Integrand size = 29, antiderivative size = 246 \[ \int \frac {\sqrt {d+e x} \left (a+b x+c x^2\right )}{\sqrt {f+g x}} \, dx=\frac {\left (c \left (5 e^2 f^2+2 d e f g+d^2 g^2\right )+2 e g (4 a e g-b (3 e f+d g))\right ) \sqrt {d+e x} \sqrt {f+g x}}{8 e^2 g^3}-\frac {(5 c e f+7 c d g-6 b e g) (d+e x)^{3/2} \sqrt {f+g x}}{12 e^2 g^2}+\frac {c (d+e x)^{5/2} \sqrt {f+g x}}{3 e^2 g}-\frac {(e f-d g) \left (c \left (5 e^2 f^2+2 d e f g+d^2 g^2\right )+2 e g (4 a e g-b (3 e f+d g))\right ) \text {arctanh}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{8 e^{5/2} g^{7/2}} \]
-1/8*(-d*g+e*f)*(c*(d^2*g^2+2*d*e*f*g+5*e^2*f^2)+2*e*g*(4*a*e*g-b*(d*g+3*e *f)))*arctanh(g^(1/2)*(e*x+d)^(1/2)/e^(1/2)/(g*x+f)^(1/2))/e^(5/2)/g^(7/2) -1/12*(-6*b*e*g+7*c*d*g+5*c*e*f)*(e*x+d)^(3/2)*(g*x+f)^(1/2)/e^2/g^2+1/3*c *(e*x+d)^(5/2)*(g*x+f)^(1/2)/e^2/g+1/8*(c*(d^2*g^2+2*d*e*f*g+5*e^2*f^2)+2* e*g*(4*a*e*g-b*(d*g+3*e*f)))*(e*x+d)^(1/2)*(g*x+f)^(1/2)/e^2/g^3
Time = 3.55 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {d+e x} \left (a+b x+c x^2\right )}{\sqrt {f+g x}} \, dx=\frac {\sqrt {d+e x} \sqrt {f+g x} \left (6 e g (4 a e g+b (-3 e f+d g+2 e g x))+c \left (-3 d^2 g^2+2 d e g (-2 f+g x)+e^2 \left (15 f^2-10 f g x+8 g^2 x^2\right )\right )\right )}{24 e^2 g^3}+\frac {(e f-d g) \left (c \left (5 e^2 f^2+2 d e f g+d^2 g^2\right )+2 e g (4 a e g-b (3 e f+d g))\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {g} \left (\sqrt {d-\frac {e f}{g}}-\sqrt {d+e x}\right )}\right )}{4 e^{5/2} g^{7/2}} \]
(Sqrt[d + e*x]*Sqrt[f + g*x]*(6*e*g*(4*a*e*g + b*(-3*e*f + d*g + 2*e*g*x)) + c*(-3*d^2*g^2 + 2*d*e*g*(-2*f + g*x) + e^2*(15*f^2 - 10*f*g*x + 8*g^2*x ^2))))/(24*e^2*g^3) + ((e*f - d*g)*(c*(5*e^2*f^2 + 2*d*e*f*g + d^2*g^2) + 2*e*g*(4*a*e*g - b*(3*e*f + d*g)))*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/(Sqrt[g ]*(Sqrt[d - (e*f)/g] - Sqrt[d + e*x]))])/(4*e^(5/2)*g^(7/2))
Time = 0.33 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.85, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {1194, 27, 90, 60, 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.
| \(\displaystyle \int \frac {\sqrt {d+e x} \left (a+b x+c x^2\right )}{\sqrt {f+g x}} \, dx\) |
| \(\Big \downarrow \) 1194 |
| \(\displaystyle \frac {\int \frac {\sqrt {d+e x} \left (6 a g e^2-(5 c e f+7 c d g-6 b e g) x e-c d (5 e f+d g)\right )}{2 \sqrt {f+g x}}dx}{3 e^2 g}+\frac {c (d+e x)^{5/2} \sqrt {f+g x}}{3 e^2 g}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {d+e x} \left (6 a g e^2-(5 c e f+7 c d g-6 b e g) x e-c d (5 e f+d g)\right )}{\sqrt {f+g x}}dx}{6 e^2 g}+\frac {c (d+e x)^{5/2} \sqrt {f+g x}}{3 e^2 g}\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {\frac {3 \left (2 e g (4 a e g-b (d g+3 e f))+c \left (d^2 g^2+2 d e f g+5 e^2 f^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {f+g x}}dx}{4 g}-\frac {(d+e x)^{3/2} \sqrt {f+g x} (-6 b e g+7 c d g+5 c e f)}{2 g}}{6 e^2 g}+\frac {c (d+e x)^{5/2} \sqrt {f+g x}}{3 e^2 g}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\frac {3 \left (2 e g (4 a e g-b (d g+3 e f))+c \left (d^2 g^2+2 d e f g+5 e^2 f^2\right )\right ) \left (\frac {\sqrt {d+e x} \sqrt {f+g x}}{g}-\frac {(e f-d g) \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x}}dx}{2 g}\right )}{4 g}-\frac {(d+e x)^{3/2} \sqrt {f+g x} (-6 b e g+7 c d g+5 c e f)}{2 g}}{6 e^2 g}+\frac {c (d+e x)^{5/2} \sqrt {f+g x}}{3 e^2 g}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {\frac {3 \left (2 e g (4 a e g-b (d g+3 e f))+c \left (d^2 g^2+2 d e f g+5 e^2 f^2\right )\right ) \left (\frac {\sqrt {d+e x} \sqrt {f+g x}}{g}-\frac {(e f-d g) \int \frac {1}{e-\frac {g (d+e x)}{f+g x}}d\frac {\sqrt {d+e x}}{\sqrt {f+g x}}}{g}\right )}{4 g}-\frac {(d+e x)^{3/2} \sqrt {f+g x} (-6 b e g+7 c d g+5 c e f)}{2 g}}{6 e^2 g}+\frac {c (d+e x)^{5/2} \sqrt {f+g x}}{3 e^2 g}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {3 \left (\frac {\sqrt {d+e x} \sqrt {f+g x}}{g}-\frac {(e f-d g) \text {arctanh}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{\sqrt {e} g^{3/2}}\right ) \left (2 e g (4 a e g-b (d g+3 e f))+c \left (d^2 g^2+2 d e f g+5 e^2 f^2\right )\right )}{4 g}-\frac {(d+e x)^{3/2} \sqrt {f+g x} (-6 b e g+7 c d g+5 c e f)}{2 g}}{6 e^2 g}+\frac {c (d+e x)^{5/2} \sqrt {f+g x}}{3 e^2 g}\) |
(c*(d + e*x)^(5/2)*Sqrt[f + g*x])/(3*e^2*g) + (-1/2*((5*c*e*f + 7*c*d*g - 6*b*e*g)*(d + e*x)^(3/2)*Sqrt[f + g*x])/g + (3*(c*(5*e^2*f^2 + 2*d*e*f*g + d^2*g^2) + 2*e*g*(4*a*e*g - b*(3*e*f + d*g)))*((Sqrt[d + e*x]*Sqrt[f + g* x])/g - ((e*f - d*g)*ArcTanh[(Sqrt[g]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[f + g*x ])])/(Sqrt[e]*g^(3/2))))/(4*g))/(6*e^2*g)
3.9.36.3.1 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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, 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*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
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] :> Simp[c^p*(d + e*x)^(m + 2*p)*((f + g*x )^(n + 1)/(g*e^(2*p)*(m + n + 2*p + 1))), x] + Simp[1/(g*e^(2*p)*(m + n + 2 *p + 1)) Int[(d + e*x)^m*(f + g*x)^n*ExpandToSum[g*(m + n + 2*p + 1)*(e^( 2*p)*(a + b*x + c*x^2)^p - c^p*(d + e*x)^(2*p)) - c^p*(e*f - d*g)*(m + 2*p) *(d + e*x)^(2*p - 1), x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ [p, 0] && !IntegerQ[m] && !IntegerQ[n] && NeQ[m + n + 2*p + 1, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(762\) vs. \(2(214)=428\).
Time = 0.49 (sec) , antiderivative size = 763, normalized size of antiderivative = 3.10
| method | result | size |
| default | \(\frac {\sqrt {e x +d}\, \sqrt {g x +f}\, \left (16 c \,e^{2} g^{2} x^{2} \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}+24 \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 ) a d \,e^{2} g^{3}-24 \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 ) a \,e^{3} f \,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 ) b \,d^{2} e \,g^{3}-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 ) b d \,e^{2} f \,g^{2}+18 \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 ) b \,e^{3} f^{2} 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^{3} g^{3}+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 f \,g^{2}+9 \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^{2} f^{2} g -15 \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^{3} f^{3}+24 \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}\, b \,e^{2} g^{2} x +4 \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}\, c d e \,g^{2} x -20 \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}\, c \,e^{2} f g x +48 \sqrt {e g}\, \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, a \,e^{2} g^{2}+12 \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}\, b d e \,g^{2}-36 \sqrt {e g}\, \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, b \,e^{2} f g -6 \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}\, c \,d^{2} g^{2}-8 \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}\, c d e f g +30 \sqrt {e g}\, \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, c \,e^{2} f^{2}\right )}{48 g^{3} \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, e^{2} \sqrt {e g}}\) | \(763\) |
1/48*(e*x+d)^(1/2)*(g*x+f)^(1/2)*(16*c*e^2*g^2*x^2*((g*x+f)*(e*x+d))^(1/2) *(e*g)^(1/2)+24*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))*a*d*e^2*g^3-24*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))*a*e^3*f*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))*b*d^2*e*g^3-12*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))*b*d*e^ 2*f*g^2+18*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))*b*e^3*f^2*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^3*g^3+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*f*g^2+9*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^2*f^2*g -15*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^3*f^3+24*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)*b*e^2*g^2*x+4*((g*x +f)*(e*x+d))^(1/2)*(e*g)^(1/2)*c*d*e*g^2*x-20*((g*x+f)*(e*x+d))^(1/2)*(e*g )^(1/2)*c*e^2*f*g*x+48*(e*g)^(1/2)*((g*x+f)*(e*x+d))^(1/2)*a*e^2*g^2+12*(( g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)*b*d*e*g^2-36*(e*g)^(1/2)*((g*x+f)*(e*x+d ))^(1/2)*b*e^2*f*g-6*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)*c*d^2*g^2-8*((g*x +f)*(e*x+d))^(1/2)*(e*g)^(1/2)*c*d*e*f*g+30*(e*g)^(1/2)*((g*x+f)*(e*x+d))^ (1/2)*c*e^2*f^2)/g^3/((g*x+f)*(e*x+d))^(1/2)/e^2/(e*g)^(1/2)
Time = 0.54 (sec) , antiderivative size = 576, normalized size of antiderivative = 2.34 \[ \int \frac {\sqrt {d+e x} \left (a+b x+c x^2\right )}{\sqrt {f+g x}} \, dx=\left [-\frac {3 \, {\left (5 \, c e^{3} f^{3} - 3 \, {\left (c d e^{2} + 2 \, b e^{3}\right )} f^{2} g - {\left (c d^{2} e - 4 \, b d e^{2} - 8 \, a e^{3}\right )} f g^{2} - {\left (c d^{3} - 2 \, b d^{2} e + 8 \, a d e^{2}\right )} g^{3}\right )} \sqrt {e g} \log \left (8 \, e^{2} g^{2} x^{2} + e^{2} f^{2} + 6 \, d e f g + d^{2} g^{2} + 4 \, {\left (2 \, e g x + e f + d g\right )} \sqrt {e g} \sqrt {e x + d} \sqrt {g x + f} + 8 \, {\left (e^{2} f g + d e g^{2}\right )} x\right ) - 4 \, {\left (8 \, c e^{3} g^{3} x^{2} + 15 \, c e^{3} f^{2} g - 2 \, {\left (2 \, c d e^{2} + 9 \, b e^{3}\right )} f g^{2} - 3 \, {\left (c d^{2} e - 2 \, b d e^{2} - 8 \, a e^{3}\right )} g^{3} - 2 \, {\left (5 \, c e^{3} f g^{2} - {\left (c d e^{2} + 6 \, b e^{3}\right )} g^{3}\right )} x\right )} \sqrt {e x + d} \sqrt {g x + f}}{96 \, e^{3} g^{4}}, \frac {3 \, {\left (5 \, c e^{3} f^{3} - 3 \, {\left (c d e^{2} + 2 \, b e^{3}\right )} f^{2} g - {\left (c d^{2} e - 4 \, b d e^{2} - 8 \, a e^{3}\right )} f g^{2} - {\left (c d^{3} - 2 \, b d^{2} e + 8 \, a d e^{2}\right )} g^{3}\right )} \sqrt {-e g} \arctan \left (\frac {{\left (2 \, e g x + e f + d g\right )} \sqrt {-e g} \sqrt {e x + d} \sqrt {g x + f}}{2 \, {\left (e^{2} g^{2} x^{2} + d e f g + {\left (e^{2} f g + d e g^{2}\right )} x\right )}}\right ) + 2 \, {\left (8 \, c e^{3} g^{3} x^{2} + 15 \, c e^{3} f^{2} g - 2 \, {\left (2 \, c d e^{2} + 9 \, b e^{3}\right )} f g^{2} - 3 \, {\left (c d^{2} e - 2 \, b d e^{2} - 8 \, a e^{3}\right )} g^{3} - 2 \, {\left (5 \, c e^{3} f g^{2} - {\left (c d e^{2} + 6 \, b e^{3}\right )} g^{3}\right )} x\right )} \sqrt {e x + d} \sqrt {g x + f}}{48 \, e^{3} g^{4}}\right ] \]
[-1/96*(3*(5*c*e^3*f^3 - 3*(c*d*e^2 + 2*b*e^3)*f^2*g - (c*d^2*e - 4*b*d*e^ 2 - 8*a*e^3)*f*g^2 - (c*d^3 - 2*b*d^2*e + 8*a*d*e^2)*g^3)*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*(8*c*e^3* g^3*x^2 + 15*c*e^3*f^2*g - 2*(2*c*d*e^2 + 9*b*e^3)*f*g^2 - 3*(c*d^2*e - 2* b*d*e^2 - 8*a*e^3)*g^3 - 2*(5*c*e^3*f*g^2 - (c*d*e^2 + 6*b*e^3)*g^3)*x)*sq rt(e*x + d)*sqrt(g*x + f))/(e^3*g^4), 1/48*(3*(5*c*e^3*f^3 - 3*(c*d*e^2 + 2*b*e^3)*f^2*g - (c*d^2*e - 4*b*d*e^2 - 8*a*e^3)*f*g^2 - (c*d^3 - 2*b*d^2* e + 8*a*d*e^2)*g^3)*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*(8*c*e^3*g^3*x^2 + 15*c*e^3*f^2*g - 2*(2*c*d*e^2 + 9*b*e^3)*f*g^2 - 3*(c*d^2*e - 2*b*d*e^2 - 8*a*e^3)*g^3 - 2*(5*c*e^3*f*g^2 - (c*d*e^2 + 6* b*e^3)*g^3)*x)*sqrt(e*x + d)*sqrt(g*x + f))/(e^3*g^4)]
\[ \int \frac {\sqrt {d+e x} \left (a+b x+c x^2\right )}{\sqrt {f+g x}} \, dx=\int \frac {\sqrt {d + e x} \left (a + b x + c x^{2}\right )}{\sqrt {f + g x}}\, dx \]
Exception generated. \[ \int \frac {\sqrt {d+e x} \left (a+b x+c x^2\right )}{\sqrt {f+g x}} \, dx=\text {Exception raised: ValueError} \]
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
Time = 0.30 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.27 \[ \int \frac {\sqrt {d+e x} \left (a+b x+c x^2\right )}{\sqrt {f+g x}} \, dx=\frac {{\left (\sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g} \sqrt {e x + d} {\left (2 \, {\left (e x + d\right )} {\left (\frac {4 \, {\left (e x + d\right )} c}{e^{3} g} - \frac {5 \, c e^{7} f g^{3} + 7 \, c d e^{6} g^{4} - 6 \, b e^{7} g^{4}}{e^{9} g^{5}}\right )} + \frac {3 \, {\left (5 \, c e^{8} f^{2} g^{2} + 2 \, c d e^{7} f g^{3} - 6 \, b e^{8} f g^{3} + c d^{2} e^{6} g^{4} - 2 \, b d e^{7} g^{4} + 8 \, a e^{8} g^{4}\right )}}{e^{9} g^{5}}\right )} + \frac {3 \, {\left (5 \, c e^{3} f^{3} - 3 \, c d e^{2} f^{2} g - 6 \, b e^{3} f^{2} g - c d^{2} e f g^{2} + 4 \, b d e^{2} f g^{2} + 8 \, a e^{3} f g^{2} - c d^{3} g^{3} + 2 \, b d^{2} e g^{3} - 8 \, a d e^{2} g^{3}\right )} \log \left ({\left | -\sqrt {e g} \sqrt {e x + d} + \sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g} \right |}\right )}{\sqrt {e g} e^{2} g^{3}}\right )} e}{24 \, {\left | e \right |}} \]
1/24*(sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(e*x + d)*(2*(e*x + d)*(4*(e *x + d)*c/(e^3*g) - (5*c*e^7*f*g^3 + 7*c*d*e^6*g^4 - 6*b*e^7*g^4)/(e^9*g^5 )) + 3*(5*c*e^8*f^2*g^2 + 2*c*d*e^7*f*g^3 - 6*b*e^8*f*g^3 + c*d^2*e^6*g^4 - 2*b*d*e^7*g^4 + 8*a*e^8*g^4)/(e^9*g^5)) + 3*(5*c*e^3*f^3 - 3*c*d*e^2*f^2 *g - 6*b*e^3*f^2*g - c*d^2*e*f*g^2 + 4*b*d*e^2*f*g^2 + 8*a*e^3*f*g^2 - c*d ^3*g^3 + 2*b*d^2*e*g^3 - 8*a*d*e^2*g^3)*log(abs(-sqrt(e*g)*sqrt(e*x + d) + sqrt(e^2*f + (e*x + d)*e*g - d*e*g)))/(sqrt(e*g)*e^2*g^3))*e/abs(e)
Time = 109.82 (sec) , antiderivative size = 1832, normalized size of antiderivative = 7.45 \[ \int \frac {\sqrt {d+e x} \left (a+b x+c x^2\right )}{\sqrt {f+g x}} \, dx=\text {Too large to display} \]
(((2*a*d*g + 2*a*e*f)*((d + e*x)^(1/2) - d^(1/2))^3)/(g^2*((f + g*x)^(1/2) - f^(1/2))^3) + ((2*a*e^2*f + 2*a*d*e*g)*((d + e*x)^(1/2) - d^(1/2)))/(g^ 3*((f + g*x)^(1/2) - f^(1/2))) - (8*a*d^(1/2)*e*f^(1/2)*((d + e*x)^(1/2) - d^(1/2))^2)/(g^2*((f + g*x)^(1/2) - f^(1/2))^2))/(((d + e*x)^(1/2) - d^(1 /2))^4/((f + g*x)^(1/2) - f^(1/2))^4 + e^2/g^2 - (2*e*((d + e*x)^(1/2) - d ^(1/2))^2)/(g*((f + g*x)^(1/2) - f^(1/2))^2)) - ((((d + e*x)^(1/2) - d^(1/ 2))*((c*d^3*e^3*g^3)/4 - (5*c*e^6*f^3)/4 + (3*c*d*e^5*f^2*g)/4 + (c*d^2*e^ 4*f*g^2)/4))/(g^9*((f + g*x)^(1/2) - f^(1/2))) - (((d + e*x)^(1/2) - d^(1/ 2))^5*((33*c*e^4*f^3)/2 + (19*c*d^3*e*g^3)/2 + (313*c*d*e^3*f^2*g)/2 + (27 5*c*d^2*e^2*f*g^2)/2))/(g^7*((f + g*x)^(1/2) - f^(1/2))^5) - (((d + e*x)^( 1/2) - d^(1/2))^7*((19*c*d^3*g^3)/2 + (33*c*e^3*f^3)/2 + (313*c*d*e^2*f^2* g)/2 + (275*c*d^2*e*f*g^2)/2))/(g^6*((f + g*x)^(1/2) - f^(1/2))^7) - (((d + e*x)^(1/2) - d^(1/2))^3*((17*c*d^3*e^2*g^3)/12 - (85*c*e^5*f^3)/12 + (17 *c*d*e^4*f^2*g)/4 + (91*c*d^2*e^3*f*g^2)/4))/(g^8*((f + g*x)^(1/2) - f^(1/ 2))^3) + (((d + e*x)^(1/2) - d^(1/2))^11*((c*d^3*g^3)/4 - (5*c*e^3*f^3)/4 + (3*c*d*e^2*f^2*g)/4 + (c*d^2*e*f*g^2)/4))/(e^2*g^4*((f + g*x)^(1/2) - f^ (1/2))^11) - (((d + e*x)^(1/2) - d^(1/2))^9*((17*c*d^3*g^3)/12 - (85*c*e^3 *f^3)/12 + (17*c*d*e^2*f^2*g)/4 + (91*c*d^2*e*f*g^2)/4))/(e*g^5*((f + g*x) ^(1/2) - f^(1/2))^9) + (d^(1/2)*f^(1/2)*((d + e*x)^(1/2) - d^(1/2))^6*(128 *c*e^3*f^2 + 64*c*d^2*e*g^2 + (704*c*d*e^2*f*g)/3))/(g^6*((f + g*x)^(1/...