Integrand size = 29, antiderivative size = 160 \[ \int \frac {a+b x+c x^2}{(d+e x)^{5/2} \sqrt {f+g x}} \, dx=-\frac {2 \left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{3 (e f-d g) (d+e x)^{3/2}}+\frac {2 \left (c \left (6 d e f-4 d^2 g\right )-e (3 b e f-b d g-2 a e g)\right ) \sqrt {f+g x}}{3 e^2 (e f-d g)^2 \sqrt {d+e x}}+\frac {2 c \text {arctanh}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{e^{5/2} \sqrt {g}} \]
2*c*arctanh(g^(1/2)*(e*x+d)^(1/2)/e^(1/2)/(g*x+f)^(1/2))/e^(5/2)/g^(1/2)-2 /3*(a+d*(-b*e+c*d)/e^2)*(g*x+f)^(1/2)/(-d*g+e*f)/(e*x+d)^(3/2)+2/3*(c*(-4* d^2*g+6*d*e*f)-e*(-2*a*e*g-b*d*g+3*b*e*f))*(g*x+f)^(1/2)/e^2/(-d*g+e*f)^2/ (e*x+d)^(1/2)
Time = 0.23 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.91 \[ \int \frac {a+b x+c x^2}{(d+e x)^{5/2} \sqrt {f+g x}} \, dx=\frac {2 \sqrt {f+g x} \left (c d \left (-3 d^2 g+6 e^2 f x+d e (5 f-4 g x)\right )+e^2 (b (-2 d f-3 e f x+d g x)+a (-e f+3 d g+2 e g x))\right )}{3 e^2 (e f-d g)^2 (d+e x)^{3/2}}+\frac {2 c \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {g} \sqrt {d+e x}}\right )}{e^{5/2} \sqrt {g}} \]
(2*Sqrt[f + g*x]*(c*d*(-3*d^2*g + 6*e^2*f*x + d*e*(5*f - 4*g*x)) + e^2*(b* (-2*d*f - 3*e*f*x + d*g*x) + a*(-(e*f) + 3*d*g + 2*e*g*x))))/(3*e^2*(e*f - d*g)^2*(d + e*x)^(3/2)) + (2*c*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/(Sqrt[g]*S qrt[d + e*x])])/(e^(5/2)*Sqrt[g])
Time = 0.33 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.12, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {1193, 27, 87, 66, 221}
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.
| \(\displaystyle \int \frac {a+b x+c x^2}{(d+e x)^{5/2} \sqrt {f+g x}} \, dx\) |
| \(\Big \downarrow \) 1193 |
| \(\displaystyle -\frac {2 \int \frac {-3 c \left (f-\frac {d g}{e}\right ) x e^2-(3 b e f-b d g-2 a e g) e+c d (3 e f-d g)}{2 e^2 (d+e x)^{3/2} \sqrt {f+g x}}dx}{3 (e f-d g)}-\frac {2 \sqrt {f+g x} \left (a+\frac {d (c d-b e)}{e^2}\right )}{3 (d+e x)^{3/2} (e f-d g)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {c d (3 e f-d g)-e (3 b e f-b d g-2 a e g)-3 c e (e f-d g) x}{(d+e x)^{3/2} \sqrt {f+g x}}dx}{3 e^2 (e f-d g)}-\frac {2 \sqrt {f+g x} \left (a+\frac {d (c d-b e)}{e^2}\right )}{3 (d+e x)^{3/2} (e f-d g)}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle -\frac {-3 c (e f-d g) \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x}}dx-\frac {2 \sqrt {f+g x} (2 c d (3 e f-2 d g)-e (-2 a e g-b d g+3 b e f))}{\sqrt {d+e x} (e f-d g)}}{3 e^2 (e f-d g)}-\frac {2 \sqrt {f+g x} \left (a+\frac {d (c d-b e)}{e^2}\right )}{3 (d+e x)^{3/2} (e f-d g)}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle -\frac {-6 c (e f-d g) \int \frac {1}{e-\frac {g (d+e x)}{f+g x}}d\frac {\sqrt {d+e x}}{\sqrt {f+g x}}-\frac {2 \sqrt {f+g x} (2 c d (3 e f-2 d g)-e (-2 a e g-b d g+3 b e f))}{\sqrt {d+e x} (e f-d g)}}{3 e^2 (e f-d g)}-\frac {2 \sqrt {f+g x} \left (a+\frac {d (c d-b e)}{e^2}\right )}{3 (d+e x)^{3/2} (e f-d g)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {-\frac {2 \sqrt {f+g x} (2 c d (3 e f-2 d g)-e (-2 a e g-b d g+3 b e f))}{\sqrt {d+e x} (e f-d g)}-\frac {6 c (e f-d g) \text {arctanh}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{\sqrt {e} \sqrt {g}}}{3 e^2 (e f-d g)}-\frac {2 \sqrt {f+g x} \left (a+\frac {d (c d-b e)}{e^2}\right )}{3 (d+e x)^{3/2} (e f-d g)}\) |
(-2*(a + (d*(c*d - b*e))/e^2)*Sqrt[f + g*x])/(3*(e*f - d*g)*(d + e*x)^(3/2 )) - ((-2*(2*c*d*(3*e*f - 2*d*g) - e*(3*b*e*f - b*d*g - 2*a*e*g))*Sqrt[f + g*x])/((e*f - d*g)*Sqrt[d + e*x]) - (6*c*(e*f - d*g)*ArcTanh[(Sqrt[g]*Sqr t[d + e*x])/(Sqrt[e]*Sqrt[f + g*x])])/(Sqrt[e]*Sqrt[g]))/(3*e^2*(e*f - d*g ))
3.9.39.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && !GtQ[c - a*(d/b), 0]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x + c*x^2)^p, d + e*x, x], R = PolynomialRemainder[(a + b*x + c*x^2)^p, d + e*x, x]}, Simp[R*(d + e*x)^(m + 1)*((f + g*x)^(n + 1)/((m + 1)*(e*f - d*g)) ), x] + Simp[1/((m + 1)*(e*f - d*g)) Int[(d + e*x)^(m + 1)*(f + g*x)^n*Ex pandToSum[(m + 1)*(e*f - d*g)*Qx - g*R*(m + n + 2), x], x], x]] /; FreeQ[{a , b, c, d, e, f, g, n}, x] && IGtQ[p, 0] && ILtQ[2*m, -2] && !IntegerQ[n] && !(EqQ[m, -2] && EqQ[p, 1] && EqQ[2*c*d - b*e, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(772\) vs. \(2(136)=272\).
Time = 0.48 (sec) , antiderivative size = 773, normalized size of antiderivative = 4.83
| method | result | size |
| default | \(\frac {\sqrt {g x +f}\, \left (3 \ln \left (\frac {2 e g x +2 \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) c \,d^{2} e^{2} g^{2} x^{2}-6 \ln \left (\frac {2 e g x +2 \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) c d \,e^{3} f g \,x^{2}+3 \ln \left (\frac {2 e g x +2 \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) c \,e^{4} f^{2} x^{2}+6 \ln \left (\frac {2 e g x +2 \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) c \,d^{3} e \,g^{2} x -12 \ln \left (\frac {2 e g x +2 \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) c \,d^{2} e^{2} f g x +6 \ln \left (\frac {2 e g x +2 \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) c d \,e^{3} f^{2} x +3 \ln \left (\frac {2 e g x +2 \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) c \,d^{4} g^{2}-6 \ln \left (\frac {2 e g x +2 \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) c \,d^{3} e f g +3 \ln \left (\frac {2 e g x +2 \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) c \,d^{2} e^{2} f^{2}+4 a \,e^{3} g x \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}+2 b d \,e^{2} g x \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}-6 b \,e^{3} f x \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}-8 c \,d^{2} e g x \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}+12 c d \,e^{2} f x \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}+6 a d \,e^{2} g \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}-2 a \,e^{3} f \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}-4 b d \,e^{2} f \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}-6 c \,d^{3} g \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}+10 c \,d^{2} e f \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}\right )}{3 \sqrt {e g}\, \left (d g -e f \right )^{2} \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, e^{2} \left (e x +d \right )^{\frac {3}{2}}}\) | \(773\) |
1/3*(g*x+f)^(1/2)*(3*ln(1/2*(2*e*g*x+2*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2) +d*g+e*f)/(e*g)^(1/2))*c*d^2*e^2*g^2*x^2-6*ln(1/2*(2*e*g*x+2*((g*x+f)*(e*x +d))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*c*d*e^3*f*g*x^2+3*ln(1/2*(2*e *g*x+2*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*c*e^4*f^2 *x^2+6*ln(1/2*(2*e*g*x+2*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g )^(1/2))*c*d^3*e*g^2*x-12*ln(1/2*(2*e*g*x+2*((g*x+f)*(e*x+d))^(1/2)*(e*g)^ (1/2)+d*g+e*f)/(e*g)^(1/2))*c*d^2*e^2*f*g*x+6*ln(1/2*(2*e*g*x+2*((g*x+f)*( e*x+d))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*c*d*e^3*f^2*x+3*ln(1/2*(2* e*g*x+2*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*c*d^4*g^ 2-6*ln(1/2*(2*e*g*x+2*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^( 1/2))*c*d^3*e*f*g+3*ln(1/2*(2*e*g*x+2*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+ d*g+e*f)/(e*g)^(1/2))*c*d^2*e^2*f^2+4*a*e^3*g*x*((g*x+f)*(e*x+d))^(1/2)*(e *g)^(1/2)+2*b*d*e^2*g*x*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)-6*b*e^3*f*x*(( g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)-8*c*d^2*e*g*x*((g*x+f)*(e*x+d))^(1/2)*(e *g)^(1/2)+12*c*d*e^2*f*x*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+6*a*d*e^2*g*( (g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)-2*a*e^3*f*((g*x+f)*(e*x+d))^(1/2)*(e*g) ^(1/2)-4*b*d*e^2*f*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)-6*c*d^3*g*((g*x+f)* (e*x+d))^(1/2)*(e*g)^(1/2)+10*c*d^2*e*f*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2 ))/(e*g)^(1/2)/(d*g-e*f)^2/((g*x+f)*(e*x+d))^(1/2)/e^2/(e*x+d)^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 389 vs. \(2 (137) = 274\).
Time = 3.31 (sec) , antiderivative size = 792, normalized size of antiderivative = 4.95 \[ \int \frac {a+b x+c x^2}{(d+e x)^{5/2} \sqrt {f+g x}} \, dx=\left [\frac {3 \, {\left (c d^{2} e^{2} f^{2} - 2 \, c d^{3} e f g + c d^{4} g^{2} + {\left (c e^{4} f^{2} - 2 \, c d e^{3} f g + c d^{2} e^{2} g^{2}\right )} x^{2} + 2 \, {\left (c d e^{3} f^{2} - 2 \, c d^{2} e^{2} f g + c d^{3} e g^{2}\right )} x\right )} \sqrt {e g} \log \left (8 \, e^{2} g^{2} x^{2} + e^{2} f^{2} + 6 \, d e f g + d^{2} g^{2} + 4 \, {\left (2 \, e g x + e f + d g\right )} \sqrt {e g} \sqrt {e x + d} \sqrt {g x + f} + 8 \, {\left (e^{2} f g + d e g^{2}\right )} x\right ) + 4 \, {\left ({\left (5 \, c d^{2} e^{2} - 2 \, b d e^{3} - a e^{4}\right )} f g - 3 \, {\left (c d^{3} e - a d e^{3}\right )} g^{2} + {\left (3 \, {\left (2 \, c d e^{3} - b e^{4}\right )} f g - {\left (4 \, c d^{2} e^{2} - b d e^{3} - 2 \, a e^{4}\right )} g^{2}\right )} x\right )} \sqrt {e x + d} \sqrt {g x + f}}{6 \, {\left (d^{2} e^{5} f^{2} g - 2 \, d^{3} e^{4} f g^{2} + d^{4} e^{3} g^{3} + {\left (e^{7} f^{2} g - 2 \, d e^{6} f g^{2} + d^{2} e^{5} g^{3}\right )} x^{2} + 2 \, {\left (d e^{6} f^{2} g - 2 \, d^{2} e^{5} f g^{2} + d^{3} e^{4} g^{3}\right )} x\right )}}, -\frac {3 \, {\left (c d^{2} e^{2} f^{2} - 2 \, c d^{3} e f g + c d^{4} g^{2} + {\left (c e^{4} f^{2} - 2 \, c d e^{3} f g + c d^{2} e^{2} g^{2}\right )} x^{2} + 2 \, {\left (c d e^{3} f^{2} - 2 \, c d^{2} e^{2} f g + c d^{3} e g^{2}\right )} x\right )} \sqrt {-e g} \arctan \left (\frac {{\left (2 \, e g x + e f + d g\right )} \sqrt {-e g} \sqrt {e x + d} \sqrt {g x + f}}{2 \, {\left (e^{2} g^{2} x^{2} + d e f g + {\left (e^{2} f g + d e g^{2}\right )} x\right )}}\right ) - 2 \, {\left ({\left (5 \, c d^{2} e^{2} - 2 \, b d e^{3} - a e^{4}\right )} f g - 3 \, {\left (c d^{3} e - a d e^{3}\right )} g^{2} + {\left (3 \, {\left (2 \, c d e^{3} - b e^{4}\right )} f g - {\left (4 \, c d^{2} e^{2} - b d e^{3} - 2 \, a e^{4}\right )} g^{2}\right )} x\right )} \sqrt {e x + d} \sqrt {g x + f}}{3 \, {\left (d^{2} e^{5} f^{2} g - 2 \, d^{3} e^{4} f g^{2} + d^{4} e^{3} g^{3} + {\left (e^{7} f^{2} g - 2 \, d e^{6} f g^{2} + d^{2} e^{5} g^{3}\right )} x^{2} + 2 \, {\left (d e^{6} f^{2} g - 2 \, d^{2} e^{5} f g^{2} + d^{3} e^{4} g^{3}\right )} x\right )}}\right ] \]
[1/6*(3*(c*d^2*e^2*f^2 - 2*c*d^3*e*f*g + c*d^4*g^2 + (c*e^4*f^2 - 2*c*d*e^ 3*f*g + c*d^2*e^2*g^2)*x^2 + 2*(c*d*e^3*f^2 - 2*c*d^2*e^2*f*g + c*d^3*e*g^ 2)*x)*sqrt(e*g)*log(8*e^2*g^2*x^2 + e^2*f^2 + 6*d*e*f*g + d^2*g^2 + 4*(2*e *g*x + e*f + d*g)*sqrt(e*g)*sqrt(e*x + d)*sqrt(g*x + f) + 8*(e^2*f*g + d*e *g^2)*x) + 4*((5*c*d^2*e^2 - 2*b*d*e^3 - a*e^4)*f*g - 3*(c*d^3*e - a*d*e^3 )*g^2 + (3*(2*c*d*e^3 - b*e^4)*f*g - (4*c*d^2*e^2 - b*d*e^3 - 2*a*e^4)*g^2 )*x)*sqrt(e*x + d)*sqrt(g*x + f))/(d^2*e^5*f^2*g - 2*d^3*e^4*f*g^2 + d^4*e ^3*g^3 + (e^7*f^2*g - 2*d*e^6*f*g^2 + d^2*e^5*g^3)*x^2 + 2*(d*e^6*f^2*g - 2*d^2*e^5*f*g^2 + d^3*e^4*g^3)*x), -1/3*(3*(c*d^2*e^2*f^2 - 2*c*d^3*e*f*g + c*d^4*g^2 + (c*e^4*f^2 - 2*c*d*e^3*f*g + c*d^2*e^2*g^2)*x^2 + 2*(c*d*e^3 *f^2 - 2*c*d^2*e^2*f*g + c*d^3*e*g^2)*x)*sqrt(-e*g)*arctan(1/2*(2*e*g*x + e*f + d*g)*sqrt(-e*g)*sqrt(e*x + d)*sqrt(g*x + f)/(e^2*g^2*x^2 + d*e*f*g + (e^2*f*g + d*e*g^2)*x)) - 2*((5*c*d^2*e^2 - 2*b*d*e^3 - a*e^4)*f*g - 3*(c *d^3*e - a*d*e^3)*g^2 + (3*(2*c*d*e^3 - b*e^4)*f*g - (4*c*d^2*e^2 - b*d*e^ 3 - 2*a*e^4)*g^2)*x)*sqrt(e*x + d)*sqrt(g*x + f))/(d^2*e^5*f^2*g - 2*d^3*e ^4*f*g^2 + d^4*e^3*g^3 + (e^7*f^2*g - 2*d*e^6*f*g^2 + d^2*e^5*g^3)*x^2 + 2 *(d*e^6*f^2*g - 2*d^2*e^5*f*g^2 + d^3*e^4*g^3)*x)]
\[ \int \frac {a+b x+c x^2}{(d+e x)^{5/2} \sqrt {f+g x}} \, dx=\int \frac {a + b x + c x^{2}}{\left (d + e x\right )^{\frac {5}{2}} \sqrt {f + g x}}\, dx \]
Exception generated. \[ \int \frac {a+b x+c x^2}{(d+e x)^{5/2} \sqrt {f+g x}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Leaf count of result is larger than twice the leaf count of optimal. 491 vs. \(2 (137) = 274\).
Time = 0.38 (sec) , antiderivative size = 491, normalized size of antiderivative = 3.07 \[ \int \frac {a+b x+c x^2}{(d+e x)^{5/2} \sqrt {f+g x}} \, dx=-\frac {c \log \left ({\left (\sqrt {e g} \sqrt {e x + d} - \sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g}\right )}^{2}\right )}{\sqrt {e g} e {\left | e \right |}} + \frac {4 \, {\left (6 \, c d e^{4} f^{2} g - 3 \, b e^{5} f^{2} g - 10 \, c d^{2} e^{3} f g^{2} + 4 \, b d e^{4} f g^{2} + 2 \, a e^{5} f g^{2} + 4 \, c d^{3} e^{2} g^{3} - b d^{2} e^{3} g^{3} - 2 \, a d e^{4} g^{3} - 12 \, {\left (\sqrt {e g} \sqrt {e x + d} - \sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g}\right )}^{2} c d e^{2} f g + 6 \, {\left (\sqrt {e g} \sqrt {e x + d} - \sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g}\right )}^{2} b e^{3} f g + 6 \, {\left (\sqrt {e g} \sqrt {e x + d} - \sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g}\right )}^{2} c d^{2} e g^{2} - 6 \, {\left (\sqrt {e g} \sqrt {e x + d} - \sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g}\right )}^{2} a e^{3} g^{2} + 6 \, {\left (\sqrt {e g} \sqrt {e x + d} - \sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g}\right )}^{4} c d g - 3 \, {\left (\sqrt {e g} \sqrt {e x + d} - \sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g}\right )}^{4} b e g\right )}}{3 \, {\left (e^{2} f - d e g - {\left (\sqrt {e g} \sqrt {e x + d} - \sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g}\right )}^{2}\right )}^{3} \sqrt {e g} {\left | e \right |}} \]
-c*log((sqrt(e*g)*sqrt(e*x + d) - sqrt(e^2*f + (e*x + d)*e*g - d*e*g))^2)/ (sqrt(e*g)*e*abs(e)) + 4/3*(6*c*d*e^4*f^2*g - 3*b*e^5*f^2*g - 10*c*d^2*e^3 *f*g^2 + 4*b*d*e^4*f*g^2 + 2*a*e^5*f*g^2 + 4*c*d^3*e^2*g^3 - b*d^2*e^3*g^3 - 2*a*d*e^4*g^3 - 12*(sqrt(e*g)*sqrt(e*x + d) - sqrt(e^2*f + (e*x + d)*e* g - d*e*g))^2*c*d*e^2*f*g + 6*(sqrt(e*g)*sqrt(e*x + d) - sqrt(e^2*f + (e*x + d)*e*g - d*e*g))^2*b*e^3*f*g + 6*(sqrt(e*g)*sqrt(e*x + d) - sqrt(e^2*f + (e*x + d)*e*g - d*e*g))^2*c*d^2*e*g^2 - 6*(sqrt(e*g)*sqrt(e*x + d) - sqr t(e^2*f + (e*x + d)*e*g - d*e*g))^2*a*e^3*g^2 + 6*(sqrt(e*g)*sqrt(e*x + d) - sqrt(e^2*f + (e*x + d)*e*g - d*e*g))^4*c*d*g - 3*(sqrt(e*g)*sqrt(e*x + d) - sqrt(e^2*f + (e*x + d)*e*g - d*e*g))^4*b*e*g)/((e^2*f - d*e*g - (sqrt (e*g)*sqrt(e*x + d) - sqrt(e^2*f + (e*x + d)*e*g - d*e*g))^2)^3*sqrt(e*g)* abs(e))
Timed out. \[ \int \frac {a+b x+c x^2}{(d+e x)^{5/2} \sqrt {f+g x}} \, dx=\int \frac {c\,x^2+b\,x+a}{\sqrt {f+g\,x}\,{\left (d+e\,x\right )}^{5/2}} \,d x \]