Integrand size = 40, antiderivative size = 341 \[ \int \frac {x^2}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}} \, dx=\frac {2 x^2}{7 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}-\frac {8 \left (2 a d e \left (c d^2+2 a e^2\right )+\left (2 c^2 d^4+a c d^2 e^2+3 a^2 e^4\right ) x\right )}{35 e \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}+\frac {16 \left (3 c^2 d^4+14 a c d^2 e^2+7 a^2 e^4\right ) \left (c d^2+a e^2+2 c d e x\right )}{105 e \left (c d^2-a e^2\right )^5 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {128 c d \left (3 c^2 d^4+14 a c d^2 e^2+7 a^2 e^4\right ) \left (c d^2+a e^2+2 c d e x\right )}{105 \left (c d^2-a e^2\right )^7 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \]
2/7*x^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)^(5/2)-8/3 5*(2*a*d*e*(2*a*e^2+c*d^2)+(3*a^2*e^4+a*c*d^2*e^2+2*c^2*d^4)*x)/e/(-a*e^2+ c*d^2)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)+16/105*(7*a^2*e^4+14*a*c* d^2*e^2+3*c^2*d^4)*(2*c*d*e*x+a*e^2+c*d^2)/e/(-a*e^2+c*d^2)^5/(a*d*e+(a*e^ 2+c*d^2)*x+c*d*e*x^2)^(3/2)-128/105*c*d*(7*a^2*e^4+14*a*c*d^2*e^2+3*c^2*d^ 4)*(2*c*d*e*x+a*e^2+c*d^2)/(-a*e^2+c*d^2)^7/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x ^2)^(1/2)
Time = 0.35 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.29 \[ \int \frac {x^2}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}} \, dx=-\frac {2 \sqrt {(a e+c d x) (d+e x)} \left (-15 d^2 e^4 (a e+c d x)^6+84 c d^3 e^3 (a e+c d x)^5 (d+e x)+42 a d e^5 (a e+c d x)^5 (d+e x)-210 c^2 d^4 e^2 (a e+c d x)^4 (d+e x)^2-280 a c d^2 e^4 (a e+c d x)^4 (d+e x)^2-35 a^2 e^6 (a e+c d x)^4 (d+e x)^2+420 c^3 d^5 e (a e+c d x)^3 (d+e x)^3+1260 a c^2 d^3 e^3 (a e+c d x)^3 (d+e x)^3+420 a^2 c d e^5 (a e+c d x)^3 (d+e x)^3+105 c^4 d^6 (a e+c d x)^2 (d+e x)^4+840 a c^3 d^4 e^2 (a e+c d x)^2 (d+e x)^4+630 a^2 c^2 d^2 e^4 (a e+c d x)^2 (d+e x)^4-70 a c^4 d^5 e (a e+c d x) (d+e x)^5-140 a^2 c^3 d^3 e^3 (a e+c d x) (d+e x)^5+21 a^2 c^4 d^4 e^2 (d+e x)^6\right )}{105 \left (c d^2-a e^2\right )^7 (a e+c d x)^3 (d+e x)^4} \]
(-2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-15*d^2*e^4*(a*e + c*d*x)^6 + 84*c*d^3* e^3*(a*e + c*d*x)^5*(d + e*x) + 42*a*d*e^5*(a*e + c*d*x)^5*(d + e*x) - 210 *c^2*d^4*e^2*(a*e + c*d*x)^4*(d + e*x)^2 - 280*a*c*d^2*e^4*(a*e + c*d*x)^4 *(d + e*x)^2 - 35*a^2*e^6*(a*e + c*d*x)^4*(d + e*x)^2 + 420*c^3*d^5*e*(a*e + c*d*x)^3*(d + e*x)^3 + 1260*a*c^2*d^3*e^3*(a*e + c*d*x)^3*(d + e*x)^3 + 420*a^2*c*d*e^5*(a*e + c*d*x)^3*(d + e*x)^3 + 105*c^4*d^6*(a*e + c*d*x)^2 *(d + e*x)^4 + 840*a*c^3*d^4*e^2*(a*e + c*d*x)^2*(d + e*x)^4 + 630*a^2*c^2 *d^2*e^4*(a*e + c*d*x)^2*(d + e*x)^4 - 70*a*c^4*d^5*e*(a*e + c*d*x)*(d + e *x)^5 - 140*a^2*c^3*d^3*e^3*(a*e + c*d*x)*(d + e*x)^5 + 21*a^2*c^4*d^4*e^2 *(d + e*x)^6))/(105*(c*d^2 - a*e^2)^7*(a*e + c*d*x)^3*(d + e*x)^4)
Time = 0.53 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1244, 27, 1159, 1089, 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.
| \(\displaystyle \int \frac {x^2}{(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}} \, dx\) |
| \(\Big \downarrow \) 1244 |
| \(\displaystyle -\frac {2 \int -\frac {e^2 \left (2 a d e-\left (3 c d^2+7 a e^2\right ) x\right )}{2 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{7/2}}dx}{7 e^3 \left (c d^2-a e^2\right )}-\frac {2 d x}{7 e (d+e x) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {2 a d e-\left (3 c d^2+7 a e^2\right ) x}{\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{7/2}}dx}{7 e \left (c d^2-a e^2\right )}-\frac {2 d x}{7 e (d+e x) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 1159 |
| \(\displaystyle \frac {-\frac {8 \left (7 a^2 e^4+14 a c d^2 e^2+3 c^2 d^4\right ) \int \frac {1}{\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{5/2}}dx}{5 \left (c d^2-a e^2\right )^2}-\frac {2 \left (x \left (7 a^2 e^4+14 a c d^2 e^2+3 c^2 d^4\right )+8 a d e \left (2 a e^2+c d^2\right )\right )}{5 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}}{7 e \left (c d^2-a e^2\right )}-\frac {2 d x}{7 e (d+e x) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 1089 |
| \(\displaystyle \frac {-\frac {8 \left (7 a^2 e^4+14 a c d^2 e^2+3 c^2 d^4\right ) \left (-\frac {8 c d e \int \frac {1}{\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{3 \left (c d^2-a e^2\right )^2}-\frac {2 \left (a e^2+c d^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\right )}{5 \left (c d^2-a e^2\right )^2}-\frac {2 \left (x \left (7 a^2 e^4+14 a c d^2 e^2+3 c^2 d^4\right )+8 a d e \left (2 a e^2+c d^2\right )\right )}{5 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}}{7 e \left (c d^2-a e^2\right )}-\frac {2 d x}{7 e (d+e x) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 1088 |
| \(\displaystyle \frac {-\frac {2 \left (x \left (7 a^2 e^4+14 a c d^2 e^2+3 c^2 d^4\right )+8 a d e \left (2 a e^2+c d^2\right )\right )}{5 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}-\frac {8 \left (7 a^2 e^4+14 a c d^2 e^2+3 c^2 d^4\right ) \left (\frac {16 c d e \left (a e^2+c d^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {2 \left (a e^2+c d^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\right )}{5 \left (c d^2-a e^2\right )^2}}{7 e \left (c d^2-a e^2\right )}-\frac {2 d x}{7 e (d+e x) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}\) |
(-2*d*x)/(7*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)^(5/2)) + ((-2*(8*a*d*e*(c*d^2 + 2*a*e^2) + (3*c^2*d^4 + 14*a*c*d^2*e ^2 + 7*a^2*e^4)*x))/(5*(c*d^2 - a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d* e*x^2)^(5/2)) - (8*(3*c^2*d^4 + 14*a*c*d^2*e^2 + 7*a^2*e^4)*((-2*(c*d^2 + a*e^2 + 2*c*d*e*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)) + (16*c*d*e*(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*(c*d^2 - a*e^2)^2))/(7 *e*(c*d^2 - a*e^2))
3.5.88.3.1 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[((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[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[2*c*((2*p + 3)/((p + 1)*(b^2 - 4*a*c))) Int[(a + b*x + c*x^2)^(p + 1), x], x] /; Fre eQ[{a, b, c}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[3*p])
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]
Leaf count of result is larger than twice the leaf count of optimal. \(662\) vs. \(2(325)=650\).
Time = 0.66 (sec) , antiderivative size = 663, normalized size of antiderivative = 1.94
| method | result | size |
| gosper | \(-\frac {2 \left (c d x +a e \right ) \left (-896 a^{2} c^{4} d^{4} e^{8} x^{6}-1792 a \,c^{5} d^{6} e^{6} x^{6}-384 c^{6} d^{8} e^{4} x^{6}-2240 a^{3} c^{3} d^{3} e^{9} x^{5}-7616 a^{2} c^{4} d^{5} e^{7} x^{5}-7232 a \,c^{5} d^{7} e^{5} x^{5}-1344 c^{6} d^{9} e^{3} x^{5}-1680 a^{4} c^{2} d^{2} e^{10} x^{4}-11200 a^{3} c^{3} d^{4} e^{8} x^{4}-20320 a^{2} c^{4} d^{6} e^{6} x^{4}-11200 a \,c^{5} d^{8} e^{4} x^{4}-1680 c^{6} d^{10} e^{2} x^{4}-280 a^{5} c d \,e^{11} x^{3}-6440 a^{4} c^{2} d^{3} e^{9} x^{3}-21680 a^{3} c^{3} d^{5} e^{7} x^{3}-24080 a^{2} c^{4} d^{7} e^{5} x^{3}-8120 a \,c^{5} d^{9} e^{3} x^{3}-840 c^{6} d^{11} e \,x^{3}+35 a^{6} e^{12} x^{2}-910 a^{5} c \,d^{2} e^{10} x^{2}-9295 a^{4} c^{2} d^{4} e^{8} x^{2}-20020 a^{3} c^{3} d^{6} e^{6} x^{2}-13195 a^{2} c^{4} d^{8} e^{4} x^{2}-2590 a \,c^{5} d^{10} e^{2} x^{2}-105 c^{6} d^{12} x^{2}+28 a^{6} d \,e^{11} x -764 a^{5} c \,d^{3} e^{9} x -6440 a^{4} c^{2} d^{5} e^{7} x -8120 a^{3} c^{3} d^{7} e^{5} x -2996 a^{2} c^{4} d^{9} e^{3} x -140 a \,c^{5} d^{11} e x +8 a^{6} d^{2} e^{10}-224 a^{5} c \,d^{4} e^{8}-1680 a^{4} c^{2} d^{6} e^{6}-1120 a^{3} c^{3} d^{8} e^{4}-56 a^{2} c^{4} d^{10} e^{2}\right )}{105 \left (a^{7} e^{14}-7 a^{6} c \,d^{2} e^{12}+21 a^{5} c^{2} d^{4} e^{10}-35 a^{4} c^{3} d^{6} e^{8}+35 a^{3} c^{4} d^{8} e^{6}-21 a^{2} c^{5} d^{10} e^{4}+7 a \,c^{6} d^{12} e^{2}-c^{7} d^{14}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {7}{2}}}\) | \(663\) |
| trager | \(-\frac {2 \left (-896 a^{2} c^{4} d^{4} e^{8} x^{6}-1792 a \,c^{5} d^{6} e^{6} x^{6}-384 c^{6} d^{8} e^{4} x^{6}-2240 a^{3} c^{3} d^{3} e^{9} x^{5}-7616 a^{2} c^{4} d^{5} e^{7} x^{5}-7232 a \,c^{5} d^{7} e^{5} x^{5}-1344 c^{6} d^{9} e^{3} x^{5}-1680 a^{4} c^{2} d^{2} e^{10} x^{4}-11200 a^{3} c^{3} d^{4} e^{8} x^{4}-20320 a^{2} c^{4} d^{6} e^{6} x^{4}-11200 a \,c^{5} d^{8} e^{4} x^{4}-1680 c^{6} d^{10} e^{2} x^{4}-280 a^{5} c d \,e^{11} x^{3}-6440 a^{4} c^{2} d^{3} e^{9} x^{3}-21680 a^{3} c^{3} d^{5} e^{7} x^{3}-24080 a^{2} c^{4} d^{7} e^{5} x^{3}-8120 a \,c^{5} d^{9} e^{3} x^{3}-840 c^{6} d^{11} e \,x^{3}+35 a^{6} e^{12} x^{2}-910 a^{5} c \,d^{2} e^{10} x^{2}-9295 a^{4} c^{2} d^{4} e^{8} x^{2}-20020 a^{3} c^{3} d^{6} e^{6} x^{2}-13195 a^{2} c^{4} d^{8} e^{4} x^{2}-2590 a \,c^{5} d^{10} e^{2} x^{2}-105 c^{6} d^{12} x^{2}+28 a^{6} d \,e^{11} x -764 a^{5} c \,d^{3} e^{9} x -6440 a^{4} c^{2} d^{5} e^{7} x -8120 a^{3} c^{3} d^{7} e^{5} x -2996 a^{2} c^{4} d^{9} e^{3} x -140 a \,c^{5} d^{11} e x +8 a^{6} d^{2} e^{10}-224 a^{5} c \,d^{4} e^{8}-1680 a^{4} c^{2} d^{6} e^{6}-1120 a^{3} c^{3} d^{8} e^{4}-56 a^{2} c^{4} d^{10} e^{2}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{105 \left (a^{6} e^{12}-6 a^{5} c \,d^{2} e^{10}+15 a^{4} c^{2} d^{4} e^{8}-20 a^{3} c^{3} d^{6} e^{6}+15 a^{2} c^{4} d^{8} e^{4}-6 a \,c^{5} d^{10} e^{2}+c^{6} d^{12}\right ) \left (c d x +a e \right )^{3} \left (e^{2} a -c \,d^{2}\right ) \left (e x +d \right )^{4}}\) | \(671\) |
| default | \(\frac {-\frac {1}{5 c d e {\left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )}^{\frac {5}{2}}}-\frac {\left (e^{2} a +c \,d^{2}\right ) \left (\frac {\frac {4}{5} c d e x +\frac {2}{5} e^{2} a +\frac {2}{5} c \,d^{2}}{\left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) {\left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )}^{\frac {5}{2}}}+\frac {16 c d e \left (\frac {\frac {4}{3} c d e x +\frac {2}{3} e^{2} a +\frac {2}{3} c \,d^{2}}{\left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) {\left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )}^{\frac {3}{2}}}+\frac {16 c d e \left (2 c d e x +e^{2} a +c \,d^{2}\right )}{3 \left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right )^{2} \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}\right )}{5 \left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right )}\right )}{2 c d e}}{e}-\frac {d \left (\frac {\frac {4}{5} c d e x +\frac {2}{5} e^{2} a +\frac {2}{5} c \,d^{2}}{\left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) {\left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )}^{\frac {5}{2}}}+\frac {16 c d e \left (\frac {\frac {4}{3} c d e x +\frac {2}{3} e^{2} a +\frac {2}{3} c \,d^{2}}{\left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) {\left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )}^{\frac {3}{2}}}+\frac {16 c d e \left (2 c d e x +e^{2} a +c \,d^{2}\right )}{3 \left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right )^{2} \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}\right )}{5 \left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right )}\right )}{e^{2}}+\frac {d^{2} \left (-\frac {2}{7 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}-\frac {12 c d e \left (-\frac {2 \left (2 c d e \left (x +\frac {d}{e}\right )+e^{2} a -c \,d^{2}\right )}{5 \left (e^{2} a -c \,d^{2}\right )^{2} \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}-\frac {16 c d e \left (-\frac {2 \left (2 c d e \left (x +\frac {d}{e}\right )+e^{2} a -c \,d^{2}\right )}{3 \left (e^{2} a -c \,d^{2}\right )^{2} \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {16 c d e \left (2 c d e \left (x +\frac {d}{e}\right )+e^{2} a -c \,d^{2}\right )}{3 \left (e^{2} a -c \,d^{2}\right )^{4} \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}\right )}{5 \left (e^{2} a -c \,d^{2}\right )^{2}}\right )}{7 \left (e^{2} a -c \,d^{2}\right )}\right )}{e^{3}}\) | \(933\) |
-2/105*(c*d*x+a*e)*(-896*a^2*c^4*d^4*e^8*x^6-1792*a*c^5*d^6*e^6*x^6-384*c^ 6*d^8*e^4*x^6-2240*a^3*c^3*d^3*e^9*x^5-7616*a^2*c^4*d^5*e^7*x^5-7232*a*c^5 *d^7*e^5*x^5-1344*c^6*d^9*e^3*x^5-1680*a^4*c^2*d^2*e^10*x^4-11200*a^3*c^3* d^4*e^8*x^4-20320*a^2*c^4*d^6*e^6*x^4-11200*a*c^5*d^8*e^4*x^4-1680*c^6*d^1 0*e^2*x^4-280*a^5*c*d*e^11*x^3-6440*a^4*c^2*d^3*e^9*x^3-21680*a^3*c^3*d^5* e^7*x^3-24080*a^2*c^4*d^7*e^5*x^3-8120*a*c^5*d^9*e^3*x^3-840*c^6*d^11*e*x^ 3+35*a^6*e^12*x^2-910*a^5*c*d^2*e^10*x^2-9295*a^4*c^2*d^4*e^8*x^2-20020*a^ 3*c^3*d^6*e^6*x^2-13195*a^2*c^4*d^8*e^4*x^2-2590*a*c^5*d^10*e^2*x^2-105*c^ 6*d^12*x^2+28*a^6*d*e^11*x-764*a^5*c*d^3*e^9*x-6440*a^4*c^2*d^5*e^7*x-8120 *a^3*c^3*d^7*e^5*x-2996*a^2*c^4*d^9*e^3*x-140*a*c^5*d^11*e*x+8*a^6*d^2*e^1 0-224*a^5*c*d^4*e^8-1680*a^4*c^2*d^6*e^6-1120*a^3*c^3*d^8*e^4-56*a^2*c^4*d ^10*e^2)/(a^7*e^14-7*a^6*c*d^2*e^12+21*a^5*c^2*d^4*e^10-35*a^4*c^3*d^6*e^8 +35*a^3*c^4*d^8*e^6-21*a^2*c^5*d^10*e^4+7*a*c^6*d^12*e^2-c^7*d^14)/(c*d*e* x^2+a*e^2*x+c*d^2*x+a*d*e)^(7/2)
Leaf count of result is larger than twice the leaf count of optimal. 1540 vs. \(2 (325) = 650\).
Time = 115.68 (sec) , antiderivative size = 1540, normalized size of antiderivative = 4.52 \[ \int \frac {x^2}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}} \, dx=\text {Too large to display} \]
-2/105*(56*a^2*c^4*d^10*e^2 + 1120*a^3*c^3*d^8*e^4 + 1680*a^4*c^2*d^6*e^6 + 224*a^5*c*d^4*e^8 - 8*a^6*d^2*e^10 + 128*(3*c^6*d^8*e^4 + 14*a*c^5*d^6*e ^6 + 7*a^2*c^4*d^4*e^8)*x^6 + 64*(21*c^6*d^9*e^3 + 113*a*c^5*d^7*e^5 + 119 *a^2*c^4*d^5*e^7 + 35*a^3*c^3*d^3*e^9)*x^5 + 80*(21*c^6*d^10*e^2 + 140*a*c ^5*d^8*e^4 + 254*a^2*c^4*d^6*e^6 + 140*a^3*c^3*d^4*e^8 + 21*a^4*c^2*d^2*e^ 10)*x^4 + 40*(21*c^6*d^11*e + 203*a*c^5*d^9*e^3 + 602*a^2*c^4*d^7*e^5 + 54 2*a^3*c^3*d^5*e^7 + 161*a^4*c^2*d^3*e^9 + 7*a^5*c*d*e^11)*x^3 + 5*(21*c^6* d^12 + 518*a*c^5*d^10*e^2 + 2639*a^2*c^4*d^8*e^4 + 4004*a^3*c^3*d^6*e^6 + 1859*a^4*c^2*d^4*e^8 + 182*a^5*c*d^2*e^10 - 7*a^6*e^12)*x^2 + 4*(35*a*c^5* d^11*e + 749*a^2*c^4*d^9*e^3 + 2030*a^3*c^3*d^7*e^5 + 1610*a^4*c^2*d^5*e^7 + 191*a^5*c*d^3*e^9 - 7*a^6*d*e^11)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(a^3*c^7*d^18*e^3 - 7*a^4*c^6*d^16*e^5 + 21*a^5*c^5*d^14*e^7 - 3 5*a^6*c^4*d^12*e^9 + 35*a^7*c^3*d^10*e^11 - 21*a^8*c^2*d^8*e^13 + 7*a^9*c* d^6*e^15 - a^10*d^4*e^17 + (c^10*d^17*e^4 - 7*a*c^9*d^15*e^6 + 21*a^2*c^8* d^13*e^8 - 35*a^3*c^7*d^11*e^10 + 35*a^4*c^6*d^9*e^12 - 21*a^5*c^5*d^7*e^1 4 + 7*a^6*c^4*d^5*e^16 - a^7*c^3*d^3*e^18)*x^7 + (4*c^10*d^18*e^3 - 25*a*c ^9*d^16*e^5 + 63*a^2*c^8*d^14*e^7 - 77*a^3*c^7*d^12*e^9 + 35*a^4*c^6*d^10* e^11 + 21*a^5*c^5*d^8*e^13 - 35*a^6*c^4*d^6*e^15 + 17*a^7*c^3*d^4*e^17 - 3 *a^8*c^2*d^2*e^19)*x^6 + 3*(2*c^10*d^19*e^2 - 10*a*c^9*d^17*e^4 + 15*a^2*c ^8*d^15*e^6 + 7*a^3*c^7*d^13*e^8 - 49*a^4*c^6*d^11*e^10 + 63*a^5*c^5*d^...
Timed out. \[ \int \frac {x^2}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {x^2}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {x^2}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}} \, dx=\int { \frac {x^{2}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {7}{2}} {\left (e x + d\right )}} \,d x } \]
Time = 17.00 (sec) , antiderivative size = 11469, normalized size of antiderivative = 33.63 \[ \int \frac {x^2}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}} \, dx=\text {Too large to display} \]
((6*c^3*d^5 + 36*a*c^2*d^3*e^2 - 10*a^2*c*d*e^4)/(105*(a*e^2 - c*d^2)^6) - x*((16*c^2*d^2*e)/(105*(a*e^2 - c*d^2)^5) - (8*c^2*d^2*e*(a*e^2 + c*d^2)) /(105*(a*e^2 - c*d^2)^6)) + (8*a*c^2*d^3*e^2)/(105*(a*e^2 - c*d^2)^6))/(x* (a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2) + (x*((a*((64*c^5*d^5*e^4*(a*e^ 2 + c*d^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d* e^5)) - (64*c^5*d^5*e^4*(5*a*e^2 - 3*c*d^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d ^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/c - ((a*e^2 + c*d^2)*(((a*e^2 + c *d^2)*((64*c^5*d^5*e^4*(a*e^2 + c*d^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (64*c^5*d^5*e^4*(5*a*e^2 - 3*c*d^2))/( 105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d* e) - (32*c^4*d^4*e^3*(7*c^2*d^4 - 9*a^2*e^4 + 18*a*c*d^2*e^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (128*a*c^5*d^6* e^5)/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (32*c^4*d^4*e^3*(a*e^2 + c*d^2)*(5*a*e^2 - 3*c*d^2))/(105*(a*e^2 - c*d^2) ^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) + (2*c^2*d^2*e^2 *(60*c^4*d^7 - 204*a*c^3*d^5*e^2 - 156*a^2*c^2*d^3*e^4 + 44*a^3*c*d*e^6))/ (105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16* c^3*d^3*e^2*(a*e^2 + c*d^2)*(7*c^2*d^4 - 9*a^2*e^4 + 18*a*c*d^2*e^2))/(105 *(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))) - (a*(((a *e^2 + c*d^2)*((64*c^5*d^5*e^4*(a*e^2 + c*d^2))/(105*(a*e^2 - c*d^2)^6*...