Integrand size = 23, antiderivative size = 276 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a+b x+c x^2\right )^2} \, dx=-\frac {\sqrt {x} (A b-2 a B-(b B-2 A c) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\left (b B-2 A c-\frac {b^2 B-4 A b c+4 a B c}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {c} \left (b^2-4 a c\right ) \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (b B-2 A c+\frac {b^2 B-4 A b c+4 a B c}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {c} \left (b^2-4 a c\right ) \sqrt {b+\sqrt {b^2-4 a c}}} \]
-(A*b-2*B*a-(-2*A*c+B*b)*x)*x^(1/2)/(-4*a*c+b^2)/(c*x^2+b*x+a)+1/2*arctan( 2^(1/2)*c^(1/2)*x^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2))*(B*b-2*A*c+(4*A*b*c- 4*B*a*c-B*b^2)/(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)*2^(1/2)/c^(1/2)/(b-(-4*a*c +b^2)^(1/2))^(1/2)+1/2*arctan(2^(1/2)*c^(1/2)*x^(1/2)/(b+(-4*a*c+b^2)^(1/2 ))^(1/2))*(B*b-2*A*c+(-4*A*b*c+4*B*a*c+B*b^2)/(-4*a*c+b^2)^(1/2))/(-4*a*c+ b^2)*2^(1/2)/c^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)
Time = 1.32 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a+b x+c x^2\right )^2} \, dx=\frac {\sqrt {x} (-A b+2 a B+b B x-2 A c x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\left (-b^2 B+4 A b c-4 a B c+b B \sqrt {b^2-4 a c}-2 A c \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {c} \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (b^2 B-4 A b c+4 a B c+b B \sqrt {b^2-4 a c}-2 A c \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {c} \left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}} \]
(Sqrt[x]*(-(A*b) + 2*a*B + b*B*x - 2*A*c*x))/((b^2 - 4*a*c)*(a + b*x + c*x ^2)) + ((-(b^2*B) + 4*A*b*c - 4*a*B*c + b*B*Sqrt[b^2 - 4*a*c] - 2*A*c*Sqrt [b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c] ]])/(Sqrt[2]*Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (( b^2*B - 4*A*b*c + 4*a*B*c + b*B*Sqrt[b^2 - 4*a*c] - 2*A*c*Sqrt[b^2 - 4*a*c ])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2] *Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]])
Time = 0.48 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {1234, 27, 1197, 1480, 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.
| \(\displaystyle \int \frac {\sqrt {x} (A+B x)}{\left (a+b x+c x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1234 |
| \(\displaystyle -\frac {\int -\frac {A b-2 a B+(b B-2 A c) x}{2 \sqrt {x} \left (c x^2+b x+a\right )}dx}{b^2-4 a c}-\frac {\sqrt {x} (-2 a B-x (b B-2 A c)+A b)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {A b-2 a B+(b B-2 A c) x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{2 \left (b^2-4 a c\right )}-\frac {\sqrt {x} (-2 a B-x (b B-2 A c)+A b)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle \frac {\int \frac {A b-2 a B+(b B-2 A c) x}{c x^2+b x+a}d\sqrt {x}}{b^2-4 a c}-\frac {\sqrt {x} (-2 a B-x (b B-2 A c)+A b)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {\frac {1}{2} \left (-\frac {4 a B c-4 A b c+b^2 B}{\sqrt {b^2-4 a c}}-2 A c+b B\right ) \int \frac {1}{\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )+c x}d\sqrt {x}+\frac {1}{2} \left (\frac {4 a B c-4 A b c+b^2 B}{\sqrt {b^2-4 a c}}-2 A c+b B\right ) \int \frac {1}{\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )+c x}d\sqrt {x}}{b^2-4 a c}-\frac {\sqrt {x} (-2 a B-x (b B-2 A c)+A b)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\left (-\frac {4 a B c-4 A b c+b^2 B}{\sqrt {b^2-4 a c}}-2 A c+b B\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (\frac {4 a B c-4 A b c+b^2 B}{\sqrt {b^2-4 a c}}-2 A c+b B\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} \sqrt {c} \sqrt {\sqrt {b^2-4 a c}+b}}}{b^2-4 a c}-\frac {\sqrt {x} (-2 a B-x (b B-2 A c)+A b)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\) |
-((Sqrt[x]*(A*b - 2*a*B - (b*B - 2*A*c)*x))/((b^2 - 4*a*c)*(a + b*x + c*x^ 2))) + (((b*B - 2*A*c - (b^2*B - 4*A*b*c + 4*a*B*c)/Sqrt[b^2 - 4*a*c])*Arc Tan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[ c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + ((b*B - 2*A*c + (b^2*B - 4*A*b*c + 4*a*B *c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(b^2 - 4*a*c)
3.11.19.3.1 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_)^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[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)), x_Symbol] :> Simp[2 Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; Fr eeQ[{a, b, c, d, e, f, g}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*( (f*b - 2*a*g + (2*c*f - b*g)*x)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*Simp[g *(2*a*e*m + b*d*(2*p + 3)) - f*(b*e*m + 2*c*d*(2*p + 3)) - e*(2*c*f - b*g)* (m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1 ] && GtQ[m, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
Time = 0.37 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.07
| method | result | size |
| derivativedivides | \(\frac {\frac {2 \left (2 A c -B b \right ) x^{\frac {3}{2}}}{8 a c -2 b^{2}}+\frac {2 \left (A b -2 B a \right ) \sqrt {x}}{8 a c -2 b^{2}}}{c \,x^{2}+b x +a}+\frac {4 c \left (-\frac {\left (2 A c \sqrt {-4 a c +b^{2}}-4 A b c -B b \sqrt {-4 a c +b^{2}}+4 B a c +B \,b^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (2 A c \sqrt {-4 a c +b^{2}}+4 A b c -B b \sqrt {-4 a c +b^{2}}-4 B a c -B \,b^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{4 a c -b^{2}}\) | \(295\) |
| default | \(\frac {\frac {2 \left (2 A c -B b \right ) x^{\frac {3}{2}}}{8 a c -2 b^{2}}+\frac {2 \left (A b -2 B a \right ) \sqrt {x}}{8 a c -2 b^{2}}}{c \,x^{2}+b x +a}+\frac {4 c \left (-\frac {\left (2 A c \sqrt {-4 a c +b^{2}}-4 A b c -B b \sqrt {-4 a c +b^{2}}+4 B a c +B \,b^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (2 A c \sqrt {-4 a c +b^{2}}+4 A b c -B b \sqrt {-4 a c +b^{2}}-4 B a c -B \,b^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{4 a c -b^{2}}\) | \(295\) |
2*(1/2*(2*A*c-B*b)/(4*a*c-b^2)*x^(3/2)+1/2*(A*b-2*B*a)/(4*a*c-b^2)*x^(1/2) )/(c*x^2+b*x+a)+4/(4*a*c-b^2)*c*(-1/8*(2*A*c*(-4*a*c+b^2)^(1/2)-4*A*b*c-B* b*(-4*a*c+b^2)^(1/2)+4*B*a*c+B*b^2)/c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4* a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2) )*c)^(1/2))+1/8*(2*A*c*(-4*a*c+b^2)^(1/2)+4*A*b*c-B*b*(-4*a*c+b^2)^(1/2)-4 *B*a*c-B*b^2)/c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2 )*arctan(c*x^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 3462 vs. \(2 (235) = 470\).
Time = 2.64 (sec) , antiderivative size = 3462, normalized size of antiderivative = 12.54 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
-1/2*(sqrt(1/2)*(a*b^2 - 4*a^2*c + (b^2*c - 4*a*c^2)*x^2 + (b^3 - 4*a*b*c) *x)*sqrt(-(B^2*a*b^3 - 4*(4*A*B*a^2 - 3*A^2*a*b)*c^2 + (12*B^2*a^2*b - 12* A*B*a*b^2 + A^2*b^3)*c + (a*b^6*c - 12*a^2*b^4*c^2 + 48*a^3*b^2*c^3 - 64*a ^4*c^4)*sqrt((B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^2*b^6*c^2 - 12*a^3*b^4 *c^3 + 48*a^4*b^2*c^4 - 64*a^5*c^5)))/(a*b^6*c - 12*a^2*b^4*c^2 + 48*a^3*b ^2*c^3 - 64*a^4*c^4))*log(sqrt(1/2)*(2*B^3*a^2*b^4 - A*B^2*a*b^5 - 16*(2*A ^2*B*a^3 - A^3*a^2*b)*c^3 + 8*(4*B^3*a^4 - 2*A*B^2*a^3*b + 2*A^2*B*a^2*b^2 - A^3*a*b^3)*c^2 - (16*B^3*a^3*b^2 - 8*A*B^2*a^2*b^3 + 2*A^2*B*a*b^4 - A^ 3*b^5)*c + (192*B*a^4*b^3*c^3 + 256*A*a^5*c^5 - 128*(2*B*a^5*b + A*a^4*b^2 )*c^4 - 8*(6*B*a^3*b^5 - A*a^2*b^6)*c^2 + (4*B*a^2*b^7 - A*a*b^8)*c)*sqrt( (B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^2*b^6*c^2 - 12*a^3*b^4*c^3 + 48*a^4 *b^2*c^4 - 64*a^5*c^5)))*sqrt(-(B^2*a*b^3 - 4*(4*A*B*a^2 - 3*A^2*a*b)*c^2 + (12*B^2*a^2*b - 12*A*B*a*b^2 + A^2*b^3)*c + (a*b^6*c - 12*a^2*b^4*c^2 + 48*a^3*b^2*c^3 - 64*a^4*c^4)*sqrt((B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^2 *b^6*c^2 - 12*a^3*b^4*c^3 + 48*a^4*b^2*c^4 - 64*a^5*c^5)))/(a*b^6*c - 12*a ^2*b^4*c^2 + 48*a^3*b^2*c^3 - 64*a^4*c^4)) - 2*(3*B^4*a^2*b^2 - A*B^3*a*b^ 3 - 4*A^4*a*c^3 + 3*(4*A^3*B*a*b - A^4*b^2)*c^2 + (4*B^4*a^3 - 12*A*B^3*a^ 2*b + A^3*B*b^3)*c)*sqrt(x)) - sqrt(1/2)*(a*b^2 - 4*a^2*c + (b^2*c - 4*a*c ^2)*x^2 + (b^3 - 4*a*b*c)*x)*sqrt(-(B^2*a*b^3 - 4*(4*A*B*a^2 - 3*A^2*a*b)* c^2 + (12*B^2*a^2*b - 12*A*B*a*b^2 + A^2*b^3)*c + (a*b^6*c - 12*a^2*b^4...
Timed out. \[ \int \frac {\sqrt {x} (A+B x)}{\left (a+b x+c x^2\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {x} (A+B x)}{\left (a+b x+c x^2\right )^2} \, dx=\int { \frac {{\left (B x + A\right )} \sqrt {x}}{{\left (c x^{2} + b x + a\right )}^{2}} \,d x } \]
-((2*B*a*c - A*b*c)*x^(5/2) + (B*a*b - (b^2 - 2*a*c)*A)*x^(3/2))/(a^2*b^2 - 4*a^3*c + (a*b^2*c - 4*a^2*c^2)*x^2 + (a*b^3 - 4*a^2*b*c)*x) + integrate (1/2*((2*B*a*c - A*b*c)*x^(3/2) + (3*B*a*b - (b^2 + 2*a*c)*A)*sqrt(x))/(a^ 2*b^2 - 4*a^3*c + (a*b^2*c - 4*a^2*c^2)*x^2 + (a*b^3 - 4*a^2*b*c)*x), x)
Leaf count of result is larger than twice the leaf count of optimal. 3781 vs. \(2 (235) = 470\).
Time = 1.06 (sec) , antiderivative size = 3781, normalized size of antiderivative = 13.70 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
(B*b*x^(3/2) - 2*A*c*x^(3/2) + 2*B*a*sqrt(x) - A*b*sqrt(x))/((c*x^2 + b*x + a)*(b^2 - 4*a*c)) + 1/8*(2*(2*b^2*c^3 - 8*a*c^4 - sqrt(2)*sqrt(b^2 - 4*a *c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sq rt(b*c + sqrt(b^2 - 4*a*c)*c)*a*c^2 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt( b^2 - 4*a*c)*c)*c^3 - 2*(b^2 - 4*a*c)*c^3)*(b^2 - 4*a*c)^2*A - (2*b^3*c^2 - 8*a*b*c^3 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^ 3 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c + 2* sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c - sqrt(2)* sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b*c^2 - 2*(b^2 - 4*a*c)* b*c^2)*(b^2 - 4*a*c)^2*B + 2*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^5* c - 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^2 - 2*sqrt(2)*sqrt(b *c + sqrt(b^2 - 4*a*c)*c)*b^4*c^2 - 2*b^5*c^2 + 16*sqrt(2)*sqrt(b*c + sqrt (b^2 - 4*a*c)*c)*a^2*b*c^3 + 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b ^2*c^3 + sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c^3 + 16*a*b^3*c^3 - 4*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^4 - 32*a^2*b*c^4 + 2*(b^2 - 4*a*c)*b^3*c^2 - 8*(b^2 - 4*a*c)*a*b*c^3)*A*abs(b^2 - 4*a*c) - 4*(sqrt(2 )*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4*c - 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^2 - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3* c^2 - 2*a*b^4*c^2 + 16*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*c^3 ...
Time = 14.20 (sec) , antiderivative size = 9434, normalized size of antiderivative = 34.18 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
((x^(1/2)*(A*b - 2*B*a))/(4*a*c - b^2) + (x^(3/2)*(2*A*c - B*b))/(4*a*c - b^2))/(a + b*x + c*x^2) - atan(((((4*A*b^7*c^2 + 512*B*a^4*c^5 - 48*A*a*b^ 5*c^3 - 256*A*a^3*b*c^5 - 8*B*a*b^6*c^2 + 192*A*a^2*b^3*c^4 + 96*B*a^2*b^4 *c^3 - 384*B*a^3*b^2*c^4)/(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c) - (2*x^(1/2)*(-(B^2*a*b^9 - B^2*a*(-(4*a*c - b^2)^9)^(1/2) + A^2*b^9*c + A^2*c*(-(4*a*c - b^2)^9)^(1/2) - 96*A^2*a^2*b^5*c^3 + 512*A^2*a^3*b^3*c^4 - 96*B^2*a^3*b^5*c^2 + 512*B^2*a^4*b^3*c^3 + 1024*A*B*a^5*c^5 - 768*A^2*a^ 4*b*c^5 - 768*B^2*a^5*b*c^4 + 128*A*B*a^2*b^6*c^2 - 384*A*B*a^3*b^4*c^3 - 12*A*B*a*b^8*c)/(8*(4096*a^7*c^7 - 24*a^2*b^10*c^2 + 240*a^3*b^8*c^3 - 128 0*a^4*b^6*c^4 + 3840*a^5*b^4*c^5 - 6144*a^6*b^2*c^6 + a*b^12*c)))^(1/2)*(4 *b^7*c^2 - 48*a*b^5*c^3 - 256*a^3*b*c^5 + 192*a^2*b^3*c^4))/(b^4 + 16*a^2* c^2 - 8*a*b^2*c))*(-(B^2*a*b^9 - B^2*a*(-(4*a*c - b^2)^9)^(1/2) + A^2*b^9* c + A^2*c*(-(4*a*c - b^2)^9)^(1/2) - 96*A^2*a^2*b^5*c^3 + 512*A^2*a^3*b^3* c^4 - 96*B^2*a^3*b^5*c^2 + 512*B^2*a^4*b^3*c^3 + 1024*A*B*a^5*c^5 - 768*A^ 2*a^4*b*c^5 - 768*B^2*a^5*b*c^4 + 128*A*B*a^2*b^6*c^2 - 384*A*B*a^3*b^4*c^ 3 - 12*A*B*a*b^8*c)/(8*(4096*a^7*c^7 - 24*a^2*b^10*c^2 + 240*a^3*b^8*c^3 - 1280*a^4*b^6*c^4 + 3840*a^5*b^4*c^5 - 6144*a^6*b^2*c^6 + a*b^12*c)))^(1/2 ) - (2*x^(1/2)*(B^2*b^4*c - 8*A^2*a*c^4 + 10*A^2*b^2*c^3 + 8*B^2*a^2*c^3 - 6*A*B*b^3*c^2 + 2*B^2*a*b^2*c^2 - 8*A*B*a*b*c^3))/(b^4 + 16*a^2*c^2 - 8*a *b^2*c))*(-(B^2*a*b^9 - B^2*a*(-(4*a*c - b^2)^9)^(1/2) + A^2*b^9*c + A^...