Integrand size = 40, antiderivative size = 393 \[ \int \frac {x^2}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=-\frac {2 a^2 e^2}{3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {4 a e \left (3 c d^2+a e^2\right )}{3 c^2 d^2 \left (c d^2-a e^2\right )^2 (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {2 \left (c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 c^2 d^2 \left (c d^2-a e^2\right )^3 (d+e x)^3}+\frac {8 \left (c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{15 c d \left (c d^2-a e^2\right )^4 (d+e x)^2}+\frac {16 \left (c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{15 \left (c d^2-a e^2\right )^5 (d+e x)} \] Output:
-2/3*a^2*e^2/c^2/d^2/(-a*e^2+c*d^2)/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x ^2)^(3/2)+4/3*a*e*(a*e^2+3*c*d^2)/c^2/d^2/(-a*e^2+c*d^2)^2/(e*x+d)^2/(a*d* e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+2/5*(5*a^2*e^4+10*a*c*d^2*e^2+c^2*d^4)* (a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^2/d^2/(-a*e^2+c*d^2)^3/(e*x+d)^3 +8/15*(5*a^2*e^4+10*a*c*d^2*e^2+c^2*d^4)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2) ^(1/2)/c/d/(-a*e^2+c*d^2)^4/(e*x+d)^2+16/15*(5*a^2*e^4+10*a*c*d^2*e^2+c^2* d^4)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e^2+c*d^2)^5/(e*x+d)
Time = 0.33 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.60 \[ \int \frac {x^2}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {2 \left (c^4 d^6 x^2 \left (15 d^2+20 d e x+8 e^2 x^2\right )+a^4 e^6 \left (8 d^2+20 d e x+15 e^2 x^2\right )+4 a^3 c d e^4 \left (20 d^3+53 d^2 e x+45 d e^2 x^2+15 e^3 x^3\right )+4 a c^3 d^4 e x \left (15 d^3+45 d^2 e x+53 d e^2 x^2+20 e^3 x^3\right )+2 a^2 c^2 d^2 e^2 \left (20 d^4+110 d^3 e x+189 d^2 e^2 x^2+110 d e^3 x^3+20 e^4 x^4\right )\right )}{15 \left (c d^2-a e^2\right )^5 (d+e x) ((a e+c d x) (d+e x))^{3/2}} \] Input:
Integrate[x^2/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]
Output:
(2*(c^4*d^6*x^2*(15*d^2 + 20*d*e*x + 8*e^2*x^2) + a^4*e^6*(8*d^2 + 20*d*e* x + 15*e^2*x^2) + 4*a^3*c*d*e^4*(20*d^3 + 53*d^2*e*x + 45*d*e^2*x^2 + 15*e ^3*x^3) + 4*a*c^3*d^4*e*x*(15*d^3 + 45*d^2*e*x + 53*d*e^2*x^2 + 20*e^3*x^3 ) + 2*a^2*c^2*d^2*e^2*(20*d^4 + 110*d^3*e*x + 189*d^2*e^2*x^2 + 110*d*e^3* x^3 + 20*e^4*x^4)))/(15*(c*d^2 - a*e^2)^5*(d + e*x)*((a*e + c*d*x)*(d + e* x))^(3/2))
Time = 0.74 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.67, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1244, 27, 1159, 1088}
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.
| \(\displaystyle \int \frac {x^2}{(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1244 |
| \(\displaystyle -\frac {2 \int -\frac {e^2 \left (2 a d e-\left (c d^2+5 a e^2\right ) x\right )}{2 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{5/2}}dx}{5 e^3 \left (c d^2-a e^2\right )}-\frac {2 d x}{5 e (d+e x) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {2 a d e-\left (c d^2+5 a e^2\right ) x}{\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{5/2}}dx}{5 e \left (c d^2-a e^2\right )}-\frac {2 d x}{5 e (d+e x) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1159 |
| \(\displaystyle \frac {-\frac {4 \left (5 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right ) \int \frac {1}{\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{3 \left (c d^2-a e^2\right )^2}-\frac {2 \left (x \left (5 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right )+4 a d e \left (3 a e^2+c d^2\right )\right )}{3 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}}{5 e \left (c d^2-a e^2\right )}-\frac {2 d x}{5 e (d+e x) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1088 |
| \(\displaystyle \frac {\frac {8 \left (5 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right ) \left (a e^2+c d^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {2 \left (x \left (5 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right )+4 a d e \left (3 a e^2+c d^2\right )\right )}{3 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}}{5 e \left (c d^2-a e^2\right )}-\frac {2 d x}{5 e (d+e x) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
Input:
Int[x^2/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]
Output:
(-2*d*x)/(5*e*(c*d^2 - a*e^2)*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e *x^2)^(3/2)) + ((-2*(4*a*d*e*(c*d^2 + 3*a*e^2) + (c^2*d^4 + 10*a*c*d^2*e^2 + 5*a^2*e^4)*x))/(3*(c*d^2 - a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e* x^2)^(3/2)) + (8*(c^2*d^4 + 10*a*c*d^2*e^2 + 5*a^2*e^4)*(c*d^2 + a*e^2 + 2 *c*d*e*x))/(3*(c*d^2 - a*e^2)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2 ]))/(5*e*(c*d^2 - a*e^2))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[-2*((b + 2*c*x)/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[((b*d - 2*a*e + (2*c*d - b*e)*x)/((p + 1)*(b^2 - 4*a*c)))*(a + b* x + c*x^2)^(p + 1), x] - Simp[(2*p + 3)*((2*c*d - b*e)/((p + 1)*(b^2 - 4*a* c))) Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] & & LtQ[p, -1] && NeQ[p, -3/2]
Int[(((f_.) + (g_.)*(x_))^(n_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_))/((d _) + (e_.)*(x_)), x_Symbol] :> Simp[(-(e*f - d*g))*(f + g*x)^(n - 1)*((a + b*x + c*x^2)^(p + 1)/(p*(2*c*d - b*e)*(d + e*x))), x] + Simp[1/(p*e^2*(2*c* d - b*e)) Int[(f + g*x)^(n - 2)*(a + b*x + c*x^2)^p*Simp[b*e*g*((-e)*f + d*g + e*f*n - d*g*n - e*f*p) + c*(d^2*g^2*(n - 1) - d*e*f*g*n + e^2*f^2*(2* p + 1)) - e*g*(b*e*g*p - c*(e*f*n - d*g*n + 2*e*f*p))*x, x], x], x] /; Free Q[{a, b, c, d, e, f, g}, x] && IGtQ[n, 1] && LtQ[p, -1] && EqQ[c*d^2 - b*d* e + a*e^2, 0]
Time = 2.68 (sec) , antiderivative size = 366, normalized size of antiderivative = 0.93
| method | result | size |
| gosper | \(-\frac {2 \left (c d x +a e \right ) \left (40 a^{2} c^{2} d^{2} e^{6} x^{4}+80 a \,c^{3} d^{4} e^{4} x^{4}+8 c^{4} d^{6} e^{2} x^{4}+60 a^{3} c d \,e^{7} x^{3}+220 a^{2} c^{2} d^{3} e^{5} x^{3}+212 a \,c^{3} d^{5} e^{3} x^{3}+20 c^{4} d^{7} e \,x^{3}+15 a^{4} e^{8} x^{2}+180 a^{3} c \,d^{2} e^{6} x^{2}+378 a^{2} c^{2} d^{4} e^{4} x^{2}+180 a \,c^{3} d^{6} e^{2} x^{2}+15 c^{4} d^{8} x^{2}+20 a^{4} d \,e^{7} x +212 a^{3} c \,d^{3} e^{5} x +220 a^{2} c^{2} d^{5} e^{3} x +60 a \,c^{3} d^{7} e x +8 a^{4} d^{2} e^{6}+80 a^{3} c \,d^{4} e^{4}+40 a^{2} c^{2} d^{6} e^{2}\right )}{15 \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right ) \left (c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}\) | \(366\) |
| orering | \(-\frac {2 \left (40 a^{2} c^{2} d^{2} e^{6} x^{4}+80 a \,c^{3} d^{4} e^{4} x^{4}+8 c^{4} d^{6} e^{2} x^{4}+60 a^{3} c d \,e^{7} x^{3}+220 a^{2} c^{2} d^{3} e^{5} x^{3}+212 a \,c^{3} d^{5} e^{3} x^{3}+20 c^{4} d^{7} e \,x^{3}+15 a^{4} e^{8} x^{2}+180 a^{3} c \,d^{2} e^{6} x^{2}+378 a^{2} c^{2} d^{4} e^{4} x^{2}+180 a \,c^{3} d^{6} e^{2} x^{2}+15 c^{4} d^{8} x^{2}+20 a^{4} d \,e^{7} x +212 a^{3} c \,d^{3} e^{5} x +220 a^{2} c^{2} d^{5} e^{3} x +60 a \,c^{3} d^{7} e x +8 a^{4} d^{2} e^{6}+80 a^{3} c \,d^{4} e^{4}+40 a^{2} c^{2} d^{6} e^{2}\right ) \left (c d x +a e \right )}{15 \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {5}{2}}}\) | \(367\) |
| trager | \(-\frac {2 \left (40 a^{2} c^{2} d^{2} e^{6} x^{4}+80 a \,c^{3} d^{4} e^{4} x^{4}+8 c^{4} d^{6} e^{2} x^{4}+60 a^{3} c d \,e^{7} x^{3}+220 a^{2} c^{2} d^{3} e^{5} x^{3}+212 a \,c^{3} d^{5} e^{3} x^{3}+20 c^{4} d^{7} e \,x^{3}+15 a^{4} e^{8} x^{2}+180 a^{3} c \,d^{2} e^{6} x^{2}+378 a^{2} c^{2} d^{4} e^{4} x^{2}+180 a \,c^{3} d^{6} e^{2} x^{2}+15 c^{4} d^{8} x^{2}+20 a^{4} d \,e^{7} x +212 a^{3} c \,d^{3} e^{5} x +220 a^{2} c^{2} d^{5} e^{3} x +60 a \,c^{3} d^{7} e x +8 a^{4} d^{2} e^{6}+80 a^{3} c \,d^{4} e^{4}+40 a^{2} c^{2} d^{6} e^{2}\right ) \sqrt {c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e}}{15 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \left (c d x +a e \right )^{2} \left (a \,e^{2}-c \,d^{2}\right ) \left (e x +d \right )^{3}}\) | \(374\) |
| default | \(\frac {-\frac {1}{3 d e c {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {3}{2}}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (\frac {\frac {4}{3} c d x e +\frac {2}{3} a \,e^{2}+\frac {2}{3} c \,d^{2}}{\left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {3}{2}}}+\frac {16 d e c \left (2 c d x e +a \,e^{2}+c \,d^{2}\right )}{3 \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right )^{2} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}\right )}{2 d e c}}{e}+\frac {d^{2} \left (-\frac {2}{5 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {8 d e c \left (-\frac {2 \left (2 d e c \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right )}{3 \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {16 d e c \left (2 d e c \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right )}{3 \left (a \,e^{2}-c \,d^{2}\right )^{4} \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}\right )}{5 \left (a \,e^{2}-c \,d^{2}\right )}\right )}{e^{3}}-\frac {d \left (\frac {\frac {4}{3} c d x e +\frac {2}{3} a \,e^{2}+\frac {2}{3} c \,d^{2}}{\left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {3}{2}}}+\frac {16 d e c \left (2 c d x e +a \,e^{2}+c \,d^{2}\right )}{3 \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right )^{2} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}\right )}{e^{2}}\) | \(621\) |
Input:
int(x^2/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2),x,method=_RETURNVE RBOSE)
Output:
-2/15*(c*d*x+a*e)*(40*a^2*c^2*d^2*e^6*x^4+80*a*c^3*d^4*e^4*x^4+8*c^4*d^6*e ^2*x^4+60*a^3*c*d*e^7*x^3+220*a^2*c^2*d^3*e^5*x^3+212*a*c^3*d^5*e^3*x^3+20 *c^4*d^7*e*x^3+15*a^4*e^8*x^2+180*a^3*c*d^2*e^6*x^2+378*a^2*c^2*d^4*e^4*x^ 2+180*a*c^3*d^6*e^2*x^2+15*c^4*d^8*x^2+20*a^4*d*e^7*x+212*a^3*c*d^3*e^5*x+ 220*a^2*c^2*d^5*e^3*x+60*a*c^3*d^7*e*x+8*a^4*d^2*e^6+80*a^3*c*d^4*e^4+40*a ^2*c^2*d^6*e^2)/(a^5*e^10-5*a^4*c*d^2*e^8+10*a^3*c^2*d^4*e^6-10*a^2*c^3*d^ 6*e^4+5*a*c^4*d^8*e^2-c^5*d^10)/(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(5/2)
Leaf count of result is larger than twice the leaf count of optimal. 820 vs. \(2 (373) = 746\).
Time = 15.57 (sec) , antiderivative size = 820, normalized size of antiderivative = 2.09 \[ \int \frac {x^2}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (40 \, a^{2} c^{2} d^{6} e^{2} + 80 \, a^{3} c d^{4} e^{4} + 8 \, a^{4} d^{2} e^{6} + 8 \, {\left (c^{4} d^{6} e^{2} + 10 \, a c^{3} d^{4} e^{4} + 5 \, a^{2} c^{2} d^{2} e^{6}\right )} x^{4} + 4 \, {\left (5 \, c^{4} d^{7} e + 53 \, a c^{3} d^{5} e^{3} + 55 \, a^{2} c^{2} d^{3} e^{5} + 15 \, a^{3} c d e^{7}\right )} x^{3} + 3 \, {\left (5 \, c^{4} d^{8} + 60 \, a c^{3} d^{6} e^{2} + 126 \, a^{2} c^{2} d^{4} e^{4} + 60 \, a^{3} c d^{2} e^{6} + 5 \, a^{4} e^{8}\right )} x^{2} + 4 \, {\left (15 \, a c^{3} d^{7} e + 55 \, a^{2} c^{2} d^{5} e^{3} + 53 \, a^{3} c d^{3} e^{5} + 5 \, a^{4} d e^{7}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{15 \, {\left (a^{2} c^{5} d^{13} e^{2} - 5 \, a^{3} c^{4} d^{11} e^{4} + 10 \, a^{4} c^{3} d^{9} e^{6} - 10 \, a^{5} c^{2} d^{7} e^{8} + 5 \, a^{6} c d^{5} e^{10} - a^{7} d^{3} e^{12} + {\left (c^{7} d^{12} e^{3} - 5 \, a c^{6} d^{10} e^{5} + 10 \, a^{2} c^{5} d^{8} e^{7} - 10 \, a^{3} c^{4} d^{6} e^{9} + 5 \, a^{4} c^{3} d^{4} e^{11} - a^{5} c^{2} d^{2} e^{13}\right )} x^{5} + {\left (3 \, c^{7} d^{13} e^{2} - 13 \, a c^{6} d^{11} e^{4} + 20 \, a^{2} c^{5} d^{9} e^{6} - 10 \, a^{3} c^{4} d^{7} e^{8} - 5 \, a^{4} c^{3} d^{5} e^{10} + 7 \, a^{5} c^{2} d^{3} e^{12} - 2 \, a^{6} c d e^{14}\right )} x^{4} + {\left (3 \, c^{7} d^{14} e - 9 \, a c^{6} d^{12} e^{3} + a^{2} c^{5} d^{10} e^{5} + 25 \, a^{3} c^{4} d^{8} e^{7} - 35 \, a^{4} c^{3} d^{6} e^{9} + 17 \, a^{5} c^{2} d^{4} e^{11} - a^{6} c d^{2} e^{13} - a^{7} e^{15}\right )} x^{3} + {\left (c^{7} d^{15} + a c^{6} d^{13} e^{2} - 17 \, a^{2} c^{5} d^{11} e^{4} + 35 \, a^{3} c^{4} d^{9} e^{6} - 25 \, a^{4} c^{3} d^{7} e^{8} - a^{5} c^{2} d^{5} e^{10} + 9 \, a^{6} c d^{3} e^{12} - 3 \, a^{7} d e^{14}\right )} x^{2} + {\left (2 \, a c^{6} d^{14} e - 7 \, a^{2} c^{5} d^{12} e^{3} + 5 \, a^{3} c^{4} d^{10} e^{5} + 10 \, a^{4} c^{3} d^{8} e^{7} - 20 \, a^{5} c^{2} d^{6} e^{9} + 13 \, a^{6} c d^{4} e^{11} - 3 \, a^{7} d^{2} e^{13}\right )} x\right )}} \] Input:
integrate(x^2/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm ="fricas")
Output:
2/15*(40*a^2*c^2*d^6*e^2 + 80*a^3*c*d^4*e^4 + 8*a^4*d^2*e^6 + 8*(c^4*d^6*e ^2 + 10*a*c^3*d^4*e^4 + 5*a^2*c^2*d^2*e^6)*x^4 + 4*(5*c^4*d^7*e + 53*a*c^3 *d^5*e^3 + 55*a^2*c^2*d^3*e^5 + 15*a^3*c*d*e^7)*x^3 + 3*(5*c^4*d^8 + 60*a* c^3*d^6*e^2 + 126*a^2*c^2*d^4*e^4 + 60*a^3*c*d^2*e^6 + 5*a^4*e^8)*x^2 + 4* (15*a*c^3*d^7*e + 55*a^2*c^2*d^5*e^3 + 53*a^3*c*d^3*e^5 + 5*a^4*d*e^7)*x)* sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(a^2*c^5*d^13*e^2 - 5*a^3*c^4* d^11*e^4 + 10*a^4*c^3*d^9*e^6 - 10*a^5*c^2*d^7*e^8 + 5*a^6*c*d^5*e^10 - a^ 7*d^3*e^12 + (c^7*d^12*e^3 - 5*a*c^6*d^10*e^5 + 10*a^2*c^5*d^8*e^7 - 10*a^ 3*c^4*d^6*e^9 + 5*a^4*c^3*d^4*e^11 - a^5*c^2*d^2*e^13)*x^5 + (3*c^7*d^13*e ^2 - 13*a*c^6*d^11*e^4 + 20*a^2*c^5*d^9*e^6 - 10*a^3*c^4*d^7*e^8 - 5*a^4*c ^3*d^5*e^10 + 7*a^5*c^2*d^3*e^12 - 2*a^6*c*d*e^14)*x^4 + (3*c^7*d^14*e - 9 *a*c^6*d^12*e^3 + a^2*c^5*d^10*e^5 + 25*a^3*c^4*d^8*e^7 - 35*a^4*c^3*d^6*e ^9 + 17*a^5*c^2*d^4*e^11 - a^6*c*d^2*e^13 - a^7*e^15)*x^3 + (c^7*d^15 + a* c^6*d^13*e^2 - 17*a^2*c^5*d^11*e^4 + 35*a^3*c^4*d^9*e^6 - 25*a^4*c^3*d^7*e ^8 - a^5*c^2*d^5*e^10 + 9*a^6*c*d^3*e^12 - 3*a^7*d*e^14)*x^2 + (2*a*c^6*d^ 14*e - 7*a^2*c^5*d^12*e^3 + 5*a^3*c^4*d^10*e^5 + 10*a^4*c^3*d^8*e^7 - 20*a ^5*c^2*d^6*e^9 + 13*a^6*c*d^4*e^11 - 3*a^7*d^2*e^13)*x)
Timed out. \[ \int \frac {x^2}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(x**2/(e*x+d)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)
Output:
Timed out
Exception generated. \[ \int \frac {x^2}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^2/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm ="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e*(a*e^2-c*d^2)>0)', see `assume ?` for mor
\[ \int \frac {x^2}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\int { \frac {x^{2}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}} {\left (e x + d\right )}} \,d x } \] Input:
integrate(x^2/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm ="giac")
Output:
integrate(x^2/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)*(e*x + d)), x )
Time = 7.06 (sec) , antiderivative size = 3099, normalized size of antiderivative = 7.89 \[ \int \frac {x^2}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:
int(x^2/((d + e*x)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)),x)
Output:
(((6*a*e^2 - 10*c*d^2)/(15*(a*e^2 - c*d^2)^4) - (4*c*d^2)/(5*(a*e^2 - c*d^ 2)^4))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x) - (((d*((e *(2*a*e^3 - 2*c*d^2*e))/(5*(a*e^2 - c*d^2)^3*(3*a*e^3 - 3*c*d^2*e)) - (4*c *d^2*e^2)/(5*(a*e^2 - c*d^2)^3*(3*a*e^3 - 3*c*d^2*e))))/e + (e*(2*c*d^3 + 2*a*d*e^2))/(5*(a*e^2 - c*d^2)^3*(3*a*e^3 - 3*c*d^2*e)))*(x*(a*e^2 + c*d^2 ) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^2 + ((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*(x*((((12*c^3*d^3*e^2)/(5*(a*e^2 - c*d^2)^2*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (4*c^3*d^3*e^2*(a*e^2 + c*d^2))/(5*(a*e^ 2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)))*(a*e^2 + c*d^2) )/(c*d*e) - (6*c^2*d^2*e*(a*e^2 + c*d^2))/(5*(a*e^2 - c*d^2)^2*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (8*a*c^3*d^4*e^3)/(5*(a*e^2 - c*d^2)^3 *(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (2*c^2*d^2*e*(46*a^2*e^4 + 4*c^2*d^4 + 66*a*c*d^2*e^2))/(15*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d ^3*e^3 + a^2*c*d*e^5))) + (a*((12*c^3*d^3*e^2)/(5*(a*e^2 - c*d^2)^2*(c^3*d ^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (4*c^3*d^3*e^2*(a*e^2 + c*d^2))/( 5*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/c - (c* d*(a*e^2 + c*d^2)*(46*a^2*e^4 + 4*c^2*d^4 + 66*a*c*d^2*e^2))/(15*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/((a*e + c*d*x)*(d + e*x)) + ((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*(x*((a*(((a*e^2 + c*d^2)*((4*c^4*d^4*e^3*(a*e^2 + c*d^2))/(15*(a*e^2 - c*d^2)^3*(c^3*d^5...
Time = 8.04 (sec) , antiderivative size = 1349, normalized size of antiderivative = 3.43 \[ \int \frac {x^2}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:
int(x^2/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x)
Output:
(2*(40*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a**3*d**3*e**5 + 120*sqrt (e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a**3*d**2*e**6*x + 120*sqrt(e)*sqrt( d)*sqrt(c)*sqrt(a*e + c*d*x)*a**3*d*e**7*x**2 + 40*sqrt(e)*sqrt(d)*sqrt(c) *sqrt(a*e + c*d*x)*a**3*e**8*x**3 + 80*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a**2*c*d**5*e**3 + 280*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a* *2*c*d**4*e**4*x + 360*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a**2*c*d* *3*e**5*x**2 + 200*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a**2*c*d**2*e **6*x**3 + 40*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a**2*c*d*e**7*x**4 + 8*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a*c**2*d**7*e + 104*sqrt(e) *sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a*c**2*d**6*e**2*x + 264*sqrt(e)*sqrt(d )*sqrt(c)*sqrt(a*e + c*d*x)*a*c**2*d**5*e**3*x**2 + 248*sqrt(e)*sqrt(d)*sq rt(c)*sqrt(a*e + c*d*x)*a*c**2*d**4*e**4*x**3 + 80*sqrt(e)*sqrt(d)*sqrt(c) *sqrt(a*e + c*d*x)*a*c**2*d**3*e**5*x**4 + 8*sqrt(e)*sqrt(d)*sqrt(c)*sqrt( a*e + c*d*x)*c**3*d**8*x + 24*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*c* *3*d**7*e*x**2 + 24*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*c**3*d**6*e* *2*x**3 + 8*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*c**3*d**5*e**3*x**4 - 8*sqrt(d + e*x)*a**4*d**2*e**7 - 20*sqrt(d + e*x)*a**4*d*e**8*x - 15*sqr t(d + e*x)*a**4*e**9*x**2 - 80*sqrt(d + e*x)*a**3*c*d**4*e**5 - 212*sqrt(d + e*x)*a**3*c*d**3*e**6*x - 180*sqrt(d + e*x)*a**3*c*d**2*e**7*x**2 - 60* sqrt(d + e*x)*a**3*c*d*e**8*x**3 - 40*sqrt(d + e*x)*a**2*c**2*d**6*e**3...