Integrand size = 23, antiderivative size = 381 \[ \int \frac {A+B x}{x^{9/2} \left (a+b x+c x^2\right )} \, dx=-\frac {2 A}{7 a x^{7/2}}+\frac {2 (A b-a B)}{5 a^2 x^{5/2}}-\frac {2 \left (A b^2-a b B-a A c\right )}{3 a^3 x^{3/2}}-\frac {2 \left (a B \left (b^2-a c\right )-A \left (b^3-2 a b c\right )\right )}{a^4 \sqrt {x}}-\frac {\sqrt {2} \sqrt {c} \left (a B \left (b^2-a c\right )-A \left (b^3-2 a b c\right )+\frac {a b B \left (b^2-3 a c\right )-A \left (b^4-4 a b^2 c+2 a^2 c^2\right )}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{a^4 \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \sqrt {c} \left (a B \left (b^2-a c\right )-A \left (b^3-2 a b c\right )-\frac {a b B \left (b^2-3 a c\right )-A \left (b^4-4 a b^2 c+2 a^2 c^2\right )}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{a^4 \sqrt {b+\sqrt {b^2-4 a c}}} \] Output:
-2/7*A/a/x^(7/2)+2/5*(A*b-B*a)/a^2/x^(5/2)-2/3*(-A*a*c+A*b^2-B*a*b)/a^3/x^ (3/2)-2*(a*B*(-a*c+b^2)-A*(-2*a*b*c+b^3))/a^4/x^(1/2)-2^(1/2)*c^(1/2)*(a*B *(-a*c+b^2)-A*(-2*a*b*c+b^3)+(a*b*B*(-3*a*c+b^2)-A*(2*a^2*c^2-4*a*b^2*c+b^ 4))/(-4*a*c+b^2)^(1/2))*arctan(2^(1/2)*c^(1/2)*x^(1/2)/(b-(-4*a*c+b^2)^(1/ 2))^(1/2))/a^4/(b-(-4*a*c+b^2)^(1/2))^(1/2)-2^(1/2)*c^(1/2)*(a*B*(-a*c+b^2 )-A*(-2*a*b*c+b^3)-(a*b*B*(-3*a*c+b^2)-A*(2*a^2*c^2-4*a*b^2*c+b^4))/(-4*a* c+b^2)^(1/2))*arctan(2^(1/2)*c^(1/2)*x^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)) /a^4/(b+(-4*a*c+b^2)^(1/2))^(1/2)
Time = 2.14 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.17 \[ \int \frac {A+B x}{x^{9/2} \left (a+b x+c x^2\right )} \, dx=\frac {\frac {210 A b^3 x^3-6 a^3 (5 A+7 B x)+14 a^2 x (5 B x (b+3 c x)+A (3 b+5 c x))-70 a b x^2 (3 b B x+A (b+6 c x))}{x^{7/2}}+\frac {105 \sqrt {2} \sqrt {c} \left (a B \left (-b^3+3 a b c-b^2 \sqrt {b^2-4 a c}+a c \sqrt {b^2-4 a c}\right )+A \left (b^4-4 a b^2 c+2 a^2 c^2+b^3 \sqrt {b^2-4 a c}-2 a b c \sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {105 \sqrt {2} \sqrt {c} \left (a B \left (b^3-3 a b c-b^2 \sqrt {b^2-4 a c}+a c \sqrt {b^2-4 a c}\right )+A \left (-b^4+4 a b^2 c-2 a^2 c^2+b^3 \sqrt {b^2-4 a c}-2 a b c \sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}}}{105 a^4} \] Input:
Integrate[(A + B*x)/(x^(9/2)*(a + b*x + c*x^2)),x]
Output:
((210*A*b^3*x^3 - 6*a^3*(5*A + 7*B*x) + 14*a^2*x*(5*B*x*(b + 3*c*x) + A*(3 *b + 5*c*x)) - 70*a*b*x^2*(3*b*B*x + A*(b + 6*c*x)))/x^(7/2) + (105*Sqrt[2 ]*Sqrt[c]*(a*B*(-b^3 + 3*a*b*c - b^2*Sqrt[b^2 - 4*a*c] + a*c*Sqrt[b^2 - 4* a*c]) + A*(b^4 - 4*a*b^2*c + 2*a^2*c^2 + b^3*Sqrt[b^2 - 4*a*c] - 2*a*b*c*S qrt[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]]) + (105*Sqrt[2]*Sqr t[c]*(a*B*(b^3 - 3*a*b*c - b^2*Sqrt[b^2 - 4*a*c] + a*c*Sqrt[b^2 - 4*a*c]) + A*(-b^4 + 4*a*b^2*c - 2*a^2*c^2 + b^3*Sqrt[b^2 - 4*a*c] - 2*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[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(105*a^4)
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.
| \(\displaystyle \int \frac {A+B x}{x^{9/2} \left (a+b x+c x^2\right )} \, dx\) |
| \(\Big \downarrow \) 1198 |
| \(\displaystyle \frac {\int -\frac {A b-a B+A c x}{x^{7/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 A}{7 a x^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {A b-a B+A c x}{x^{7/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 A}{7 a x^{7/2}}\) |
| \(\Big \downarrow \) 1198 |
| \(\displaystyle -\frac {\frac {\int -\frac {A b^2-a B b-a A c+(A b-a B) c x}{x^{5/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{5 a x^{5/2}}}{a}-\frac {2 A}{7 a x^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\int -\frac {a b B-A \left (b^2-a c\right )-(A b-a B) c x}{x^{5/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{5 a x^{5/2}}}{a}-\frac {2 A}{7 a x^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\int -\frac {A b^2-a B b-a A c+(A b-a B) c x}{x^{5/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{5 a x^{5/2}}}{a}-\frac {2 A}{7 a x^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\int -\frac {a b B-A \left (b^2-a c\right )-(A b-a B) c x}{x^{5/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{5 a x^{5/2}}}{a}-\frac {2 A}{7 a x^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\int -\frac {A b^2-a B b-a A c+(A b-a B) c x}{x^{5/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{5 a x^{5/2}}}{a}-\frac {2 A}{7 a x^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\int -\frac {a b B-A \left (b^2-a c\right )-(A b-a B) c x}{x^{5/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{5 a x^{5/2}}}{a}-\frac {2 A}{7 a x^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\int -\frac {A b^2-a B b-a A c+(A b-a B) c x}{x^{5/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{5 a x^{5/2}}}{a}-\frac {2 A}{7 a x^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\int -\frac {a b B-A \left (b^2-a c\right )-(A b-a B) c x}{x^{5/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{5 a x^{5/2}}}{a}-\frac {2 A}{7 a x^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\int -\frac {A b^2-a B b-a A c+(A b-a B) c x}{x^{5/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{5 a x^{5/2}}}{a}-\frac {2 A}{7 a x^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\int -\frac {a b B-A \left (b^2-a c\right )-(A b-a B) c x}{x^{5/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{5 a x^{5/2}}}{a}-\frac {2 A}{7 a x^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\int -\frac {A b^2-a B b-a A c+(A b-a B) c x}{x^{5/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{5 a x^{5/2}}}{a}-\frac {2 A}{7 a x^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\int -\frac {a b B-A \left (b^2-a c\right )-(A b-a B) c x}{x^{5/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{5 a x^{5/2}}}{a}-\frac {2 A}{7 a x^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\int -\frac {A b^2-a B b-a A c+(A b-a B) c x}{x^{5/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{5 a x^{5/2}}}{a}-\frac {2 A}{7 a x^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\int -\frac {a b B-A \left (b^2-a c\right )-(A b-a B) c x}{x^{5/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{5 a x^{5/2}}}{a}-\frac {2 A}{7 a x^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\int -\frac {A b^2-a B b-a A c+(A b-a B) c x}{x^{5/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{5 a x^{5/2}}}{a}-\frac {2 A}{7 a x^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\int -\frac {a b B-A \left (b^2-a c\right )-(A b-a B) c x}{x^{5/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{5 a x^{5/2}}}{a}-\frac {2 A}{7 a x^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\int -\frac {A b^2-a B b-a A c+(A b-a B) c x}{x^{5/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{5 a x^{5/2}}}{a}-\frac {2 A}{7 a x^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\int -\frac {a b B-A \left (b^2-a c\right )-(A b-a B) c x}{x^{5/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{5 a x^{5/2}}}{a}-\frac {2 A}{7 a x^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\int -\frac {A b^2-a B b-a A c+(A b-a B) c x}{x^{5/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{5 a x^{5/2}}}{a}-\frac {2 A}{7 a x^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\int -\frac {a b B-A \left (b^2-a c\right )-(A b-a B) c x}{x^{5/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{5 a x^{5/2}}}{a}-\frac {2 A}{7 a x^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\int -\frac {A b^2-a B b-a A c+(A b-a B) c x}{x^{5/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{5 a x^{5/2}}}{a}-\frac {2 A}{7 a x^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\int -\frac {a b B-A \left (b^2-a c\right )-(A b-a B) c x}{x^{5/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{5 a x^{5/2}}}{a}-\frac {2 A}{7 a x^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\int -\frac {A b^2-a B b-a A c+(A b-a B) c x}{x^{5/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{5 a x^{5/2}}}{a}-\frac {2 A}{7 a x^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\int -\frac {a b B-A \left (b^2-a c\right )-(A b-a B) c x}{x^{5/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{5 a x^{5/2}}}{a}-\frac {2 A}{7 a x^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\int -\frac {A b^2-a B b-a A c+(A b-a B) c x}{x^{5/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{5 a x^{5/2}}}{a}-\frac {2 A}{7 a x^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\int -\frac {a b B-A \left (b^2-a c\right )-(A b-a B) c x}{x^{5/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{5 a x^{5/2}}}{a}-\frac {2 A}{7 a x^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\int -\frac {A b^2-a B b-a A c+(A b-a B) c x}{x^{5/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{5 a x^{5/2}}}{a}-\frac {2 A}{7 a x^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\int -\frac {a b B-A \left (b^2-a c\right )-(A b-a B) c x}{x^{5/2} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 (A b-a B)}{5 a x^{5/2}}}{a}-\frac {2 A}{7 a x^{7/2}}\) |
Input:
Int[(A + B*x)/(x^(9/2)*(a + b*x + c*x^2)),x]
Output:
$Aborted
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 = 1.18 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.08
| method | result | size |
| derivativedivides | \(\frac {8 c \left (-\frac {\left (-2 A a b c \sqrt {-4 a c +b^{2}}+A \,b^{3} \sqrt {-4 a c +b^{2}}+2 a^{2} A \,c^{2}-4 A a \,b^{2} c +A \,b^{4}+B \,a^{2} c \sqrt {-4 a c +b^{2}}-B a \,b^{2} \sqrt {-4 a c +b^{2}}+3 a^{2} b B c -B a \,b^{3}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (-2 A a b c \sqrt {-4 a c +b^{2}}+A \,b^{3} \sqrt {-4 a c +b^{2}}-2 a^{2} A \,c^{2}+4 A a \,b^{2} c -A \,b^{4}+B \,a^{2} c \sqrt {-4 a c +b^{2}}-B a \,b^{2} \sqrt {-4 a c +b^{2}}-3 a^{2} b B c +B a \,b^{3}\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {x}\, c \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{a^{4}}-\frac {2 A}{7 a \,x^{\frac {7}{2}}}-\frac {2 \left (-A b +B a \right )}{5 a^{2} x^{\frac {5}{2}}}-\frac {2 \left (-A a c +b^{2} A -a b B \right )}{3 a^{3} x^{\frac {3}{2}}}-\frac {2 \left (2 A a b c -A \,b^{3}-B \,a^{2} c +B a \,b^{2}\right )}{a^{4} \sqrt {x}}\) | \(412\) |
| default | \(\frac {8 c \left (-\frac {\left (-2 A a b c \sqrt {-4 a c +b^{2}}+A \,b^{3} \sqrt {-4 a c +b^{2}}+2 a^{2} A \,c^{2}-4 A a \,b^{2} c +A \,b^{4}+B \,a^{2} c \sqrt {-4 a c +b^{2}}-B a \,b^{2} \sqrt {-4 a c +b^{2}}+3 a^{2} b B c -B a \,b^{3}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (-2 A a b c \sqrt {-4 a c +b^{2}}+A \,b^{3} \sqrt {-4 a c +b^{2}}-2 a^{2} A \,c^{2}+4 A a \,b^{2} c -A \,b^{4}+B \,a^{2} c \sqrt {-4 a c +b^{2}}-B a \,b^{2} \sqrt {-4 a c +b^{2}}-3 a^{2} b B c +B a \,b^{3}\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {x}\, c \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{a^{4}}-\frac {2 A}{7 a \,x^{\frac {7}{2}}}-\frac {2 \left (-A b +B a \right )}{5 a^{2} x^{\frac {5}{2}}}-\frac {2 \left (-A a c +b^{2} A -a b B \right )}{3 a^{3} x^{\frac {3}{2}}}-\frac {2 \left (2 A a b c -A \,b^{3}-B \,a^{2} c +B a \,b^{2}\right )}{a^{4} \sqrt {x}}\) | \(412\) |
| risch | \(-\frac {2 \left (210 A a b c \,x^{3}-105 A \,b^{3} x^{3}-105 B \,a^{2} c \,x^{3}+105 B a \,b^{2} x^{3}-35 A \,a^{2} c \,x^{2}+35 A a \,b^{2} x^{2}-35 B \,a^{2} b \,x^{2}-21 A \,a^{2} b x +21 B \,a^{3} x +15 a^{3} A \right )}{105 a^{4} x^{\frac {7}{2}}}-\frac {8 c \left (-\frac {\left (2 A a b c \sqrt {-4 a c +b^{2}}-A \,b^{3} \sqrt {-4 a c +b^{2}}-2 a^{2} A \,c^{2}+4 A a \,b^{2} c -A \,b^{4}-B \,a^{2} c \sqrt {-4 a c +b^{2}}+B a \,b^{2} \sqrt {-4 a c +b^{2}}-3 a^{2} b B c +B a \,b^{3}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (2 A a b c \sqrt {-4 a c +b^{2}}-A \,b^{3} \sqrt {-4 a c +b^{2}}+2 a^{2} A \,c^{2}-4 A a \,b^{2} c +A \,b^{4}-B \,a^{2} c \sqrt {-4 a c +b^{2}}+B a \,b^{2} \sqrt {-4 a c +b^{2}}+3 a^{2} b B c -B a \,b^{3}\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {x}\, c \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{a^{4}}\) | \(429\) |
Input:
int((B*x+A)/x^(9/2)/(c*x^2+b*x+a),x,method=_RETURNVERBOSE)
Output:
8/a^4*c*(-1/8*(-2*A*a*b*c*(-4*a*c+b^2)^(1/2)+A*b^3*(-4*a*c+b^2)^(1/2)+2*a^ 2*A*c^2-4*A*a*b^2*c+A*b^4+B*a^2*c*(-4*a*c+b^2)^(1/2)-B*a*b^2*(-4*a*c+b^2)^ (1/2)+3*a^2*b*B*c-B*a*b^3)/(-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*a*b*c*(-4*a*c+b^2)^(1/2)+A*b^3*(-4*a*c+b^2)^(1/2)-2*a^2*A*c^2+4 *A*a*b^2*c-A*b^4+B*a^2*c*(-4*a*c+b^2)^(1/2)-B*a*b^2*(-4*a*c+b^2)^(1/2)-3*a ^2*b*B*c+B*a*b^3)/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1 /2)*arctan(x^(1/2)*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)))-2/7*A/a/x^ (7/2)-2/5*(-A*b+B*a)/a^2/x^(5/2)-2/3*(-A*a*c+A*b^2-B*a*b)/a^3/x^(3/2)-2*(2 *A*a*b*c-A*b^3-B*a^2*c+B*a*b^2)/a^4/x^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 10514 vs. \(2 (326) = 652\).
Time = 6.37 (sec) , antiderivative size = 10514, normalized size of antiderivative = 27.60 \[ \int \frac {A+B x}{x^{9/2} \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \] Input:
integrate((B*x+A)/x^(9/2)/(c*x^2+b*x+a),x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {A+B x}{x^{9/2} \left (a+b x+c x^2\right )} \, dx=\text {Timed out} \] Input:
integrate((B*x+A)/x**(9/2)/(c*x**2+b*x+a),x)
Output:
Timed out
\[ \int \frac {A+B x}{x^{9/2} \left (a+b x+c x^2\right )} \, dx=\int { \frac {B x + A}{{\left (c x^{2} + b x + a\right )} x^{\frac {9}{2}}} \,d x } \] Input:
integrate((B*x+A)/x^(9/2)/(c*x^2+b*x+a),x, algorithm="maxima")
Output:
-2/105*(15*A*a^4/x^(7/2) - 105*((b^4 - 3*a*b^2*c + a^2*c^2)*A - (a*b^3 - 2 *a^2*b*c)*B)*sqrt(x) - 105*((a*b^3 - 2*a^2*b*c)*A - (a^2*b^2 - a^3*c)*B)/s qrt(x) - 35*(B*a^3*b - (a^2*b^2 - a^3*c)*A)/x^(3/2) + 21*(B*a^4 - A*a^3*b) /x^(5/2))/a^5 - integrate((((b^4*c - 3*a*b^2*c^2 + a^2*c^3)*A - (a*b^3*c - 2*a^2*b*c^2)*B)*x^(3/2) + ((b^5 - 4*a*b^3*c + 3*a^2*b*c^2)*A - (a*b^4 - 3 *a^2*b^2*c + a^3*c^2)*B)*sqrt(x))/(a^5*c*x^2 + a^5*b*x + a^6), x)
Leaf count of result is larger than twice the leaf count of optimal. 4090 vs. \(2 (326) = 652\).
Time = 1.06 (sec) , antiderivative size = 4090, normalized size of antiderivative = 10.73 \[ \int \frac {A+B x}{x^{9/2} \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \] Input:
integrate((B*x+A)/x^(9/2)/(c*x^2+b*x+a),x, algorithm="giac")
Output:
1/2*((sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^8 - 11*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^6*c - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b ^7*c - 2*b^8*c + 41*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c^2 + 14*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^5*c^2 + sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^6*c^2 + 22*a*b^6*c^2 + 2*b^7*c^2 - 56*sqrt(2)*sqrt( b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^2*c^3 - 26*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c^3 - 7*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4*c^ 3 - 82*a^2*b^4*c^3 - 18*a*b^5*c^3 + 16*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c )*c)*a^4*c^4 + 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b*c^4 + 13*sq rt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^4 + 112*a^3*b^2*c^4 + 50*a ^2*b^3*c^4 - 4*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*c^5 - 32*a^4*c^ 5 - 40*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 + 9*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 - 2 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c^2 - 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4*c^2 - s qrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^5*c^2 + 20*sqrt (2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b*c^3 + 10*sqrt( 2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^3 + 5*sqrt( 2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^3 - 5*sqrt...
Time = 15.33 (sec) , antiderivative size = 17910, normalized size of antiderivative = 47.01 \[ \int \frac {A+B x}{x^{9/2} \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \] Input:
int((A + B*x)/(x^(9/2)*(a + b*x + c*x^2)),x)
Output:
((2*x^3*(A*b^3 - B*a*b^2 + B*a^2*c - 2*A*a*b*c))/a^4 - (2*A)/(7*a) + (2*x^ 2*(A*a*c - A*b^2 + B*a*b))/(3*a^3) + (2*x*(A*b - B*a))/(5*a^2))/x^(7/2) + atan((((-(A^2*b^11 + B^2*a^2*b^9 + A^2*b^8*(-(4*a*c - b^2)^3)^(1/2) - 2*A* B*a*b^10 + 63*A^2*a^2*b^7*c^2 - 138*A^2*a^3*b^5*c^3 + 129*A^2*a^4*b^3*c^4 + A^2*a^4*c^4*(-(4*a*c - b^2)^3)^(1/2) + B^2*a^2*b^6*(-(4*a*c - b^2)^3)^(1 /2) + 42*B^2*a^4*b^5*c^2 - 63*B^2*a^5*b^3*c^3 - B^2*a^5*c^3*(-(4*a*c - b^2 )^3)^(1/2) + 16*A*B*a^6*c^5 - 13*A^2*a*b^9*c - 36*A^2*a^5*b*c^5 - 11*B^2*a ^3*b^7*c + 28*B^2*a^6*b*c^4 + 15*A^2*a^2*b^4*c^2*(-(4*a*c - b^2)^3)^(1/2) - 10*A^2*a^3*b^2*c^3*(-(4*a*c - b^2)^3)^(1/2) + 6*B^2*a^4*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 104*A*B*a^3*b^6*c^2 + 192*A*B*a^4*b^4*c^3 - 132*A*B*a^5 *b^2*c^4 - 7*A^2*a*b^6*c*(-(4*a*c - b^2)^3)^(1/2) - 5*B^2*a^3*b^4*c*(-(4*a *c - b^2)^3)^(1/2) - 2*A*B*a*b^7*(-(4*a*c - b^2)^3)^(1/2) + 24*A*B*a^2*b^8 *c + 12*A*B*a^2*b^5*c*(-(4*a*c - b^2)^3)^(1/2) + 8*A*B*a^4*b*c^3*(-(4*a*c - b^2)^3)^(1/2) - 20*A*B*a^3*b^3*c^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^9*b^4 + 16*a^11*c^2 - 8*a^10*b^2*c)))^(1/2)*(32*A*a^19*c^5 + x^(1/2)*(32*a^21*b *c^3 - 8*a^20*b^3*c^2)*(-(A^2*b^11 + B^2*a^2*b^9 + A^2*b^8*(-(4*a*c - b^2) ^3)^(1/2) - 2*A*B*a*b^10 + 63*A^2*a^2*b^7*c^2 - 138*A^2*a^3*b^5*c^3 + 129* A^2*a^4*b^3*c^4 + A^2*a^4*c^4*(-(4*a*c - b^2)^3)^(1/2) + B^2*a^2*b^6*(-(4* a*c - b^2)^3)^(1/2) + 42*B^2*a^4*b^5*c^2 - 63*B^2*a^5*b^3*c^3 - B^2*a^5*c^ 3*(-(4*a*c - b^2)^3)^(1/2) + 16*A*B*a^6*c^5 - 13*A^2*a*b^9*c - 36*A^2*a...
Time = 0.46 (sec) , antiderivative size = 896, normalized size of antiderivative = 2.35 \[ \int \frac {A+B x}{x^{9/2} \left (a+b x+c x^2\right )} \, dx =\text {Too large to display} \] Input:
int((B*x+A)/x^(9/2)/(c*x^2+b*x+a),x)
Output:
(126*sqrt(x)*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt (a) - b) - 2*sqrt(x)*sqrt(c))/sqrt(2*sqrt(c)*sqrt(a) + b))*a*b*c**2*x**3 - 42*sqrt(x)*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt( a) - b) - 2*sqrt(x)*sqrt(c))/sqrt(2*sqrt(c)*sqrt(a) + b))*b**3*c*x**3 - 84 *sqrt(x)*sqrt(c)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(x)*sqrt(c))/sqrt(2*sqrt(c)*sqrt(a) + b))*a**2*c**2*x**3 + 42 *sqrt(x)*sqrt(c)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(x)*sqrt(c))/sqrt(2*sqrt(c)*sqrt(a) + b))*a*b**2*c*x**3 - 126 *sqrt(x)*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) + 2*sqrt(x)*sqrt(c))/sqrt(2*sqrt(c)*sqrt(a) + b))*a*b*c**2*x**3 + 42* sqrt(x)*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) + 2*sqrt(x)*sqrt(c))/sqrt(2*sqrt(c)*sqrt(a) + b))*b**3*c*x**3 + 84*sqr t(x)*sqrt(c)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) + 2*sqrt(x)*sqrt(c))/sqrt(2*sqrt(c)*sqrt(a) + b))*a**2*c**2*x**3 - 42*sqr t(x)*sqrt(c)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) + 2*sqrt(x)*sqrt(c))/sqrt(2*sqrt(c)*sqrt(a) + b))*a*b**2*c*x**3 - 63*sqrt (x)*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) - b)*log( - sqrt(x)*sqrt(2*sqrt(c)*sqrt (a) - b) + sqrt(a) + sqrt(c)*x)*a*b*c**2*x**3 + 21*sqrt(x)*sqrt(a)*sqrt(2* sqrt(c)*sqrt(a) - b)*log( - sqrt(x)*sqrt(2*sqrt(c)*sqrt(a) - b) + sqrt(a) + sqrt(c)*x)*b**3*c*x**3 + 63*sqrt(x)*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) - ...