Integrand size = 27, antiderivative size = 150 \[ \int \frac {A+B x}{(d+e x)^2 \sqrt {a+b x+c x^2}} \, dx=\frac {(B d-A e) \sqrt {a+b x+c x^2}}{\left (c d^2-b d e+a e^2\right ) (d+e x)}-\frac {(b B d-2 A c d+A b e-2 a B e) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{2 \left (c d^2-b d e+a e^2\right )^{3/2}} \]
-1/2*(A*b*e-2*A*c*d-2*B*a*e+B*b*d)*arctanh(1/2*(b*d-2*a*e+(-b*e+2*c*d)*x)/ (a*e^2-b*d*e+c*d^2)^(1/2)/(c*x^2+b*x+a)^(1/2))/(a*e^2-b*d*e+c*d^2)^(3/2)+( -A*e+B*d)*(c*x^2+b*x+a)^(1/2)/(a*e^2-b*d*e+c*d^2)/(e*x+d)
Time = 0.69 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.07 \[ \int \frac {A+B x}{(d+e x)^2 \sqrt {a+b x+c x^2}} \, dx=\frac {\frac {(B d-A e) \left (c d^2+e (-b d+a e)\right ) \sqrt {a+x (b+c x)}}{d+e x}+\sqrt {-c d^2+b d e-a e^2} (-b (B d+A e)+2 (A c d+a B e)) \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+x (b+c x)}}{\sqrt {-c d^2+e (b d-a e)}}\right )}{\left (c d^2+e (-b d+a e)\right )^2} \]
(((B*d - A*e)*(c*d^2 + e*(-(b*d) + a*e))*Sqrt[a + x*(b + c*x)])/(d + e*x) + Sqrt[-(c*d^2) + b*d*e - a*e^2]*(-(b*(B*d + A*e)) + 2*(A*c*d + a*B*e))*Ar cTan[(Sqrt[c]*(d + e*x) - e*Sqrt[a + x*(b + c*x)])/Sqrt[-(c*d^2) + e*(b*d - a*e)]])/(c*d^2 + e*(-(b*d) + a*e))^2
Time = 0.30 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {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.
\(\displaystyle \int \frac {A+B x}{(d+e x)^2 \sqrt {a+b x+c x^2}} \, dx\) |
\(\Big \downarrow \) 1228 |
\(\displaystyle \frac {\sqrt {a+b x+c x^2} (B d-A e)}{(d+e x) \left (a e^2-b d e+c d^2\right )}-\frac {(-2 a B e+A b e-2 A c d+b B d) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{2 \left (a e^2-b d e+c d^2\right )}\) |
\(\Big \downarrow \) 1154 |
\(\displaystyle \frac {(-2 a B e+A b e-2 A c d+b B d) \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{a e^2-b d e+c d^2}+\frac {\sqrt {a+b x+c x^2} (B d-A e)}{(d+e x) \left (a e^2-b d e+c d^2\right )}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\sqrt {a+b x+c x^2} (B d-A e)}{(d+e x) \left (a e^2-b d e+c d^2\right )}-\frac {(-2 a B e+A b e-2 A c d+b B d) \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{3/2}}\) |
((B*d - A*e)*Sqrt[a + b*x + c*x^2])/((c*d^2 - b*d*e + a*e^2)*(d + e*x)) - ((b*B*d - 2*A*c*d + A*b*e - 2*a*B*e)*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)* x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(2*(c*d^2 - b*d *e + a*e^2)^(3/2))
3.25.70.3.1 Defintions of rubi rules used
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]
Leaf count of result is larger than twice the leaf count of optimal. \(436\) vs. \(2(138)=276\).
Time = 0.51 (sec) , antiderivative size = 437, normalized size of antiderivative = 2.91
method | result | size |
default | \(-\frac {B \ln \left (\frac {\frac {2 e^{2} a -2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}+\frac {\left (A e -B d \right ) \left (-\frac {e^{2} \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}{\left (e^{2} a -b d e +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}+\frac {\left (b e -2 c d \right ) e \ln \left (\frac {\frac {2 e^{2} a -2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 \left (e^{2} a -b d e +c \,d^{2}\right ) \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}\right )}{e^{3}}\) | \(437\) |
-B/e^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e- 2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d )/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))+(A*e-B*d)/e^3*(-1/(a* e^2-b*d*e+c*d^2)*e^2/(x+d/e)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d *e+c*d^2)/e^2)^(1/2)+1/2*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)/((a*e^2-b*d*e+c *d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a *e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d *e+c*d^2)/e^2)^(1/2))/(x+d/e)))
Leaf count of result is larger than twice the leaf count of optimal. 353 vs. \(2 (138) = 276\).
Time = 1.28 (sec) , antiderivative size = 748, normalized size of antiderivative = 4.99 \[ \int \frac {A+B x}{(d+e x)^2 \sqrt {a+b x+c x^2}} \, dx=\left [\frac {{\left ({\left (B b - 2 \, A c\right )} d^{2} - {\left (2 \, B a - A b\right )} d e + {\left ({\left (B b - 2 \, A c\right )} d e - {\left (2 \, B a - A b\right )} e^{2}\right )} x\right )} \sqrt {c d^{2} - b d e + a e^{2}} \log \left (\frac {8 \, a b d e - 8 \, a^{2} e^{2} - {\left (b^{2} + 4 \, a c\right )} d^{2} - {\left (8 \, c^{2} d^{2} - 8 \, b c d e + {\left (b^{2} + 4 \, a c\right )} e^{2}\right )} x^{2} + 4 \, \sqrt {c d^{2} - b d e + a e^{2}} \sqrt {c x^{2} + b x + a} {\left (b d - 2 \, a e + {\left (2 \, c d - b e\right )} x\right )} - 2 \, {\left (4 \, b c d^{2} + 4 \, a b e^{2} - {\left (3 \, b^{2} + 4 \, a c\right )} d e\right )} x}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 4 \, {\left (B c d^{3} - A a e^{3} - {\left (B b + A c\right )} d^{2} e + {\left (B a + A b\right )} d e^{2}\right )} \sqrt {c x^{2} + b x + a}}{4 \, {\left (c^{2} d^{5} - 2 \, b c d^{4} e - 2 \, a b d^{2} e^{3} + a^{2} d e^{4} + {\left (b^{2} + 2 \, a c\right )} d^{3} e^{2} + {\left (c^{2} d^{4} e - 2 \, b c d^{3} e^{2} - 2 \, a b d e^{4} + a^{2} e^{5} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{3}\right )} x\right )}}, -\frac {{\left ({\left (B b - 2 \, A c\right )} d^{2} - {\left (2 \, B a - A b\right )} d e + {\left ({\left (B b - 2 \, A c\right )} d e - {\left (2 \, B a - A b\right )} e^{2}\right )} x\right )} \sqrt {-c d^{2} + b d e - a e^{2}} \arctan \left (-\frac {\sqrt {-c d^{2} + b d e - a e^{2}} \sqrt {c x^{2} + b x + a} {\left (b d - 2 \, a e + {\left (2 \, c d - b e\right )} x\right )}}{2 \, {\left (a c d^{2} - a b d e + a^{2} e^{2} + {\left (c^{2} d^{2} - b c d e + a c e^{2}\right )} x^{2} + {\left (b c d^{2} - b^{2} d e + a b e^{2}\right )} x\right )}}\right ) - 2 \, {\left (B c d^{3} - A a e^{3} - {\left (B b + A c\right )} d^{2} e + {\left (B a + A b\right )} d e^{2}\right )} \sqrt {c x^{2} + b x + a}}{2 \, {\left (c^{2} d^{5} - 2 \, b c d^{4} e - 2 \, a b d^{2} e^{3} + a^{2} d e^{4} + {\left (b^{2} + 2 \, a c\right )} d^{3} e^{2} + {\left (c^{2} d^{4} e - 2 \, b c d^{3} e^{2} - 2 \, a b d e^{4} + a^{2} e^{5} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{3}\right )} x\right )}}\right ] \]
[1/4*(((B*b - 2*A*c)*d^2 - (2*B*a - A*b)*d*e + ((B*b - 2*A*c)*d*e - (2*B*a - A*b)*e^2)*x)*sqrt(c*d^2 - b*d*e + a*e^2)*log((8*a*b*d*e - 8*a^2*e^2 - ( b^2 + 4*a*c)*d^2 - (8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2 + 4*sqr t(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e )*x) - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)/(e^2*x^2 + 2*d*e *x + d^2)) + 4*(B*c*d^3 - A*a*e^3 - (B*b + A*c)*d^2*e + (B*a + A*b)*d*e^2) *sqrt(c*x^2 + b*x + a))/(c^2*d^5 - 2*b*c*d^4*e - 2*a*b*d^2*e^3 + a^2*d*e^4 + (b^2 + 2*a*c)*d^3*e^2 + (c^2*d^4*e - 2*b*c*d^3*e^2 - 2*a*b*d*e^4 + a^2* e^5 + (b^2 + 2*a*c)*d^2*e^3)*x), -1/2*(((B*b - 2*A*c)*d^2 - (2*B*a - A*b)* d*e + ((B*b - 2*A*c)*d*e - (2*B*a - A*b)*e^2)*x)*sqrt(-c*d^2 + b*d*e - a*e ^2)*arctan(-1/2*sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e)*x)/(a*c*d^2 - a*b*d*e + a^2*e^2 + (c^2*d^2 - b*c*d*e + a*c*e^2)*x^2 + (b*c*d^2 - b^2*d*e + a*b*e^2)*x)) - 2*(B*c*d^3 - A*a*e^3 - (B*b + A*c)*d^2*e + (B*a + A*b)*d*e^2)*sqrt(c*x^2 + b*x + a))/(c^2*d^5 - 2*b*c*d^4*e - 2*a*b*d^2*e^3 + a^2*d*e^4 + (b^2 + 2*a*c)*d^3*e^2 + (c^2*d ^4*e - 2*b*c*d^3*e^2 - 2*a*b*d*e^4 + a^2*e^5 + (b^2 + 2*a*c)*d^2*e^3)*x)]
\[ \int \frac {A+B x}{(d+e x)^2 \sqrt {a+b x+c x^2}} \, dx=\int \frac {A + B x}{\left (d + e x\right )^{2} \sqrt {a + b x + c x^{2}}}\, dx \]
Exception generated. \[ \int \frac {A+B x}{(d+e x)^2 \sqrt {a+b x+c x^2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*e^2-b*d*e>0)', see `assume?` f or more de
\[ \int \frac {A+B x}{(d+e x)^2 \sqrt {a+b x+c x^2}} \, dx=\int { \frac {B x + A}{\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {A+B x}{(d+e x)^2 \sqrt {a+b x+c x^2}} \, dx=\int \frac {A+B\,x}{{\left (d+e\,x\right )}^2\,\sqrt {c\,x^2+b\,x+a}} \,d x \]