Integrand size = 38, antiderivative size = 209 \[ \int \frac {x}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 d}{5 e \left (c d^2-a e^2\right ) (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {2 \left (c d^2+5 a e^2\right )}{15 e \left (c d^2-a e^2\right )^2 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {8 c d \left (c d^2+5 a e^2\right ) \left (c d^2+a e^2+2 c d e x\right )}{15 e \left (c d^2-a e^2\right )^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \] Output:
-2/5*d/e/(-a*e^2+c*d^2)/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)- 2/15*(5*a*e^2+c*d^2)/e/(-a*e^2+c*d^2)^2/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d *e*x^2)^(1/2)+8/15*c*d*(5*a*e^2+c*d^2)*(2*c*d*e*x+a*e^2+c*d^2)/e/(-a*e^2+c *d^2)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)
Time = 0.23 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.74 \[ \int \frac {x}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {2 \left (-a^3 e^5 (2 d+5 e x)+c^3 d^4 x \left (15 d^2+20 d e x+8 e^2 x^2\right )+a^2 c d e^3 \left (20 d^2+49 d e x+20 e^2 x^2\right )+a c^2 d^2 e \left (30 d^3+85 d^2 e x+104 d e^2 x^2+40 e^3 x^3\right )\right )}{15 \left (c d^2-a e^2\right )^4 (d+e x)^2 \sqrt {(a e+c d x) (d+e x)}} \] Input:
Integrate[x/((d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]
Output:
(2*(-(a^3*e^5*(2*d + 5*e*x)) + c^3*d^4*x*(15*d^2 + 20*d*e*x + 8*e^2*x^2) + a^2*c*d*e^3*(20*d^2 + 49*d*e*x + 20*e^2*x^2) + a*c^2*d^2*e*(30*d^3 + 85*d ^2*e*x + 104*d*e^2*x^2 + 40*e^3*x^3)))/(15*(c*d^2 - a*e^2)^4*(d + e*x)^2*S qrt[(a*e + c*d*x)*(d + e*x)])
Time = 0.63 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.02, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {1220, 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.
| \(\displaystyle \int \frac {x}{(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1220 |
| \(\displaystyle -\frac {\left (5 a e^2+c d^2\right ) \int \frac {1}{(d+e x) \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{5 e \left (c d^2-a e^2\right )}-\frac {2 d}{5 e (d+e x)^2 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle -\frac {\left (5 a e^2+c d^2\right ) \left (\frac {4 c d \int \frac {1}{\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{3 \left (c d^2-a e^2\right )}+\frac {2}{3 (d+e x) \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{5 e \left (c d^2-a e^2\right )}-\frac {2 d}{5 e (d+e x)^2 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 1088 |
| \(\displaystyle -\frac {2 d}{5 e (d+e x)^2 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {\left (5 a e^2+c d^2\right ) \left (\frac {2}{3 (d+e x) \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {8 c d \left (a e^2+c d^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{5 e \left (c d^2-a e^2\right )}\) |
Input:
Int[x/((d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]
Output:
(-2*d)/(5*e*(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]) - ((c*d^2 + 5*a*e^2)*(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*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 = 2.37 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.17
| method | result | size |
| gosper | \(-\frac {2 \left (c d x +a e \right ) \left (-40 x^{3} a \,c^{2} d^{2} e^{4}-8 x^{3} c^{3} d^{4} e^{2}-20 x^{2} a^{2} c d \,e^{5}-104 x^{2} a \,c^{2} d^{3} e^{3}-20 x^{2} c^{3} d^{5} e +5 e^{6} a^{3} x -49 d^{2} e^{4} a^{2} c x -85 d^{4} e^{2} a \,c^{2} x -15 d^{6} c^{3} x +2 a^{3} d \,e^{5}-20 d^{3} e^{3} c \,a^{2}-30 a \,c^{2} d^{5} e \right )}{15 \left (e x +d \right ) \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \left (c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}\) | \(244\) |
| orering | \(-\frac {2 \left (-40 x^{3} a \,c^{2} d^{2} e^{4}-8 x^{3} c^{3} d^{4} e^{2}-20 x^{2} a^{2} c d \,e^{5}-104 x^{2} a \,c^{2} d^{3} e^{3}-20 x^{2} c^{3} d^{5} e +5 e^{6} a^{3} x -49 d^{2} e^{4} a^{2} c x -85 d^{4} e^{2} a \,c^{2} x -15 d^{6} c^{3} x +2 a^{3} d \,e^{5}-20 d^{3} e^{3} c \,a^{2}-30 a \,c^{2} d^{5} e \right ) \left (c d x +a e \right )}{15 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \left (e x +d \right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {3}{2}}}\) | \(245\) |
| trager | \(-\frac {2 \left (-40 x^{3} a \,c^{2} d^{2} e^{4}-8 x^{3} c^{3} d^{4} e^{2}-20 x^{2} a^{2} c d \,e^{5}-104 x^{2} a \,c^{2} d^{3} e^{3}-20 x^{2} c^{3} d^{5} e +5 e^{6} a^{3} x -49 d^{2} e^{4} a^{2} c x -85 d^{4} e^{2} a \,c^{2} x -15 d^{6} c^{3} x +2 a^{3} d \,e^{5}-20 d^{3} e^{3} c \,a^{2}-30 a \,c^{2} d^{5} e \right ) \sqrt {c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e}}{15 \left (c d x +a e \right ) \left (a \,e^{2}-c \,d^{2}\right ) \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} a \,c^{2}-d^{6} c^{3}\right ) \left (e x +d \right )^{3}}\) | \(247\) |
| default | \(\frac {-\frac {2}{3 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}+\frac {8 d e c \left (2 d e c \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right )}{3 \left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}}{e^{2}}-\frac {d \left (-\frac {2}{5 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2} \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}-\frac {6 d e c \left (-\frac {2}{3 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}+\frac {8 d e c \left (2 d e c \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right )}{3 \left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}\right )}{5 \left (a \,e^{2}-c \,d^{2}\right )}\right )}{e^{3}}\) | \(375\) |
Input:
int(x/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2),x,method=_RETURNVE RBOSE)
Output:
-2/15*(c*d*x+a*e)*(-40*a*c^2*d^2*e^4*x^3-8*c^3*d^4*e^2*x^3-20*a^2*c*d*e^5* x^2-104*a*c^2*d^3*e^3*x^2-20*c^3*d^5*e*x^2+5*a^3*e^6*x-49*a^2*c*d^2*e^4*x- 85*a*c^2*d^4*e^2*x-15*c^3*d^6*x+2*a^3*d*e^5-20*a^2*c*d^3*e^3-30*a*c^2*d^5* e)/(e*x+d)/(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)/(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. 517 vs. \(2 (197) = 394\).
Time = 3.60 (sec) , antiderivative size = 517, normalized size of antiderivative = 2.47 \[ \int \frac {x}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {2 \, {\left (30 \, a c^{2} d^{5} e + 20 \, a^{2} c d^{3} e^{3} - 2 \, a^{3} d e^{5} + 8 \, {\left (c^{3} d^{4} e^{2} + 5 \, a c^{2} d^{2} e^{4}\right )} x^{3} + 4 \, {\left (5 \, c^{3} d^{5} e + 26 \, a c^{2} d^{3} e^{3} + 5 \, a^{2} c d e^{5}\right )} x^{2} + {\left (15 \, c^{3} d^{6} + 85 \, a c^{2} d^{4} e^{2} + 49 \, a^{2} c d^{2} e^{4} - 5 \, a^{3} e^{6}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{15 \, {\left (a c^{4} d^{11} e - 4 \, a^{2} c^{3} d^{9} e^{3} + 6 \, a^{3} c^{2} d^{7} e^{5} - 4 \, a^{4} c d^{5} e^{7} + a^{5} d^{3} e^{9} + {\left (c^{5} d^{9} e^{3} - 4 \, a c^{4} d^{7} e^{5} + 6 \, a^{2} c^{3} d^{5} e^{7} - 4 \, a^{3} c^{2} d^{3} e^{9} + a^{4} c d e^{11}\right )} x^{4} + {\left (3 \, c^{5} d^{10} e^{2} - 11 \, a c^{4} d^{8} e^{4} + 14 \, a^{2} c^{3} d^{6} e^{6} - 6 \, a^{3} c^{2} d^{4} e^{8} - a^{4} c d^{2} e^{10} + a^{5} e^{12}\right )} x^{3} + 3 \, {\left (c^{5} d^{11} e - 3 \, a c^{4} d^{9} e^{3} + 2 \, a^{2} c^{3} d^{7} e^{5} + 2 \, a^{3} c^{2} d^{5} e^{7} - 3 \, a^{4} c d^{3} e^{9} + a^{5} d e^{11}\right )} x^{2} + {\left (c^{5} d^{12} - a c^{4} d^{10} e^{2} - 6 \, a^{2} c^{3} d^{8} e^{4} + 14 \, a^{3} c^{2} d^{6} e^{6} - 11 \, a^{4} c d^{4} e^{8} + 3 \, a^{5} d^{2} e^{10}\right )} x\right )}} \] Input:
integrate(x/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm ="fricas")
Output:
2/15*(30*a*c^2*d^5*e + 20*a^2*c*d^3*e^3 - 2*a^3*d*e^5 + 8*(c^3*d^4*e^2 + 5 *a*c^2*d^2*e^4)*x^3 + 4*(5*c^3*d^5*e + 26*a*c^2*d^3*e^3 + 5*a^2*c*d*e^5)*x ^2 + (15*c^3*d^6 + 85*a*c^2*d^4*e^2 + 49*a^2*c*d^2*e^4 - 5*a^3*e^6)*x)*sqr t(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(a*c^4*d^11*e - 4*a^2*c^3*d^9*e^3 + 6*a^3*c^2*d^7*e^5 - 4*a^4*c*d^5*e^7 + a^5*d^3*e^9 + (c^5*d^9*e^3 - 4*a* c^4*d^7*e^5 + 6*a^2*c^3*d^5*e^7 - 4*a^3*c^2*d^3*e^9 + a^4*c*d*e^11)*x^4 + (3*c^5*d^10*e^2 - 11*a*c^4*d^8*e^4 + 14*a^2*c^3*d^6*e^6 - 6*a^3*c^2*d^4*e^ 8 - a^4*c*d^2*e^10 + a^5*e^12)*x^3 + 3*(c^5*d^11*e - 3*a*c^4*d^9*e^3 + 2*a ^2*c^3*d^7*e^5 + 2*a^3*c^2*d^5*e^7 - 3*a^4*c*d^3*e^9 + a^5*d*e^11)*x^2 + ( c^5*d^12 - a*c^4*d^10*e^2 - 6*a^2*c^3*d^8*e^4 + 14*a^3*c^2*d^6*e^6 - 11*a^ 4*c*d^4*e^8 + 3*a^5*d^2*e^10)*x)
\[ \int \frac {x}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {x}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{2}}\, dx \] Input:
integrate(x/(e*x+d)**2/(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)**2), x)
Exception generated. \[ \int \frac {x}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x/(e*x+d)^2/(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
Leaf count of result is larger than twice the leaf count of optimal. 4167 vs. \(2 (197) = 394\).
Time = 0.23 (sec) , antiderivative size = 4167, normalized size of antiderivative = 19.94 \[ \int \frac {x}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate(x/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm ="giac")
Output:
2/15*(15*a*c^2*d^2*e^3*abs(e)/((c^4*d^8*sgn(1/(e*x + d))*sgn(e) - 4*a*c^3* d^6*e^2*sgn(1/(e*x + d))*sgn(e) + 6*a^2*c^2*d^4*e^4*sgn(1/(e*x + d))*sgn(e ) - 4*a^3*c*d^2*e^6*sgn(1/(e*x + d))*sgn(e) + a^4*e^8*sgn(1/(e*x + d))*sgn (e))*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))) - 8*(c^3*d^4*e*abs (e) + 5*a*c^2*d^2*e^3*abs(e))*sgn(1/(e*x + d))*sgn(e)/(sqrt(c*d*e)*c^4*d^8 - 4*sqrt(c*d*e)*a*c^3*d^6*e^2 + 6*sqrt(c*d*e)*a^2*c^2*d^4*e^4 - 4*sqrt(c* d*e)*a^3*c*d^2*e^6 + sqrt(c*d*e)*a^4*e^8) + (15*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*c^18*d^35*e^10*abs(e)*sgn(1/(e*x + d))^4*sgn(e)^4 - 210*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a*c^17*d^33*e^12*a bs(e)*sgn(1/(e*x + d))^4*sgn(e)^4 + 1320*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^2*c^16*d^31*e^14*abs(e)*sgn(1/(e*x + d))^4*sgn(e)^4 - 4 800*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^3*c^15*d^29*e^16*a bs(e)*sgn(1/(e*x + d))^4*sgn(e)^4 + 10500*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^4*c^14*d^27*e^18*abs(e)*sgn(1/(e*x + d))^4*sgn(e)^4 - 10920*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^5*c^13*d^25*e^20 *abs(e)*sgn(1/(e*x + d))^4*sgn(e)^4 - 10920*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^6*c^12*d^23*e^22*abs(e)*sgn(1/(e*x + d))^4*sgn(e)^4 + 68640*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^7*c^11*d^21*e^ 24*abs(e)*sgn(1/(e*x + d))^4*sgn(e)^4 - 150150*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^8*c^10*d^19*e^26*abs(e)*sgn(1/(e*x + d))^4*sgn...
Time = 7.09 (sec) , antiderivative size = 1844, normalized size of antiderivative = 8.82 \[ \int \frac {x}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:
int(x/((d + e*x)^2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)),x)
Output:
(((e*(2*a*e^2 + 2*c*d^2))/(5*(a*e^2 - c*d^2)^2*(3*a*e^3 - 3*c*d^2*e)) - (4 *c*d^2*e)/(5*(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*((8*c^3*d^4*e)/(15*(a*e^2 - c*d^2)^5) - (16*c^2*d^2*e*(a*e^2 + c*d^2))/(15*(a*e^2 - c*d^2)^5)))/e + ( 2*c*d*(a*e^2 + c*d^2)^2)/(5*(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*((e^2*(10*c^2*d^3 + 6*a*c*d*e^2))/(5 *(a*e^2 - c*d^2)^2*(3*a^2*e^5 + 3*c^2*d^4*e - 6*a*c*d^2*e^3)) - (4*c^2*d^3 *e^2)/(5*(a*e^2 - c*d^2)^2*(3*a^2*e^5 + 3*c^2*d^4*e - 6*a*c*d^2*e^3))))/e - (12*a^2*e^5)/(5*(a*e^2 - c*d^2)^2*(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 + ((x*(a* e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((a*(((a*e^2 + c*d^2)*((8*c^5*d^5* e^3*(a*e^2 + c*d^2))/(15*(a*e^2 - c*d^2)^4*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16*c^5*d^5*e^3*(3*a*e^2 + 2*c*d^2))/(15*(a*e^2 - c*d^2)^4 *(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) - (16*a*c^5*d^6*e^ 4)/(15*(a*e^2 - c*d^2)^4*(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^2*(30*c^4*d^6 + 16*a*c^3*d^4*e^2 + 10*a^2*c^2*d^2*e^4))/(15*(a* e^2 - c*d^2)^4*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (8*c^4*d^4*e ^2*(a*e^2 + c*d^2)*(3*a*e^2 + 2*c*d^2))/(15*(a*e^2 - c*d^2)^4*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/c - x*((a*((8*c^5*d^5*e^3*(a*e^2 + c*d^ 2))/(15*(a*e^2 - c*d^2)^4*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) ...
Time = 0.27 (sec) , antiderivative size = 713, normalized size of antiderivative = 3.41 \[ \int \frac {x}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {-\frac {16 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a c \,d^{4} e^{2}}{3}-16 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a c \,d^{3} e^{3} x -16 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a c \,d^{2} e^{4} x^{2}-\frac {16 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a c d \,e^{5} x^{3}}{3}-\frac {16 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, c^{2} d^{6}}{15}-\frac {16 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, c^{2} d^{5} e x}{5}-\frac {16 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, c^{2} d^{4} e^{2} x^{2}}{5}-\frac {16 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, c^{2} d^{3} e^{3} x^{3}}{15}-\frac {4 \sqrt {e x +d}\, a^{3} d \,e^{6}}{15}-\frac {2 \sqrt {e x +d}\, a^{3} e^{7} x}{3}+\frac {8 \sqrt {e x +d}\, a^{2} c \,d^{3} e^{4}}{3}+\frac {98 \sqrt {e x +d}\, a^{2} c \,d^{2} e^{5} x}{15}+\frac {8 \sqrt {e x +d}\, a^{2} c d \,e^{6} x^{2}}{3}+4 \sqrt {e x +d}\, a \,c^{2} d^{5} e^{2}+\frac {34 \sqrt {e x +d}\, a \,c^{2} d^{4} e^{3} x}{3}+\frac {208 \sqrt {e x +d}\, a \,c^{2} d^{3} e^{4} x^{2}}{15}+\frac {16 \sqrt {e x +d}\, a \,c^{2} d^{2} e^{5} x^{3}}{3}+2 \sqrt {e x +d}\, c^{3} d^{6} e x +\frac {8 \sqrt {e x +d}\, c^{3} d^{5} e^{2} x^{2}}{3}+\frac {16 \sqrt {e x +d}\, c^{3} d^{4} e^{3} x^{3}}{15}}{\sqrt {c d x +a e}\, e \left (a^{4} e^{11} x^{3}-4 a^{3} c \,d^{2} e^{9} x^{3}+6 a^{2} c^{2} d^{4} e^{7} x^{3}-4 a \,c^{3} d^{6} e^{5} x^{3}+c^{4} d^{8} e^{3} x^{3}+3 a^{4} d \,e^{10} x^{2}-12 a^{3} c \,d^{3} e^{8} x^{2}+18 a^{2} c^{2} d^{5} e^{6} x^{2}-12 a \,c^{3} d^{7} e^{4} x^{2}+3 c^{4} d^{9} e^{2} x^{2}+3 a^{4} d^{2} e^{9} x -12 a^{3} c \,d^{4} e^{7} x +18 a^{2} c^{2} d^{6} e^{5} x -12 a \,c^{3} d^{8} e^{3} x +3 c^{4} d^{10} e x +a^{4} d^{3} e^{8}-4 a^{3} c \,d^{5} e^{6}+6 a^{2} c^{2} d^{7} e^{4}-4 a \,c^{3} d^{9} e^{2}+c^{4} d^{11}\right )} \] Input:
int(x/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)
Output:
(2*( - 40*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a*c*d**4*e**2 - 120*sq rt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a*c*d**3*e**3*x - 120*sqrt(e)*sqrt (d)*sqrt(c)*sqrt(a*e + c*d*x)*a*c*d**2*e**4*x**2 - 40*sqrt(e)*sqrt(d)*sqrt (c)*sqrt(a*e + c*d*x)*a*c*d*e**5*x**3 - 8*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*c**2*d**6 - 24*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*c**2*d* *5*e*x - 24*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*c**2*d**4*e**2*x**2 - 8*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*c**2*d**3*e**3*x**3 - 2*sqrt (d + e*x)*a**3*d*e**6 - 5*sqrt(d + e*x)*a**3*e**7*x + 20*sqrt(d + e*x)*a** 2*c*d**3*e**4 + 49*sqrt(d + e*x)*a**2*c*d**2*e**5*x + 20*sqrt(d + e*x)*a** 2*c*d*e**6*x**2 + 30*sqrt(d + e*x)*a*c**2*d**5*e**2 + 85*sqrt(d + e*x)*a*c **2*d**4*e**3*x + 104*sqrt(d + e*x)*a*c**2*d**3*e**4*x**2 + 40*sqrt(d + e* x)*a*c**2*d**2*e**5*x**3 + 15*sqrt(d + e*x)*c**3*d**6*e*x + 20*sqrt(d + e* x)*c**3*d**5*e**2*x**2 + 8*sqrt(d + e*x)*c**3*d**4*e**3*x**3))/(15*sqrt(a* e + c*d*x)*e*(a**4*d**3*e**8 + 3*a**4*d**2*e**9*x + 3*a**4*d*e**10*x**2 + a**4*e**11*x**3 - 4*a**3*c*d**5*e**6 - 12*a**3*c*d**4*e**7*x - 12*a**3*c*d **3*e**8*x**2 - 4*a**3*c*d**2*e**9*x**3 + 6*a**2*c**2*d**7*e**4 + 18*a**2* c**2*d**6*e**5*x + 18*a**2*c**2*d**5*e**6*x**2 + 6*a**2*c**2*d**4*e**7*x** 3 - 4*a*c**3*d**9*e**2 - 12*a*c**3*d**8*e**3*x - 12*a*c**3*d**7*e**4*x**2 - 4*a*c**3*d**6*e**5*x**3 + c**4*d**11 + 3*c**4*d**10*e*x + 3*c**4*d**9*e* *2*x**2 + c**4*d**8*e**3*x**3))