Integrand size = 46, antiderivative size = 351 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=-\frac {(e f-d g) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 g (c d f-a e g) \sqrt {d+e x} (f+g x)^3}-\frac {\left (6 a e^2 g-c d (e f+5 d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 g (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}-\frac {c d \left (6 a e^2 g-c d (e f+5 d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 g (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}-\frac {c^2 d^2 \left (6 a e^2 g-c d (e f+5 d g)\right ) \arctan \left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{8 g^{3/2} (c d f-a e g)^{7/2}} \]
-1/8*c^2*d^2*(6*a*e^2*g-c*d*(5*d*g+e*f))*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)^(7/2)-1/3*(-d*g+e*f)*(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)^3/(e*x+d)^(1/2)-1/12*(6*a*e^2*g-c*d*(5*d*g+e*f))*(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)^2/(e*x+d)^(1/ 2)-1/8*c*d*(6*a*e^2*g-c*d*(5*d*g+e*f))*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^( 1/2)/g/(-a*e*g+c*d*f)^3/(g*x+f)/(e*x+d)^(1/2)
Time = 1.03 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.79 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {c^2 d^2 \sqrt {d+e x} \left (\frac {\sqrt {g} (a e+c d x) \left (4 a^2 e^2 g^2 (2 d g+e (f+3 g x))-2 a c d e g \left (d g (13 f+5 g x)+e \left (8 f^2+25 f g x+9 g^2 x^2\right )\right )+c^2 d^2 \left (e f \left (-3 f^2+8 f g x+3 g^2 x^2\right )+d g \left (33 f^2+40 f g x+15 g^2 x^2\right )\right )\right )}{c^2 d^2 (c d f-a e g)^3 (f+g x)^3}+\frac {3 \left (-6 a e^2 g+c d (e f+5 d g)\right ) \sqrt {a e+c d x} \arctan \left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )}{(c d f-a e g)^{7/2}}\right )}{24 g^{3/2} \sqrt {(a e+c d x) (d+e x)}} \]
(c^2*d^2*Sqrt[d + e*x]*((Sqrt[g]*(a*e + c*d*x)*(4*a^2*e^2*g^2*(2*d*g + e*( f + 3*g*x)) - 2*a*c*d*e*g*(d*g*(13*f + 5*g*x) + e*(8*f^2 + 25*f*g*x + 9*g^ 2*x^2)) + c^2*d^2*(e*f*(-3*f^2 + 8*f*g*x + 3*g^2*x^2) + d*g*(33*f^2 + 40*f *g*x + 15*g^2*x^2))))/(c^2*d^2*(c*d*f - a*e*g)^3*(f + g*x)^3) + (3*(-6*a*e ^2*g + c*d*(e*f + 5*d*g))*Sqrt[a*e + c*d*x]*ArcTan[(Sqrt[g]*Sqrt[a*e + c*d *x])/Sqrt[c*d*f - a*e*g]])/(c*d*f - a*e*g)^(7/2)))/(24*g^(3/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])
Time = 0.66 (sec) , antiderivative size = 332, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.109, Rules used = {1257, 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.
| \(\displaystyle \int \frac {(d+e x)^{3/2}}{(f+g x)^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \, dx\) |
| \(\Big \downarrow \) 1257 |
| \(\displaystyle -\frac {\left (6 a e^2 g-c d (5 d g+e f)\right ) \int \frac {\sqrt {d+e x}}{(f+g x)^3 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{6 g (c d f-a e g)}-\frac {(e f-d g) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 g \sqrt {d+e x} (f+g x)^3 (c d f-a e g)}\) |
| \(\Big \downarrow \) 1254 |
| \(\displaystyle -\frac {\left (6 a e^2 g-c d (5 d g+e f)\right ) \left (\frac {3 c d \int \frac {\sqrt {d+e x}}{(f+g x)^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 \sqrt {d+e x} (f+g x)^2 (c d f-a e g)}\right )}{6 g (c d f-a e g)}-\frac {(e f-d g) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 g \sqrt {d+e x} (f+g x)^3 (c d f-a e g)}\) |
| \(\Big \downarrow \) 1254 |
| \(\displaystyle -\frac {\left (6 a e^2 g-c d (5 d g+e f)\right ) \left (\frac {3 c d \left (\frac {c d \int \frac {\sqrt {d+e x}}{(f+g x) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} (f+g x) (c d f-a e g)}\right )}{4 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 \sqrt {d+e x} (f+g x)^2 (c d f-a e g)}\right )}{6 g (c d f-a e g)}-\frac {(e f-d g) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 g \sqrt {d+e x} (f+g x)^3 (c d f-a e g)}\) |
| \(\Big \downarrow \) 1255 |
| \(\displaystyle -\frac {\left (6 a e^2 g-c d (5 d g+e f)\right ) \left (\frac {3 c d \left (\frac {c d e^2 \int \frac {1}{(c d f-a e g) e^2+\frac {g \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right ) e^2}{d+e x}}d\frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {d+e x}}}{c d f-a e g}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} (f+g x) (c d f-a e g)}\right )}{4 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 \sqrt {d+e x} (f+g x)^2 (c d f-a e g)}\right )}{6 g (c d f-a e g)}-\frac {(e f-d g) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 g \sqrt {d+e x} (f+g x)^3 (c d f-a e g)}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {\left (6 a e^2 g-c d (5 d g+e f)\right ) \left (\frac {3 c d \left (\frac {c d \arctan \left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{\sqrt {g} (c d f-a e g)^{3/2}}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} (f+g x) (c d f-a e g)}\right )}{4 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 \sqrt {d+e x} (f+g x)^2 (c d f-a e g)}\right )}{6 g (c d f-a e g)}-\frac {(e f-d g) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 g \sqrt {d+e x} (f+g x)^3 (c d f-a e g)}\) |
-1/3*((e*f - d*g)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(g*(c*d*f - a*e*g)*Sqrt[d + e*x]*(f + g*x)^3) - ((6*a*e^2*g - c*d*(e*f + 5*d*g))*(Sqr t[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*(c*d*f - a*e*g))
3.8.91.3.1 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[(-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]
Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e^2*(e*f - d*g)*(d + e*x)^(m - 2)*( f + g*x)^(n + 1)*((a + b*x + c*x^2)^(p + 1)/(g*(n + 1)*(c*e*f + c*d*g - b*e *g))), x] - Simp[e*((b*e*g*(n + 1) + c*e*f*(p + 1) - c*d*g*(2*n + p + 3))/( g*(n + 1)*(c*e*f + c*d*g - b*e*g))) Int[(d + e*x)^(m - 1)*(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 - 1, 0] && LtQ[n, -1] && Integer Q[2*p]
Leaf count of result is larger than twice the leaf count of optimal. \(1131\) vs. \(2(319)=638\).
Time = 0.56 (sec) , antiderivative size = 1132, normalized size of antiderivative = 3.23
-1/24*((c*d*x+a*e)*(e*x+d))^(1/2)*(40*c^2*d^3*f*g^2*x*(c*d*x+a*e)^(1/2)*(( a*e*g-c*d*f)*g)^(1/2)+18*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/ 2))*a*c^2*d^2*e^2*g^4*x^3-3*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^ (1/2))*c^3*d^3*e*f*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*e*f^2*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*e*f^3*g*x+18*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f) *g)^(1/2))*a*c^2*d^2*e^2*f^3*g+8*a^2*d*e^2*g^3*(c*d*x+a*e)^(1/2)*((a*e*g-c *d*f)*g)^(1/2)+4*a^2*e^3*f*g^2*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+3 3*c^2*d^3*f^2*g*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-3*c^2*d^2*e*f^3* (c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-45*arctanh(g*(c*d*x+a*e)^(1/2)/( (a*e*g-c*d*f)*g)^(1/2))*c^3*d^4*f*g^3*x^2-45*arctanh(g*(c*d*x+a*e)^(1/2)/( (a*e*g-c*d*f)*g)^(1/2))*c^3*d^4*f^2*g^2*x-50*a*c*d*e^2*f*g^2*x*(c*d*x+a*e) ^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-15*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d* f)*g)^(1/2))*c^3*d^4*g^4*x^3-15*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f) *g)^(1/2))*c^3*d^4*f^3*g-3*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^( 1/2))*c^3*d^3*e*f^4+54*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2) )*a*c^2*d^2*e^2*f*g^3*x^2-26*a*c*d^2*e*f*g^2*(c*d*x+a*e)^(1/2)*((a*e*g-c*d *f)*g)^(1/2)-16*a*c*d*e^2*f^2*g*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+ 54*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*a*c^2*d^2*e^2*f^2* g^2*x-18*a*c*d*e^2*g^3*x^2*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+3*...
Leaf count of result is larger than twice the leaf count of optimal. 1347 vs. \(2 (319) = 638\).
Time = 0.96 (sec) , antiderivative size = 2736, normalized size of antiderivative = 7.79 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\text {Too large to display} \]
integrate((e*x+d)^(3/2)/(g*x+f)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2), x, algorithm="fricas")
[-1/48*(3*(c^3*d^4*e*f^4 + (5*c^3*d^5 - 6*a*c^2*d^3*e^2)*f^3*g + (c^3*d^3* e^2*f*g^3 + (5*c^3*d^4*e - 6*a*c^2*d^2*e^3)*g^4)*x^4 + (3*c^3*d^3*e^2*f^2* g^2 + 2*(8*c^3*d^4*e - 9*a*c^2*d^2*e^3)*f*g^3 + (5*c^3*d^5 - 6*a*c^2*d^3*e ^2)*g^4)*x^3 + 3*(c^3*d^3*e^2*f^3*g + 6*(c^3*d^4*e - a*c^2*d^2*e^3)*f^2*g^ 2 + (5*c^3*d^5 - 6*a*c^2*d^3*e^2)*f*g^3)*x^2 + (c^3*d^3*e^2*f^4 + 2*(4*c^3 *d^4*e - 3*a*c^2*d^2*e^3)*f^3*g + 3*(5*c^3*d^5 - 6*a*c^2*d^3*e^2)*f^2*g^2) *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*e*f^4*g + 8*a^3*d*e^3*g^5 - (33*c^3*d^4 - 13*a*c^2*d^2* e^2)*f^3*g^2 + (59*a*c^2*d^3*e - 20*a^2*c*d*e^3)*f^2*g^3 - 2*(17*a^2*c*d^2 *e^2 - 2*a^3*e^4)*f*g^4 - 3*(c^3*d^3*e*f^2*g^3 + (5*c^3*d^4 - 7*a*c^2*d^2* e^2)*f*g^4 - (5*a*c^2*d^3*e - 6*a^2*c*d*e^3)*g^5)*x^2 - 2*(4*c^3*d^3*e*f^3 *g^2 + (20*c^3*d^4 - 29*a*c^2*d^2*e^2)*f^2*g^3 - (25*a*c^2*d^3*e - 31*a^2* c*d*e^3)*f*g^4 + (5*a^2*c*d^2*e^2 - 6*a^3*e^4)*g^5)*x)*sqrt(c*d*e*x^2 + a* d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(c^4*d^5*f^7*g^2 - 4*a*c^3*d^4*e*f ^6*g^3 + 6*a^2*c^2*d^3*e^2*f^5*g^4 - 4*a^3*c*d^2*e^3*f^4*g^5 + a^4*d*e^4*f ^3*g^6 + (c^4*d^4*e*f^4*g^5 - 4*a*c^3*d^3*e^2*f^3*g^6 + 6*a^2*c^2*d^2*e^3* f^2*g^7 - 4*a^3*c*d*e^4*f*g^8 + a^4*e^5*g^9)*x^4 + (3*c^4*d^4*e*f^5*g^4 + a^4*d*e^4*g^9 + (c^4*d^5 - 12*a*c^3*d^3*e^2)*f^4*g^5 - 2*(2*a*c^3*d^4*e...
Timed out. \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\text {Timed out} \]
\[ \int \frac {(d+e x)^{3/2}}{(f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (g x + f\right )}^{4}} \,d x } \]
integrate((e*x+d)^(3/2)/(g*x+f)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2), x, algorithm="maxima")
Leaf count of result is larger than twice the leaf count of optimal. 1879 vs. \(2 (319) = 638\).
Time = 0.74 (sec) , antiderivative size = 1879, normalized size of antiderivative = 5.35 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\text {Too large to display} \]
integrate((e*x+d)^(3/2)/(g*x+f)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2), x, algorithm="giac")
1/24*e^3*(3*(c^4*d^4*e*f + 5*c^4*d^5*g - 6*a*c^3*d^3*e^2*g)*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^3*d^3* e*f^3*g*abs(e) - 3*a*c^2*d^2*e^2*f^2*g^2*abs(e) + 3*a^2*c*d*e^3*f*g^3*abs( e) - a^3*e^4*g^4*abs(e))*sqrt(c*d*f*g - a*e*g^2)*e) - (3*sqrt((e*x + d)*c* d*e - c*d^2*e + a*e^3)*c^6*d^6*e^5*f^3 - 33*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c^6*d^7*e^4*f^2*g + 24*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a *c^5*d^5*e^6*f^2*g + 66*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a*c^5*d^6* e^5*f*g^2 - 57*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a^2*c^4*d^4*e^7*f*g ^2 - 33*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a^2*c^4*d^5*e^6*g^3 + 30*s qrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a^3*c^3*d^3*e^8*g^3 - 8*((e*x + d)* c*d*e - c*d^2*e + a*e^3)^(3/2)*c^5*d^5*e^3*f^2*g - 40*((e*x + d)*c*d*e - c *d^2*e + a*e^3)^(3/2)*c^5*d^6*e^2*f*g^2 + 56*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*c^4*d^4*e^4*f*g^2 + 40*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^ (3/2)*a*c^4*d^5*e^3*g^3 - 48*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^2 *c^3*d^3*e^5*g^3 - 3*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*c^4*d^4*e*f *g^2 - 15*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*c^4*d^5*g^3 + 18*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a*c^3*d^3*e^2*g^3)/((c^3*d^3*e*f^3*g* abs(e) - 3*a*c^2*d^2*e^2*f^2*g^2*abs(e) + 3*a^2*c*d*e^3*f*g^3*abs(e) - a^3 *e^4*g^4*abs(e))*(c*d*e^2*f - a*e^3*g + ((e*x + d)*c*d*e - c*d^2*e + a*e^3 )*g)^3))/(c*d) - 1/24*(3*c^3*d^3*e^4*f^3*arctan(sqrt(-c*d^2*e + a*e^3)*...
Timed out. \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {{\left (d+e\,x\right )}^{3/2}}{{\left (f+g\,x\right )}^4\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}} \,d x \]