Integrand size = 23, antiderivative size = 307 \[ \int \frac {A+B x}{x^{7/2} \left (a+b x+c x^2\right )} \, dx=-\frac {2 A}{5 a x^{5/2}}+\frac {2 (A b-a B)}{3 a^2 x^{3/2}}-\frac {2 \left (A b^2-a b B-a A c\right )}{a^3 \sqrt {x}}-\frac {\sqrt {2} \sqrt {c} \left (A b^2-a b B-a A c-\frac {a B \left (b^2-2 a c\right )-A \left (b^3-3 a b c\right )}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{a^3 \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \sqrt {c} \left (A b^2-a b B-a A c+\frac {a B \left (b^2-2 a c\right )-A \left (b^3-3 a b c\right )}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{a^3 \sqrt {b+\sqrt {b^2-4 a c}}} \]
-2/5*A/a/x^(5/2)+2/3*(A*b-B*a)/a^2/x^(3/2)-2*(-A*a*c+A*b^2-B*a*b)/a^3/x^(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)*c ^(1/2)*(A*b^2-a*b*B-A*a*c+(-a*B*(-2*a*c+b^2)+A*(-3*a*b*c+b^3))/(-4*a*c+b^2 )^(1/2))/a^3/(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)*c^(1/2)*(A*b^2-a*b*B-A*a*c+(a*B*(-2*a *c+b^2)-A*(-3*a*b*c+b^3))/(-4*a*c+b^2)^(1/2))/a^3/(b+(-4*a*c+b^2)^(1/2))^( 1/2)
Time = 1.00 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.15 \[ \int \frac {A+B x}{x^{7/2} \left (a+b x+c x^2\right )} \, dx=\frac {\frac {-30 A b^2 x^2-2 a^2 (3 A+5 B x)+10 a x (3 b B x+A (b+3 c x))}{x^{5/2}}+\frac {15 \sqrt {2} \sqrt {c} \left (a B \left (b^2-2 a c+b \sqrt {b^2-4 a c}\right )-A \left (b^3-3 a b c+b^2 \sqrt {b^2-4 a c}-a c \sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {15 \sqrt {2} \sqrt {c} \left (a B \left (-b^2+2 a c+b \sqrt {b^2-4 a c}\right )+A \left (b^3-3 a b c-b^2 \sqrt {b^2-4 a c}+a c \sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}}}{15 a^3} \]
((-30*A*b^2*x^2 - 2*a^2*(3*A + 5*B*x) + 10*a*x*(3*b*B*x + A*(b + 3*c*x)))/ x^(5/2) + (15*Sqrt[2]*Sqrt[c]*(a*B*(b^2 - 2*a*c + b*Sqrt[b^2 - 4*a*c]) - A *(b^3 - 3*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - a*c*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*Sqrt[2]*Sqrt[c]*(a*B*(-b^2 + 2*a*c + b* Sqrt[b^2 - 4*a*c]) + A*(b^3 - 3*a*b*c - b^2*Sqrt[b^2 - 4*a*c] + a*c*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*a^3)
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.
| \(\displaystyle \int \frac {A+B x}{x^{7/2} \left (a+b x+c x^2\right )} \, dx\) |
| \(\Big \downarrow \) 1198 |
| \(\displaystyle \frac {\int -\frac {A b-a B+A c x}{x^{5/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {A b-a B+A c x}{x^{5/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 1198 |
| \(\displaystyle -\frac {\frac {\int -\frac {A b^2-a B b-a A c+(A b-a B) c x}{x^{3/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{3 a x^{3/2}}}{a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\int -\frac {a b B-A \left (b^2-a c\right )-(A b-a B) c x}{x^{3/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{3 a x^{3/2}}}{a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\int -\frac {A b^2-a B b-a A c+(A b-a B) c x}{x^{3/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{3 a x^{3/2}}}{a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\int -\frac {a b B-A \left (b^2-a c\right )-(A b-a B) c x}{x^{3/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{3 a x^{3/2}}}{a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\int -\frac {A b^2-a B b-a A c+(A b-a B) c x}{x^{3/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{3 a x^{3/2}}}{a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\int -\frac {a b B-A \left (b^2-a c\right )-(A b-a B) c x}{x^{3/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{3 a x^{3/2}}}{a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\int -\frac {A b^2-a B b-a A c+(A b-a B) c x}{x^{3/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{3 a x^{3/2}}}{a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\int -\frac {a b B-A \left (b^2-a c\right )-(A b-a B) c x}{x^{3/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{3 a x^{3/2}}}{a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\int -\frac {A b^2-a B b-a A c+(A b-a B) c x}{x^{3/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{3 a x^{3/2}}}{a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\int -\frac {a b B-A \left (b^2-a c\right )-(A b-a B) c x}{x^{3/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{3 a x^{3/2}}}{a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\int -\frac {A b^2-a B b-a A c+(A b-a B) c x}{x^{3/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{3 a x^{3/2}}}{a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\int -\frac {a b B-A \left (b^2-a c\right )-(A b-a B) c x}{x^{3/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{3 a x^{3/2}}}{a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\int -\frac {A b^2-a B b-a A c+(A b-a B) c x}{x^{3/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{3 a x^{3/2}}}{a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\int -\frac {a b B-A \left (b^2-a c\right )-(A b-a B) c x}{x^{3/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{3 a x^{3/2}}}{a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\int -\frac {A b^2-a B b-a A c+(A b-a B) c x}{x^{3/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{3 a x^{3/2}}}{a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\int -\frac {a b B-A \left (b^2-a c\right )-(A b-a B) c x}{x^{3/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{3 a x^{3/2}}}{a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\int -\frac {A b^2-a B b-a A c+(A b-a B) c x}{x^{3/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{3 a x^{3/2}}}{a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\int -\frac {a b B-A \left (b^2-a c\right )-(A b-a B) c x}{x^{3/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{3 a x^{3/2}}}{a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\int -\frac {A b^2-a B b-a A c+(A b-a B) c x}{x^{3/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{3 a x^{3/2}}}{a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\int -\frac {a b B-A \left (b^2-a c\right )-(A b-a B) c x}{x^{3/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{3 a x^{3/2}}}{a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\int -\frac {A b^2-a B b-a A c+(A b-a B) c x}{x^{3/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{3 a x^{3/2}}}{a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\int -\frac {a b B-A \left (b^2-a c\right )-(A b-a B) c x}{x^{3/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{3 a x^{3/2}}}{a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\int -\frac {A b^2-a B b-a A c+(A b-a B) c x}{x^{3/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{3 a x^{3/2}}}{a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\int -\frac {a b B-A \left (b^2-a c\right )-(A b-a B) c x}{x^{3/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{3 a x^{3/2}}}{a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\int -\frac {A b^2-a B b-a A c+(A b-a B) c x}{x^{3/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{3 a x^{3/2}}}{a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\int -\frac {a b B-A \left (b^2-a c\right )-(A b-a B) c x}{x^{3/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{3 a x^{3/2}}}{a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\int -\frac {A b^2-a B b-a A c+(A b-a B) c x}{x^{3/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{3 a x^{3/2}}}{a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\int -\frac {a b B-A \left (b^2-a c\right )-(A b-a B) c x}{x^{3/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{3 a x^{3/2}}}{a}-\frac {2 A}{5 a x^{5/2}}\) |
3.11.15.3.1 Defintions of rubi rules used
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*f - d*g)*((d + e*x)^(m + 1)/((m + 1)*(c *d^2 - b*d*e + a*e^2))), x] + Simp[1/(c*d^2 - b*d*e + a*e^2) Int[(d + e*x )^(m + 1)*(Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x]/(a + b*x + c*x^ 2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && FractionQ[m] && LtQ[m, -1 ]
Time = 0.40 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.02
| method | result | size |
| derivativedivides | \(\frac {8 c \left (-\frac {\left (A \sqrt {-4 a c +b^{2}}\, a c -A \sqrt {-4 a c +b^{2}}\, b^{2}+3 A a b c -A \,b^{3}+a b B \sqrt {-4 a c +b^{2}}-2 B \,a^{2} c +B a \,b^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (A \sqrt {-4 a c +b^{2}}\, a c -A \sqrt {-4 a c +b^{2}}\, b^{2}-3 A a b c +A \,b^{3}+a b B \sqrt {-4 a c +b^{2}}+2 B \,a^{2} c -B a \,b^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{a^{3}}-\frac {2 A}{5 a \,x^{\frac {5}{2}}}-\frac {2 \left (-A b +B a \right )}{3 a^{2} x^{\frac {3}{2}}}-\frac {2 \left (-A a c +A \,b^{2}-a b B \right )}{a^{3} \sqrt {x}}\) | \(314\) |
| default | \(\frac {8 c \left (-\frac {\left (A \sqrt {-4 a c +b^{2}}\, a c -A \sqrt {-4 a c +b^{2}}\, b^{2}+3 A a b c -A \,b^{3}+a b B \sqrt {-4 a c +b^{2}}-2 B \,a^{2} c +B a \,b^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (A \sqrt {-4 a c +b^{2}}\, a c -A \sqrt {-4 a c +b^{2}}\, b^{2}-3 A a b c +A \,b^{3}+a b B \sqrt {-4 a c +b^{2}}+2 B \,a^{2} c -B a \,b^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{a^{3}}-\frac {2 A}{5 a \,x^{\frac {5}{2}}}-\frac {2 \left (-A b +B a \right )}{3 a^{2} x^{\frac {3}{2}}}-\frac {2 \left (-A a c +A \,b^{2}-a b B \right )}{a^{3} \sqrt {x}}\) | \(314\) |
| risch | \(-\frac {2 \left (-15 a A c \,x^{2}+15 A \,b^{2} x^{2}-15 B a b \,x^{2}-5 a A b x +5 a^{2} B x +3 A \,a^{2}\right )}{15 a^{3} x^{\frac {5}{2}}}+\frac {8 c \left (-\frac {\left (A \sqrt {-4 a c +b^{2}}\, a c -A \sqrt {-4 a c +b^{2}}\, b^{2}+3 A a b c -A \,b^{3}+a b B \sqrt {-4 a c +b^{2}}-2 B \,a^{2} c +B a \,b^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (A \sqrt {-4 a c +b^{2}}\, a c -A \sqrt {-4 a c +b^{2}}\, b^{2}-3 A a b c +A \,b^{3}+a b B \sqrt {-4 a c +b^{2}}+2 B \,a^{2} c -B a \,b^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{a^{3}}\) | \(318\) |
8/a^3*c*(-1/8*(A*(-4*a*c+b^2)^(1/2)*a*c-A*(-4*a*c+b^2)^(1/2)*b^2+3*A*a*b*c -A*b^3+a*b*B*(-4*a*c+b^2)^(1/2)-2*B*a^2*c+B*a*b^2)/(-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*(A*(-4*a*c+b^2)^(1/2)*a*c-A*(-4*a*c+b^2)^(1/2 )*b^2-3*A*a*b*c+A*b^3+a*b*B*(-4*a*c+b^2)^(1/2)+2*B*a^2*c-B*a*b^2)/(-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)))-2/5*A/a/x^(5/2)-2/3*(-A*b+B*a)/a^2/x ^(3/2)-2*(-A*a*c+A*b^2-B*a*b)/a^3/x^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 7971 vs. \(2 (252) = 504\).
Time = 6.81 (sec) , antiderivative size = 7971, normalized size of antiderivative = 25.96 \[ \int \frac {A+B x}{x^{7/2} \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {A+B x}{x^{7/2} \left (a+b x+c x^2\right )} \, dx=\text {Timed out} \]
\[ \int \frac {A+B x}{x^{7/2} \left (a+b x+c x^2\right )} \, dx=\int { \frac {B x + A}{{\left (c x^{2} + b x + a\right )} x^{\frac {7}{2}}} \,d x } \]
-2/15*(3*A*a^3/x^(5/2) + 15*((b^3 - 2*a*b*c)*A - (a*b^2 - a^2*c)*B)*sqrt(x ) - 15*(B*a^2*b - (a*b^2 - a^2*c)*A)/sqrt(x) + 5*(B*a^3 - A*a^2*b)/x^(3/2) )/a^4 - integrate(-(((b^3*c - 2*a*b*c^2)*A - (a*b^2*c - a^2*c^2)*B)*x^(3/2 ) + ((b^4 - 3*a*b^2*c + a^2*c^2)*A - (a*b^3 - 2*a^2*b*c)*B)*sqrt(x))/(a^4* c*x^2 + a^4*b*x + a^5), x)
Leaf count of result is larger than twice the leaf count of optimal. 5013 vs. \(2 (252) = 504\).
Time = 1.34 (sec) , antiderivative size = 5013, normalized size of antiderivative = 16.33 \[ \int \frac {A+B x}{x^{7/2} \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \]
-1/4*((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)*A*a^2 - (2*a*b^5*c^2 - 16*a^2*b^3*c^3 + 3 2*a^3*b*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a* b^5 + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^3* c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4*c - 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b*c^2 - 8 *sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^2 - s qrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^2 + 4*sqr t(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^3 - 2*(b^2 - 4*a*c)*a*b^3*c^2 + 8*(b^2 - 4*a*c)*a^2*b*c^3)*B*a^2 + 2*(sqrt(2)*sqrt...
Time = 12.90 (sec) , antiderivative size = 13983, normalized size of antiderivative = 45.55 \[ \int \frac {A+B x}{x^{7/2} \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \]
atan(((x^(1/2)*(16*A^2*a^12*c^6 - 16*B^2*a^13*c^5 - 8*A^2*a^9*b^6*c^3 + 48 *A^2*a^10*b^4*c^4 - 72*A^2*a^11*b^2*c^5 - 8*B^2*a^11*b^4*c^3 + 32*B^2*a^12 *b^2*c^4 + 16*A*B*a^10*b^5*c^3 - 80*A*B*a^11*b^3*c^4 + 80*A*B*a^12*b*c^5) + (-(A^2*b^9 + B^2*a^2*b^7 + A^2*b^6*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^ 8 + 42*A^2*a^2*b^5*c^2 - 63*A^2*a^3*b^3*c^3 - A^2*a^3*c^3*(-(4*a*c - b^2)^ 3)^(1/2) + B^2*a^2*b^4*(-(4*a*c - b^2)^3)^(1/2) + 25*B^2*a^4*b^3*c^2 + B^2 *a^4*c^2*(-(4*a*c - b^2)^3)^(1/2) - 16*A*B*a^5*c^4 - 11*A^2*a*b^7*c + 28*A ^2*a^4*b*c^4 - 9*B^2*a^3*b^5*c - 20*B^2*a^5*b*c^3 + 6*A^2*a^2*b^2*c^2*(-(4 *a*c - b^2)^3)^(1/2) - 66*A*B*a^3*b^4*c^2 + 76*A*B*a^4*b^2*c^3 - 5*A^2*a*b ^4*c*(-(4*a*c - b^2)^3)^(1/2) - 3*B^2*a^3*b^2*c*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^5*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a^2*b^6*c + 8*A*B*a^2*b^3*c *(-(4*a*c - b^2)^3)^(1/2) - 6*A*B*a^3*b*c^2*(-(4*a*c - b^2)^3)^(1/2))/(2*( a^7*b^4 + 16*a^9*c^2 - 8*a^8*b^2*c)))^(1/2)*(x^(1/2)*(32*a^16*b*c^3 - 8*a^ 15*b^3*c^2)*(-(A^2*b^9 + B^2*a^2*b^7 + A^2*b^6*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^8 + 42*A^2*a^2*b^5*c^2 - 63*A^2*a^3*b^3*c^3 - A^2*a^3*c^3*(-(4*a *c - b^2)^3)^(1/2) + B^2*a^2*b^4*(-(4*a*c - b^2)^3)^(1/2) + 25*B^2*a^4*b^3 *c^2 + B^2*a^4*c^2*(-(4*a*c - b^2)^3)^(1/2) - 16*A*B*a^5*c^4 - 11*A^2*a*b^ 7*c + 28*A^2*a^4*b*c^4 - 9*B^2*a^3*b^5*c - 20*B^2*a^5*b*c^3 + 6*A^2*a^2*b^ 2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 66*A*B*a^3*b^4*c^2 + 76*A*B*a^4*b^2*c^3 - 5*A^2*a*b^4*c*(-(4*a*c - b^2)^3)^(1/2) - 3*B^2*a^3*b^2*c*(-(4*a*c - b^...