Integrand size = 23, antiderivative size = 231 \[ \int \frac {A+B x}{x^3 \left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 \left (A b^2-a b B-2 a A c+(A b-2 a B) c x\right )}{a \left (b^2-4 a c\right ) x^2 \sqrt {a+b x+c x^2}}-\frac {\left (5 A b^2-4 a b B-12 a A c\right ) \sqrt {a+b x+c x^2}}{2 a^2 \left (b^2-4 a c\right ) x^2}-\frac {\left (4 a B \left (3 b^2-8 a c\right )-A \left (15 b^3-52 a b c\right )\right ) \sqrt {a+b x+c x^2}}{4 a^3 \left (b^2-4 a c\right ) x}-\frac {3 \left (5 A b^2-4 a b B-4 a A c\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 a^{7/2}} \]
-3/8*(-4*A*a*c+5*A*b^2-4*B*a*b)*arctanh(1/2*(b*x+2*a)/a^(1/2)/(c*x^2+b*x+a )^(1/2))/a^(7/2)+2*(A*b^2-a*b*B-2*A*a*c+(A*b-2*B*a)*c*x)/a/(-4*a*c+b^2)/x^ 2/(c*x^2+b*x+a)^(1/2)-1/2*(-12*A*a*c+5*A*b^2-4*B*a*b)*(c*x^2+b*x+a)^(1/2)/ a^2/(-4*a*c+b^2)/x^2-1/4*(4*a*B*(-8*a*c+3*b^2)-A*(-52*a*b*c+15*b^3))*(c*x^ 2+b*x+a)^(1/2)/a^3/(-4*a*c+b^2)/x
Time = 1.10 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.91 \[ \int \frac {A+B x}{x^3 \left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {\frac {\sqrt {a} \left (-8 a^3 c (A+2 B x)-15 A b^3 x^2 (b+c x)+2 a^2 \left (A \left (b^2+10 b c x-12 c^2 x^2\right )+2 B x \left (b^2-10 b c x-8 c^2 x^2\right )\right )+a b x \left (12 b B x (b+c x)+A \left (-5 b^2+62 b c x+52 c^2 x^2\right )\right )\right )}{x^2 \sqrt {a+x (b+c x)}}-3 \left (b^2-4 a c\right ) \left (5 A b^2-4 a b B-4 a A c\right ) \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{4 a^{7/2} \left (-b^2+4 a c\right )} \]
((Sqrt[a]*(-8*a^3*c*(A + 2*B*x) - 15*A*b^3*x^2*(b + c*x) + 2*a^2*(A*(b^2 + 10*b*c*x - 12*c^2*x^2) + 2*B*x*(b^2 - 10*b*c*x - 8*c^2*x^2)) + a*b*x*(12* b*B*x*(b + c*x) + A*(-5*b^2 + 62*b*c*x + 52*c^2*x^2))))/(x^2*Sqrt[a + x*(b + c*x)]) - 3*(b^2 - 4*a*c)*(5*A*b^2 - 4*a*b*B - 4*a*A*c)*ArcTanh[(Sqrt[c] *x - Sqrt[a + x*(b + c*x)])/Sqrt[a]])/(4*a^(7/2)*(-b^2 + 4*a*c))
Time = 0.50 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.03, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {1235, 27, 1237, 27, 25, 1228, 1154, 219}
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.
| \(\displaystyle \int \frac {A+B x}{x^3 \left (a+b x+c x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle \frac {2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{a x^2 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {2 \int -\frac {5 A b^2-4 a B b-12 a A c+4 (A b-2 a B) c x}{2 x^3 \sqrt {c x^2+b x+a}}dx}{a \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {5 A b^2-4 a B b-12 a A c+4 (A b-2 a B) c x}{x^3 \sqrt {c x^2+b x+a}}dx}{a \left (b^2-4 a c\right )}+\frac {2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{a x^2 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {-\frac {\int -\frac {4 a B \left (3 b^2-8 a c\right )-2 A \left (\frac {15 b^3}{2}-26 a b c\right )-2 c \left (5 A b^2-4 a B b-12 a A c\right ) x}{2 x^2 \sqrt {c x^2+b x+a}}dx}{2 a}-\frac {\sqrt {a+b x+c x^2} \left (-12 a A c-4 a b B+5 A b^2\right )}{2 a x^2}}{a \left (b^2-4 a c\right )}+\frac {2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{a x^2 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int -\frac {15 A b^3-12 a B b^2-52 a A c b+32 a^2 B c+2 c \left (5 A b^2-4 a B b-12 a A c\right ) x}{x^2 \sqrt {c x^2+b x+a}}dx}{4 a}-\frac {\sqrt {a+b x+c x^2} \left (-12 a A c-4 a b B+5 A b^2\right )}{2 a x^2}}{a \left (b^2-4 a c\right )}+\frac {2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{a x^2 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\int \frac {15 A b^3-12 a B b^2-52 a A c b+32 a^2 B c+2 c \left (5 A b^2-4 a B b-12 a A c\right ) x}{x^2 \sqrt {c x^2+b x+a}}dx}{4 a}-\frac {\sqrt {a+b x+c x^2} \left (-12 a A c-4 a b B+5 A b^2\right )}{2 a x^2}}{a \left (b^2-4 a c\right )}+\frac {2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{a x^2 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle \frac {-\frac {\frac {\sqrt {a+b x+c x^2} \left (4 a B \left (3 b^2-8 a c\right )-A \left (15 b^3-52 a b c\right )\right )}{a x}-\frac {3 \left (b^2-4 a c\right ) \left (-4 a A c-4 a b B+5 A b^2\right ) \int \frac {1}{x \sqrt {c x^2+b x+a}}dx}{2 a}}{4 a}-\frac {\sqrt {a+b x+c x^2} \left (-12 a A c-4 a b B+5 A b^2\right )}{2 a x^2}}{a \left (b^2-4 a c\right )}+\frac {2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{a x^2 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {-\frac {\frac {3 \left (b^2-4 a c\right ) \left (-4 a A c-4 a b B+5 A b^2\right ) \int \frac {1}{4 a-\frac {(2 a+b x)^2}{c x^2+b x+a}}d\frac {2 a+b x}{\sqrt {c x^2+b x+a}}}{a}+\frac {\sqrt {a+b x+c x^2} \left (4 a B \left (3 b^2-8 a c\right )-A \left (15 b^3-52 a b c\right )\right )}{a x}}{4 a}-\frac {\sqrt {a+b x+c x^2} \left (-12 a A c-4 a b B+5 A b^2\right )}{2 a x^2}}{a \left (b^2-4 a c\right )}+\frac {2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{a x^2 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {-\frac {\frac {3 \left (b^2-4 a c\right ) \left (-4 a A c-4 a b B+5 A b^2\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 a^{3/2}}+\frac {\sqrt {a+b x+c x^2} \left (4 a B \left (3 b^2-8 a c\right )-A \left (15 b^3-52 a b c\right )\right )}{a x}}{4 a}-\frac {\sqrt {a+b x+c x^2} \left (-12 a A c-4 a b B+5 A b^2\right )}{2 a x^2}}{a \left (b^2-4 a c\right )}+\frac {2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{a x^2 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
(2*(A*b^2 - a*b*B - 2*a*A*c + (A*b - 2*a*B)*c*x))/(a*(b^2 - 4*a*c)*x^2*Sqr t[a + b*x + c*x^2]) + (-1/2*((5*A*b^2 - 4*a*b*B - 12*a*A*c)*Sqrt[a + b*x + c*x^2])/(a*x^2) - (((4*a*B*(3*b^2 - 8*a*c) - A*(15*b^3 - 52*a*b*c))*Sqrt[ a + b*x + c*x^2])/(a*x) + (3*(b^2 - 4*a*c)*(5*A*b^2 - 4*a*b*B - 4*a*A*c)*A rcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(2*a^(3/2)))/(4*a)) /(a*(b^2 - 4*a*c))
3.10.68.3.1 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[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e *f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^ (m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x ] && EqQ[Simplify[m + 2*p + 3], 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 *a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m *(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] )
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b* x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1) *(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[ (c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 ] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Time = 0.83 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.61
| method | result | size |
| risch | \(-\frac {\sqrt {c \,x^{2}+b x +a}\, \left (-7 A b x +4 a B x +2 a A \right )}{4 a^{3} x^{2}}-\frac {-\frac {14 A \,b^{3} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {16 B \,a^{2} c \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {8 B a \,b^{2} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {8 A a b c \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\left (4 A a \,c^{2}-7 A \,b^{2} c +4 B a b c \right ) \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )+3 a \left (4 A a c -5 A \,b^{2}+4 a b B \right ) \left (\frac {1}{a \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{a \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {\ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{a^{\frac {3}{2}}}\right )}{8 a^{3}}\) | \(372\) |
| default | \(A \left (-\frac {1}{2 a \,x^{2} \sqrt {c \,x^{2}+b x +a}}-\frac {5 b \left (-\frac {1}{a x \sqrt {c \,x^{2}+b x +a}}-\frac {3 b \left (\frac {1}{a \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{a \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {\ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{a^{\frac {3}{2}}}\right )}{2 a}-\frac {4 c \left (2 c x +b \right )}{a \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{4 a}-\frac {3 c \left (\frac {1}{a \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{a \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {\ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{a^{\frac {3}{2}}}\right )}{2 a}\right )+B \left (-\frac {1}{a x \sqrt {c \,x^{2}+b x +a}}-\frac {3 b \left (\frac {1}{a \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{a \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {\ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{a^{\frac {3}{2}}}\right )}{2 a}-\frac {4 c \left (2 c x +b \right )}{a \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )\) | \(426\) |
-1/4*(c*x^2+b*x+a)^(1/2)*(-7*A*b*x+4*B*a*x+2*A*a)/a^3/x^2-1/8/a^3*(-14*A*b ^3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+16*B*a^2*c*(2*c*x+b)/(4*a*c-b ^2)/(c*x^2+b*x+a)^(1/2)+8*B*a*b^2*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2 )-8*A*a*b*c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+(4*A*a*c^2-7*A*b^2*c +4*B*a*b*c)*(-1/c/(c*x^2+b*x+a)^(1/2)-b/c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x +a)^(1/2))+3*a*(4*A*a*c-5*A*b^2+4*B*a*b)*(1/a/(c*x^2+b*x+a)^(1/2)-b/a*(2*c *x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-1/a^(3/2)*ln((2*a+b*x+2*a^(1/2)*(c*x ^2+b*x+a)^(1/2))/x)))
Leaf count of result is larger than twice the leaf count of optimal. 430 vs. \(2 (212) = 424\).
Time = 0.74 (sec) , antiderivative size = 869, normalized size of antiderivative = 3.76 \[ \int \frac {A+B x}{x^3 \left (a+b x+c x^2\right )^{3/2}} \, dx=\left [-\frac {3 \, {\left ({\left (16 \, A a^{2} c^{3} + 8 \, {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} c^{2} - {\left (4 \, B a b^{3} - 5 \, A b^{4}\right )} c\right )} x^{4} - {\left (4 \, B a b^{4} - 5 \, A b^{5} - 16 \, A a^{2} b c^{2} - 8 \, {\left (2 \, B a^{2} b^{2} - 3 \, A a b^{3}\right )} c\right )} x^{3} - {\left (4 \, B a^{2} b^{3} - 5 \, A a b^{4} - 16 \, A a^{3} c^{2} - 8 \, {\left (2 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} c\right )} x^{2}\right )} \sqrt {a} \log \left (-\frac {8 \, a b x + {\left (b^{2} + 4 \, a c\right )} x^{2} + 4 \, \sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{2}}\right ) + 4 \, {\left (2 \, A a^{3} b^{2} - 8 \, A a^{4} c - {\left (4 \, {\left (8 \, B a^{3} - 13 \, A a^{2} b\right )} c^{2} - 3 \, {\left (4 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} c\right )} x^{3} + {\left (12 \, B a^{2} b^{3} - 15 \, A a b^{4} - 24 \, A a^{3} c^{2} - 2 \, {\left (20 \, B a^{3} b - 31 \, A a^{2} b^{2}\right )} c\right )} x^{2} + {\left (4 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3} - 4 \, {\left (4 \, B a^{4} - 5 \, A a^{3} b\right )} c\right )} x\right )} \sqrt {c x^{2} + b x + a}}{16 \, {\left ({\left (a^{4} b^{2} c - 4 \, a^{5} c^{2}\right )} x^{4} + {\left (a^{4} b^{3} - 4 \, a^{5} b c\right )} x^{3} + {\left (a^{5} b^{2} - 4 \, a^{6} c\right )} x^{2}\right )}}, \frac {3 \, {\left ({\left (16 \, A a^{2} c^{3} + 8 \, {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} c^{2} - {\left (4 \, B a b^{3} - 5 \, A b^{4}\right )} c\right )} x^{4} - {\left (4 \, B a b^{4} - 5 \, A b^{5} - 16 \, A a^{2} b c^{2} - 8 \, {\left (2 \, B a^{2} b^{2} - 3 \, A a b^{3}\right )} c\right )} x^{3} - {\left (4 \, B a^{2} b^{3} - 5 \, A a b^{4} - 16 \, A a^{3} c^{2} - 8 \, {\left (2 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} c\right )} x^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{2} + a b x + a^{2}\right )}}\right ) - 2 \, {\left (2 \, A a^{3} b^{2} - 8 \, A a^{4} c - {\left (4 \, {\left (8 \, B a^{3} - 13 \, A a^{2} b\right )} c^{2} - 3 \, {\left (4 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} c\right )} x^{3} + {\left (12 \, B a^{2} b^{3} - 15 \, A a b^{4} - 24 \, A a^{3} c^{2} - 2 \, {\left (20 \, B a^{3} b - 31 \, A a^{2} b^{2}\right )} c\right )} x^{2} + {\left (4 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3} - 4 \, {\left (4 \, B a^{4} - 5 \, A a^{3} b\right )} c\right )} x\right )} \sqrt {c x^{2} + b x + a}}{8 \, {\left ({\left (a^{4} b^{2} c - 4 \, a^{5} c^{2}\right )} x^{4} + {\left (a^{4} b^{3} - 4 \, a^{5} b c\right )} x^{3} + {\left (a^{5} b^{2} - 4 \, a^{6} c\right )} x^{2}\right )}}\right ] \]
[-1/16*(3*((16*A*a^2*c^3 + 8*(2*B*a^2*b - 3*A*a*b^2)*c^2 - (4*B*a*b^3 - 5* A*b^4)*c)*x^4 - (4*B*a*b^4 - 5*A*b^5 - 16*A*a^2*b*c^2 - 8*(2*B*a^2*b^2 - 3 *A*a*b^3)*c)*x^3 - (4*B*a^2*b^3 - 5*A*a*b^4 - 16*A*a^3*c^2 - 8*(2*B*a^3*b - 3*A*a^2*b^2)*c)*x^2)*sqrt(a)*log(-(8*a*b*x + (b^2 + 4*a*c)*x^2 + 4*sqrt( c*x^2 + b*x + a)*(b*x + 2*a)*sqrt(a) + 8*a^2)/x^2) + 4*(2*A*a^3*b^2 - 8*A* a^4*c - (4*(8*B*a^3 - 13*A*a^2*b)*c^2 - 3*(4*B*a^2*b^2 - 5*A*a*b^3)*c)*x^3 + (12*B*a^2*b^3 - 15*A*a*b^4 - 24*A*a^3*c^2 - 2*(20*B*a^3*b - 31*A*a^2*b^ 2)*c)*x^2 + (4*B*a^3*b^2 - 5*A*a^2*b^3 - 4*(4*B*a^4 - 5*A*a^3*b)*c)*x)*sqr t(c*x^2 + b*x + a))/((a^4*b^2*c - 4*a^5*c^2)*x^4 + (a^4*b^3 - 4*a^5*b*c)*x ^3 + (a^5*b^2 - 4*a^6*c)*x^2), 1/8*(3*((16*A*a^2*c^3 + 8*(2*B*a^2*b - 3*A* a*b^2)*c^2 - (4*B*a*b^3 - 5*A*b^4)*c)*x^4 - (4*B*a*b^4 - 5*A*b^5 - 16*A*a^ 2*b*c^2 - 8*(2*B*a^2*b^2 - 3*A*a*b^3)*c)*x^3 - (4*B*a^2*b^3 - 5*A*a*b^4 - 16*A*a^3*c^2 - 8*(2*B*a^3*b - 3*A*a^2*b^2)*c)*x^2)*sqrt(-a)*arctan(1/2*sqr t(c*x^2 + b*x + a)*(b*x + 2*a)*sqrt(-a)/(a*c*x^2 + a*b*x + a^2)) - 2*(2*A* a^3*b^2 - 8*A*a^4*c - (4*(8*B*a^3 - 13*A*a^2*b)*c^2 - 3*(4*B*a^2*b^2 - 5*A *a*b^3)*c)*x^3 + (12*B*a^2*b^3 - 15*A*a*b^4 - 24*A*a^3*c^2 - 2*(20*B*a^3*b - 31*A*a^2*b^2)*c)*x^2 + (4*B*a^3*b^2 - 5*A*a^2*b^3 - 4*(4*B*a^4 - 5*A*a^ 3*b)*c)*x)*sqrt(c*x^2 + b*x + a))/((a^4*b^2*c - 4*a^5*c^2)*x^4 + (a^4*b^3 - 4*a^5*b*c)*x^3 + (a^5*b^2 - 4*a^6*c)*x^2)]
\[ \int \frac {A+B x}{x^3 \left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {A + B x}{x^{3} \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {A+B x}{x^3 \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Leaf count of result is larger than twice the leaf count of optimal. 467 vs. \(2 (212) = 424\).
Time = 0.29 (sec) , antiderivative size = 467, normalized size of antiderivative = 2.02 \[ \int \frac {A+B x}{x^3 \left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left (\frac {{\left (B a^{4} b^{2} c - A a^{3} b^{3} c - 2 \, B a^{5} c^{2} + 3 \, A a^{4} b c^{2}\right )} x}{a^{6} b^{2} - 4 \, a^{7} c} + \frac {B a^{4} b^{3} - A a^{3} b^{4} - 3 \, B a^{5} b c + 4 \, A a^{4} b^{2} c - 2 \, A a^{5} c^{2}}{a^{6} b^{2} - 4 \, a^{7} c}\right )}}{\sqrt {c x^{2} + b x + a}} - \frac {3 \, {\left (4 \, B a b - 5 \, A b^{2} + 4 \, A a c\right )} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + b x + a}}{\sqrt {-a}}\right )}{4 \, \sqrt {-a} a^{3}} + \frac {4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} B a b - 7 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A b^{2} + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a c + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} B a^{2} \sqrt {c} - 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} A a b \sqrt {c} - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} B a^{2} b + 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a b^{2} + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{2} c - 8 \, B a^{3} \sqrt {c} + 16 \, A a^{2} b \sqrt {c}}{4 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} - a\right )}^{2} a^{3}} \]
-2*((B*a^4*b^2*c - A*a^3*b^3*c - 2*B*a^5*c^2 + 3*A*a^4*b*c^2)*x/(a^6*b^2 - 4*a^7*c) + (B*a^4*b^3 - A*a^3*b^4 - 3*B*a^5*b*c + 4*A*a^4*b^2*c - 2*A*a^5 *c^2)/(a^6*b^2 - 4*a^7*c))/sqrt(c*x^2 + b*x + a) - 3/4*(4*B*a*b - 5*A*b^2 + 4*A*a*c)*arctan(-(sqrt(c)*x - sqrt(c*x^2 + b*x + a))/sqrt(-a))/(sqrt(-a) *a^3) + 1/4*(4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a*b - 7*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*b^2 + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3 *A*a*c + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*a^2*sqrt(c) - 8*(sqrt(c )*x - sqrt(c*x^2 + b*x + a))^2*A*a*b*sqrt(c) - 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a^2*b + 9*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a*b^2 + 4*(sq rt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^2*c - 8*B*a^3*sqrt(c) + 16*A*a^2*b*sq rt(c))/(((sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2 - a)^2*a^3)
Timed out. \[ \int \frac {A+B x}{x^3 \left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {A+B\,x}{x^3\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \]