Integrand size = 46, antiderivative size = 335 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^5} \, dx=-\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 g^2 \sqrt {d+e x} (f+g x)^3}+\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 g^2 (c d f-a e g) \sqrt {d+e x} (f+g x)^2}+\frac {3 c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 g^2 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 g (d+e x)^{3/2} (f+g x)^4}+\frac {3 c^4 d^4 \arctan \left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{64 g^{5/2} (c d f-a e g)^{5/2}} \]
-1/4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/g/(e*x+d)^(3/2)/(g*x+f)^4+3/6 4*c^4*d^4*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^(5/2)/(-a*e*g+c*d*f)^(5/2)-1/8*c*d*(a*d*e+(a* e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/g^2/(g*x+f)^3/(e*x+d)^(1/2)+1/32*c^2*d^2*(a* d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/g^2/(-a*e*g+c*d*f)/(g*x+f)^2/(e*x+d)^ (1/2)+3/64*c^3*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/g^2/(-a*e*g+c*d *f)^2/(g*x+f)/(e*x+d)^(1/2)
Time = 1.10 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.72 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^5} \, dx=\frac {c^4 d^4 ((a e+c d x) (d+e x))^{3/2} \left (\frac {\sqrt {g} \left (-16 a^3 e^3 g^3+24 a^2 c d e^2 g^2 (f-g x)-2 a c^2 d^2 e g \left (f^2-22 f g x+g^2 x^2\right )+c^3 d^3 \left (-3 f^3-11 f^2 g x+11 f g^2 x^2+3 g^3 x^3\right )\right )}{c^4 d^4 (c d f-a e g)^2 (a e+c d x) (f+g x)^4}+\frac {3 \arctan \left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )}{(c d f-a e g)^{5/2} (a e+c d x)^{3/2}}\right )}{64 g^{5/2} (d+e x)^{3/2}} \]
(c^4*d^4*((a*e + c*d*x)*(d + e*x))^(3/2)*((Sqrt[g]*(-16*a^3*e^3*g^3 + 24*a ^2*c*d*e^2*g^2*(f - g*x) - 2*a*c^2*d^2*e*g*(f^2 - 22*f*g*x + g^2*x^2) + c^ 3*d^3*(-3*f^3 - 11*f^2*g*x + 11*f*g^2*x^2 + 3*g^3*x^3)))/(c^4*d^4*(c*d*f - a*e*g)^2*(a*e + c*d*x)*(f + g*x)^4) + (3*ArcTan[(Sqrt[g]*Sqrt[a*e + c*d*x ])/Sqrt[c*d*f - a*e*g]])/((c*d*f - a*e*g)^(5/2)*(a*e + c*d*x)^(3/2))))/(64 *g^(5/2)*(d + e*x)^(3/2))
Time = 0.70 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.03, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {1249, 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.
| \(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^5} \, dx\) |
| \(\Big \downarrow \) 1249 |
| \(\displaystyle \frac {3 c d \int \frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {d+e x} (f+g x)^4}dx}{8 g}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 g (d+e x)^{3/2} (f+g x)^4}\) |
| \(\Big \downarrow \) 1249 |
| \(\displaystyle \frac {3 c d \left (\frac {c d \int \frac {\sqrt {d+e x}}{(f+g x)^3 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{6 g}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 g \sqrt {d+e x} (f+g x)^3}\right )}{8 g}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 g (d+e x)^{3/2} (f+g x)^4}\) |
| \(\Big \downarrow \) 1254 |
| \(\displaystyle \frac {3 c d \left (\frac {c d \left (\frac {3 c d \int \frac {\sqrt {d+e x}}{(f+g x)^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 \sqrt {d+e x} (f+g x)^2 (c d f-a e g)}\right )}{6 g}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 g \sqrt {d+e x} (f+g x)^3}\right )}{8 g}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 g (d+e x)^{3/2} (f+g x)^4}\) |
| \(\Big \downarrow \) 1254 |
| \(\displaystyle \frac {3 c d \left (\frac {c d \left (\frac {3 c d \left (\frac {c d \int \frac {\sqrt {d+e x}}{(f+g x) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} (f+g x) (c d f-a e g)}\right )}{4 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 \sqrt {d+e x} (f+g x)^2 (c d f-a e g)}\right )}{6 g}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 g \sqrt {d+e x} (f+g x)^3}\right )}{8 g}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 g (d+e x)^{3/2} (f+g x)^4}\) |
| \(\Big \downarrow \) 1255 |
| \(\displaystyle \frac {3 c d \left (\frac {c d \left (\frac {3 c d \left (\frac {c d e^2 \int \frac {1}{(c d f-a e g) e^2+\frac {g \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right ) e^2}{d+e x}}d\frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {d+e x}}}{c d f-a e g}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} (f+g x) (c d f-a e g)}\right )}{4 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 \sqrt {d+e x} (f+g x)^2 (c d f-a e g)}\right )}{6 g}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 g \sqrt {d+e x} (f+g x)^3}\right )}{8 g}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 g (d+e x)^{3/2} (f+g x)^4}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {3 c d \left (\frac {c d \left (\frac {3 c d \left (\frac {c d \arctan \left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{\sqrt {g} (c d f-a e g)^{3/2}}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} (f+g x) (c d f-a e g)}\right )}{4 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 \sqrt {d+e x} (f+g x)^2 (c d f-a e g)}\right )}{6 g}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 g \sqrt {d+e x} (f+g x)^3}\right )}{8 g}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 g (d+e x)^{3/2} (f+g x)^4}\) |
-1/4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(g*(d + e*x)^(3/2)*(f + g*x)^4) + (3*c*d*(-1/3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(g*Sqr t[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)))/(8*g)
3.7.98.3.1 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]
Leaf count of result is larger than twice the leaf count of optimal. \(654\) vs. \(2(297)=594\).
Time = 0.57 (sec) , antiderivative size = 655, normalized size of antiderivative = 1.96
| method | result | size |
| default | \(-\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (3 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{4} d^{4} g^{4} x^{4}+12 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{4} d^{4} f \,g^{3} x^{3}+18 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{4} d^{4} f^{2} g^{2} x^{2}+12 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{4} d^{4} f^{3} g x -3 c^{3} d^{3} g^{3} x^{3} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+3 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{4} d^{4} f^{4}+2 a \,c^{2} d^{2} e \,g^{3} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}-11 c^{3} d^{3} f \,g^{2} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+24 a^{2} c d \,e^{2} g^{3} x \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}-44 a \,c^{2} d^{2} e f \,g^{2} x \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+11 c^{3} d^{3} f^{2} g x \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+16 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a^{3} e^{3} g^{3}-24 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a^{2} c d \,e^{2} f \,g^{2}+2 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a \,c^{2} d^{2} e \,f^{2} g +3 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, c^{3} d^{3} f^{3}\right )}{64 \sqrt {e x +d}\, \sqrt {\left (a e g -c d f \right ) g}\, \left (g x +f \right )^{4} g^{2} \left (a e g -c d f \right )^{2} \sqrt {c d x +a e}}\) | \(655\) |
-1/64*((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^4*d^4*g^4*x^4+12*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d *f)*g)^(1/2))*c^4*d^4*f*g^3*x^3+18*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d *f)*g)^(1/2))*c^4*d^4*f^2*g^2*x^2+12*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c *d*f)*g)^(1/2))*c^4*d^4*f^3*g*x-3*c^3*d^3*g^3*x^3*(c*d*x+a*e)^(1/2)*((a*e* g-c*d*f)*g)^(1/2)+3*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c ^4*d^4*f^4+2*a*c^2*d^2*e*g^3*x^2*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2) -11*c^3*d^3*f*g^2*x^2*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+24*a^2*c*d *e^2*g^3*x*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-44*a*c^2*d^2*e*f*g^2* x*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+11*c^3*d^3*f^2*g*x*(c*d*x+a*e) ^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+16*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2 )*a^3*e^3*g^3-24*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)*a^2*c*d*e^2*f*g ^2+2*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)*a*c^2*d^2*e*f^2*g+3*(c*d*x+ a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)*c^3*d^3*f^3)/(e*x+d)^(1/2)/((a*e*g-c*d* f)*g)^(1/2)/(g*x+f)^4/g^2/(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. 1098 vs. \(2 (297) = 594\).
Time = 0.80 (sec) , antiderivative size = 2238, normalized size of antiderivative = 6.68 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^5} \, dx=\text {Too large to display} \]
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2)/(g*x+f)^5, x, algorithm="fricas")
[-1/128*(3*(c^4*d^4*e*g^4*x^5 + c^4*d^5*f^4 + (4*c^4*d^4*e*f*g^3 + c^4*d^5 *g^4)*x^4 + 2*(3*c^4*d^4*e*f^2*g^2 + 2*c^4*d^5*f*g^3)*x^3 + 2*(2*c^4*d^4*e *f^3*g + 3*c^4*d^5*f^2*g^2)*x^2 + (c^4*d^4*e*f^4 + 4*c^4*d^5*f^3*g)*x)*sqr t(-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)*sq rt(-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^4*d^4*f^4*g - a*c^3*d^3*e*f^3*g^2 - 26*a^2*c^2*d^2*e^2*f^2*g^3 + 40* a^3*c*d*e^3*f*g^4 - 16*a^4*e^4*g^5 - 3*(c^4*d^4*f*g^4 - a*c^3*d^3*e*g^5)*x ^3 - (11*c^4*d^4*f^2*g^3 - 13*a*c^3*d^3*e*f*g^4 + 2*a^2*c^2*d^2*e^2*g^5)*x ^2 + (11*c^4*d^4*f^3*g^2 - 55*a*c^3*d^3*e*f^2*g^3 + 68*a^2*c^2*d^2*e^2*f*g ^4 - 24*a^3*c*d*e^3*g^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sq rt(e*x + d))/(c^3*d^4*f^7*g^3 - 3*a*c^2*d^3*e*f^6*g^4 + 3*a^2*c*d^2*e^2*f^ 5*g^5 - a^3*d*e^3*f^4*g^6 + (c^3*d^3*e*f^3*g^7 - 3*a*c^2*d^2*e^2*f^2*g^8 + 3*a^2*c*d*e^3*f*g^9 - a^3*e^4*g^10)*x^5 + (4*c^3*d^3*e*f^4*g^6 - a^3*d*e^ 3*g^10 + (c^3*d^4 - 12*a*c^2*d^2*e^2)*f^3*g^7 - 3*(a*c^2*d^3*e - 4*a^2*c*d *e^3)*f^2*g^8 + (3*a^2*c*d^2*e^2 - 4*a^3*e^4)*f*g^9)*x^4 + 2*(3*c^3*d^3*e* f^5*g^5 - 2*a^3*d*e^3*f*g^9 + (2*c^3*d^4 - 9*a*c^2*d^2*e^2)*f^4*g^6 - 3*(2 *a*c^2*d^3*e - 3*a^2*c*d*e^3)*f^3*g^7 + 3*(2*a^2*c*d^2*e^2 - a^3*e^4)*f^2* g^8)*x^3 + 2*(2*c^3*d^3*e*f^6*g^4 - 3*a^3*d*e^3*f^2*g^8 + 3*(c^3*d^4 - 2*a *c^2*d^2*e^2)*f^5*g^5 - 3*(3*a*c^2*d^3*e - 2*a^2*c*d*e^3)*f^4*g^6 + (9*...
Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^5} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^5} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{5}} \,d x } \]
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2)/(g*x+f)^5, x, algorithm="maxima")
Leaf count of result is larger than twice the leaf count of optimal. 1785 vs. \(2 (297) = 594\).
Time = 1.21 (sec) , antiderivative size = 1785, normalized size of antiderivative = 5.33 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^5} \, dx=\text {Too large to display} \]
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2)/(g*x+f)^5, x, algorithm="giac")
3/64*c^4*d^4*abs(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 - 2*a*c*d*e*f*g^3 + a^2*e^2*g^4) *sqrt(c*d*f*g - a*e*g^2)*e) - 1/64*(3*c^4*d^4*e^4*f^4*abs(e)*arctan(sqrt(- c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e)) - 12*c^4*d^5*e^3*f^3*g*abs (e)*arctan(sqrt(-c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e)) + 18*c^4* d^6*e^2*f^2*g^2*abs(e)*arctan(sqrt(-c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g - a*e *g^2)*e)) - 12*c^4*d^7*e*f*g^3*abs(e)*arctan(sqrt(-c*d^2*e + a*e^3)*g/(sqr t(c*d*f*g - a*e*g^2)*e)) + 3*c^4*d^8*g^4*abs(e)*arctan(sqrt(-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^3*d^3*e^3*f^3*abs(e) + 11*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f* g - a*e*g^2)*c^3*d^4*e^2*f^2*g*abs(e) - 2*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d* f*g - a*e*g^2)*a*c^2*d^2*e^4*f^2*g*abs(e) + 11*sqrt(-c*d^2*e + a*e^3)*sqrt (c*d*f*g - a*e*g^2)*c^3*d^5*e*f*g^2*abs(e) - 44*sqrt(-c*d^2*e + a*e^3)*sqr t(c*d*f*g - a*e*g^2)*a*c^2*d^3*e^3*f*g^2*abs(e) + 24*sqrt(-c*d^2*e + a*e^3 )*sqrt(c*d*f*g - a*e*g^2)*a^2*c*d*e^5*f*g^2*abs(e) - 3*sqrt(-c*d^2*e + a*e ^3)*sqrt(c*d*f*g - a*e*g^2)*c^3*d^6*g^3*abs(e) - 2*sqrt(-c*d^2*e + a*e^3)* sqrt(c*d*f*g - a*e*g^2)*a*c^2*d^4*e^2*g^3*abs(e) + 24*sqrt(-c*d^2*e + a*e^ 3)*sqrt(c*d*f*g - a*e*g^2)*a^2*c*d^2*e^4*g^3*abs(e) - 16*sqrt(-c*d^2*e + a *e^3)*sqrt(c*d*f*g - a*e*g^2)*a^3*e^6*g^3*abs(e))/(sqrt(c*d*f*g - a*e*g^2) *c^2*d^2*e^5*f^6*g^2 - 4*sqrt(c*d*f*g - a*e*g^2)*c^2*d^3*e^4*f^5*g^3 - ...
Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^5} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}}{{\left (f+g\,x\right )}^5\,{\left (d+e\,x\right )}^{3/2}} \,d x \]