Integrand size = 46, antiderivative size = 277 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^4} \, dx=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 g \sqrt {d+e x} (f+g x)^3}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 g (c d f-a e g) \sqrt {d+e x} (f+g x)^2}+\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 g (c d f-a e g)^2 \sqrt {d+e x} (f+g x)}+\frac {c^3 d^3 \arctan \left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{8 g^{3/2} (c d f-a e g)^{5/2}} \]
1/8*c^3*d^3*arctan(g^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e*g +c*d*f)^(1/2)/(e*x+d)^(1/2))/g^(3/2)/(-a*e*g+c*d*f)^(5/2)-1/3*(a*d*e+(a*e^ 2+c*d^2)*x+c*d*e*x^2)^(1/2)/g/(g*x+f)^3/(e*x+d)^(1/2)+1/12*c*d*(a*d*e+(a*e ^2+c*d^2)*x+c*d*e*x^2)^(1/2)/g/(-a*e*g+c*d*f)/(g*x+f)^2/(e*x+d)^(1/2)+1/8* c^2*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/g/(-a*e*g+c*d*f)^2/(g*x+f) /(e*x+d)^(1/2)
Time = 0.72 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.73 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^4} \, dx=\frac {\sqrt {a e+c d x} \sqrt {d+e x} \left (\sqrt {g} \sqrt {c d f-a e g} \sqrt {a e+c d x} \left (-8 a^2 e^2 g^2-2 a c d e g (-7 f+g x)+c^2 d^2 \left (-3 f^2+8 f g x+3 g^2 x^2\right )\right )+3 c^3 d^3 (f+g x)^3 \arctan \left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )\right )}{24 g^{3/2} (c d f-a e g)^{5/2} \sqrt {(a e+c d x) (d+e x)} (f+g x)^3} \]
(Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*(Sqrt[g]*Sqrt[c*d*f - a*e*g]*Sqrt[a*e + c *d*x]*(-8*a^2*e^2*g^2 - 2*a*c*d*e*g*(-7*f + g*x) + c^2*d^2*(-3*f^2 + 8*f*g *x + 3*g^2*x^2)) + 3*c^3*d^3*(f + g*x)^3*ArcTan[(Sqrt[g]*Sqrt[a*e + c*d*x] )/Sqrt[c*d*f - a*e*g]]))/(24*g^(3/2)*(c*d*f - a*e*g)^(5/2)*Sqrt[(a*e + c*d *x)*(d + e*x)]*(f + g*x)^3)
Time = 0.58 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.109, Rules used = {1249, 1254, 1254, 1255, 218}
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.
| \(\displaystyle \int \frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} (f+g x)^4} \, dx\) |
| \(\Big \downarrow \) 1249 |
| \(\displaystyle \frac {c d \int \frac {\sqrt {d+e x}}{(f+g x)^3 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{6 g}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 g \sqrt {d+e x} (f+g x)^3}\) |
| \(\Big \downarrow \) 1254 |
| \(\displaystyle \frac {c d \left (\frac {3 c d \int \frac {\sqrt {d+e x}}{(f+g x)^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 \sqrt {d+e x} (f+g x)^2 (c d f-a e g)}\right )}{6 g}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 g \sqrt {d+e x} (f+g x)^3}\) |
| \(\Big \downarrow \) 1254 |
| \(\displaystyle \frac {c d \left (\frac {3 c d \left (\frac {c d \int \frac {\sqrt {d+e x}}{(f+g x) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} (f+g x) (c d f-a e g)}\right )}{4 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 \sqrt {d+e x} (f+g x)^2 (c d f-a e g)}\right )}{6 g}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 g \sqrt {d+e x} (f+g x)^3}\) |
| \(\Big \downarrow \) 1255 |
| \(\displaystyle \frac {c d \left (\frac {3 c d \left (\frac {c d e^2 \int \frac {1}{(c d f-a e g) e^2+\frac {g \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right ) e^2}{d+e x}}d\frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {d+e x}}}{c d f-a e g}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} (f+g x) (c d f-a e g)}\right )}{4 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 \sqrt {d+e x} (f+g x)^2 (c d f-a e g)}\right )}{6 g}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 g \sqrt {d+e x} (f+g x)^3}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {c d \left (\frac {3 c d \left (\frac {c d \arctan \left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{\sqrt {g} (c d f-a e g)^{3/2}}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} (f+g x) (c d f-a e g)}\right )}{4 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 \sqrt {d+e x} (f+g x)^2 (c d f-a e g)}\right )}{6 g}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 g \sqrt {d+e x} (f+g x)^3}\) |
-1/3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(g*Sqrt[d + e*x]*(f + g*x )^3) + (c*d*(Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(2*(c*d*f - a*e*g )*Sqrt[d + e*x]*(f + g*x)^2) + (3*c*d*(Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c* d*e*x^2]/((c*d*f - a*e*g)*Sqrt[d + e*x]*(f + g*x)) + (c*d*ArcTan[(Sqrt[g]* Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(Sqrt[c*d*f - a*e*g]*Sqrt[d + e*x])])/(Sqrt[g]*(c*d*f - a*e*g)^(3/2))))/(4*(c*d*f - a*e*g))))/(6*g)
3.7.87.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^m*(f + g*x)^(n + 1)*((a + b*x + c*x^2)^p/(g*(n + 1))), x] + Simp[c*(m/(e*g*(n + 1))) Int[(d + e*x) ^(m + 1)*(f + g*x)^(n + 1)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b , c, d, e, f, g}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[m + p, 0] && G tQ[p, 0] && LtQ[n, -1] && !(IntegerQ[n + p] && LeQ[n + p + 2, 0])
Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-e^2)*(d + e*x)^(m - 1)*(f + g*x)^ (n + 1)*((a + b*x + c*x^2)^(p + 1)/((n + 1)*(c*e*f + c*d*g - b*e*g))), x] - Simp[c*e*((m - n - 2)/((n + 1)*(c*e*f + c*d*g - b*e*g))) Int[(d + e*x)^m *(f + g*x)^(n + 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] && EqQ[m + p, 0] && LtQ[n, -1 ] && IntegerQ[2*p]
Int[Sqrt[(d_) + (e_.)*(x_)]/(((f_.) + (g_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[2*e^2 Subst[Int[1/(c*(e*f + d*g) - b*e *g + e^2*g*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{ a, b, c, d, e, f, g}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0]
Time = 0.53 (sec) , antiderivative size = 443, normalized size of antiderivative = 1.60
| method | result | size |
| default | \(-\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (3 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{3} d^{3} g^{3} x^{3}+9 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{3} d^{3} f \,g^{2} x^{2}+9 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{3} d^{3} f^{2} g x +3 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{3} d^{3} f^{3}-3 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, c^{2} d^{2} g^{2} x^{2}+2 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a c d e \,g^{2} x -8 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, c^{2} d^{2} f g x +8 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a^{2} e^{2} g^{2}-14 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a c d e f g +3 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, c^{2} d^{2} f^{2}\right )}{24 \sqrt {e x +d}\, \sqrt {\left (a e g -c d f \right ) g}\, \left (g x +f \right )^{3} g \left (a e g -c d f \right )^{2} \sqrt {c d x +a e}}\) | \(443\) |
-1/24*((c*d*x+a*e)*(e*x+d))^(1/2)*(3*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c *d*f)*g)^(1/2))*c^3*d^3*g^3*x^3+9*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d* f)*g)^(1/2))*c^3*d^3*f*g^2*x^2+9*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f )*g)^(1/2))*c^3*d^3*f^2*g*x+3*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g )^(1/2))*c^3*d^3*f^3-3*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)*c^2*d^2*g ^2*x^2+2*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)*a*c*d*e*g^2*x-8*(c*d*x+ a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)*c^2*d^2*f*g*x+8*(c*d*x+a*e)^(1/2)*((a*e *g-c*d*f)*g)^(1/2)*a^2*e^2*g^2-14*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2 )*a*c*d*e*f*g+3*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)*c^2*d^2*f^2)/(e* x+d)^(1/2)/((a*e*g-c*d*f)*g)^(1/2)/(g*x+f)^3/g/(a*e*g-c*d*f)^2/(c*d*x+a*e) ^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 845 vs. \(2 (245) = 490\).
Time = 0.80 (sec) , antiderivative size = 1732, normalized size of antiderivative = 6.25 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^4} \, dx=\text {Too large to display} \]
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(g*x+f)^4/(e*x+d)^(1/2), x, algorithm="fricas")
[-1/48*(3*(c^3*d^3*e*g^3*x^4 + c^3*d^4*f^3 + (3*c^3*d^3*e*f*g^2 + c^3*d^4* g^3)*x^3 + 3*(c^3*d^3*e*f^2*g + c^3*d^4*f*g^2)*x^2 + (c^3*d^3*e*f^3 + 3*c^ 3*d^4*f^2*g)*x)*sqrt(-c*d*f*g + a*e*g^2)*log(-(c*d*e*g*x^2 - c*d^2*f + 2*a *d*e*g - (c*d*e*f - (c*d^2 + 2*a*e^2)*g)*x - 2*sqrt(c*d*e*x^2 + a*d*e + (c *d^2 + a*e^2)*x)*sqrt(-c*d*f*g + a*e*g^2)*sqrt(e*x + d))/(e*g*x^2 + d*f + (e*f + d*g)*x)) + 2*(3*c^3*d^3*f^3*g - 17*a*c^2*d^2*e*f^2*g^2 + 22*a^2*c*d *e^2*f*g^3 - 8*a^3*e^3*g^4 - 3*(c^3*d^3*f*g^3 - a*c^2*d^2*e*g^4)*x^2 - 2*( 4*c^3*d^3*f^2*g^2 - 5*a*c^2*d^2*e*f*g^3 + a^2*c*d*e^2*g^4)*x)*sqrt(c*d*e*x ^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(c^3*d^4*f^6*g^2 - 3*a*c^2* d^3*e*f^5*g^3 + 3*a^2*c*d^2*e^2*f^4*g^4 - a^3*d*e^3*f^3*g^5 + (c^3*d^3*e*f ^3*g^5 - 3*a*c^2*d^2*e^2*f^2*g^6 + 3*a^2*c*d*e^3*f*g^7 - a^3*e^4*g^8)*x^4 + (3*c^3*d^3*e*f^4*g^4 - a^3*d*e^3*g^8 + (c^3*d^4 - 9*a*c^2*d^2*e^2)*f^3*g ^5 - 3*(a*c^2*d^3*e - 3*a^2*c*d*e^3)*f^2*g^6 + 3*(a^2*c*d^2*e^2 - a^3*e^4) *f*g^7)*x^3 + 3*(c^3*d^3*e*f^5*g^3 - a^3*d*e^3*f*g^7 + (c^3*d^4 - 3*a*c^2* d^2*e^2)*f^4*g^4 - 3*(a*c^2*d^3*e - a^2*c*d*e^3)*f^3*g^5 + (3*a^2*c*d^2*e^ 2 - a^3*e^4)*f^2*g^6)*x^2 + (c^3*d^3*e*f^6*g^2 - 3*a^3*d*e^3*f^2*g^6 + 3*( c^3*d^4 - a*c^2*d^2*e^2)*f^5*g^3 - 3*(3*a*c^2*d^3*e - a^2*c*d*e^3)*f^4*g^4 + (9*a^2*c*d^2*e^2 - a^3*e^4)*f^3*g^5)*x), -1/24*(3*(c^3*d^3*e*g^3*x^4 + c^3*d^4*f^3 + (3*c^3*d^3*e*f*g^2 + c^3*d^4*g^3)*x^3 + 3*(c^3*d^3*e*f^2*g + c^3*d^4*f*g^2)*x^2 + (c^3*d^3*e*f^3 + 3*c^3*d^4*f^2*g)*x)*sqrt(c*d*f*g...
Timed out. \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^4} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^4} \, dx=\int { \frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{\sqrt {e x + d} {\left (g x + f\right )}^{4}} \,d x } \]
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(g*x+f)^4/(e*x+d)^(1/2), x, algorithm="maxima")
Leaf count of result is larger than twice the leaf count of optimal. 1234 vs. \(2 (245) = 490\).
Time = 0.66 (sec) , antiderivative size = 1234, normalized size of antiderivative = 4.45 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^4} \, dx=\frac {{\left (\frac {3 \, c^{3} d^{3} e \arctan \left (\frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right )}{{\left (c^{2} d^{2} f^{2} g - 2 \, a c d e f g^{2} + a^{2} e^{2} g^{3}\right )} \sqrt {c d f g - a e g^{2}}} - \frac {3 \, c^{3} d^{3} e^{4} f^{3} \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right ) - 9 \, c^{3} d^{4} e^{3} f^{2} g \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right ) + 9 \, c^{3} d^{5} e^{2} f g^{2} \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right ) - 3 \, c^{3} d^{6} e g^{3} \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right ) - 3 \, \sqrt {-c d^{2} e + a e^{3}} \sqrt {c d f g - a e g^{2}} c^{2} d^{2} e^{3} f^{2} - 8 \, \sqrt {-c d^{2} e + a e^{3}} \sqrt {c d f g - a e g^{2}} c^{2} d^{3} e^{2} f g + 14 \, \sqrt {-c d^{2} e + a e^{3}} \sqrt {c d f g - a e g^{2}} a c d e^{4} f g + 3 \, \sqrt {-c d^{2} e + a e^{3}} \sqrt {c d f g - a e g^{2}} c^{2} d^{4} e g^{2} + 2 \, \sqrt {-c d^{2} e + a e^{3}} \sqrt {c d f g - a e g^{2}} a c d^{2} e^{3} g^{2} - 8 \, \sqrt {-c d^{2} e + a e^{3}} \sqrt {c d f g - a e g^{2}} a^{2} e^{5} g^{2}}{\sqrt {c d f g - a e g^{2}} c^{2} d^{2} e^{3} f^{5} g - 3 \, \sqrt {c d f g - a e g^{2}} c^{2} d^{3} e^{2} f^{4} g^{2} - 2 \, \sqrt {c d f g - a e g^{2}} a c d e^{4} f^{4} g^{2} + 3 \, \sqrt {c d f g - a e g^{2}} c^{2} d^{4} e f^{3} g^{3} + 6 \, \sqrt {c d f g - a e g^{2}} a c d^{2} e^{3} f^{3} g^{3} + \sqrt {c d f g - a e g^{2}} a^{2} e^{5} f^{3} g^{3} - \sqrt {c d f g - a e g^{2}} c^{2} d^{5} f^{2} g^{4} - 6 \, \sqrt {c d f g - a e g^{2}} a c d^{3} e^{2} f^{2} g^{4} - 3 \, \sqrt {c d f g - a e g^{2}} a^{2} d e^{4} f^{2} g^{4} + 2 \, \sqrt {c d f g - a e g^{2}} a c d^{4} e f g^{5} + 3 \, \sqrt {c d f g - a e g^{2}} a^{2} d^{2} e^{3} f g^{5} - \sqrt {c d f g - a e g^{2}} a^{2} d^{3} e^{2} g^{6}} - \frac {3 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c^{5} d^{5} e^{6} f^{2} - 6 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a c^{4} d^{4} e^{7} f g + 3 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a^{2} c^{3} d^{3} e^{8} g^{2} - 8 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{4} d^{4} e^{4} f g + 8 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a c^{3} d^{3} e^{5} g^{2} - 3 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} c^{3} d^{3} e^{2} g^{2}}{{\left (c^{2} d^{2} f^{2} g - 2 \, a c d e f g^{2} + a^{2} e^{2} g^{3}\right )} {\left (c d e^{2} f - a e^{3} g + {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} g\right )}^{3}}\right )} {\left | e \right |}}{24 \, e^{2}} \]
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(g*x+f)^4/(e*x+d)^(1/2), x, algorithm="giac")
1/24*(3*c^3*d^3*e*arctan(sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*g/(sqrt(c *d*f*g - a*e*g^2)*e))/((c^2*d^2*f^2*g - 2*a*c*d*e*f*g^2 + a^2*e^2*g^3)*sqr t(c*d*f*g - a*e*g^2)) - (3*c^3*d^3*e^4*f^3*arctan(sqrt(-c*d^2*e + a*e^3)*g /(sqrt(c*d*f*g - a*e*g^2)*e)) - 9*c^3*d^4*e^3*f^2*g*arctan(sqrt(-c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e)) + 9*c^3*d^5*e^2*f*g^2*arctan(sqrt(- c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e)) - 3*c^3*d^6*e*g^3*arctan(s qrt(-c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e)) - 3*sqrt(-c*d^2*e + a *e^3)*sqrt(c*d*f*g - a*e*g^2)*c^2*d^2*e^3*f^2 - 8*sqrt(-c*d^2*e + a*e^3)*s qrt(c*d*f*g - a*e*g^2)*c^2*d^3*e^2*f*g + 14*sqrt(-c*d^2*e + a*e^3)*sqrt(c* d*f*g - a*e*g^2)*a*c*d*e^4*f*g + 3*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a *e*g^2)*c^2*d^4*e*g^2 + 2*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a *c*d^2*e^3*g^2 - 8*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a^2*e^5* g^2)/(sqrt(c*d*f*g - a*e*g^2)*c^2*d^2*e^3*f^5*g - 3*sqrt(c*d*f*g - a*e*g^2 )*c^2*d^3*e^2*f^4*g^2 - 2*sqrt(c*d*f*g - a*e*g^2)*a*c*d*e^4*f^4*g^2 + 3*sq rt(c*d*f*g - a*e*g^2)*c^2*d^4*e*f^3*g^3 + 6*sqrt(c*d*f*g - a*e*g^2)*a*c*d^ 2*e^3*f^3*g^3 + sqrt(c*d*f*g - a*e*g^2)*a^2*e^5*f^3*g^3 - sqrt(c*d*f*g - a *e*g^2)*c^2*d^5*f^2*g^4 - 6*sqrt(c*d*f*g - a*e*g^2)*a*c*d^3*e^2*f^2*g^4 - 3*sqrt(c*d*f*g - a*e*g^2)*a^2*d*e^4*f^2*g^4 + 2*sqrt(c*d*f*g - a*e*g^2)*a* c*d^4*e*f*g^5 + 3*sqrt(c*d*f*g - a*e*g^2)*a^2*d^2*e^3*f*g^5 - sqrt(c*d*f*g - a*e*g^2)*a^2*d^3*e^2*g^6) - (3*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^...
Timed out. \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^4} \, dx=\int \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (f+g\,x\right )}^4\,\sqrt {d+e\,x}} \,d x \]