Integrand size = 40, antiderivative size = 504 \[ \int \frac {1}{x (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {2 c d}{a e \left (c d^2-a e^2\right ) (d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {2 \left (7 c d^2+a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 a d \left (c d^2-a e^2\right )^2 (d+e x)^4}+\frac {2 \left (35 c^2 d^4+20 a c d^2 e^2-7 a^2 e^4\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 a d^2 \left (c d^2-a e^2\right )^3 (d+e x)^3}+\frac {2 \left (105 c^3 d^6+185 a c^2 d^4 e^2-133 a^2 c d^2 e^4+35 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{105 a d^3 \left (c d^2-a e^2\right )^4 (d+e x)^2}+\frac {2 \left (105 c^4 d^8+790 a c^3 d^6 e^2-896 a^2 c^2 d^4 e^4+490 a^3 c d^2 e^6-105 a^4 e^8\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{105 a d^4 \left (c d^2-a e^2\right )^5 (d+e x)}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} (d+e x)}{\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{a^{3/2} d^{9/2} e^{3/2}} \] Output:
2*c*d/a/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) +2/7*(a*e^2+7*c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/a/d/(-a*e^2+c *d^2)^2/(e*x+d)^4+2/35*(-7*a^2*e^4+20*a*c*d^2*e^2+35*c^2*d^4)*(a*d*e+(a*e^ 2+c*d^2)*x+c*d*e*x^2)^(1/2)/a/d^2/(-a*e^2+c*d^2)^3/(e*x+d)^3+2/105*(35*a^3 *e^6-133*a^2*c*d^2*e^4+185*a*c^2*d^4*e^2+105*c^3*d^6)*(a*d*e+(a*e^2+c*d^2) *x+c*d*e*x^2)^(1/2)/a/d^3/(-a*e^2+c*d^2)^4/(e*x+d)^2+2/105*(-105*a^4*e^8+4 90*a^3*c*d^2*e^6-896*a^2*c^2*d^4*e^4+790*a*c^3*d^6*e^2+105*c^4*d^8)*(a*d*e +(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/a/d^4/(-a*e^2+c*d^2)^5/(e*x+d)-2*arctanh (a^(1/2)*e^(1/2)*(e*x+d)/d^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/ a^(3/2)/d^(9/2)/e^(3/2)
Time = 1.04 (sec) , antiderivative size = 421, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {2 \left (-\frac {\sqrt {a} \sqrt {d} \sqrt {e} (a e+c d x) \left (15 a d^3 e^6 (a e+c d x)^4-105 a c d^4 e^5 (a e+c d x)^3 (d+e x)+21 a^2 d^2 e^7 (a e+c d x)^3 (d+e x)+350 a c^2 d^5 e^4 (a e+c d x)^2 (d+e x)^2-175 a^2 c d^3 e^6 (a e+c d x)^2 (d+e x)^2+35 a^3 d e^8 (a e+c d x)^2 (d+e x)^2-1050 a c^3 d^6 e^3 (a e+c d x) (d+e x)^3+1050 a^2 c^2 d^4 e^5 (a e+c d x) (d+e x)^3-525 a^3 c d^2 e^7 (a e+c d x) (d+e x)^3+105 a^4 e^9 (a e+c d x) (d+e x)^3-105 c^5 d^9 (d+e x)^4\right )}{\left (c d^2-a e^2\right )^5 (d+e x)^2}-105 (a e+c d x)^{3/2} (d+e x)^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} \sqrt {d+e x}}{\sqrt {d} \sqrt {a e+c d x}}\right )\right )}{105 a^{3/2} d^{9/2} e^{3/2} ((a e+c d x) (d+e x))^{3/2}} \] Input:
Integrate[1/(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*(-((Sqrt[a]*Sqrt[d]*Sqrt[e]*(a*e + c*d*x)*(15*a*d^3*e^6*(a*e + c*d*x)^4 - 105*a*c*d^4*e^5*(a*e + c*d*x)^3*(d + e*x) + 21*a^2*d^2*e^7*(a*e + c*d*x )^3*(d + e*x) + 350*a*c^2*d^5*e^4*(a*e + c*d*x)^2*(d + e*x)^2 - 175*a^2*c* d^3*e^6*(a*e + c*d*x)^2*(d + e*x)^2 + 35*a^3*d*e^8*(a*e + c*d*x)^2*(d + e* x)^2 - 1050*a*c^3*d^6*e^3*(a*e + c*d*x)*(d + e*x)^3 + 1050*a^2*c^2*d^4*e^5 *(a*e + c*d*x)*(d + e*x)^3 - 525*a^3*c*d^2*e^7*(a*e + c*d*x)*(d + e*x)^3 + 105*a^4*e^9*(a*e + c*d*x)*(d + e*x)^3 - 105*c^5*d^9*(d + e*x)^4))/((c*d^2 - a*e^2)^5*(d + e*x)^2)) - 105*(a*e + c*d*x)^(3/2)*(d + e*x)^(3/2)*ArcTan h[(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])/(Sqrt[d]*Sqrt[a*e + c*d*x])]))/(105*a^(3 /2)*d^(9/2)*e^(3/2)*((a*e + c*d*x)*(d + e*x))^(3/2))
Time = 1.72 (sec) , antiderivative size = 727, normalized size of antiderivative = 1.44, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1259, 2009}
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.
| \(\displaystyle \int \frac {1}{x (d+e x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1259 |
| \(\displaystyle \int \left (-\frac {e}{d^2 (d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}-\frac {e}{d (d+e x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}-\frac {e}{d^3 (d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac {1}{d^3 x \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\text {arctanh}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{a^{3/2} d^{9/2} e^{3/2}}+\frac {2 \left (a^2 e^4+c d e x \left (a e^2+c d^2\right )+c^2 d^4\right )}{a d^4 e \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {128 c^3 d^2 e \left (a e^2+c d^2+2 c d e x\right )}{35 \left (c d^2-a e^2\right )^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {16 c^2 e \left (a e^2+c d^2+2 c d e x\right )}{5 \left (c d^2-a e^2\right )^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {32 c^2 d e}{35 (d+e x) \left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {8 c e \left (a e^2+c d^2+2 c d e x\right )}{3 d^2 \left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {4 c e}{5 d (d+e x) \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {16 c e}{35 (d+e x)^2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {2 e}{5 d^2 (d+e x)^2 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {2 e}{7 d (d+e x)^3 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {2 e}{3 d^3 (d+e x) \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
Input:
Int[1/(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*e)/(7*d*(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]) - (16*c*e)/(35*(c*d^2 - a*e^2)^2*(d + e*x)^2*Sqrt[a*d*e + (c*d^ 2 + a*e^2)*x + c*d*e*x^2]) - (2*e)/(5*d^2*(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]) - (32*c^2*d*e)/(35*(c*d^2 - a*e^2 )^3*(d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (4*c*e)/(5*d* (c*d^2 - a*e^2)^2*(d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (2*e)/(3*d^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]) + (128*c^3*d^2*e*(c*d^2 + a*e^2 + 2*c*d*e*x))/(35*(c*d^2 - a*e^ 2)^5*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (16*c^2*e*(c*d^2 + a*e ^2 + 2*c*d*e*x))/(5*(c*d^2 - a*e^2)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d *e*x^2]) + (8*c*e*(c*d^2 + a*e^2 + 2*c*d*e*x))/(3*d^2*(c*d^2 - a*e^2)^3*Sq rt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (2*(c^2*d^4 + a^2*e^4 + c*d*e *(c*d^2 + a*e^2)*x))/(a*d^4*e*(c*d^2 - a*e^2)^2*Sqrt[a*d*e + (c*d^2 + a*e^ 2)*x + c*d*e*x^2]) - ArcTanh[(2*a*d*e + (c*d^2 + a*e^2)*x)/(2*Sqrt[a]*Sqrt [d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])]/(a^(3/2)*d^(9/2) *e^(3/2))
Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x )^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[m, 0] && (ILtQ[n, 0] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0])) && !IGtQ[n, 0]
Time = 3.08 (sec) , antiderivative size = 901, normalized size of antiderivative = 1.79
| method | result | size |
| default | \(\frac {\frac {1}{a d e \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (2 c d x e +a \,e^{2}+c \,d^{2}\right )}{a d e \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {\ln \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{x}\right )}{a d e \sqrt {a d e}}}{d^{3}}-\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 )}}}{d^{3}}-\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 \,d^{2}}-\frac {-\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 )}}{e^{2} d}\) | \(901\) |
Input:
int(1/x/(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:
1/d^3*(1/a/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-(a*e^2+c*d^2)/a/d/e *(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+(a*e^2+c*d ^2)*x+c*d*x^2*e)^(1/2)-1/a/d/e/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2 *(a*d*e)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2))/x))-1/d^3*(-2/3/(a *e^2-c*d^2)/(x+d/e)/(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)+8/3*d*e* c/(a*e^2-c*d^2)^3*(2*d*e*c*(x+d/e)+a*e^2-c*d^2)/(d*e*c*(x+d/e)^2+(a*e^2-c* d^2)*(x+d/e))^(1/2))-1/e/d^2*(-2/5/(a*e^2-c*d^2)/(x+d/e)^2/(d*e*c*(x+d/e)^ 2+(a*e^2-c*d^2)*(x+d/e))^(1/2)-6/5*d*e*c/(a*e^2-c*d^2)*(-2/3/(a*e^2-c*d^2) /(x+d/e)/(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)+8/3*d*e*c/(a*e^2-c* d^2)^3*(2*d*e*c*(x+d/e)+a*e^2-c*d^2)/(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e ))^(1/2)))-1/e^2/d*(-2/7/(a*e^2-c*d^2)/(x+d/e)^3/(d*e*c*(x+d/e)^2+(a*e^2-c *d^2)*(x+d/e))^(1/2)-8/7*d*e*c/(a*e^2-c*d^2)*(-2/5/(a*e^2-c*d^2)/(x+d/e)^2 /(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)-6/5*d*e*c/(a*e^2-c*d^2)*(-2 /3/(a*e^2-c*d^2)/(x+d/e)/(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)+8/3 *d*e*c/(a*e^2-c*d^2)^3*(2*d*e*c*(x+d/e)+a*e^2-c*d^2)/(d*e*c*(x+d/e)^2+(a*e ^2-c*d^2)*(x+d/e))^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 1588 vs. \(2 (472) = 944\).
Time = 22.89 (sec) , antiderivative size = 3196, normalized size of antiderivative = 6.34 \[ \int \frac {1}{x (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate(1/x/(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:
Too large to include
\[ \int \frac {1}{x (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {1}{x \left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{3}}\, dx \] Input:
integrate(1/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(1/(x*((d + e*x)*(a*e + c*d*x))**(3/2)*(d + e*x)**3), x)
\[ \int \frac {1}{x (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{3} x} \,d x } \] Input:
integrate(1/x/(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:
integrate(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(e*x + d)^3*x), x)
\[ \int \frac {1}{x (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{3} x} \,d x } \] Input:
integrate(1/x/(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(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(e*x + d)^3*x), x)
Timed out. \[ \int \frac {1}{x (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {1}{x\,{\left (d+e\,x\right )}^3\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}} \,d x \] Input:
int(1/(x*(d + e*x)^3*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)),x)
Output:
int(1/(x*(d + e*x)^3*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)), x)
Time = 16.55 (sec) , antiderivative size = 8301, normalized size of antiderivative = 16.47 \[ \int \frac {1}{x (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
int(1/x/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)
Output:
(105*sqrt(e)*sqrt(d)*sqrt(a)*sqrt(a*e + c*d*x)*log(sqrt(e)*sqrt(a*e + c*d* x) - sqrt(2*sqrt(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + sqrt(d)*sqrt(c)*sqrt( d + e*x))*a**5*d**4*e**10 + 420*sqrt(e)*sqrt(d)*sqrt(a)*sqrt(a*e + c*d*x)* log(sqrt(e)*sqrt(a*e + c*d*x) - sqrt(2*sqrt(c)*sqrt(a)*d*e + a*e**2 + c*d* *2) + sqrt(d)*sqrt(c)*sqrt(d + e*x))*a**5*d**3*e**11*x + 630*sqrt(e)*sqrt( d)*sqrt(a)*sqrt(a*e + c*d*x)*log(sqrt(e)*sqrt(a*e + c*d*x) - sqrt(2*sqrt(c )*sqrt(a)*d*e + a*e**2 + c*d**2) + sqrt(d)*sqrt(c)*sqrt(d + e*x))*a**5*d** 2*e**12*x**2 + 420*sqrt(e)*sqrt(d)*sqrt(a)*sqrt(a*e + c*d*x)*log(sqrt(e)*s qrt(a*e + c*d*x) - sqrt(2*sqrt(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + sqrt(d) *sqrt(c)*sqrt(d + e*x))*a**5*d*e**13*x**3 + 105*sqrt(e)*sqrt(d)*sqrt(a)*sq rt(a*e + c*d*x)*log(sqrt(e)*sqrt(a*e + c*d*x) - sqrt(2*sqrt(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + sqrt(d)*sqrt(c)*sqrt(d + e*x))*a**5*e**14*x**4 - 525 *sqrt(e)*sqrt(d)*sqrt(a)*sqrt(a*e + c*d*x)*log(sqrt(e)*sqrt(a*e + c*d*x) - sqrt(2*sqrt(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + sqrt(d)*sqrt(c)*sqrt(d + e*x))*a**4*c*d**6*e**8 - 2100*sqrt(e)*sqrt(d)*sqrt(a)*sqrt(a*e + c*d*x)*lo g(sqrt(e)*sqrt(a*e + c*d*x) - sqrt(2*sqrt(c)*sqrt(a)*d*e + a*e**2 + c*d**2 ) + sqrt(d)*sqrt(c)*sqrt(d + e*x))*a**4*c*d**5*e**9*x - 3150*sqrt(e)*sqrt( d)*sqrt(a)*sqrt(a*e + c*d*x)*log(sqrt(e)*sqrt(a*e + c*d*x) - sqrt(2*sqrt(c )*sqrt(a)*d*e + a*e**2 + c*d**2) + sqrt(d)*sqrt(c)*sqrt(d + e*x))*a**4*c*d **4*e**10*x**2 - 2100*sqrt(e)*sqrt(d)*sqrt(a)*sqrt(a*e + c*d*x)*log(sqr...