Integrand size = 23, antiderivative size = 275 \[ \int \frac {x^{3/2} (A+B x)}{a+b x+c x^2} \, dx=-\frac {2 (b B-A c) \sqrt {x}}{c^2}+\frac {2 B x^{3/2}}{3 c}+\frac {\sqrt {2} \left (b^2 B-A b c-a B c-\frac {b^3 B-A b^2 c-3 a b B c+2 a A c^2}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{c^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \left (b^2 B-A b c-a B c+\frac {b^3 B-A b^2 c-3 a b B c+2 a A c^2}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{c^{5/2} \sqrt {b+\sqrt {b^2-4 a c}}} \]
2/3*B*x^(3/2)/c-2*(-A*c+B*b)*x^(1/2)/c^2+arctan(2^(1/2)*c^(1/2)*x^(1/2)/(b -(-4*a*c+b^2)^(1/2))^(1/2))*2^(1/2)*(B*b^2-A*b*c-B*a*c+(-2*A*a*c^2+A*b^2*c +3*B*a*b*c-B*b^3)/(-4*a*c+b^2)^(1/2))/c^(5/2)/(b-(-4*a*c+b^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))*2^(1/2)*(B*b ^2-A*b*c-B*a*c+(2*A*a*c^2-A*b^2*c-3*B*a*b*c+B*b^3)/(-4*a*c+b^2)^(1/2))/c^( 5/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)
Time = 0.86 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.21 \[ \int \frac {x^{3/2} (A+B x)}{a+b x+c x^2} \, dx=\frac {2 \sqrt {c} \sqrt {x} (-3 b B+3 A c+B c x)+\frac {3 \sqrt {2} \left (-b^3 B+b c \left (3 a B-A \sqrt {b^2-4 a c}\right )+b^2 \left (A c+B \sqrt {b^2-4 a c}\right )-a c \left (2 A c+B \sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {3 \sqrt {2} \left (b^3 B-b c \left (3 a B+A \sqrt {b^2-4 a c}\right )+a c \left (2 A c-B \sqrt {b^2-4 a c}\right )+b^2 \left (-A c+B \sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}}}{3 c^{5/2}} \]
(2*Sqrt[c]*Sqrt[x]*(-3*b*B + 3*A*c + B*c*x) + (3*Sqrt[2]*(-(b^3*B) + b*c*( 3*a*B - A*Sqrt[b^2 - 4*a*c]) + b^2*(A*c + B*Sqrt[b^2 - 4*a*c]) - a*c*(2*A* c + B*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b ^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (3*Sqrt[2 ]*(b^3*B - b*c*(3*a*B + A*Sqrt[b^2 - 4*a*c]) + a*c*(2*A*c - B*Sqrt[b^2 - 4 *a*c]) + b^2*(-(A*c) + B*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[ x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4 *a*c]]))/(3*c^(5/2))
Time = 0.56 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {1196, 25, 1196, 25, 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.
| \(\displaystyle \int \frac {x^{3/2} (A+B x)}{a+b x+c x^2} \, dx\) |
| \(\Big \downarrow \) 1196 |
| \(\displaystyle \frac {\int -\frac {\sqrt {x} (a B+(b B-A c) x)}{c x^2+b x+a}dx}{c}+\frac {2 B x^{3/2}}{3 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 B x^{3/2}}{3 c}-\frac {\int \frac {\sqrt {x} (a B+(b B-A c) x)}{c x^2+b x+a}dx}{c}\) |
| \(\Big \downarrow \) 1196 |
| \(\displaystyle \frac {2 B x^{3/2}}{3 c}-\frac {\frac {\int -\frac {a (b B-A c)+\left (B b^2-A c b-a B c\right ) x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{c}+\frac {2 \sqrt {x} (b B-A c)}{c}}{c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 B x^{3/2}}{3 c}-\frac {\frac {2 \sqrt {x} (b B-A c)}{c}-\frac {\int \frac {a (b B-A c)+\left (B b^2-A c b-a B c\right ) x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{c}}{c}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle \frac {2 B x^{3/2}}{3 c}-\frac {\frac {2 \sqrt {x} (b B-A c)}{c}-\frac {2 \int \frac {a (b B-A c)+\left (B b^2-A c b-a B c\right ) x}{c x^2+b x+a}d\sqrt {x}}{c}}{c}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {2 B x^{3/2}}{3 c}-\frac {\frac {2 \sqrt {x} (b B-A c)}{c}-\frac {2 \left (\frac {1}{2} \left (-\frac {2 a A c^2-3 a b B c-A b^2 c+b^3 B}{\sqrt {b^2-4 a c}}-a B c-A b c+b^2 B\right ) \int \frac {1}{\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )+c x}d\sqrt {x}+\frac {1}{2} \left (\frac {2 a A c^2-3 a b B c-A b^2 c+b^3 B}{\sqrt {b^2-4 a c}}-a B c-A b c+b^2 B\right ) \int \frac {1}{\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )+c x}d\sqrt {x}\right )}{c}}{c}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {2 B x^{3/2}}{3 c}-\frac {\frac {2 \sqrt {x} (b B-A c)}{c}-\frac {2 \left (\frac {\left (-\frac {2 a A c^2-3 a b B c-A b^2 c+b^3 B}{\sqrt {b^2-4 a c}}-a B c-A b c+b^2 B\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (\frac {2 a A c^2-3 a b B c-A b^2 c+b^3 B}{\sqrt {b^2-4 a c}}-a B c-A b c+b^2 B\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} \sqrt {c} \sqrt {\sqrt {b^2-4 a c}+b}}\right )}{c}}{c}\) |
(2*B*x^(3/2))/(3*c) - ((2*(b*B - A*c)*Sqrt[x])/c - (2*(((b^2*B - A*b*c - a *B*c - (b^3*B - A*b^2*c - 3*a*b*B*c + 2*a*A*c^2)/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*B - A*b*c - a*B*c + (b^3*B - A*b^2*c - 3*a*b*B*c + 2*a*A*c^2)/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]])))/c)/c
3.11.10.3.1 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_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[g*((d + e*x)^m/(c*m)), x] + Simp[1/c Int [(d + e*x)^(m - 1)*(Simp[c*d*f - a*e*g + (g*c*d - b*e*g + c*e*f)*x, x]/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && FractionQ[m] & & GtQ[m, 0]
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_)^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 = 0.40 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.06
| method | result | size |
| risch | \(\frac {2 \left (B c x +3 A c -3 B b \right ) \sqrt {x}}{3 c^{2}}-\frac {8 \left (\frac {\left (A b c \sqrt {-4 a c +b^{2}}-2 A a \,c^{2}+A \,b^{2} c +B a c \sqrt {-4 a c +b^{2}}-B \,b^{2} \sqrt {-4 a c +b^{2}}+3 B a b c -B \,b^{3}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}-\frac {\left (A b c \sqrt {-4 a c +b^{2}}+2 A a \,c^{2}-A \,b^{2} c +B a c \sqrt {-4 a c +b^{2}}-B \,b^{2} \sqrt {-4 a c +b^{2}}-3 B a b c +B \,b^{3}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{c}\) | \(291\) |
| derivativedivides | \(\frac {\frac {2 B c \,x^{\frac {3}{2}}}{3}+2 A c \sqrt {x}-2 \sqrt {x}\, B b}{c^{2}}+\frac {-\frac {\left (-A b c \sqrt {-4 a c +b^{2}}-2 A a \,c^{2}+A \,b^{2} c -B a c \sqrt {-4 a c +b^{2}}+B \,b^{2} \sqrt {-4 a c +b^{2}}+3 B a b c -B \,b^{3}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (-A b c \sqrt {-4 a c +b^{2}}+2 A a \,c^{2}-A \,b^{2} c -B a c \sqrt {-4 a c +b^{2}}+B \,b^{2} \sqrt {-4 a c +b^{2}}-3 B a b c +B \,b^{3}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}}{c}\) | \(298\) |
| default | \(\frac {\frac {2 B c \,x^{\frac {3}{2}}}{3}+2 A c \sqrt {x}-2 \sqrt {x}\, B b}{c^{2}}+\frac {-\frac {\left (-A b c \sqrt {-4 a c +b^{2}}-2 A a \,c^{2}+A \,b^{2} c -B a c \sqrt {-4 a c +b^{2}}+B \,b^{2} \sqrt {-4 a c +b^{2}}+3 B a b c -B \,b^{3}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (-A b c \sqrt {-4 a c +b^{2}}+2 A a \,c^{2}-A \,b^{2} c -B a c \sqrt {-4 a c +b^{2}}+B \,b^{2} \sqrt {-4 a c +b^{2}}-3 B a b c +B \,b^{3}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}}{c}\) | \(298\) |
2/3*(B*c*x+3*A*c-3*B*b)*x^(1/2)/c^2-8/c*(1/8*(A*b*c*(-4*a*c+b^2)^(1/2)-2*A *a*c^2+A*b^2*c+B*a*c*(-4*a*c+b^2)^(1/2)-B*b^2*(-4*a*c+b^2)^(1/2)+3*B*a*b*c -B*b^3)/c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arct an(c*x^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))-1/8*(A*b*c*(-4*a*c+ b^2)^(1/2)+2*A*a*c^2-A*b^2*c+B*a*c*(-4*a*c+b^2)^(1/2)-B*b^2*(-4*a*c+b^2)^( 1/2)-3*B*a*b*c+B*b^3)/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)))
Leaf count of result is larger than twice the leaf count of optimal. 5148 vs. \(2 (229) = 458\).
Time = 7.06 (sec) , antiderivative size = 5148, normalized size of antiderivative = 18.72 \[ \int \frac {x^{3/2} (A+B x)}{a+b x+c x^2} \, dx=\text {Too large to display} \]
Leaf count of result is larger than twice the leaf count of optimal. 17734 vs. \(2 (264) = 528\).
Time = 16.75 (sec) , antiderivative size = 17734, normalized size of antiderivative = 64.49 \[ \int \frac {x^{3/2} (A+B x)}{a+b x+c x^2} \, dx=\text {Too large to display} \]
Piecewise((A*a**2*log(sqrt(x) - sqrt(-a/b))/(b**3*sqrt(-a/b)) - A*a**2*log (sqrt(x) + sqrt(-a/b))/(b**3*sqrt(-a/b)) - 2*A*a*sqrt(x)/b**2 + 2*A*x**(3/ 2)/(3*b) - B*a**3*log(sqrt(x) - sqrt(-a/b))/(b**4*sqrt(-a/b)) + B*a**3*log (sqrt(x) + sqrt(-a/b))/(b**4*sqrt(-a/b)) + 2*B*a**2*sqrt(x)/b**3 - 2*B*a*x **(3/2)/(3*b**2) + 2*B*x**(5/2)/(5*b), Eq(c, 0)), (-A*b*log(sqrt(x) - sqrt (-b/c))/(c**2*sqrt(-b/c)) + A*b*log(sqrt(x) + sqrt(-b/c))/(c**2*sqrt(-b/c) ) + 2*A*sqrt(x)/c + B*b**2*log(sqrt(x) - sqrt(-b/c))/(c**3*sqrt(-b/c)) - B *b**2*log(sqrt(x) + sqrt(-b/c))/(c**3*sqrt(-b/c)) - 2*B*b*sqrt(x)/c**2 + 2 *B*x**(3/2)/(3*c), Eq(a, 0)), (-18*sqrt(2)*A*b**2*c*log(sqrt(x) - sqrt(2)* sqrt(-b/c)/2)/(24*b*c**3*sqrt(-b/c) + 48*c**4*x*sqrt(-b/c)) + 18*sqrt(2)*A *b**2*c*log(sqrt(x) + sqrt(2)*sqrt(-b/c)/2)/(24*b*c**3*sqrt(-b/c) + 48*c** 4*x*sqrt(-b/c)) + 72*A*b*c**2*sqrt(x)*sqrt(-b/c)/(24*b*c**3*sqrt(-b/c) + 4 8*c**4*x*sqrt(-b/c)) - 36*sqrt(2)*A*b*c**2*x*log(sqrt(x) - sqrt(2)*sqrt(-b /c)/2)/(24*b*c**3*sqrt(-b/c) + 48*c**4*x*sqrt(-b/c)) + 36*sqrt(2)*A*b*c**2 *x*log(sqrt(x) + sqrt(2)*sqrt(-b/c)/2)/(24*b*c**3*sqrt(-b/c) + 48*c**4*x*s qrt(-b/c)) + 96*A*c**3*x**(3/2)*sqrt(-b/c)/(24*b*c**3*sqrt(-b/c) + 48*c**4 *x*sqrt(-b/c)) + 15*sqrt(2)*B*b**3*log(sqrt(x) - sqrt(2)*sqrt(-b/c)/2)/(24 *b*c**3*sqrt(-b/c) + 48*c**4*x*sqrt(-b/c)) - 15*sqrt(2)*B*b**3*log(sqrt(x) + sqrt(2)*sqrt(-b/c)/2)/(24*b*c**3*sqrt(-b/c) + 48*c**4*x*sqrt(-b/c)) - 6 0*B*b**2*c*sqrt(x)*sqrt(-b/c)/(24*b*c**3*sqrt(-b/c) + 48*c**4*x*sqrt(-b...
\[ \int \frac {x^{3/2} (A+B x)}{a+b x+c x^2} \, dx=\int { \frac {{\left (B x + A\right )} x^{\frac {3}{2}}}{c x^{2} + b x + a} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 4399 vs. \(2 (229) = 458\).
Time = 1.02 (sec) , antiderivative size = 4399, normalized size of antiderivative = 16.00 \[ \int \frac {x^{3/2} (A+B x)}{a+b x+c x^2} \, dx=\text {Too large to display} \]
1/4*((2*b^5*c^3 - 16*a*b^3*c^4 + 32*a^2*b*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)* sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^5*c + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b *c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^2 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c^2 - 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^3 - 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sq rt(b^2 - 4*a*c)*c)*a*b^2*c^3 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b ^2 - 4*a*c)*c)*b^3*c^3 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^4 - 2*(b^2 - 4*a*c)*b^3*c^3 + 8*(b^2 - 4*a*c)*a*b*c^4)*A* c^2 - (2*b^6*c^2 - 18*a*b^4*c^3 + 48*a^2*b^2*c^4 - 32*a^3*c^5 - sqrt(2)*sq rt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^6 + 9*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4*c + 2*sqrt(2)*sqrt(b^2 - 4*a *c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^5*c - 24*sqrt(2)*sqrt(b^2 - 4*a*c)*s qrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^2 - 10*sqrt(2)*sqrt(b^2 - 4*a*c)* sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt (b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c^2 + 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b* c + sqrt(b^2 - 4*a*c)*c)*a^3*c^3 + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^3 + 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sq rt(b^2 - 4*a*c)*c)*a*b^2*c^3 - 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt (b^2 - 4*a*c)*c)*a^2*c^4 - 2*(b^2 - 4*a*c)*b^4*c^2 + 10*(b^2 - 4*a*c)*a*b^ 2*c^3 - 8*(b^2 - 4*a*c)*a^2*c^4)*B*c^2 - 2*(sqrt(2)*sqrt(b*c + sqrt(b^2...
Time = 11.37 (sec) , antiderivative size = 10204, normalized size of antiderivative = 37.11 \[ \int \frac {x^{3/2} (A+B x)}{a+b x+c x^2} \, dx=\text {Too large to display} \]
x^(1/2)*((2*A)/c - (2*B*b)/c^2) - atan(((((8*(4*A*a^2*c^5 - A*a*b^2*c^4 + B*a*b^3*c^3 - 4*B*a^2*b*c^4))/c^3 - (8*x^(1/2)*(b^3*c^5 - 4*a*b*c^6)*(-(B^ 2*b^7 + A^2*b^5*c^2 - B^2*b^4*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*b^6*c + 25* B^2*a^2*b^3*c^2 - A^2*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - B^2*a^2*c^2*(-(4* a*c - b^2)^3)^(1/2) + 16*A*B*a^3*c^4 - 9*B^2*a*b^5*c - 7*A^2*a*b^3*c^3 + 1 2*A^2*a^2*b*c^4 + A^2*a*c^3*(-(4*a*c - b^2)^3)^(1/2) - 20*B^2*a^3*b*c^3 - 36*A*B*a^2*b^2*c^3 + 3*B^2*a*b^2*c*(-(4*a*c - b^2)^3)^(1/2) + 16*A*B*a*b^4 *c^2 + 2*A*B*b^3*c*(-(4*a*c - b^2)^3)^(1/2) - 4*A*B*a*b*c^2*(-(4*a*c - b^2 )^3)^(1/2))/(2*(16*a^2*c^7 + b^4*c^5 - 8*a*b^2*c^6)))^(1/2))/c^3)*(-(B^2*b ^7 + A^2*b^5*c^2 - B^2*b^4*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*b^6*c + 25*B^2 *a^2*b^3*c^2 - A^2*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - B^2*a^2*c^2*(-(4*a*c - b^2)^3)^(1/2) + 16*A*B*a^3*c^4 - 9*B^2*a*b^5*c - 7*A^2*a*b^3*c^3 + 12*A ^2*a^2*b*c^4 + A^2*a*c^3*(-(4*a*c - b^2)^3)^(1/2) - 20*B^2*a^3*b*c^3 - 36* A*B*a^2*b^2*c^3 + 3*B^2*a*b^2*c*(-(4*a*c - b^2)^3)^(1/2) + 16*A*B*a*b^4*c^ 2 + 2*A*B*b^3*c*(-(4*a*c - b^2)^3)^(1/2) - 4*A*B*a*b*c^2*(-(4*a*c - b^2)^3 )^(1/2))/(2*(16*a^2*c^7 + b^4*c^5 - 8*a*b^2*c^6)))^(1/2) - (8*x^(1/2)*(B^2 *b^6 + 2*A^2*a^2*c^4 + A^2*b^4*c^2 - 2*B^2*a^3*c^3 - 2*A*B*b^5*c + 9*B^2*a ^2*b^2*c^2 - 6*B^2*a*b^4*c - 4*A^2*a*b^2*c^3 + 10*A*B*a*b^3*c^2 - 10*A*B*a ^2*b*c^3))/c^3)*(-(B^2*b^7 + A^2*b^5*c^2 - B^2*b^4*(-(4*a*c - b^2)^3)^(1/2 ) - 2*A*B*b^6*c + 25*B^2*a^2*b^3*c^2 - A^2*b^2*c^2*(-(4*a*c - b^2)^3)^(...