Integrand size = 23, antiderivative size = 116 \[ \int \frac {A+B x}{x^3 \sqrt {a+b x+c x^2}} \, dx=-\frac {A \sqrt {a+b x+c x^2}}{2 a x^2}+\frac {(3 A b-4 a B) \sqrt {a+b x+c x^2}}{4 a^2 x}-\frac {\left (3 A b^2-4 a b B-4 a A c\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 a^{5/2}} \]
-1/8*(-4*A*a*c+3*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^(5/2)-1/2*A*(c*x^2+b*x+a)^(1/2)/a/x^2+1/4*(3*A*b-4*B*a)*(c*x^2+ b*x+a)^(1/2)/a^2/x
Time = 0.61 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.05 \[ \int \frac {A+B x}{x^3 \sqrt {a+b x+c x^2}} \, dx=\frac {\frac {\sqrt {a} (3 A b x-2 a (A+2 B x)) \sqrt {a+x (b+c x)}}{x^2}+3 A b^2 \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )+4 a (b B+A c) \text {arctanh}\left (\frac {-\sqrt {c} x+\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{4 a^{5/2}} \]
((Sqrt[a]*(3*A*b*x - 2*a*(A + 2*B*x))*Sqrt[a + x*(b + c*x)])/x^2 + 3*A*b^2 *ArcTanh[(Sqrt[c]*x - Sqrt[a + x*(b + c*x)])/Sqrt[a]] + 4*a*(b*B + A*c)*Ar cTanh[(-(Sqrt[c]*x) + Sqrt[a + x*(b + c*x)])/Sqrt[a]])/(4*a^(5/2))
Time = 0.29 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {1237, 27, 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.
\(\displaystyle \int \frac {A+B x}{x^3 \sqrt {a+b x+c x^2}} \, dx\) |
\(\Big \downarrow \) 1237 |
\(\displaystyle -\frac {\int \frac {3 A b-4 a B+2 A c x}{2 x^2 \sqrt {c x^2+b x+a}}dx}{2 a}-\frac {A \sqrt {a+b x+c x^2}}{2 a x^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int \frac {3 A b-4 a B+2 A c x}{x^2 \sqrt {c x^2+b x+a}}dx}{4 a}-\frac {A \sqrt {a+b x+c x^2}}{2 a x^2}\) |
\(\Big \downarrow \) 1228 |
\(\displaystyle -\frac {-\frac {\left (-4 a A c-4 a b B+3 A b^2\right ) \int \frac {1}{x \sqrt {c x^2+b x+a}}dx}{2 a}-\frac {(3 A b-4 a B) \sqrt {a+b x+c x^2}}{a x}}{4 a}-\frac {A \sqrt {a+b x+c x^2}}{2 a x^2}\) |
\(\Big \downarrow \) 1154 |
\(\displaystyle -\frac {\frac {\left (-4 a A c-4 a b B+3 A b^2\right ) \int \frac {1}{4 a-\frac {(2 a+b x)^2}{c x^2+b x+a}}d\frac {2 a+b x}{\sqrt {c x^2+b x+a}}}{a}-\frac {(3 A b-4 a B) \sqrt {a+b x+c x^2}}{a x}}{4 a}-\frac {A \sqrt {a+b x+c x^2}}{2 a x^2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {\frac {\left (-4 a A c-4 a b B+3 A b^2\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 a^{3/2}}-\frac {(3 A b-4 a B) \sqrt {a+b x+c x^2}}{a x}}{4 a}-\frac {A \sqrt {a+b x+c x^2}}{2 a x^2}\) |
-1/2*(A*Sqrt[a + b*x + c*x^2])/(a*x^2) - (-(((3*A*b - 4*a*B)*Sqrt[a + b*x + c*x^2])/(a*x)) + ((3*A*b^2 - 4*a*b*B - 4*a*A*c)*ArcTanh[(2*a + b*x)/(2*S qrt[a]*Sqrt[a + b*x + c*x^2])])/(2*a^(3/2)))/(4*a)
3.10.57.3.1 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[(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 = 0.48 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.76
method | result | size |
risch | \(-\frac {\sqrt {c \,x^{2}+b x +a}\, \left (-3 A b x +4 a B x +2 a A \right )}{4 a^{2} x^{2}}+\frac {\left (4 A a c -3 A \,b^{2}+4 a b B \right ) \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{8 a^{\frac {5}{2}}}\) | \(88\) |
default | \(A \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{2 a \,x^{2}}-\frac {3 b \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{a x}+\frac {b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{4 a}+\frac {c \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )+B \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{a x}+\frac {b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )\) | \(180\) |
-1/4*(c*x^2+b*x+a)^(1/2)*(-3*A*b*x+4*B*a*x+2*A*a)/a^2/x^2+1/8*(4*A*a*c-3*A *b^2+4*B*a*b)/a^(5/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)
Time = 0.42 (sec) , antiderivative size = 235, normalized size of antiderivative = 2.03 \[ \int \frac {A+B x}{x^3 \sqrt {a+b x+c x^2}} \, dx=\left [\frac {{\left (4 \, B a b - 3 \, A b^{2} + 4 \, A a c\right )} \sqrt {a} x^{2} \log \left (-\frac {8 \, a b x + {\left (b^{2} + 4 \, a c\right )} x^{2} + 4 \, \sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{2}}\right ) - 4 \, {\left (2 \, A a^{2} + {\left (4 \, B a^{2} - 3 \, A a b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{16 \, a^{3} x^{2}}, -\frac {{\left (4 \, B a b - 3 \, A b^{2} + 4 \, A a c\right )} \sqrt {-a} x^{2} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{2} + a b x + a^{2}\right )}}\right ) + 2 \, {\left (2 \, A a^{2} + {\left (4 \, B a^{2} - 3 \, A a b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{8 \, a^{3} x^{2}}\right ] \]
[1/16*((4*B*a*b - 3*A*b^2 + 4*A*a*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*A*a^2 + (4*B*a^2 - 3*A*a*b)*x)*sqrt(c*x^2 + b*x + a))/(a^3*x^2), -1/8*(( 4*B*a*b - 3*A*b^2 + 4*A*a*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)) + 2*(2*A*a^2 + (4*B*a^2 - 3 *A*a*b)*x)*sqrt(c*x^2 + b*x + a))/(a^3*x^2)]
\[ \int \frac {A+B x}{x^3 \sqrt {a+b x+c x^2}} \, dx=\int \frac {A + B x}{x^{3} \sqrt {a + b x + c x^{2}}}\, dx \]
Exception generated. \[ \int \frac {A+B x}{x^3 \sqrt {a+b x+c x^2}} \, dx=\text {Exception raised: ValueError} \]
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. 303 vs. \(2 (98) = 196\).
Time = 0.30 (sec) , antiderivative size = 303, normalized size of antiderivative = 2.61 \[ \int \frac {A+B x}{x^3 \sqrt {a+b x+c x^2}} \, dx=-\frac {{\left (4 \, B a b - 3 \, A b^{2} + 4 \, A a c\right )} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + b x + a}}{\sqrt {-a}}\right )}{4 \, \sqrt {-a} a^{2}} + \frac {4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} B a b - 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A b^{2} + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a c + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} B a^{2} \sqrt {c} - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} B a^{2} b + 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a b^{2} + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{2} c - 8 \, B a^{3} \sqrt {c} + 8 \, A a^{2} b \sqrt {c}}{4 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} - a\right )}^{2} a^{2}} \]
-1/4*(4*B*a*b - 3*A*b^2 + 4*A*a*c)*arctan(-(sqrt(c)*x - sqrt(c*x^2 + b*x + a))/sqrt(-a))/(sqrt(-a)*a^2) + 1/4*(4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)) ^3*B*a*b - 3*(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) - 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a^2*b + 5*(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*a^2)
Timed out. \[ \int \frac {A+B x}{x^3 \sqrt {a+b x+c x^2}} \, dx=\int \frac {A+B\,x}{x^3\,\sqrt {c\,x^2+b\,x+a}} \,d x \]