Integrand size = 23, antiderivative size = 179 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^3} \, dx=-\frac {3 (A b+2 a B-(b B+2 A c) x) \sqrt {a+b x+c x^2}}{4 x}-\frac {(A-B x) \left (a+b x+c x^2\right )^{3/2}}{2 x^2}-\frac {3 \left (4 a b B+A \left (b^2+4 a c\right )\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {a}}+\frac {3 \left (b^2 B+4 A b c+4 a B c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {c}} \]
-1/2*(-B*x+A)*(c*x^2+b*x+a)^(3/2)/x^2-3/8*(4*a*b*B+A*(4*a*c+b^2))*arctanh( 1/2*(b*x+2*a)/a^(1/2)/(c*x^2+b*x+a)^(1/2))/a^(1/2)+3/8*(4*A*b*c+4*B*a*c+B* b^2)*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(1/2)-3/4*(A*b+2 *B*a-(2*A*c+B*b)*x)*(c*x^2+b*x+a)^(1/2)/x
Time = 1.02 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.89 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^3} \, dx=\frac {1}{8} \left (\frac {2 \sqrt {a+x (b+c x)} (-2 a (A+2 B x)+x (B x (5 b+2 c x)+A (-5 b+4 c x)))}{x^2}-\frac {6 \left (4 a b B+A \left (b^2+4 a c\right )\right ) \text {arctanh}\left (\frac {-\sqrt {c} x+\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {3 \left (b^2 B+4 A b c+4 a B c\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{\sqrt {c}}\right ) \]
((2*Sqrt[a + x*(b + c*x)]*(-2*a*(A + 2*B*x) + x*(B*x*(5*b + 2*c*x) + A*(-5 *b + 4*c*x))))/x^2 - (6*(4*a*b*B + A*(b^2 + 4*a*c))*ArcTanh[(-(Sqrt[c]*x) + Sqrt[a + x*(b + c*x)])/Sqrt[a]])/Sqrt[a] - (3*(b^2*B + 4*A*b*c + 4*a*B*c )*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/Sqrt[c])/8
Time = 0.43 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.02, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {1230, 27, 1230, 25, 1269, 1092, 219, 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.
| \(\displaystyle \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^3} \, dx\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle -\frac {3}{8} \int -\frac {2 (A b+2 a B+(b B+2 A c) x) \sqrt {c x^2+b x+a}}{x^2}dx-\frac {(A-B x) \left (a+b x+c x^2\right )^{3/2}}{2 x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{4} \int \frac {(A b+2 a B+(b B+2 A c) x) \sqrt {c x^2+b x+a}}{x^2}dx-\frac {(A-B x) \left (a+b x+c x^2\right )^{3/2}}{2 x^2}\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle \frac {3}{4} \left (-\frac {1}{2} \int -\frac {4 a b B+A \left (b^2+4 a c\right )+\left (B b^2+4 A c b+4 a B c\right ) x}{x \sqrt {c x^2+b x+a}}dx-\frac {\sqrt {a+b x+c x^2} (2 a B-x (2 A c+b B)+A b)}{x}\right )-\frac {(A-B x) \left (a+b x+c x^2\right )^{3/2}}{2 x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3}{4} \left (\frac {1}{2} \int \frac {4 a b B+A \left (b^2+4 a c\right )+\left (B b^2+4 A c b+4 a B c\right ) x}{x \sqrt {c x^2+b x+a}}dx-\frac {\sqrt {a+b x+c x^2} (2 a B-x (2 A c+b B)+A b)}{x}\right )-\frac {(A-B x) \left (a+b x+c x^2\right )^{3/2}}{2 x^2}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {3}{4} \left (\frac {1}{2} \left (\left (4 a B c+4 A b c+b^2 B\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx+\left (A \left (4 a c+b^2\right )+4 a b B\right ) \int \frac {1}{x \sqrt {c x^2+b x+a}}dx\right )-\frac {\sqrt {a+b x+c x^2} (2 a B-x (2 A c+b B)+A b)}{x}\right )-\frac {(A-B x) \left (a+b x+c x^2\right )^{3/2}}{2 x^2}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {3}{4} \left (\frac {1}{2} \left (\left (A \left (4 a c+b^2\right )+4 a b B\right ) \int \frac {1}{x \sqrt {c x^2+b x+a}}dx+2 \left (4 a B c+4 A b c+b^2 B\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}\right )-\frac {\sqrt {a+b x+c x^2} (2 a B-x (2 A c+b B)+A b)}{x}\right )-\frac {(A-B x) \left (a+b x+c x^2\right )^{3/2}}{2 x^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {3}{4} \left (\frac {1}{2} \left (\left (A \left (4 a c+b^2\right )+4 a b B\right ) \int \frac {1}{x \sqrt {c x^2+b x+a}}dx+\frac {\left (4 a B c+4 A b c+b^2 B\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {c}}\right )-\frac {\sqrt {a+b x+c x^2} (2 a B-x (2 A c+b B)+A b)}{x}\right )-\frac {(A-B x) \left (a+b x+c x^2\right )^{3/2}}{2 x^2}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {3}{4} \left (\frac {1}{2} \left (\frac {\left (4 a B c+4 A b c+b^2 B\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {c}}-2 \left (A \left (4 a c+b^2\right )+4 a b B\right ) \int \frac {1}{4 a-\frac {(2 a+b x)^2}{c x^2+b x+a}}d\frac {2 a+b x}{\sqrt {c x^2+b x+a}}\right )-\frac {\sqrt {a+b x+c x^2} (2 a B-x (2 A c+b B)+A b)}{x}\right )-\frac {(A-B x) \left (a+b x+c x^2\right )^{3/2}}{2 x^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {3}{4} \left (\frac {1}{2} \left (\frac {\left (4 a B c+4 A b c+b^2 B\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {c}}-\frac {\left (A \left (4 a c+b^2\right )+4 a b B\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{\sqrt {a}}\right )-\frac {\sqrt {a+b x+c x^2} (2 a B-x (2 A c+b B)+A b)}{x}\right )-\frac {(A-B x) \left (a+b x+c x^2\right )^{3/2}}{2 x^2}\) |
-1/2*((A - B*x)*(a + b*x + c*x^2)^(3/2))/x^2 + (3*(-(((A*b + 2*a*B - (b*B + 2*A*c)*x)*Sqrt[a + b*x + c*x^2])/x) + (-(((4*a*b*B + A*(b^2 + 4*a*c))*Ar cTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/Sqrt[a]) + ((b^2*B + 4*A*b*c + 4*a*B*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])] )/Sqrt[c])/2))/4
3.10.29.3.1 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/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
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[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Simp[p/(e^2*(m + 1)*(m + 2*p + 2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1)*Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m + 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, - 1] || EqQ[p, 1] || (IntegerQ[p] && !RationalQ[m])) && NeQ[m, -1] && !ILtQ [m + 2*p + 1, 0] && (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[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Time = 0.47 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.62
| method | result | size |
| risch | \(-\frac {\sqrt {c \,x^{2}+b x +a}\, \left (5 A b x +4 a B x +2 a A \right )}{4 x^{2}}+\frac {3 B \,b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 \sqrt {c}}+\frac {B c x \sqrt {c \,x^{2}+b x +a}}{2}+\frac {5 B b \sqrt {c \,x^{2}+b x +a}}{4}+\frac {3 B a \sqrt {c}\, \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2}+A c \sqrt {c \,x^{2}+b x +a}+\frac {3 A b \sqrt {c}\, \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2}-\frac {3 \sqrt {a}\, \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) A c}{2}-\frac {3 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) A \,b^{2}}{8 \sqrt {a}}-\frac {3 \sqrt {a}\, \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) b B}{2}\) | \(290\) |
| default | \(A \left (-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {b \left (-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{a x}+\frac {3 b \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{3}+\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{2}+a \left (\sqrt {c \,x^{2}+b x +a}+\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 \sqrt {c}}-\sqrt {a}\, \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )\right )\right )}{2 a}+\frac {4 c \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{a}\right )}{4 a}+\frac {3 c \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{3}+\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{2}+a \left (\sqrt {c \,x^{2}+b x +a}+\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 \sqrt {c}}-\sqrt {a}\, \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )\right )\right )}{2 a}\right )+B \left (-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{a x}+\frac {3 b \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{3}+\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{2}+a \left (\sqrt {c \,x^{2}+b x +a}+\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 \sqrt {c}}-\sqrt {a}\, \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )\right )\right )}{2 a}+\frac {4 c \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{a}\right )\) | \(797\) |
-1/4*(c*x^2+b*x+a)^(1/2)*(5*A*b*x+4*B*a*x+2*A*a)/x^2+3/8*B*b^2*ln((1/2*b+c *x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)+1/2*B*c*x*(c*x^2+b*x+a)^(1/2)+5/4 *B*b*(c*x^2+b*x+a)^(1/2)+3/2*B*a*c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x +a)^(1/2))+A*c*(c*x^2+b*x+a)^(1/2)+3/2*A*b*c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+ (c*x^2+b*x+a)^(1/2))-3/2*a^(1/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2) )/x)*A*c-3/8/a^(1/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)*A*b^2-3 /2*a^(1/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)*b*B
Time = 1.17 (sec) , antiderivative size = 921, normalized size of antiderivative = 5.15 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^3} \, dx=\left [\frac {3 \, {\left (B a b^{2} + 4 \, {\left (B a^{2} + A a b\right )} c\right )} \sqrt {c} x^{2} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) + 3 \, {\left (4 \, A a c^{2} + {\left (4 \, B a b + A b^{2}\right )} c\right )} \sqrt {a} x^{2} \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 \, B a c^{2} x^{3} - 2 \, A a^{2} c - {\left (4 \, B a^{2} + 5 \, A a b\right )} c x + {\left (5 \, B a b c + 4 \, A a c^{2}\right )} x^{2}\right )} \sqrt {c x^{2} + b x + a}}{16 \, a c x^{2}}, -\frac {6 \, {\left (B a b^{2} + 4 \, {\left (B a^{2} + A a b\right )} c\right )} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - 3 \, {\left (4 \, A a c^{2} + {\left (4 \, B a b + A b^{2}\right )} c\right )} \sqrt {a} x^{2} \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 \, B a c^{2} x^{3} - 2 \, A a^{2} c - {\left (4 \, B a^{2} + 5 \, A a b\right )} c x + {\left (5 \, B a b c + 4 \, A a c^{2}\right )} x^{2}\right )} \sqrt {c x^{2} + b x + a}}{16 \, a c x^{2}}, \frac {6 \, {\left (4 \, A a c^{2} + {\left (4 \, B a b + A b^{2}\right )} c\right )} \sqrt {-a} x^{2} \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 ) + 3 \, {\left (B a b^{2} + 4 \, {\left (B a^{2} + A a b\right )} c\right )} \sqrt {c} x^{2} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (2 \, B a c^{2} x^{3} - 2 \, A a^{2} c - {\left (4 \, B a^{2} + 5 \, A a b\right )} c x + {\left (5 \, B a b c + 4 \, A a c^{2}\right )} x^{2}\right )} \sqrt {c x^{2} + b x + a}}{16 \, a c x^{2}}, \frac {3 \, {\left (4 \, A a c^{2} + {\left (4 \, B a b + A b^{2}\right )} c\right )} \sqrt {-a} x^{2} \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 ) - 3 \, {\left (B a b^{2} + 4 \, {\left (B a^{2} + A a b\right )} c\right )} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (2 \, B a c^{2} x^{3} - 2 \, A a^{2} c - {\left (4 \, B a^{2} + 5 \, A a b\right )} c x + {\left (5 \, B a b c + 4 \, A a c^{2}\right )} x^{2}\right )} \sqrt {c x^{2} + b x + a}}{8 \, a c x^{2}}\right ] \]
[1/16*(3*(B*a*b^2 + 4*(B*a^2 + A*a*b)*c)*sqrt(c)*x^2*log(-8*c^2*x^2 - 8*b* c*x - b^2 - 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) + 3*(4*A* a*c^2 + (4*B*a*b + A*b^2)*c)*sqrt(a)*x^2*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*B*a*c^ 2*x^3 - 2*A*a^2*c - (4*B*a^2 + 5*A*a*b)*c*x + (5*B*a*b*c + 4*A*a*c^2)*x^2) *sqrt(c*x^2 + b*x + a))/(a*c*x^2), -1/16*(6*(B*a*b^2 + 4*(B*a^2 + A*a*b)*c )*sqrt(-c)*x^2*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)/(c^2* x^2 + b*c*x + a*c)) - 3*(4*A*a*c^2 + (4*B*a*b + A*b^2)*c)*sqrt(a)*x^2*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*B*a*c^2*x^3 - 2*A*a^2*c - (4*B*a^2 + 5*A*a*b)*c*x + (5*B*a*b*c + 4*A*a*c^2)*x^2)*sqrt(c*x^2 + b*x + a))/(a*c*x^2), 1/16*(6*(4 *A*a*c^2 + (4*B*a*b + A*b^2)*c)*sqrt(-a)*x^2*arctan(1/2*sqrt(c*x^2 + b*x + a)*(b*x + 2*a)*sqrt(-a)/(a*c*x^2 + a*b*x + a^2)) + 3*(B*a*b^2 + 4*(B*a^2 + A*a*b)*c)*sqrt(c)*x^2*log(-8*c^2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x^2 + b* x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) + 4*(2*B*a*c^2*x^3 - 2*A*a^2*c - (4*B* a^2 + 5*A*a*b)*c*x + (5*B*a*b*c + 4*A*a*c^2)*x^2)*sqrt(c*x^2 + b*x + a))/( a*c*x^2), 1/8*(3*(4*A*a*c^2 + (4*B*a*b + A*b^2)*c)*sqrt(-a)*x^2*arctan(1/2 *sqrt(c*x^2 + b*x + a)*(b*x + 2*a)*sqrt(-a)/(a*c*x^2 + a*b*x + a^2)) - 3*( B*a*b^2 + 4*(B*a^2 + A*a*b)*c)*sqrt(-c)*x^2*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) + 2*(2*B*a*c^2*x^3 - 2...
\[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^3} \, dx=\int \frac {\left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{x^{3}}\, dx \]
Exception generated. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^3} \, dx=\text {Exception raised: ValueError} \]
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. 412 vs. \(2 (152) = 304\).
Time = 0.31 (sec) , antiderivative size = 412, normalized size of antiderivative = 2.30 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^3} \, dx=\frac {1}{4} \, {\left (2 \, B c x + \frac {5 \, B b c + 4 \, A c^{2}}{c}\right )} \sqrt {c x^{2} + b x + a} + \frac {3 \, {\left (4 \, B a b + A b^{2} + 4 \, A a c\right )} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + b x + a}}{\sqrt {-a}}\right )}{4 \, \sqrt {-a}} - \frac {3 \, {\left (B b^{2} + 4 \, B a c + 4 \, A b c\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{8 \, \sqrt {c}} + \frac {4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} B a b + 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A b^{2} + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a c + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} B a^{2} \sqrt {c} + 16 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} A a b \sqrt {c} - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} B a^{2} b - 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a b^{2} + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{2} c - 8 \, B a^{3} \sqrt {c} - 8 \, A a^{2} b \sqrt {c}}{4 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} - a\right )}^{2}} \]
1/4*(2*B*c*x + (5*B*b*c + 4*A*c^2)/c)*sqrt(c*x^2 + b*x + a) + 3/4*(4*B*a*b + A*b^2 + 4*A*a*c)*arctan(-(sqrt(c)*x - sqrt(c*x^2 + b*x + a))/sqrt(-a))/ sqrt(-a) - 3/8*(B*b^2 + 4*B*a*c + 4*A*b*c)*log(abs(2*(sqrt(c)*x - sqrt(c*x ^2 + b*x + a))*sqrt(c) + b))/sqrt(c) + 1/4*(4*(sqrt(c)*x - sqrt(c*x^2 + b* x + a))^3*B*a*b + 5*(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) + 16*(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 - 3*(sqrt(c)*x - sqrt(c *x^2 + b*x + a))*A*a*b^2 + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^2*c - 8*B*a^3*sqrt(c) - 8*A*a^2*b*sqrt(c))/((sqrt(c)*x - sqrt(c*x^2 + b*x + a)) ^2 - a)^2
Timed out. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^3} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{x^3} \,d x \]