Integrand size = 40, antiderivative size = 310 \[ \int \frac {x^2}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {2 d^2}{7 e^2 \left (c d^2-a e^2\right ) (d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {4 d \left (3 c d^2-7 a e^2\right )}{35 e^2 \left (c d^2-a e^2\right )^2 (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {2 \left (c^2 d^4-14 a c d^2 e^2-35 a^2 e^4\right )}{105 e^2 \left (c d^2-a e^2\right )^3 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {8 c d \left (c^2 d^4-14 a c d^2 e^2-35 a^2 e^4\right ) \left (c d^2+a e^2+2 c d e x\right )}{105 e^2 \left (c d^2-a e^2\right )^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \] Output:
2/7*d^2/e^2/(-a*e^2+c*d^2)/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/ 2)-4/35*d*(-7*a*e^2+3*c*d^2)/e^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)-2/105*(-35*a^2*e^4-14*a*c*d^2*e^2+c^2*d^4)/e^2/( -a*e^2+c*d^2)^3/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+8/105*c*d* (-35*a^2*e^4-14*a*c*d^2*e^2+c^2*d^4)*(2*c*d*e*x+a*e^2+c*d^2)/e^2/(-a*e^2+c *d^2)^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)
Time = 0.30 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.76 \[ \int \frac {x^2}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {2 \left (c^4 d^6 x^2 \left (35 d^2+28 d e x+8 e^2 x^2\right )+a^4 e^6 \left (8 d^2+28 d e x+35 e^2 x^2\right )-4 a c^3 d^4 e x \left (35 d^3+119 d^2 e x+97 d e^2 x^2+28 e^3 x^3\right )-4 a^3 c d e^4 \left (28 d^3+97 d^2 e x+119 d e^2 x^2+35 e^3 x^3\right )-2 a^2 c^2 d^2 e^2 \left (140 d^4+518 d^3 e x+711 d^2 e^2 x^2+518 d e^3 x^3+140 e^4 x^4\right )\right )}{105 \left (c d^2-a e^2\right )^5 (d+e x)^3 \sqrt {(a e+c d x) (d+e x)}} \] Input:
Integrate[x^2/((d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)), x]
Output:
(2*(c^4*d^6*x^2*(35*d^2 + 28*d*e*x + 8*e^2*x^2) + a^4*e^6*(8*d^2 + 28*d*e* x + 35*e^2*x^2) - 4*a*c^3*d^4*e*x*(35*d^3 + 119*d^2*e*x + 97*d*e^2*x^2 + 2 8*e^3*x^3) - 4*a^3*c*d*e^4*(28*d^3 + 97*d^2*e*x + 119*d*e^2*x^2 + 35*e^3*x ^3) - 2*a^2*c^2*d^2*e^2*(140*d^4 + 518*d^3*e*x + 711*d^2*e^2*x^2 + 518*d*e ^3*x^3 + 140*e^4*x^4)))/(105*(c*d^2 - a*e^2)^5*(d + e*x)^3*Sqrt[(a*e + c*d *x)*(d + e*x)])
Time = 1.08 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.17, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1267, 27, 1220, 1129, 1129, 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.
| \(\displaystyle \int \frac {x^2}{(d+e x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1267 |
| \(\displaystyle -\frac {\int \frac {e \left (d \left (c d^2+5 a e^2\right )+e \left (7 c d^2+5 a e^2\right ) x\right )}{2 (d+e x)^3 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{3 c d e^3}-\frac {1}{3 c d e^2 (d+e x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {d \left (c d^2+5 a e^2\right )+e \left (7 c d^2+5 a e^2\right ) x}{(d+e x)^3 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{6 c d e^2}-\frac {1}{3 c d e^2 (d+e x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 1220 |
| \(\displaystyle -\frac {\frac {\left (-35 a^2 e^4-14 a c d^2 e^2+c^2 d^4\right ) \int \frac {1}{(d+e x)^2 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{7 \left (c d^2-a e^2\right )}-\frac {12 c d^3}{7 (d+e x)^3 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{6 c d e^2}-\frac {1}{3 c d e^2 (d+e x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle -\frac {\frac {\left (-35 a^2 e^4-14 a c d^2 e^2+c^2 d^4\right ) \left (\frac {6 c d \int \frac {1}{(d+e x) \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{5 \left (c d^2-a e^2\right )}+\frac {2}{5 (d+e x)^2 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{7 \left (c d^2-a e^2\right )}-\frac {12 c d^3}{7 (d+e x)^3 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{6 c d e^2}-\frac {1}{3 c d e^2 (d+e x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle -\frac {\frac {\left (-35 a^2 e^4-14 a c d^2 e^2+c^2 d^4\right ) \left (\frac {6 c d \left (\frac {4 c d \int \frac {1}{\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{3 \left (c d^2-a e^2\right )}+\frac {2}{3 (d+e x) \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{5 \left (c d^2-a e^2\right )}+\frac {2}{5 (d+e x)^2 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{7 \left (c d^2-a e^2\right )}-\frac {12 c d^3}{7 (d+e x)^3 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{6 c d e^2}-\frac {1}{3 c d e^2 (d+e x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 1088 |
| \(\displaystyle -\frac {\frac {\left (-35 a^2 e^4-14 a c d^2 e^2+c^2 d^4\right ) \left (\frac {6 c d \left (\frac {2}{3 (d+e x) \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {8 c d \left (a e^2+c d^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{5 \left (c d^2-a e^2\right )}+\frac {2}{5 (d+e x)^2 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{7 \left (c d^2-a e^2\right )}-\frac {12 c d^3}{7 (d+e x)^3 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{6 c d e^2}-\frac {1}{3 c d e^2 (d+e x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
Input:
Int[x^2/((d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]
Output:
-1/3*1/(c*d*e^2*(d + e*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - ((-12*c*d^3)/(7*(c*d^2 - a*e^2)*(d + e*x)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)* x + c*d*e*x^2]) + ((c^2*d^4 - 14*a*c*d^2*e^2 - 35*a^2*e^4)*(2/(5*(c*d^2 - a*e^2)*(d + e*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (6*c*d*( 2/(3*(c*d^2 - a*e^2)*(d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2] ) - (8*c*d*(c*d^2 + a*e^2 + 2*c*d*e*x))/(3*(c*d^2 - a*e^2)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])))/(5*(c*d^2 - a*e^2))))/(7*(c*d^2 - a*e^2) ))/(6*c*d*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_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(-e)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((m + p + 1)*(2* c*d - b*e))), x] + Simp[c*(Simplify[m + 2*p + 2]/((m + p + 1)*(2*c*d - b*e) )) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d , e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[Simplify[m + 2*p + 2], 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d*g - e*f)*(d + e*x)^m*((a + b*x + c*x ^2)^(p + 1)/((2*c*d - b*e)*(m + p + 1))), x] + Simp[(m*(g*(c*d - b*e) + c*e *f) + e*(p + 1)*(2*c*f - b*g))/(e*(2*c*d - b*e)*(m + p + 1)) Int[(d + e*x )^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ((LtQ[m, -1] && !IGtQ[m + p + 1, 0 ]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0 ]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[g^n*(d + e*x)^(m + n - 1)*((a + b *x + c*x^2)^(p + 1)/(c*e^(n - 1)*(m + n + 2*p + 1))), x] + Simp[1/(c*e^n*(m + n + 2*p + 1)) Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^n*(m + n + 2*p + 1)*(f + g*x)^n - c*g^n*(m + n + 2*p + 1)*(d + e*x)^n - g^n*(d + e*x)^(n - 2)*(b*d*e*(p + 1) + a*e^2*(m + n - 1) - c*d^2*(m + n + 2*p + 1) - e*(2*c*d - b*e)*(m + n + p)*x), x], x], x] /; FreeQ[{a, b, c, d, e, f, g , m, p}, x] && IGtQ[n, 1] && IntegerQ[m] && NeQ[m + n + 2*p + 1, 0]
Time = 2.75 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.20
| method | result | size |
| gosper | \(-\frac {2 \left (c d x +a e \right ) \left (-280 a^{2} c^{2} d^{2} e^{6} x^{4}-112 a \,c^{3} d^{4} e^{4} x^{4}+8 c^{4} d^{6} e^{2} x^{4}-140 a^{3} c d \,e^{7} x^{3}-1036 a^{2} c^{2} d^{3} e^{5} x^{3}-388 a \,c^{3} d^{5} e^{3} x^{3}+28 c^{4} d^{7} e \,x^{3}+35 a^{4} e^{8} x^{2}-476 a^{3} c \,d^{2} e^{6} x^{2}-1422 a^{2} c^{2} d^{4} e^{4} x^{2}-476 a \,c^{3} d^{6} e^{2} x^{2}+35 c^{4} d^{8} x^{2}+28 a^{4} d \,e^{7} x -388 a^{3} c \,d^{3} e^{5} x -1036 a^{2} c^{2} d^{5} e^{3} x -140 a \,c^{3} d^{7} e x +8 a^{4} d^{2} e^{6}-112 a^{3} c \,d^{4} e^{4}-280 a^{2} c^{2} d^{6} e^{2}\right )}{105 \left (e x +d \right )^{2} \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right ) \left (c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}\) | \(373\) |
| trager | \(-\frac {2 \left (-280 a^{2} c^{2} d^{2} e^{6} x^{4}-112 a \,c^{3} d^{4} e^{4} x^{4}+8 c^{4} d^{6} e^{2} x^{4}-140 a^{3} c d \,e^{7} x^{3}-1036 a^{2} c^{2} d^{3} e^{5} x^{3}-388 a \,c^{3} d^{5} e^{3} x^{3}+28 c^{4} d^{7} e \,x^{3}+35 a^{4} e^{8} x^{2}-476 a^{3} c \,d^{2} e^{6} x^{2}-1422 a^{2} c^{2} d^{4} e^{4} x^{2}-476 a \,c^{3} d^{6} e^{2} x^{2}+35 c^{4} d^{8} x^{2}+28 a^{4} d \,e^{7} x -388 a^{3} c \,d^{3} e^{5} x -1036 a^{2} c^{2} d^{5} e^{3} x -140 a \,c^{3} d^{7} e x +8 a^{4} d^{2} e^{6}-112 a^{3} c \,d^{4} e^{4}-280 a^{2} c^{2} d^{6} e^{2}\right ) \sqrt {c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e}}{105 \left (c d x +a e \right ) \left (a \,e^{2}-c \,d^{2}\right ) \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \left (e x +d \right )^{4}}\) | \(374\) |
| orering | \(-\frac {2 \left (-280 a^{2} c^{2} d^{2} e^{6} x^{4}-112 a \,c^{3} d^{4} e^{4} x^{4}+8 c^{4} d^{6} e^{2} x^{4}-140 a^{3} c d \,e^{7} x^{3}-1036 a^{2} c^{2} d^{3} e^{5} x^{3}-388 a \,c^{3} d^{5} e^{3} x^{3}+28 c^{4} d^{7} e \,x^{3}+35 a^{4} e^{8} x^{2}-476 a^{3} c \,d^{2} e^{6} x^{2}-1422 a^{2} c^{2} d^{4} e^{4} x^{2}-476 a \,c^{3} d^{6} e^{2} x^{2}+35 c^{4} d^{8} x^{2}+28 a^{4} d \,e^{7} x -388 a^{3} c \,d^{3} e^{5} x -1036 a^{2} c^{2} d^{5} e^{3} x -140 a \,c^{3} d^{7} e x +8 a^{4} d^{2} e^{6}-112 a^{3} c \,d^{4} e^{4}-280 a^{2} c^{2} d^{6} e^{2}\right ) \left (c d x +a e \right )}{105 \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right ) \left (e x +d \right )^{2} {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {3}{2}}}\) | \(374\) |
| default | \(\frac {-\frac {2}{3 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \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 )}}+\frac {8 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 )^{3} \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 )}}}{e^{3}}+\frac {d^{2} \left (-\frac {2}{7 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{3} \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 )}}-\frac {8 d e c \left (-\frac {2}{5 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2} \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 )}}-\frac {6 d e c \left (-\frac {2}{3 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \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 )}}+\frac {8 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 )^{3} \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 )}{7 \left (a \,e^{2}-c \,d^{2}\right )}\right )}{e^{5}}-\frac {2 d \left (-\frac {2}{5 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2} \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 )}}-\frac {6 d e c \left (-\frac {2}{3 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \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 )}}+\frac {8 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 )^{3} \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^{4}}\) | \(685\) |
Input:
int(x^2/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2),x,method=_RETURN VERBOSE)
Output:
-2/105*(c*d*x+a*e)*(-280*a^2*c^2*d^2*e^6*x^4-112*a*c^3*d^4*e^4*x^4+8*c^4*d ^6*e^2*x^4-140*a^3*c*d*e^7*x^3-1036*a^2*c^2*d^3*e^5*x^3-388*a*c^3*d^5*e^3* x^3+28*c^4*d^7*e*x^3+35*a^4*e^8*x^2-476*a^3*c*d^2*e^6*x^2-1422*a^2*c^2*d^4 *e^4*x^2-476*a*c^3*d^6*e^2*x^2+35*c^4*d^8*x^2+28*a^4*d*e^7*x-388*a^3*c*d^3 *e^5*x-1036*a^2*c^2*d^5*e^3*x-140*a*c^3*d^7*e*x+8*a^4*d^2*e^6-112*a^3*c*d^ 4*e^4-280*a^2*c^2*d^6*e^2)/(e*x+d)^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)^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 785 vs. \(2 (294) = 588\).
Time = 10.79 (sec) , antiderivative size = 785, normalized size of antiderivative = 2.53 \[ \int \frac {x^2}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left (280 \, a^{2} c^{2} d^{6} e^{2} + 112 \, a^{3} c d^{4} e^{4} - 8 \, a^{4} d^{2} e^{6} - 8 \, {\left (c^{4} d^{6} e^{2} - 14 \, a c^{3} d^{4} e^{4} - 35 \, a^{2} c^{2} d^{2} e^{6}\right )} x^{4} - 4 \, {\left (7 \, c^{4} d^{7} e - 97 \, a c^{3} d^{5} e^{3} - 259 \, a^{2} c^{2} d^{3} e^{5} - 35 \, a^{3} c d e^{7}\right )} x^{3} - {\left (35 \, c^{4} d^{8} - 476 \, a c^{3} d^{6} e^{2} - 1422 \, a^{2} c^{2} d^{4} e^{4} - 476 \, a^{3} c d^{2} e^{6} + 35 \, a^{4} e^{8}\right )} x^{2} + 4 \, {\left (35 \, a c^{3} d^{7} e + 259 \, a^{2} c^{2} d^{5} e^{3} + 97 \, a^{3} c d^{3} e^{5} - 7 \, 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}}{105 \, {\left (a c^{5} d^{14} e - 5 \, a^{2} c^{4} d^{12} e^{3} + 10 \, a^{3} c^{3} d^{10} e^{5} - 10 \, a^{4} c^{2} d^{8} e^{7} + 5 \, a^{5} c d^{6} e^{9} - a^{6} d^{4} e^{11} + {\left (c^{6} d^{11} e^{4} - 5 \, a c^{5} d^{9} e^{6} + 10 \, a^{2} c^{4} d^{7} e^{8} - 10 \, a^{3} c^{3} d^{5} e^{10} + 5 \, a^{4} c^{2} d^{3} e^{12} - a^{5} c d e^{14}\right )} x^{5} + {\left (4 \, c^{6} d^{12} e^{3} - 19 \, a c^{5} d^{10} e^{5} + 35 \, a^{2} c^{4} d^{8} e^{7} - 30 \, a^{3} c^{3} d^{6} e^{9} + 10 \, a^{4} c^{2} d^{4} e^{11} + a^{5} c d^{2} e^{13} - a^{6} e^{15}\right )} x^{4} + 2 \, {\left (3 \, c^{6} d^{13} e^{2} - 13 \, a c^{5} d^{11} e^{4} + 20 \, a^{2} c^{4} d^{9} e^{6} - 10 \, a^{3} c^{3} d^{7} e^{8} - 5 \, a^{4} c^{2} d^{5} e^{10} + 7 \, a^{5} c d^{3} e^{12} - 2 \, a^{6} d e^{14}\right )} x^{3} + 2 \, {\left (2 \, c^{6} d^{14} e - 7 \, a c^{5} d^{12} e^{3} + 5 \, a^{2} c^{4} d^{10} e^{5} + 10 \, a^{3} c^{3} d^{8} e^{7} - 20 \, a^{4} c^{2} d^{6} e^{9} + 13 \, a^{5} c d^{4} e^{11} - 3 \, a^{6} d^{2} e^{13}\right )} x^{2} + {\left (c^{6} d^{15} - a c^{5} d^{13} e^{2} - 10 \, a^{2} c^{4} d^{11} e^{4} + 30 \, a^{3} c^{3} d^{9} e^{6} - 35 \, a^{4} c^{2} d^{7} e^{8} + 19 \, a^{5} c d^{5} e^{10} - 4 \, a^{6} d^{3} e^{12}\right )} x\right )}} \] Input:
integrate(x^2/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorit hm="fricas")
Output:
-2/105*(280*a^2*c^2*d^6*e^2 + 112*a^3*c*d^4*e^4 - 8*a^4*d^2*e^6 - 8*(c^4*d ^6*e^2 - 14*a*c^3*d^4*e^4 - 35*a^2*c^2*d^2*e^6)*x^4 - 4*(7*c^4*d^7*e - 97* a*c^3*d^5*e^3 - 259*a^2*c^2*d^3*e^5 - 35*a^3*c*d*e^7)*x^3 - (35*c^4*d^8 - 476*a*c^3*d^6*e^2 - 1422*a^2*c^2*d^4*e^4 - 476*a^3*c*d^2*e^6 + 35*a^4*e^8) *x^2 + 4*(35*a*c^3*d^7*e + 259*a^2*c^2*d^5*e^3 + 97*a^3*c*d^3*e^5 - 7*a^4* d*e^7)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(a*c^5*d^14*e - 5*a^ 2*c^4*d^12*e^3 + 10*a^3*c^3*d^10*e^5 - 10*a^4*c^2*d^8*e^7 + 5*a^5*c*d^6*e^ 9 - a^6*d^4*e^11 + (c^6*d^11*e^4 - 5*a*c^5*d^9*e^6 + 10*a^2*c^4*d^7*e^8 - 10*a^3*c^3*d^5*e^10 + 5*a^4*c^2*d^3*e^12 - a^5*c*d*e^14)*x^5 + (4*c^6*d^12 *e^3 - 19*a*c^5*d^10*e^5 + 35*a^2*c^4*d^8*e^7 - 30*a^3*c^3*d^6*e^9 + 10*a^ 4*c^2*d^4*e^11 + a^5*c*d^2*e^13 - a^6*e^15)*x^4 + 2*(3*c^6*d^13*e^2 - 13*a *c^5*d^11*e^4 + 20*a^2*c^4*d^9*e^6 - 10*a^3*c^3*d^7*e^8 - 5*a^4*c^2*d^5*e^ 10 + 7*a^5*c*d^3*e^12 - 2*a^6*d*e^14)*x^3 + 2*(2*c^6*d^14*e - 7*a*c^5*d^12 *e^3 + 5*a^2*c^4*d^10*e^5 + 10*a^3*c^3*d^8*e^7 - 20*a^4*c^2*d^6*e^9 + 13*a ^5*c*d^4*e^11 - 3*a^6*d^2*e^13)*x^2 + (c^6*d^15 - a*c^5*d^13*e^2 - 10*a^2* c^4*d^11*e^4 + 30*a^3*c^3*d^9*e^6 - 35*a^4*c^2*d^7*e^8 + 19*a^5*c*d^5*e^10 - 4*a^6*d^3*e^12)*x)
\[ \int \frac {x^2}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {x^{2}}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{3}}\, dx \] Input:
integrate(x**2/(e*x+d)**3/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)
Output:
Integral(x**2/(((d + e*x)*(a*e + c*d*x))**(3/2)*(d + e*x)**3), x)
Exception generated. \[ \int \frac {x^2}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^2/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorit hm="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
\[ \int \frac {x^2}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int { \frac {x^{2}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{3}} \,d x } \] Input:
integrate(x^2/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorit hm="giac")
Output:
integrate(x^2/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(e*x + d)^3), x)
Time = 8.44 (sec) , antiderivative size = 5771, normalized size of antiderivative = 18.62 \[ \int \frac {x^2}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:
int(x^2/((d + e*x)^3*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)),x)
Output:
(((10*a*e^2 - 14*c*d^2)/(35*(a*e^2 - c*d^2)^2*(3*a*e^3 - 3*c*d^2*e)) - (4* c*d^2)/(7*(a*e^2 - c*d^2)^2*(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 - (((d*((16*c^2*d^2)/(35*(a*e^2 - c*d ^2)^4) - (8*c^3*d^4)/(35*(a*e^2 - c*d^2)^5)))/e - (34*c^3*d^5 - 188*a*c^2* d^3*e^2 + 82*a^2*c*d*e^4)/(105*e*(a*e^2 - c*d^2)^5))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x) - (((d*((d*((d*((8*c^4*d^5*e^3)/(35*(a *e^2 - c*d^2)^4*(3*a^2*e^5 + 3*c^2*d^4*e - 6*a*c*d^2*e^3)) - (8*c^3*d^3*e^ 3*(3*a*e^2 - c*d^2))/(35*(a*e^2 - c*d^2)^4*(3*a^2*e^5 + 3*c^2*d^4*e - 6*a* c*d^2*e^3))))/e + (e^2*(26*c^4*d^6 - 124*a*c^3*d^4*e^2 + 74*a^2*c^2*d^2*e^ 4))/(35*(a*e^2 - c*d^2)^4*(3*a^2*e^5 + 3*c^2*d^4*e - 6*a*c*d^2*e^3))))/e + (4*a*c*d*e^3*(22*c^2*d^4 - 21*a^2*e^4 + 15*a*c*d^2*e^2))/(35*(a*e^2 - c*d ^2)^4*(3*a^2*e^5 + 3*c^2*d^4*e - 6*a*c*d^2*e^3))))/e - (e^2*(84*a^3*c*d^2* e^4 - 96*a^4*e^6 + 44*a^2*c^2*d^4*e^2))/(35*(a*e^2 - c*d^2)^4*(3*a^2*e^5 + 3*c^2*d^4*e - 6*a*c*d^2*e^3)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1 /2))/(d + e*x)^2 - (((2*c*d^3 + 2*a*d*e^2)/(7*(a*e^2 - c*d^2)^2*(5*a*e^3 - 5*c*d^2*e)) + (d*((2*a*e^3 - 2*c*d^2*e)/(7*(a*e^2 - c*d^2)^2*(5*a*e^3 - 5 *c*d^2*e)) - (4*c*d^2*e)/(7*(a*e^2 - c*d^2)^2*(5*a*e^3 - 5*c*d^2*e))))/e)* (x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^3 + (((16*a^3*e^6 + 6*c^3*d^6 - 32*a*c^2*d^4*e^2 - 22*a^2*c*d^2*e^4)/(35*(a*e^2 - c*d^2)^4* (3*a*e^3 - 3*c*d^2*e)) - (d*((d*((8*c^3*d^4*e^2)/(35*(a*e^2 - c*d^2)^4*...
Time = 7.70 (sec) , antiderivative size = 1240, normalized size of antiderivative = 4.00 \[ \int \frac {x^2}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
int(x^2/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)
Output:
(2*( - 280*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a**2*c*d**5*e**4 - 11 20*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a**2*c*d**4*e**5*x - 1680*sqr t(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a**2*c*d**3*e**6*x**2 - 1120*sqrt(e )*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a**2*c*d**2*e**7*x**3 - 280*sqrt(e)*sq rt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a**2*c*d*e**8*x**4 - 112*sqrt(e)*sqrt(d)*s qrt(c)*sqrt(a*e + c*d*x)*a*c**2*d**7*e**2 - 448*sqrt(e)*sqrt(d)*sqrt(c)*sq rt(a*e + c*d*x)*a*c**2*d**6*e**3*x - 672*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a*c**2*d**5*e**4*x**2 - 448*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c* d*x)*a*c**2*d**4*e**5*x**3 - 112*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x) *a*c**2*d**3*e**6*x**4 + 8*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*c**3* d**9 + 32*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*c**3*d**8*e*x + 48*sqr t(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*c**3*d**7*e**2*x**2 + 32*sqrt(e)*sq rt(d)*sqrt(c)*sqrt(a*e + c*d*x)*c**3*d**6*e**3*x**3 + 8*sqrt(e)*sqrt(d)*sq rt(c)*sqrt(a*e + c*d*x)*c**3*d**5*e**4*x**4 - 8*sqrt(d + e*x)*a**4*d**2*e* *8 - 28*sqrt(d + e*x)*a**4*d*e**9*x - 35*sqrt(d + e*x)*a**4*e**10*x**2 + 1 12*sqrt(d + e*x)*a**3*c*d**4*e**6 + 388*sqrt(d + e*x)*a**3*c*d**3*e**7*x + 476*sqrt(d + e*x)*a**3*c*d**2*e**8*x**2 + 140*sqrt(d + e*x)*a**3*c*d*e**9 *x**3 + 280*sqrt(d + e*x)*a**2*c**2*d**6*e**4 + 1036*sqrt(d + e*x)*a**2*c* *2*d**5*e**5*x + 1422*sqrt(d + e*x)*a**2*c**2*d**4*e**6*x**2 + 1036*sqrt(d + e*x)*a**2*c**2*d**3*e**7*x**3 + 280*sqrt(d + e*x)*a**2*c**2*d**2*e**...