Integrand size = 38, antiderivative size = 284 \[ \int \frac {x}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 d}{7 e \left (c d^2-a e^2\right ) (d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {2 \left (c d^2+7 a e^2\right )}{35 e \left (c d^2-a e^2\right )^2 (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {4 c d \left (c d^2+7 a e^2\right )}{35 e \left (c d^2-a e^2\right )^3 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {16 c^2 d^2 \left (c d^2+7 a e^2\right ) \left (c d^2+a e^2+2 c d e x\right )}{35 e \left (c d^2-a e^2\right )^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \] Output:
-2/7*d/e/(-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)- 2/35*(7*a*e^2+c*d^2)/e/(-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)-4/35*c*d*(7*a*e^2+c*d^2)/e/(-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)+16/35*c^2*d^2*(7*a*e^2+c*d^2)*(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)^(1/2)
Time = 0.28 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.78 \[ \int \frac {x}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {2 \left (a^4 e^7 (2 d+7 e x)-2 a^3 c d e^5 \left (7 d^2+24 d e x+7 e^2 x^2\right )+c^4 d^5 x \left (35 d^3+70 d^2 e x+56 d e^2 x^2+16 e^3 x^3\right )+2 a^2 c^2 d^2 e^3 \left (35 d^3+119 d^2 e x+97 d e^2 x^2+28 e^3 x^3\right )+2 a c^3 d^3 e \left (35 d^4+140 d^3 e x+259 d^2 e^2 x^2+200 d e^3 x^3+56 e^4 x^4\right )\right )}{35 \left (c d^2-a e^2\right )^5 (d+e x)^3 \sqrt {(a e+c d x) (d+e x)}} \] Input:
Integrate[x/((d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]
Output:
(2*(a^4*e^7*(2*d + 7*e*x) - 2*a^3*c*d*e^5*(7*d^2 + 24*d*e*x + 7*e^2*x^2) + c^4*d^5*x*(35*d^3 + 70*d^2*e*x + 56*d*e^2*x^2 + 16*e^3*x^3) + 2*a^2*c^2*d ^2*e^3*(35*d^3 + 119*d^2*e*x + 97*d*e^2*x^2 + 28*e^3*x^3) + 2*a*c^3*d^3*e* (35*d^4 + 140*d^3*e*x + 259*d^2*e^2*x^2 + 200*d*e^3*x^3 + 56*e^4*x^4)))/(3 5*(c*d^2 - a*e^2)^5*(d + e*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)])
Time = 0.77 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {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.
| \(\displaystyle \int \frac {x}{(d+e x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1220 |
| \(\displaystyle -\frac {\left (7 a e^2+c d^2\right ) \int \frac {1}{(d+e x)^2 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{7 e \left (c d^2-a e^2\right )}-\frac {2 d}{7 e (d+e x)^3 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle -\frac {\left (7 a e^2+c d^2\right ) \left (\frac {6 c d \int \frac {1}{(d+e x) \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{5 \left (c d^2-a e^2\right )}+\frac {2}{5 (d+e x)^2 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{7 e \left (c d^2-a e^2\right )}-\frac {2 d}{7 e (d+e x)^3 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle -\frac {\left (7 a e^2+c d^2\right ) \left (\frac {6 c d \left (\frac {4 c d \int \frac {1}{\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{3 \left (c d^2-a e^2\right )}+\frac {2}{3 (d+e x) \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{5 \left (c d^2-a e^2\right )}+\frac {2}{5 (d+e x)^2 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{7 e \left (c d^2-a e^2\right )}-\frac {2 d}{7 e (d+e x)^3 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 1088 |
| \(\displaystyle -\frac {2 d}{7 e (d+e x)^3 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {\left (7 a e^2+c d^2\right ) \left (\frac {6 c d \left (\frac {2}{3 (d+e x) \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {8 c d \left (a e^2+c d^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{5 \left (c d^2-a e^2\right )}+\frac {2}{5 (d+e x)^2 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{7 e \left (c d^2-a e^2\right )}\) |
Input:
Int[x/((d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]
Output:
(-2*d)/(7*e*(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*d^2 + 7*a*e^2)*(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*e*(c*d^2 - a*e^2))
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 ]
Time = 2.79 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.24
| method | result | size |
| gosper | \(-\frac {2 \left (c d x +a e \right ) \left (112 a \,c^{3} d^{3} e^{5} x^{4}+16 c^{4} d^{5} e^{3} x^{4}+56 a^{2} c^{2} d^{2} e^{6} x^{3}+400 a \,c^{3} d^{4} e^{4} x^{3}+56 c^{4} d^{6} e^{2} x^{3}-14 a^{3} c d \,e^{7} x^{2}+194 a^{2} c^{2} d^{3} e^{5} x^{2}+518 a \,c^{3} d^{5} e^{3} x^{2}+70 c^{4} d^{7} e \,x^{2}+7 a^{4} e^{8} x -48 a^{3} c \,d^{2} e^{6} x +238 a^{2} c^{2} d^{4} e^{4} x +280 a \,c^{3} d^{6} e^{2} x +35 c^{4} d^{8} x +2 a^{4} d \,e^{7}-14 a^{3} c \,d^{3} e^{5}+70 a^{2} c^{2} d^{5} e^{3}+70 a \,c^{3} d^{7} e \right )}{35 \left (e x +d \right )^{2} \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right ) \left (c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}\) | \(351\) |
| trager | \(-\frac {2 \left (112 a \,c^{3} d^{3} e^{5} x^{4}+16 c^{4} d^{5} e^{3} x^{4}+56 a^{2} c^{2} d^{2} e^{6} x^{3}+400 a \,c^{3} d^{4} e^{4} x^{3}+56 c^{4} d^{6} e^{2} x^{3}-14 a^{3} c d \,e^{7} x^{2}+194 a^{2} c^{2} d^{3} e^{5} x^{2}+518 a \,c^{3} d^{5} e^{3} x^{2}+70 c^{4} d^{7} e \,x^{2}+7 a^{4} e^{8} x -48 a^{3} c \,d^{2} e^{6} x +238 a^{2} c^{2} d^{4} e^{4} x +280 a \,c^{3} d^{6} e^{2} x +35 c^{4} d^{8} x +2 a^{4} d \,e^{7}-14 a^{3} c \,d^{3} e^{5}+70 a^{2} c^{2} d^{5} e^{3}+70 a \,c^{3} d^{7} e \right ) \sqrt {c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e}}{35 \left (c d x +a e \right ) \left (a \,e^{2}-c \,d^{2}\right ) \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \left (e x +d \right )^{4}}\) | \(352\) |
| orering | \(-\frac {2 \left (112 a \,c^{3} d^{3} e^{5} x^{4}+16 c^{4} d^{5} e^{3} x^{4}+56 a^{2} c^{2} d^{2} e^{6} x^{3}+400 a \,c^{3} d^{4} e^{4} x^{3}+56 c^{4} d^{6} e^{2} x^{3}-14 a^{3} c d \,e^{7} x^{2}+194 a^{2} c^{2} d^{3} e^{5} x^{2}+518 a \,c^{3} d^{5} e^{3} x^{2}+70 c^{4} d^{7} e \,x^{2}+7 a^{4} e^{8} x -48 a^{3} c \,d^{2} e^{6} x +238 a^{2} c^{2} d^{4} e^{4} x +280 a \,c^{3} d^{6} e^{2} x +35 c^{4} d^{8} x +2 a^{4} d \,e^{7}-14 a^{3} c \,d^{3} e^{5}+70 a^{2} c^{2} d^{5} e^{3}+70 a \,c^{3} d^{7} e \right ) \left (c d x +a e \right )}{35 \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right ) \left (e x +d \right )^{2} {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {3}{2}}}\) | \(352\) |
| default | \(\frac {-\frac {2}{5 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2} \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}-\frac {6 d e c \left (-\frac {2}{3 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}+\frac {8 d e c \left (2 d e c \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right )}{3 \left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}\right )}{5 \left (a \,e^{2}-c \,d^{2}\right )}}{e^{3}}-\frac {d \left (-\frac {2}{7 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{3} \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}-\frac {8 d e c \left (-\frac {2}{5 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2} \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}-\frac {6 d e c \left (-\frac {2}{3 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}+\frac {8 d e c \left (2 d e c \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right )}{3 \left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}\right )}{5 \left (a \,e^{2}-c \,d^{2}\right )}\right )}{7 \left (a \,e^{2}-c \,d^{2}\right )}\right )}{e^{4}}\) | \(537\) |
Input:
int(x/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2),x,method=_RETURNVE RBOSE)
Output:
-2/35*(c*d*x+a*e)*(112*a*c^3*d^3*e^5*x^4+16*c^4*d^5*e^3*x^4+56*a^2*c^2*d^2 *e^6*x^3+400*a*c^3*d^4*e^4*x^3+56*c^4*d^6*e^2*x^3-14*a^3*c*d*e^7*x^2+194*a ^2*c^2*d^3*e^5*x^2+518*a*c^3*d^5*e^3*x^2+70*c^4*d^7*e*x^2+7*a^4*e^8*x-48*a ^3*c*d^2*e^6*x+238*a^2*c^2*d^4*e^4*x+280*a*c^3*d^6*e^2*x+35*c^4*d^8*x+2*a^ 4*d*e^7-14*a^3*c*d^3*e^5+70*a^2*c^2*d^5*e^3+70*a*c^3*d^7*e)/(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. 770 vs. \(2 (268) = 536\).
Time = 10.91 (sec) , antiderivative size = 770, normalized size of antiderivative = 2.71 \[ \int \frac {x}{(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 (70 \, a c^{3} d^{7} e + 70 \, a^{2} c^{2} d^{5} e^{3} - 14 \, a^{3} c d^{3} e^{5} + 2 \, a^{4} d e^{7} + 16 \, {\left (c^{4} d^{5} e^{3} + 7 \, a c^{3} d^{3} e^{5}\right )} x^{4} + 8 \, {\left (7 \, c^{4} d^{6} e^{2} + 50 \, a c^{3} d^{4} e^{4} + 7 \, a^{2} c^{2} d^{2} e^{6}\right )} x^{3} + 2 \, {\left (35 \, c^{4} d^{7} e + 259 \, a c^{3} d^{5} e^{3} + 97 \, a^{2} c^{2} d^{3} e^{5} - 7 \, a^{3} c d e^{7}\right )} x^{2} + {\left (35 \, c^{4} d^{8} + 280 \, a c^{3} d^{6} e^{2} + 238 \, a^{2} c^{2} d^{4} e^{4} - 48 \, a^{3} c d^{2} e^{6} + 7 \, a^{4} e^{8}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{35 \, {\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/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm ="fricas")
Output:
2/35*(70*a*c^3*d^7*e + 70*a^2*c^2*d^5*e^3 - 14*a^3*c*d^3*e^5 + 2*a^4*d*e^7 + 16*(c^4*d^5*e^3 + 7*a*c^3*d^3*e^5)*x^4 + 8*(7*c^4*d^6*e^2 + 50*a*c^3*d^ 4*e^4 + 7*a^2*c^2*d^2*e^6)*x^3 + 2*(35*c^4*d^7*e + 259*a*c^3*d^5*e^3 + 97* a^2*c^2*d^3*e^5 - 7*a^3*c*d*e^7)*x^2 + (35*c^4*d^8 + 280*a*c^3*d^6*e^2 + 2 38*a^2*c^2*d^4*e^4 - 48*a^3*c*d^2*e^6 + 7*a^4*e^8)*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^1 3 - 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)
\[ \int \frac {x}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {x}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{3}}\, dx \] Input:
integrate(x/(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/(((d + e*x)*(a*e + c*d*x))**(3/2)*(d + e*x)**3), x)
Exception generated. \[ \int \frac {x}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),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
\[ \int \frac {x}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int { \frac {x}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{3}} \,d x } \] Input:
integrate(x/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm ="giac")
Output:
integrate(x/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(e*x + d)^3), x )
Time = 7.87 (sec) , antiderivative size = 3635, normalized size of antiderivative = 12.80 \[ \int \frac {x}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:
int(x/((d + e*x)^3*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)),x)
Output:
(((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)) - (24*c^3*d^3*e^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*(70*c^4*d^6 + 24*a*c ^3*d^4*e^2 - 22*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^2*c*d*e^5*(27*a*e^2 - 35*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)))*(x*(a*e^2 + c *d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^2 + (((e*(2*a*e^2 + 2*c*d^2))/ (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)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2) )/(d + e*x)^3 + (((12*c^3*d^4 + 12*a*c^2*d^2*e^2)/(35*(a*e^2 - c*d^2)^5) - (8*c^3*d^4)/(35*(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*((8*c^3*d^4*e^2)/(35*(a*e^2 - c*d^2)^4*(3*a*e^3 - 3*c*d^2*e)) - (8*c^2*d^2*e^2*(2*a*e^2 + 3*c*d^2))/(35*(a*e^2 - c*d^2)^4 *(3*a*e^3 - 3*c*d^2*e))))/e + (e*(10*c^3*d^5 + 16*a*c^2*d^3*e^2 + 6*a^2*c* d*e^4))/(35*(a*e^2 - c*d^2)^4*(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*((e^2*(14*c^2*d^3 + 6*a*c*d* e^2))/(7*(a*e^2 - c*d^2)^2*(5*a^2*e^5 + 5*c^2*d^4*e - 10*a*c*d^2*e^3)) - ( 4*c^2*d^3*e^2)/(7*(a*e^2 - c*d^2)^2*(5*a^2*e^5 + 5*c^2*d^4*e - 10*a*c*d^2* e^3))))/e - (16*a^2*e^5)/(7*(a*e^2 - c*d^2)^2*(5*a^2*e^5 + 5*c^2*d^4*e - 1 0*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*...
Time = 0.42 (sec) , antiderivative size = 1067, normalized size of antiderivative = 3.76 \[ \int \frac {x}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
int(x/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)
Output:
(2*(112*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a*c**2*d**6*e**2 + 448*s qrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a*c**2*d**5*e**3*x + 672*sqrt(e)* sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a*c**2*d**4*e**4*x**2 + 448*sqrt(e)*sqrt (d)*sqrt(c)*sqrt(a*e + c*d*x)*a*c**2*d**3*e**5*x**3 + 112*sqrt(e)*sqrt(d)* sqrt(c)*sqrt(a*e + c*d*x)*a*c**2*d**2*e**6*x**4 + 16*sqrt(e)*sqrt(d)*sqrt( c)*sqrt(a*e + c*d*x)*c**3*d**8 + 64*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d *x)*c**3*d**7*e*x + 96*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*c**3*d**6 *e**2*x**2 + 64*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*c**3*d**5*e**3*x **3 + 16*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*c**3*d**4*e**4*x**4 - 2 *sqrt(d + e*x)*a**4*d*e**8 - 7*sqrt(d + e*x)*a**4*e**9*x + 14*sqrt(d + e*x )*a**3*c*d**3*e**6 + 48*sqrt(d + e*x)*a**3*c*d**2*e**7*x + 14*sqrt(d + e*x )*a**3*c*d*e**8*x**2 - 70*sqrt(d + e*x)*a**2*c**2*d**5*e**4 - 238*sqrt(d + e*x)*a**2*c**2*d**4*e**5*x - 194*sqrt(d + e*x)*a**2*c**2*d**3*e**6*x**2 - 56*sqrt(d + e*x)*a**2*c**2*d**2*e**7*x**3 - 70*sqrt(d + e*x)*a*c**3*d**7* e**2 - 280*sqrt(d + e*x)*a*c**3*d**6*e**3*x - 518*sqrt(d + e*x)*a*c**3*d** 5*e**4*x**2 - 400*sqrt(d + e*x)*a*c**3*d**4*e**5*x**3 - 112*sqrt(d + e*x)* a*c**3*d**3*e**6*x**4 - 35*sqrt(d + e*x)*c**4*d**8*e*x - 70*sqrt(d + e*x)* c**4*d**7*e**2*x**2 - 56*sqrt(d + e*x)*c**4*d**6*e**3*x**3 - 16*sqrt(d + e *x)*c**4*d**5*e**4*x**4))/(35*sqrt(a*e + c*d*x)*e*(a**5*d**4*e**10 + 4*a** 5*d**3*e**11*x + 6*a**5*d**2*e**12*x**2 + 4*a**5*d*e**13*x**3 + a**5*e*...