x)*x**6 + 5*sqrt(x)*x**5 + 2*sqrt(x)*x**4),x)*x**6 - 78897*sqrt(x)*int(sqrt(3*x**2 + 5*x + 2)/(3*sqrt(x)*x**5 + 5*sqrt(x)*x**4 + 2*sqrt(x)*x**3),x)*x**6)/(21060*sqrt(x)*x**6)