Integrand size = 46, antiderivative size = 463 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^7} \, dx=-\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 g^3 \sqrt {d+e x} (f+g x)^4}+\frac {c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 g^3 (c d f-a e g) \sqrt {d+e x} (f+g x)^3}+\frac {5 c^4 d^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{768 g^3 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}+\frac {5 c^5 d^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 g^3 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{12 g^2 (d+e x)^{3/2} (f+g x)^5}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 g (d+e x)^{5/2} (f+g x)^6}+\frac {5 c^6 d^6 \arctan \left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{512 g^{7/2} (c d f-a e g)^{7/2}} \]
-1/12*c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/g^2/(e*x+d)^(3/2)/(g*x+f )^5-1/6*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/g/(e*x+d)^(5/2)/(g*x+f)^6+ 5/512*c^6*d^6*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^(7/2)/(-a*e*g+c*d*f)^(7/2)-1/32*c^2*d^2*( a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/g^3/(g*x+f)^4/(e*x+d)^(1/2)+1/192*c ^3*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/g^3/(-a*e*g+c*d*f)/(g*x+f)^ 3/(e*x+d)^(1/2)+5/768*c^4*d^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/g^3/ (-a*e*g+c*d*f)^2/(g*x+f)^2/(e*x+d)^(1/2)+5/512*c^5*d^5*(a*d*e+(a*e^2+c*d^2 )*x+c*d*e*x^2)^(1/2)/g^3/(-a*e*g+c*d*f)^3/(g*x+f)/(e*x+d)^(1/2)
Time = 2.72 (sec) , antiderivative size = 370, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^7} \, dx=\frac {c^6 d^6 ((a e+c d x) (d+e x))^{5/2} \left (\frac {\sqrt {g} \left (256 a^5 e^5 g^5+640 a^4 c d e^4 g^4 (-f+g x)+16 a^3 c^2 d^2 e^3 g^3 \left (27 f^2-106 f g x+27 g^2 x^2\right )+8 a^2 c^3 d^3 e^2 g^2 \left (-f^3+159 f^2 g x-159 f g^2 x^2+g^3 x^3\right )-2 a c^4 d^4 e g \left (5 f^4+28 f^3 g x-594 f^2 g^2 x^2+28 f g^3 x^3+5 g^4 x^4\right )+c^5 d^5 \left (-15 f^5-85 f^4 g x-198 f^3 g^2 x^2+198 f^2 g^3 x^3+85 f g^4 x^4+15 g^5 x^5\right )\right )}{c^6 d^6 (c d f-a e g)^3 (a e+c d x)^2 (f+g x)^6}+\frac {15 \arctan \left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )}{(c d f-a e g)^{7/2} (a e+c d x)^{5/2}}\right )}{1536 g^{7/2} (d+e x)^{5/2}} \]
(c^6*d^6*((a*e + c*d*x)*(d + e*x))^(5/2)*((Sqrt[g]*(256*a^5*e^5*g^5 + 640* a^4*c*d*e^4*g^4*(-f + g*x) + 16*a^3*c^2*d^2*e^3*g^3*(27*f^2 - 106*f*g*x + 27*g^2*x^2) + 8*a^2*c^3*d^3*e^2*g^2*(-f^3 + 159*f^2*g*x - 159*f*g^2*x^2 + g^3*x^3) - 2*a*c^4*d^4*e*g*(5*f^4 + 28*f^3*g*x - 594*f^2*g^2*x^2 + 28*f*g^ 3*x^3 + 5*g^4*x^4) + c^5*d^5*(-15*f^5 - 85*f^4*g*x - 198*f^3*g^2*x^2 + 198 *f^2*g^3*x^3 + 85*f*g^4*x^4 + 15*g^5*x^5)))/(c^6*d^6*(c*d*f - a*e*g)^3*(a* e + c*d*x)^2*(f + g*x)^6) + (15*ArcTan[(Sqrt[g]*Sqrt[a*e + c*d*x])/Sqrt[c* d*f - a*e*g]])/((c*d*f - a*e*g)^(7/2)*(a*e + c*d*x)^(5/2))))/(1536*g^(7/2) *(d + e*x)^(5/2))
Time = 0.96 (sec) , antiderivative size = 486, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {1249, 1249, 1249, 1254, 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.
| \(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^7} \, dx\) |
| \(\Big \downarrow \) 1249 |
| \(\displaystyle \frac {5 c d \int \frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^6}dx}{12 g}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 g (d+e x)^{5/2} (f+g x)^6}\) |
| \(\Big \downarrow \) 1249 |
| \(\displaystyle \frac {5 c d \left (\frac {3 c d \int \frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {d+e x} (f+g x)^5}dx}{10 g}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 g (d+e x)^{3/2} (f+g x)^5}\right )}{12 g}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 g (d+e x)^{5/2} (f+g x)^6}\) |
| \(\Big \downarrow \) 1249 |
| \(\displaystyle \frac {5 c d \left (\frac {3 c d \left (\frac {c d \int \frac {\sqrt {d+e x}}{(f+g x)^4 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 g}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 g \sqrt {d+e x} (f+g x)^4}\right )}{10 g}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 g (d+e x)^{3/2} (f+g x)^5}\right )}{12 g}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 g (d+e x)^{5/2} (f+g x)^6}\) |
| \(\Big \downarrow \) 1254 |
| \(\displaystyle \frac {5 c d \left (\frac {3 c d \left (\frac {c d \left (\frac {5 c d \int \frac {\sqrt {d+e x}}{(f+g x)^3 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{6 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt {d+e x} (f+g x)^3 (c d f-a e g)}\right )}{8 g}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 g \sqrt {d+e x} (f+g x)^4}\right )}{10 g}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 g (d+e x)^{3/2} (f+g x)^5}\right )}{12 g}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 g (d+e x)^{5/2} (f+g x)^6}\) |
| \(\Big \downarrow \) 1254 |
| \(\displaystyle \frac {5 c d \left (\frac {3 c d \left (\frac {c d \left (\frac {5 c d \left (\frac {3 c d \int \frac {\sqrt {d+e x}}{(f+g x)^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 \sqrt {d+e x} (f+g x)^2 (c d f-a e g)}\right )}{6 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt {d+e x} (f+g x)^3 (c d f-a e g)}\right )}{8 g}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 g \sqrt {d+e x} (f+g x)^4}\right )}{10 g}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 g (d+e x)^{3/2} (f+g x)^5}\right )}{12 g}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 g (d+e x)^{5/2} (f+g x)^6}\) |
| \(\Big \downarrow \) 1254 |
| \(\displaystyle \frac {5 c d \left (\frac {3 c d \left (\frac {c d \left (\frac {5 c d \left (\frac {3 c d \left (\frac {c d \int \frac {\sqrt {d+e x}}{(f+g x) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} (f+g x) (c d f-a e g)}\right )}{4 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 \sqrt {d+e x} (f+g x)^2 (c d f-a e g)}\right )}{6 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt {d+e x} (f+g x)^3 (c d f-a e g)}\right )}{8 g}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 g \sqrt {d+e x} (f+g x)^4}\right )}{10 g}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 g (d+e x)^{3/2} (f+g x)^5}\right )}{12 g}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 g (d+e x)^{5/2} (f+g x)^6}\) |
| \(\Big \downarrow \) 1255 |
| \(\displaystyle \frac {5 c d \left (\frac {3 c d \left (\frac {c d \left (\frac {5 c d \left (\frac {3 c d \left (\frac {c d e^2 \int \frac {1}{(c d f-a e g) e^2+\frac {g \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right ) e^2}{d+e x}}d\frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {d+e x}}}{c d f-a e g}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} (f+g x) (c d f-a e g)}\right )}{4 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 \sqrt {d+e x} (f+g x)^2 (c d f-a e g)}\right )}{6 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt {d+e x} (f+g x)^3 (c d f-a e g)}\right )}{8 g}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 g \sqrt {d+e x} (f+g x)^4}\right )}{10 g}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 g (d+e x)^{3/2} (f+g x)^5}\right )}{12 g}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 g (d+e x)^{5/2} (f+g x)^6}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {5 c d \left (\frac {3 c d \left (\frac {c d \left (\frac {5 c d \left (\frac {3 c d \left (\frac {c d \arctan \left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{\sqrt {g} (c d f-a e g)^{3/2}}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} (f+g x) (c d f-a e g)}\right )}{4 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 \sqrt {d+e x} (f+g x)^2 (c d f-a e g)}\right )}{6 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt {d+e x} (f+g x)^3 (c d f-a e g)}\right )}{8 g}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 g \sqrt {d+e x} (f+g x)^4}\right )}{10 g}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 g (d+e x)^{3/2} (f+g x)^5}\right )}{12 g}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 g (d+e x)^{5/2} (f+g x)^6}\) |
-1/6*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(g*(d + e*x)^(5/2)*(f + g*x)^6) + (5*c*d*(-1/5*(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)^5) + (3*c*d*(-1/4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(g*Sqrt[d + e*x]*(f + g*x)^4) + (c*d*(Sqrt[a*d*e + (c*d^2 + a *e^2)*x + c*d*e*x^2]/(3*(c*d*f - a*e*g)*Sqrt[d + e*x]*(f + g*x)^3) + (5*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*(c*d*f - a*e*g)) ))/(8*g)))/(10*g)))/(12*g)
3.8.11.3.1 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. \(1250\) vs. \(2(413)=826\).
Time = 0.58 (sec) , antiderivative size = 1251, normalized size of antiderivative = 2.70
1/1536*((c*d*x+a*e)*(e*x+d))^(1/2)*(-1188*a*c^4*d^4*e*f^2*g^3*x^2*(c*d*x+a *e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-15*c^5*d^5*g^5*x^5*(c*d*x+a*e)^(1/2)*((a *e*g-c*d*f)*g)^(1/2)+90*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2 ))*c^6*d^6*f^5*g*x+225*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2) )*c^6*d^6*f^4*g^2*x^2+225*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1 /2))*c^6*d^6*f^2*g^4*x^4+300*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g) ^(1/2))*c^6*d^6*f^3*g^3*x^3+90*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)* g)^(1/2))*c^6*d^6*f*g^5*x^5+15*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)* g)^(1/2))*c^6*d^6*f^6-256*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*a^5*e^ 5*g^5+15*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*c^5*d^5*f^5+15*arctanh( g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^6*d^6*g^6*x^6-432*a^3*c^2*d ^2*e^3*g^5*x^2*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-640*a^4*c*d*e^4*g ^5*x*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+10*a*c^4*d^4*e*g^5*x^4*(c*d *x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+640*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a *e)^(1/2)*a^4*c*d*e^4*f*g^4-432*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)* a^3*c^2*d^2*e^3*f^2*g^3+8*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*a^2*c^ 3*d^3*e^2*f^3*g^2+10*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*a*c^4*d^4*e *f^4*g-85*c^5*d^5*f*g^4*x^4*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-198* c^5*d^5*f^2*g^3*x^3*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+198*c^5*d^5* f^3*g^2*x^2*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+85*c^5*d^5*f^4*g*...
Leaf count of result is larger than twice the leaf count of optimal. 1915 vs. \(2 (413) = 826\).
Time = 3.97 (sec) , antiderivative size = 3872, normalized size of antiderivative = 8.36 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^7} \, dx=\text {Too large to display} \]
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2)/(g*x+f)^7, x, algorithm="fricas")
[1/3072*(15*(c^6*d^6*e*g^6*x^7 + c^6*d^7*f^6 + (6*c^6*d^6*e*f*g^5 + c^6*d^ 7*g^6)*x^6 + 3*(5*c^6*d^6*e*f^2*g^4 + 2*c^6*d^7*f*g^5)*x^5 + 5*(4*c^6*d^6* e*f^3*g^3 + 3*c^6*d^7*f^2*g^4)*x^4 + 5*(3*c^6*d^6*e*f^4*g^2 + 4*c^6*d^7*f^ 3*g^3)*x^3 + 3*(2*c^6*d^6*e*f^5*g + 5*c^6*d^7*f^4*g^2)*x^2 + (c^6*d^6*e*f^ 6 + 6*c^6*d^7*f^5*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*(15*c^6*d^6*f^6*g - 5*a*c^5*d^5*e*f^5*g^2 - 2* a^2*c^4*d^4*e^2*f^4*g^3 - 440*a^3*c^3*d^3*e^3*f^3*g^4 + 1072*a^4*c^2*d^2*e ^4*f^2*g^5 - 896*a^5*c*d*e^5*f*g^6 + 256*a^6*e^6*g^7 - 15*(c^6*d^6*f*g^6 - a*c^5*d^5*e*g^7)*x^5 - 5*(17*c^6*d^6*f^2*g^5 - 19*a*c^5*d^5*e*f*g^6 + 2*a ^2*c^4*d^4*e^2*g^7)*x^4 - 2*(99*c^6*d^6*f^3*g^4 - 127*a*c^5*d^5*e*f^2*g^5 + 32*a^2*c^4*d^4*e^2*f*g^6 - 4*a^3*c^3*d^3*e^3*g^7)*x^3 + 6*(33*c^6*d^6*f^ 4*g^3 - 231*a*c^5*d^5*e*f^3*g^4 + 410*a^2*c^4*d^4*e^2*f^2*g^5 - 284*a^3*c^ 3*d^3*e^3*f*g^6 + 72*a^4*c^2*d^2*e^4*g^7)*x^2 + (85*c^6*d^6*f^5*g^2 - 29*a *c^5*d^5*e*f^4*g^3 - 1328*a^2*c^4*d^4*e^2*f^3*g^4 + 2968*a^3*c^3*d^3*e^3*f ^2*g^5 - 2336*a^4*c^2*d^2*e^4*f*g^6 + 640*a^5*c*d*e^5*g^7)*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^10*g^4 - 4*a*c^3 *d^4*e*f^9*g^5 + 6*a^2*c^2*d^3*e^2*f^8*g^6 - 4*a^3*c*d^2*e^3*f^7*g^7 + a^4 *d*e^4*f^6*g^8 + (c^4*d^4*e*f^4*g^10 - 4*a*c^3*d^3*e^2*f^3*g^11 + 6*a^2...
Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^7} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^7} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{7}} \,d x } \]
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2)/(g*x+f)^7, x, algorithm="maxima")
Leaf count of result is larger than twice the leaf count of optimal. 3412 vs. \(2 (413) = 826\).
Time = 7.62 (sec) , antiderivative size = 3412, normalized size of antiderivative = 7.37 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^7} \, dx=\text {Too large to display} \]
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2)/(g*x+f)^7, x, algorithm="giac")
5/512*c^6*d^6*abs(e)*arctan(sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*g/(sqr t(c*d*f*g - a*e*g^2)*e))/((c^3*d^3*f^3*g^3 - 3*a*c^2*d^2*e*f^2*g^4 + 3*a^2 *c*d*e^2*f*g^5 - a^3*e^3*g^6)*sqrt(c*d*f*g - a*e*g^2)*e) - 1/1536*(15*c^6* d^6*e^6*f^6*abs(e)*arctan(sqrt(-c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g - a*e*g^2 )*e)) - 90*c^6*d^7*e^5*f^5*g*abs(e)*arctan(sqrt(-c*d^2*e + a*e^3)*g/(sqrt( c*d*f*g - a*e*g^2)*e)) + 225*c^6*d^8*e^4*f^4*g^2*abs(e)*arctan(sqrt(-c*d^2 *e + a*e^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e)) - 300*c^6*d^9*e^3*f^3*g^3*abs(e )*arctan(sqrt(-c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e)) + 225*c^6*d ^10*e^2*f^2*g^4*abs(e)*arctan(sqrt(-c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g - a*e *g^2)*e)) - 90*c^6*d^11*e*f*g^5*abs(e)*arctan(sqrt(-c*d^2*e + a*e^3)*g/(sq rt(c*d*f*g - a*e*g^2)*e)) + 15*c^6*d^12*g^6*abs(e)*arctan(sqrt(-c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e)) - 15*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d *f*g - a*e*g^2)*c^5*d^5*e^5*f^5*abs(e) + 85*sqrt(-c*d^2*e + a*e^3)*sqrt(c* d*f*g - a*e*g^2)*c^5*d^6*e^4*f^4*g*abs(e) - 10*sqrt(-c*d^2*e + a*e^3)*sqrt (c*d*f*g - a*e*g^2)*a*c^4*d^4*e^6*f^4*g*abs(e) - 198*sqrt(-c*d^2*e + a*e^3 )*sqrt(c*d*f*g - a*e*g^2)*c^5*d^7*e^3*f^3*g^2*abs(e) + 56*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a*c^4*d^5*e^5*f^3*g^2*abs(e) - 8*sqrt(-c*d^ 2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a^2*c^3*d^3*e^7*f^3*g^2*abs(e) - 198* sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*c^5*d^8*e^2*f^2*g^3*abs(e) + 1188*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a*c^4*d^6*e^4*f^2...
Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^7} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{{\left (f+g\,x\right )}^7\,{\left (d+e\,x\right )}^{5/2}} \,d x \]