Integrand size = 23, antiderivative size = 318 \[ \int \frac {(1+x)^{3/2} \left (1-x+x^2\right )^{3/2}}{x^2} \, dx=-\frac {\sqrt {1+x} \sqrt {1-x+x^2}}{x}+\frac {2}{7} x^2 \sqrt {1+x} \sqrt {1-x+x^2}+\frac {27 \sqrt {1+x} \sqrt {1-x+x^2}}{7 \left (1+\sqrt {3}+x\right )}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (1+x)^{3/2} \sqrt {1-x+x^2} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} E\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{14 \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \left (1+x^3\right )}+\frac {9 \sqrt {2} 3^{3/4} (1+x)^{3/2} \sqrt {1-x+x^2} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{7 \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \left (1+x^3\right )} \] Output:
-(1+x)^(1/2)*(x^2-x+1)^(1/2)/x+2/7*x^2*(1+x)^(1/2)*(x^2-x+1)^(1/2)+27*(1+x )^(1/2)*(x^2-x+1)^(1/2)/(7+7*3^(1/2)+7*x)-27/14*3^(1/4)*(1/2*6^(1/2)-1/2*2 ^(1/2))*(1+x)^(3/2)*(x^2-x+1)^(1/2)*((x^2-x+1)/(1+x+3^(1/2))^2)^(1/2)*Elli pticE((1+x-3^(1/2))/(1+x+3^(1/2)),I*3^(1/2)+2*I)/((1+x)/(1+x+3^(1/2))^2)^( 1/2)/(x^3+1)+9/7*2^(1/2)*3^(3/4)*(1+x)^(3/2)*(x^2-x+1)^(1/2)*((x^2-x+1)/(1 +x+3^(1/2))^2)^(1/2)*EllipticF((1+x-3^(1/2))/(1+x+3^(1/2)),I*3^(1/2)+2*I)/ ((1+x)/(1+x+3^(1/2))^2)^(1/2)/(x^3+1)
Result contains complex when optimal does not.
Time = 10.61 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.77 \[ \int \frac {(1+x)^{3/2} \left (1-x+x^2\right )^{3/2}}{x^2} \, dx=\frac {\sqrt {1+x} \left (\frac {4 \left (1-x+x^2\right ) \left (-7+2 x^3\right )}{x}-\frac {27 \sqrt {2} \sqrt {\frac {-i+\sqrt {3}+2 i x}{-3 i+\sqrt {3}}} \left (\left (-3 i+\sqrt {3}\right ) E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {-\frac {i (1+x)}{3 i+\sqrt {3}}}\right )|\frac {3 i+\sqrt {3}}{3 i-\sqrt {3}}\right )-\left (-i+\sqrt {3}\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {-\frac {i (1+x)}{3 i+\sqrt {3}}}\right ),\frac {3 i+\sqrt {3}}{3 i-\sqrt {3}}\right )\right )}{\sqrt {-\frac {i (1+x)}{i+\sqrt {3}-2 i x}}}\right )}{28 \sqrt {1-x+x^2}} \] Input:
Integrate[((1 + x)^(3/2)*(1 - x + x^2)^(3/2))/x^2,x]
Output:
(Sqrt[1 + x]*((4*(1 - x + x^2)*(-7 + 2*x^3))/x - (27*Sqrt[2]*Sqrt[(-I + Sq rt[3] + (2*I)*x)/(-3*I + Sqrt[3])]*((-3*I + Sqrt[3])*EllipticE[I*ArcSinh[S qrt[2]*Sqrt[((-I)*(1 + x))/(3*I + Sqrt[3])]], (3*I + Sqrt[3])/(3*I - Sqrt[ 3])] - (-I + Sqrt[3])*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[((-I)*(1 + x))/(3*I + Sqrt[3])]], (3*I + Sqrt[3])/(3*I - Sqrt[3])]))/Sqrt[((-I)*(1 + x))/(I + Sqrt[3] - (2*I)*x)]))/(28*Sqrt[1 - x + x^2])
Time = 0.68 (sec) , antiderivative size = 308, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {1210, 809, 811, 832, 759, 2416}
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.
| \(\displaystyle \int \frac {(x+1)^{3/2} \left (x^2-x+1\right )^{3/2}}{x^2} \, dx\) |
| \(\Big \downarrow \) 1210 |
| \(\displaystyle \frac {\sqrt {x+1} \sqrt {x^2-x+1} \int \frac {\left (x^3+1\right )^{3/2}}{x^2}dx}{\sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 809 |
| \(\displaystyle \frac {\sqrt {x+1} \sqrt {x^2-x+1} \left (\frac {9}{2} \int x \sqrt {x^3+1}dx-\frac {\left (x^3+1\right )^{3/2}}{x}\right )}{\sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 811 |
| \(\displaystyle \frac {\sqrt {x+1} \sqrt {x^2-x+1} \left (\frac {9}{2} \left (\frac {3}{7} \int \frac {x}{\sqrt {x^3+1}}dx+\frac {2}{7} \sqrt {x^3+1} x^2\right )-\frac {\left (x^3+1\right )^{3/2}}{x}\right )}{\sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 832 |
| \(\displaystyle \frac {\sqrt {x+1} \sqrt {x^2-x+1} \left (\frac {9}{2} \left (\frac {3}{7} \left (\int \frac {x-\sqrt {3}+1}{\sqrt {x^3+1}}dx-\left (1-\sqrt {3}\right ) \int \frac {1}{\sqrt {x^3+1}}dx\right )+\frac {2}{7} \sqrt {x^3+1} x^2\right )-\frac {\left (x^3+1\right )^{3/2}}{x}\right )}{\sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {\sqrt {x+1} \sqrt {x^2-x+1} \left (\frac {9}{2} \left (\frac {3}{7} \left (\int \frac {x-\sqrt {3}+1}{\sqrt {x^3+1}}dx-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}\right )+\frac {2}{7} \sqrt {x^3+1} x^2\right )-\frac {\left (x^3+1\right )^{3/2}}{x}\right )}{\sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 2416 |
| \(\displaystyle \frac {\sqrt {x+1} \sqrt {x^2-x+1} \left (\frac {9}{2} \left (\frac {3}{7} \left (-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{\sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}+\frac {2 \sqrt {x^3+1}}{x+\sqrt {3}+1}\right )+\frac {2}{7} \sqrt {x^3+1} x^2\right )-\frac {\left (x^3+1\right )^{3/2}}{x}\right )}{\sqrt {x^3+1}}\) |
Input:
Int[((1 + x)^(3/2)*(1 - x + x^2)^(3/2))/x^2,x]
Output:
(Sqrt[1 + x]*Sqrt[1 - x + x^2]*(-((1 + x^3)^(3/2)/x) + (9*((2*x^2*Sqrt[1 + x^3])/7 + (3*((2*Sqrt[1 + x^3])/(1 + Sqrt[3] + x) - (3^(1/4)*Sqrt[2 - Sqr t[3]]*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*EllipticE[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]])/(Sqrt[(1 + x)/(1 + Sqr t[3] + x)^2]*Sqrt[1 + x^3]) - (2*(1 - Sqrt[3])*Sqrt[2 + Sqrt[3]]*(1 + x)*S qrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*EllipticF[ArcSin[(1 - Sqrt[3] + x)/ (1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]])/(3^(1/4)*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x^3])))/7))/2))/Sqrt[1 + x^3]
Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 + Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s *x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[s* ((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] & & PosQ[a]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* x)^(m + 1)*((a + b*x^n)^p/(c*(m + 1))), x] - Simp[b*n*(p/(c^n*(m + 1))) I nt[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && IGtQ [n, 0] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + n*p + n + 1)/n, 0] && IntB inomialQ[a, b, c, n, m, p, x]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* x)^(m + 1)*((a + b*x^n)^p/(c*(m + n*p + 1))), x] + Simp[a*n*(p/(m + n*p + 1 )) Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && I GtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m , p, x]
Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] ], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 - Sqrt[3]))*(s/r) Int[1/Sqrt[a + b*x ^3], x], x] + Simp[1/r Int[((1 - Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x ]] /; FreeQ[{a, b}, x] && PosQ[a]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^FracPart[p]*((a + b*x + c*x^2)^FracPart[p]/(a*d + c*e*x^3)^FracPart[p]) Int[(f + g*x)^n*(a*d + c* e*x^3)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[b*d + a*e, 0] && EqQ[c*d + b*e, 0] && EqQ[m, p]
Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N umer[Simplify[(1 - Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 - Sqrt[3])*(d/c) ]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 + Sqrt[3])*s + r*x))), x] - S imp[3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( (1 + Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[s*((s + r*x)/((1 + Sqrt [3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3]) *s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && Eq Q[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
Time = 2.85 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.72
| method | result | size |
| risch | \(\frac {\sqrt {x +1}\, \sqrt {x^{2}-x +1}\, \left (2 x^{3}-7\right )}{7 x}+\frac {27 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (\left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )+\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )\right ) \sqrt {\left (x +1\right ) \left (x^{2}-x +1\right )}}{7 \sqrt {x^{3}+1}\, \sqrt {x +1}\, \sqrt {x^{2}-x +1}}\) | \(230\) |
| elliptic | \(\frac {\sqrt {x +1}\, \sqrt {x^{2}-x +1}\, \sqrt {\left (x +1\right ) \left (x^{2}-x +1\right )}\, \left (-\frac {\sqrt {x^{3}+1}}{x}+\frac {27 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (\left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )+\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )\right )}{7 \sqrt {x^{3}+1}}+\frac {2 x^{2} \sqrt {x^{3}+1}}{7}\right )}{x^{3}+1}\) | \(235\) |
| default | \(\frac {\sqrt {x +1}\, \sqrt {x^{2}-x +1}\, \left (27 i \sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{i \sqrt {3}-3}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}, \sqrt {-\frac {i \sqrt {3}-3}{i \sqrt {3}+3}}\right ) \sqrt {3}\, x +4 x^{6}+81 \sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{i \sqrt {3}-3}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}, \sqrt {-\frac {i \sqrt {3}-3}{i \sqrt {3}+3}}\right ) x -162 \sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{i \sqrt {3}-3}}\, \operatorname {EllipticE}\left (\sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}, \sqrt {-\frac {i \sqrt {3}-3}{i \sqrt {3}+3}}\right ) x -10 x^{3}-14\right )}{14 x \left (x^{3}+1\right )}\) | \(368\) |
Input:
int((x+1)^(3/2)*(x^2-x+1)^(3/2)/x^2,x,method=_RETURNVERBOSE)
Output:
1/7*(x+1)^(1/2)*(x^2-x+1)^(1/2)*(2*x^3-7)/x+27/7*(3/2-1/2*I*3^(1/2))*((x+1 )/(3/2-1/2*I*3^(1/2)))^(1/2)*((x-1/2-1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^ (1/2)*((x-1/2+1/2*I*3^(1/2))/(-3/2+1/2*I*3^(1/2)))^(1/2)/(x^3+1)^(1/2)*((- 3/2-1/2*I*3^(1/2))*EllipticE(((x+1)/(3/2-1/2*I*3^(1/2)))^(1/2),((-3/2+1/2* I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2))+(1/2+1/2*I*3^(1/2))*EllipticF(((x+ 1)/(3/2-1/2*I*3^(1/2)))^(1/2),((-3/2+1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^ (1/2)))*((x+1)*(x^2-x+1))^(1/2)/(x+1)^(1/2)/(x^2-x+1)^(1/2)
Time = 0.07 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.12 \[ \int \frac {(1+x)^{3/2} \left (1-x+x^2\right )^{3/2}}{x^2} \, dx=\frac {{\left (2 \, x^{3} - 7\right )} \sqrt {x^{2} - x + 1} \sqrt {x + 1} - 27 \, x {\rm weierstrassZeta}\left (0, -4, {\rm weierstrassPInverse}\left (0, -4, x\right )\right )}{7 \, x} \] Input:
integrate((1+x)^(3/2)*(x^2-x+1)^(3/2)/x^2,x, algorithm="fricas")
Output:
1/7*((2*x^3 - 7)*sqrt(x^2 - x + 1)*sqrt(x + 1) - 27*x*weierstrassZeta(0, - 4, weierstrassPInverse(0, -4, x)))/x
\[ \int \frac {(1+x)^{3/2} \left (1-x+x^2\right )^{3/2}}{x^2} \, dx=\int \frac {\left (x + 1\right )^{\frac {3}{2}} \left (x^{2} - x + 1\right )^{\frac {3}{2}}}{x^{2}}\, dx \] Input:
integrate((1+x)**(3/2)*(x**2-x+1)**(3/2)/x**2,x)
Output:
Integral((x + 1)**(3/2)*(x**2 - x + 1)**(3/2)/x**2, x)
\[ \int \frac {(1+x)^{3/2} \left (1-x+x^2\right )^{3/2}}{x^2} \, dx=\int { \frac {{\left (x^{2} - x + 1\right )}^{\frac {3}{2}} {\left (x + 1\right )}^{\frac {3}{2}}}{x^{2}} \,d x } \] Input:
integrate((1+x)^(3/2)*(x^2-x+1)^(3/2)/x^2,x, algorithm="maxima")
Output:
integrate((x^2 - x + 1)^(3/2)*(x + 1)^(3/2)/x^2, x)
\[ \int \frac {(1+x)^{3/2} \left (1-x+x^2\right )^{3/2}}{x^2} \, dx=\int { \frac {{\left (x^{2} - x + 1\right )}^{\frac {3}{2}} {\left (x + 1\right )}^{\frac {3}{2}}}{x^{2}} \,d x } \] Input:
integrate((1+x)^(3/2)*(x^2-x+1)^(3/2)/x^2,x, algorithm="giac")
Output:
integrate((x^2 - x + 1)^(3/2)*(x + 1)^(3/2)/x^2, x)
Timed out. \[ \int \frac {(1+x)^{3/2} \left (1-x+x^2\right )^{3/2}}{x^2} \, dx=\int \frac {{\left (x+1\right )}^{3/2}\,{\left (x^2-x+1\right )}^{3/2}}{x^2} \,d x \] Input:
int(((x + 1)^(3/2)*(x^2 - x + 1)^(3/2))/x^2,x)
Output:
int(((x + 1)^(3/2)*(x^2 - x + 1)^(3/2))/x^2, x)
\[ \int \frac {(1+x)^{3/2} \left (1-x+x^2\right )^{3/2}}{x^2} \, dx=\frac {2 \sqrt {x +1}\, \sqrt {x^{2}-x +1}\, x^{3}+20 \sqrt {x +1}\, \sqrt {x^{2}-x +1}+27 \left (\int \frac {\sqrt {x +1}\, \sqrt {x^{2}-x +1}}{x^{5}+x^{2}}d x \right ) x}{7 x} \] Input:
int((1+x)^(3/2)*(x^2-x+1)^(3/2)/x^2,x)
Output:
(2*sqrt(x + 1)*sqrt(x**2 - x + 1)*x**3 + 20*sqrt(x + 1)*sqrt(x**2 - x + 1) + 27*int((sqrt(x + 1)*sqrt(x**2 - x + 1))/(x**5 + x**2),x)*x)/(7*x)