Integrand size = 23, antiderivative size = 347 \[ \int \frac {x^{5/2} (A+B x)}{a+b x+c x^2} \, dx=\frac {2 \left (b^2 B-A b c-a B c\right ) \sqrt {x}}{c^3}-\frac {2 (b B-A c) x^{3/2}}{3 c^2}+\frac {2 B x^{5/2}}{5 c}-\frac {\sqrt {2} \left (b^3 B-A b^2 c-2 a b B c+a A c^2-\frac {b^4 B-A b^3 c-4 a b^2 B c+3 a A b c^2+2 a^2 B c^2}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{c^{7/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \left (b^3 B-A b^2 c-2 a b B c+a A c^2+\frac {b^4 B-A b^3 c-4 a b^2 B c+3 a A b c^2+2 a^2 B c^2}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{c^{7/2} \sqrt {b+\sqrt {b^2-4 a c}}} \]
-2/3*(-A*c+B*b)*x^(3/2)/c^2+2/5*B*x^(5/2)/c+2*(-A*b*c-B*a*c+B*b^2)*x^(1/2) /c^3-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^3-A*b^2*c-2*B*a*b*c+A*a*c^2+(-3*A*a*b*c^2+A*b^3*c-2*B*a^2*c^2+4*B*a*b ^2*c-B*b^4)/(-4*a*c+b^2)^(1/2))/c^(7/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)-arcta n(2^(1/2)*c^(1/2)*x^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2))*2^(1/2)*(B*b^3-A*b ^2*c-2*B*a*b*c+A*a*c^2+(3*A*a*b*c^2-A*b^3*c+2*B*a^2*c^2-4*B*a*b^2*c+B*b^4) /(-4*a*c+b^2)^(1/2))/c^(7/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)
Time = 0.96 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.18 \[ \int \frac {x^{5/2} (A+B x)}{a+b x+c x^2} \, dx=\frac {2 \sqrt {c} \sqrt {x} \left (15 b^2 B-5 b c (3 A+B x)+c (-15 a B+c x (5 A+3 B x))\right )-\frac {15 \sqrt {2} \left (-b^4 B+b^2 c \left (4 a B-A \sqrt {b^2-4 a c}\right )+a c^2 \left (-2 a B+A \sqrt {b^2-4 a c}\right )+b^3 \left (A c+B \sqrt {b^2-4 a c}\right )-a b c \left (3 A c+2 B \sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {15 \sqrt {2} \left (b^4 B+a c^2 \left (2 a B+A \sqrt {b^2-4 a c}\right )-b^2 c \left (4 a B+A \sqrt {b^2-4 a c}\right )+a b c \left (3 A c-2 B \sqrt {b^2-4 a c}\right )+b^3 \left (-A c+B \sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}}}{15 c^{7/2}} \]
(2*Sqrt[c]*Sqrt[x]*(15*b^2*B - 5*b*c*(3*A + B*x) + c*(-15*a*B + c*x*(5*A + 3*B*x))) - (15*Sqrt[2]*(-(b^4*B) + b^2*c*(4*a*B - A*Sqrt[b^2 - 4*a*c]) + a*c^2*(-2*a*B + A*Sqrt[b^2 - 4*a*c]) + b^3*(A*c + B*Sqrt[b^2 - 4*a*c]) - a *b*c*(3*A*c + 2*B*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqr t[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (15*Sqrt[2]*(b^4*B + a*c^2*(2*a*B + A*Sqrt[b^2 - 4*a*c]) - b^2*c*(4*a*B + A*Sqrt[b^2 - 4*a*c]) + a*b*c*(3*A*c - 2*B*Sqrt[b^2 - 4*a*c]) + b^3*(-(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]]))/(15*c^(7 /2))
Time = 0.79 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {1196, 25, 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.
| \(\displaystyle \int \frac {x^{5/2} (A+B x)}{a+b x+c x^2} \, dx\) |
| \(\Big \downarrow \) 1196 |
| \(\displaystyle \frac {\int -\frac {x^{3/2} (a B+(b B-A c) x)}{c x^2+b x+a}dx}{c}+\frac {2 B x^{5/2}}{5 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 B x^{5/2}}{5 c}-\frac {\int \frac {x^{3/2} (a B+(b B-A c) x)}{c x^2+b x+a}dx}{c}\) |
| \(\Big \downarrow \) 1196 |
| \(\displaystyle \frac {2 B x^{5/2}}{5 c}-\frac {\frac {\int -\frac {\sqrt {x} \left (a (b B-A c)+\left (B b^2-A c b-a B c\right ) x\right )}{c x^2+b x+a}dx}{c}+\frac {2 x^{3/2} (b B-A c)}{3 c}}{c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 B x^{5/2}}{5 c}-\frac {\frac {2 x^{3/2} (b B-A c)}{3 c}-\frac {\int \frac {\sqrt {x} \left (a (b B-A c)+\left (B b^2-A c b-a B c\right ) x\right )}{c x^2+b x+a}dx}{c}}{c}\) |
| \(\Big \downarrow \) 1196 |
| \(\displaystyle \frac {2 B x^{5/2}}{5 c}-\frac {\frac {2 x^{3/2} (b B-A c)}{3 c}-\frac {\frac {\int -\frac {a \left (B b^2-A c b-a B c\right )+\left (B b^3-A c b^2-2 a B c b+a A c^2\right ) x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{c}+\frac {2 \sqrt {x} \left (-a B c-A b c+b^2 B\right )}{c}}{c}}{c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 B x^{5/2}}{5 c}-\frac {\frac {2 x^{3/2} (b B-A c)}{3 c}-\frac {\frac {2 \sqrt {x} \left (-a B c-A b c+b^2 B\right )}{c}-\frac {\int \frac {a \left (B b^2-A c b-a B c\right )+\left (B b^3-A c b^2-2 a B c b+a A c^2\right ) x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{c}}{c}}{c}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle \frac {2 B x^{5/2}}{5 c}-\frac {\frac {2 x^{3/2} (b B-A c)}{3 c}-\frac {\frac {2 \sqrt {x} \left (-a B c-A b c+b^2 B\right )}{c}-\frac {2 \int \frac {a \left (B b^2-A c b-a B c\right )+\left (B b^3-A c b^2-2 a B c b+a A c^2\right ) x}{c x^2+b x+a}d\sqrt {x}}{c}}{c}}{c}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {2 B x^{5/2}}{5 c}-\frac {\frac {2 x^{3/2} (b B-A c)}{3 c}-\frac {\frac {2 \sqrt {x} \left (-a B c-A b c+b^2 B\right )}{c}-\frac {2 \left (\frac {1}{2} \left (-\frac {2 a^2 B c^2+3 a A b c^2-4 a b^2 B c-A b^3 c+b^4 B}{\sqrt {b^2-4 a c}}+a A c^2-2 a b B c-A b^2 c+b^3 B\right ) \int \frac {1}{\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )+c x}d\sqrt {x}+\frac {1}{2} \left (\frac {2 a^2 B c^2+3 a A b c^2-4 a b^2 B c-A b^3 c+b^4 B}{\sqrt {b^2-4 a c}}+a A c^2-2 a b B c-A b^2 c+b^3 B\right ) \int \frac {1}{\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )+c x}d\sqrt {x}\right )}{c}}{c}}{c}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {2 B x^{5/2}}{5 c}-\frac {\frac {2 x^{3/2} (b B-A c)}{3 c}-\frac {\frac {2 \sqrt {x} \left (-a B c-A b c+b^2 B\right )}{c}-\frac {2 \left (\frac {\left (-\frac {2 a^2 B c^2+3 a A b c^2-4 a b^2 B c-A b^3 c+b^4 B}{\sqrt {b^2-4 a c}}+a A c^2-2 a b B c-A b^2 c+b^3 B\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (\frac {2 a^2 B c^2+3 a A b c^2-4 a b^2 B c-A b^3 c+b^4 B}{\sqrt {b^2-4 a c}}+a A c^2-2 a b B c-A b^2 c+b^3 B\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} \sqrt {c} \sqrt {\sqrt {b^2-4 a c}+b}}\right )}{c}}{c}}{c}\) |
(2*B*x^(5/2))/(5*c) - ((2*(b*B - A*c)*x^(3/2))/(3*c) - ((2*(b^2*B - A*b*c - a*B*c)*Sqrt[x])/c - (2*(((b^3*B - A*b^2*c - 2*a*b*B*c + a*A*c^2 - (b^4*B - A*b^3*c - 4*a*b^2*B*c + 3*a*A*b*c^2 + 2*a^2*B*c^2)/Sqrt[b^2 - 4*a*c])*A rcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqr t[c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + ((b^3*B - A*b^2*c - 2*a*b*B*c + a*A*c^ 2 + (b^4*B - A*b^3*c - 4*a*b^2*B*c + 3*a*A*b*c^2 + 2*a^2*B*c^2)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(S qrt[2]*Sqrt[c]*Sqrt[b + Sqrt[b^2 - 4*a*c]])))/c)/c)/c
3.11.9.3.1 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 = 381, normalized size of antiderivative = 1.10
| method | result | size |
| risch | \(-\frac {2 \left (-3 B \,c^{2} x^{2}-5 A \,c^{2} x +5 B b c x +15 A b c +15 B a c -15 B \,b^{2}\right ) \sqrt {x}}{15 c^{3}}-\frac {8 \left (\frac {\left (A a \,c^{2} \sqrt {-4 a c +b^{2}}-A \,b^{2} c \sqrt {-4 a c +b^{2}}+3 a A b \,c^{2}-A \,b^{3} c -2 B a b c \sqrt {-4 a c +b^{2}}+B \,b^{3} \sqrt {-4 a c +b^{2}}+2 a^{2} B \,c^{2}-4 a \,b^{2} B c +b^{4} B \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}-\frac {\left (A a \,c^{2} \sqrt {-4 a c +b^{2}}-A \,b^{2} c \sqrt {-4 a c +b^{2}}-3 a A b \,c^{2}+A \,b^{3} c -2 B a b c \sqrt {-4 a c +b^{2}}+B \,b^{3} \sqrt {-4 a c +b^{2}}-2 a^{2} B \,c^{2}+4 a \,b^{2} B c -b^{4} B \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{c^{2}}\) | \(381\) |
| derivativedivides | \(-\frac {2 \left (-\frac {B \,c^{2} x^{\frac {5}{2}}}{5}-\frac {A \,c^{2} x^{\frac {3}{2}}}{3}+\frac {x^{\frac {3}{2}} B b c}{3}+\sqrt {x}\, A b c +a B c \sqrt {x}-b^{2} B \sqrt {x}\right )}{c^{3}}+\frac {-\frac {\left (-A a \,c^{2} \sqrt {-4 a c +b^{2}}+A \,b^{2} c \sqrt {-4 a c +b^{2}}+3 a A b \,c^{2}-A \,b^{3} c +2 B a b c \sqrt {-4 a c +b^{2}}-B \,b^{3} \sqrt {-4 a c +b^{2}}+2 a^{2} B \,c^{2}-4 a \,b^{2} B c +b^{4} B \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (-A a \,c^{2} \sqrt {-4 a c +b^{2}}+A \,b^{2} c \sqrt {-4 a c +b^{2}}-3 a A b \,c^{2}+A \,b^{3} c +2 B a b c \sqrt {-4 a c +b^{2}}-B \,b^{3} \sqrt {-4 a c +b^{2}}-2 a^{2} B \,c^{2}+4 a \,b^{2} B c -b^{4} B \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}}{c^{2}}\) | \(391\) |
| default | \(-\frac {2 \left (-\frac {B \,c^{2} x^{\frac {5}{2}}}{5}-\frac {A \,c^{2} x^{\frac {3}{2}}}{3}+\frac {x^{\frac {3}{2}} B b c}{3}+\sqrt {x}\, A b c +a B c \sqrt {x}-b^{2} B \sqrt {x}\right )}{c^{3}}+\frac {-\frac {\left (-A a \,c^{2} \sqrt {-4 a c +b^{2}}+A \,b^{2} c \sqrt {-4 a c +b^{2}}+3 a A b \,c^{2}-A \,b^{3} c +2 B a b c \sqrt {-4 a c +b^{2}}-B \,b^{3} \sqrt {-4 a c +b^{2}}+2 a^{2} B \,c^{2}-4 a \,b^{2} B c +b^{4} B \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (-A a \,c^{2} \sqrt {-4 a c +b^{2}}+A \,b^{2} c \sqrt {-4 a c +b^{2}}-3 a A b \,c^{2}+A \,b^{3} c +2 B a b c \sqrt {-4 a c +b^{2}}-B \,b^{3} \sqrt {-4 a c +b^{2}}-2 a^{2} B \,c^{2}+4 a \,b^{2} B c -b^{4} B \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}}{c^{2}}\) | \(391\) |
-2/15*(-3*B*c^2*x^2-5*A*c^2*x+5*B*b*c*x+15*A*b*c+15*B*a*c-15*B*b^2)*x^(1/2 )/c^3-8/c^2*(1/8*(A*a*c^2*(-4*a*c+b^2)^(1/2)-A*b^2*c*(-4*a*c+b^2)^(1/2)+3* a*A*b*c^2-A*b^3*c-2*B*a*b*c*(-4*a*c+b^2)^(1/2)+B*b^3*(-4*a*c+b^2)^(1/2)+2* a^2*B*c^2-4*a*b^2*B*c+b^4*B)/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) )-1/8*(A*a*c^2*(-4*a*c+b^2)^(1/2)-A*b^2*c*(-4*a*c+b^2)^(1/2)-3*a*A*b*c^2+A *b^3*c-2*B*a*b*c*(-4*a*c+b^2)^(1/2)+B*b^3*(-4*a*c+b^2)^(1/2)-2*a^2*B*c^2+4 *a*b^2*B*c-b^4*B)/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. 7707 vs. \(2 (297) = 594\).
Time = 15.59 (sec) , antiderivative size = 7707, normalized size of antiderivative = 22.21 \[ \int \frac {x^{5/2} (A+B x)}{a+b x+c x^2} \, dx=\text {Too large to display} \]
Leaf count of result is larger than twice the leaf count of optimal. 40613 vs. \(2 (347) = 694\).
Time = 70.03 (sec) , antiderivative size = 40613, normalized size of antiderivative = 117.04 \[ \int \frac {x^{5/2} (A+B x)}{a+b x+c x^2} \, dx=\text {Too large to display} \]
Piecewise((-A*a**3*log(sqrt(x) - sqrt(-a/b))/(b**4*sqrt(-a/b)) + A*a**3*lo g(sqrt(x) + sqrt(-a/b))/(b**4*sqrt(-a/b)) + 2*A*a**2*sqrt(x)/b**3 - 2*A*a* x**(3/2)/(3*b**2) + 2*A*x**(5/2)/(5*b) + B*a**4*log(sqrt(x) - sqrt(-a/b))/ (b**5*sqrt(-a/b)) - B*a**4*log(sqrt(x) + sqrt(-a/b))/(b**5*sqrt(-a/b)) - 2 *B*a**3*sqrt(x)/b**4 + 2*B*a**2*x**(3/2)/(3*b**3) - 2*B*a*x**(5/2)/(5*b**2 ) + 2*B*x**(7/2)/(7*b), Eq(c, 0)), (A*b**2*log(sqrt(x) - sqrt(-b/c))/(c**3 *sqrt(-b/c)) - A*b**2*log(sqrt(x) + sqrt(-b/c))/(c**3*sqrt(-b/c)) - 2*A*b* sqrt(x)/c**2 + 2*A*x**(3/2)/(3*c) - B*b**3*log(sqrt(x) - sqrt(-b/c))/(c**4 *sqrt(-b/c)) + B*b**3*log(sqrt(x) + sqrt(-b/c))/(c**4*sqrt(-b/c)) + 2*B*b* *2*sqrt(x)/c**3 - 2*B*b*x**(3/2)/(3*c**2) + 2*B*x**(5/2)/(5*c), Eq(a, 0)), (150*sqrt(2)*A*b**3*c*log(sqrt(x) - sqrt(2)*sqrt(-b/c)/2)/(240*b*c**4*sqr t(-b/c) + 480*c**5*x*sqrt(-b/c)) - 150*sqrt(2)*A*b**3*c*log(sqrt(x) + sqrt (2)*sqrt(-b/c)/2)/(240*b*c**4*sqrt(-b/c) + 480*c**5*x*sqrt(-b/c)) - 600*A* b**2*c**2*sqrt(x)*sqrt(-b/c)/(240*b*c**4*sqrt(-b/c) + 480*c**5*x*sqrt(-b/c )) + 300*sqrt(2)*A*b**2*c**2*x*log(sqrt(x) - sqrt(2)*sqrt(-b/c)/2)/(240*b* c**4*sqrt(-b/c) + 480*c**5*x*sqrt(-b/c)) - 300*sqrt(2)*A*b**2*c**2*x*log(s qrt(x) + sqrt(2)*sqrt(-b/c)/2)/(240*b*c**4*sqrt(-b/c) + 480*c**5*x*sqrt(-b /c)) - 800*A*b*c**3*x**(3/2)*sqrt(-b/c)/(240*b*c**4*sqrt(-b/c) + 480*c**5* x*sqrt(-b/c)) + 320*A*c**4*x**(5/2)*sqrt(-b/c)/(240*b*c**4*sqrt(-b/c) + 48 0*c**5*x*sqrt(-b/c)) - 105*sqrt(2)*B*b**4*log(sqrt(x) - sqrt(2)*sqrt(-b...
\[ \int \frac {x^{5/2} (A+B x)}{a+b x+c x^2} \, dx=\int { \frac {{\left (B x + A\right )} x^{\frac {5}{2}}}{c x^{2} + b x + a} \,d x } \]
2/15*(3*B*c*x^(5/2) - 5*(B*b - A*c)*x^(3/2))/c^2 - integrate(((A*b*c - (b^ 2 - a*c)*B)*x^(3/2) - (B*a*b - A*a*c)*sqrt(x))/(c^3*x^2 + b*c^2*x + a*c^2) , x)
Leaf count of result is larger than twice the leaf count of optimal. 5319 vs. \(2 (297) = 594\).
Time = 1.04 (sec) , antiderivative size = 5319, normalized size of antiderivative = 15.33 \[ \int \frac {x^{5/2} (A+B x)}{a+b x+c x^2} \, dx=\text {Too large to display} \]
-1/4*((2*b^6*c^3 - 18*a*b^4*c^4 + 48*a^2*b^2*c^5 - 32*a^3*c^6 - sqrt(2)*sq rt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^6*c + 9*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4*c^2 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^5*c^2 - 24*sqrt(2)*sqrt(b^2 - 4* a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^3 - 10*sqrt(2)*sqrt(b^2 - 4 *a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^3 - sqrt(2)*sqrt(b^2 - 4*a*c )*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c^3 + 16*sqrt(2)*sqrt(b^2 - 4*a*c)*s qrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*c^4 + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt( b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^4 + 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b* c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^4 - 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^5 - 2*(b^2 - 4*a*c)*b^4*c^3 + 10*(b^2 - 4*a*c )*a*b^2*c^4 - 8*(b^2 - 4*a*c)*a^2*c^5)*A*c^2 - (2*b^7*c^2 - 20*a*b^5*c^3 + 64*a^2*b^3*c^4 - 64*a^3*b*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt (b^2 - 4*a*c)*c)*b^7 + 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^5*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a* c)*c)*b^6*c - 32*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c) *a^2*b^3*c^2 - 12*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^2 + 32*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b *c^3 + 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2...
Time = 11.88 (sec) , antiderivative size = 14120, normalized size of antiderivative = 40.69 \[ \int \frac {x^{5/2} (A+B x)}{a+b x+c x^2} \, dx=\text {Too large to display} \]
x^(3/2)*((2*A)/(3*c) - (2*B*b)/(3*c^2)) - x^(1/2)*((b*((2*A)/c - (2*B*b)/c ^2))/c + (2*B*a)/c^2) + atan(((((8*(4*B*a^3*c^6 - A*a*b^3*c^5 + 4*A*a^2*b* c^6 + B*a*b^4*c^4 - 5*B*a^2*b^2*c^5))/c^5 - (8*x^(1/2)*(b^3*c^7 - 4*a*b*c^ 8)*(-(B^2*b^9 + A^2*b^7*c^2 + B^2*b^6*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*b^8 *c + 25*A^2*a^2*b^3*c^4 + A^2*a^2*c^4*(-(4*a*c - b^2)^3)^(1/2) + 42*B^2*a^ 2*b^5*c^2 - 63*B^2*a^3*b^3*c^3 + A^2*b^4*c^2*(-(4*a*c - b^2)^3)^(1/2) - B^ 2*a^3*c^3*(-(4*a*c - b^2)^3)^(1/2) - 16*A*B*a^4*c^5 - 11*B^2*a*b^7*c - 9*A ^2*a*b^5*c^3 - 20*A^2*a^3*b*c^5 + 28*B^2*a^4*b*c^4 + 6*B^2*a^2*b^2*c^2*(-( 4*a*c - b^2)^3)^(1/2) - 66*A*B*a^2*b^4*c^3 + 76*A*B*a^3*b^2*c^4 - 5*B^2*a* b^4*c*(-(4*a*c - b^2)^3)^(1/2) - 3*A^2*a*b^2*c^3*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a*b^6*c^2 - 2*A*B*b^5*c*(-(4*a*c - b^2)^3)^(1/2) + 8*A*B*a*b^3*c^ 2*(-(4*a*c - b^2)^3)^(1/2) - 6*A*B*a^2*b*c^3*(-(4*a*c - b^2)^3)^(1/2))/(2* (16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2))/c^5)*(-(B^2*b^9 + A^2*b^7*c^ 2 + B^2*b^6*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*b^8*c + 25*A^2*a^2*b^3*c^4 + A^2*a^2*c^4*(-(4*a*c - b^2)^3)^(1/2) + 42*B^2*a^2*b^5*c^2 - 63*B^2*a^3*b^3 *c^3 + A^2*b^4*c^2*(-(4*a*c - b^2)^3)^(1/2) - B^2*a^3*c^3*(-(4*a*c - b^2)^ 3)^(1/2) - 16*A*B*a^4*c^5 - 11*B^2*a*b^7*c - 9*A^2*a*b^5*c^3 - 20*A^2*a^3* b*c^5 + 28*B^2*a^4*b*c^4 + 6*B^2*a^2*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 66 *A*B*a^2*b^4*c^3 + 76*A*B*a^3*b^2*c^4 - 5*B^2*a*b^4*c*(-(4*a*c - b^2)^3)^( 1/2) - 3*A^2*a*b^2*c^3*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a*b^6*c^2 - 2*...