Integrand size = 44, antiderivative size = 117 \[ \int \frac {f+g x}{(d+e x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=-\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (2 c d-b e) (d+e x)}+\frac {2 g \arctan \left (\frac {\sqrt {c} (d+e x)}{\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{\sqrt {c} e^2} \] Output:
-2*(-d*g+e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/e^2/(-b*e+2*c*d)/(e*x +d)+2*g*arctan(c^(1/2)*(e*x+d)/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2))/c^( 1/2)/e^2
Time = 0.32 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.21 \[ \int \frac {f+g x}{(d+e x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\frac {-2 \sqrt {c} (e f-d g) (-c d+b e+c e x)+2 (2 c d-b e) g \sqrt {d+e x} \sqrt {c d-b e-c e x} \arctan \left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {c} \sqrt {d+e x}}\right )}{\sqrt {c} e^2 (-2 c d+b e) \sqrt {(d+e x) (-b e+c (d-e x))}} \] Input:
Integrate[(f + g*x)/((d + e*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2]), x]
Output:
(-2*Sqrt[c]*(e*f - d*g)*(-(c*d) + b*e + c*e*x) + 2*(2*c*d - b*e)*g*Sqrt[d + e*x]*Sqrt[c*d - b*e - c*e*x]*ArcTan[Sqrt[c*d - b*e - c*e*x]/(Sqrt[c]*Sqr t[d + e*x])])/(Sqrt[c]*e^2*(-2*c*d + b*e)*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))])
Time = 0.51 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {1220, 1092, 217}
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.
\(\displaystyle \int \frac {f+g x}{(d+e x) \sqrt {-b d e-b e^2 x+c d^2-c e^2 x^2}} \, dx\) |
\(\Big \downarrow \) 1220 |
\(\displaystyle \frac {g \int \frac {1}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{e}-\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (d+e x) (2 c d-b e)}\) |
\(\Big \downarrow \) 1092 |
\(\displaystyle \frac {2 g \int \frac {1}{-\frac {(b+2 c x)^2 e^4}{-c x^2 e^2-b x e^2+d (c d-b e)}-4 c e^2}d\left (-\frac {e^2 (b+2 c x)}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}\right )}{e}-\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (d+e x) (2 c d-b e)}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {g \arctan \left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{\sqrt {c} e^2}-\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (d+e x) (2 c d-b e)}\) |
Input:
Int[(f + g*x)/((d + e*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2]),x]
Output:
(-2*(e*f - d*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(e^2*(2*c*d - b *e)*(d + e*x)) + (g*ArcTan[(e*(b + 2*c*x))/(2*Sqrt[c]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])])/(Sqrt[c]*e^2)
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[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[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d*g - e*f)*(d + e*x)^m*((a + b*x + c*x ^2)^(p + 1)/((2*c*d - b*e)*(m + p + 1))), x] + Simp[(m*(g*(c*d - b*e) + c*e *f) + e*(p + 1)*(2*c*f - b*g))/(e*(2*c*d - b*e)*(m + p + 1)) 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[c*d^2 - b*d*e + a*e^2, 0] && ((LtQ[m, -1] && !IGtQ[m + p + 1, 0 ]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0 ]
Time = 2.60 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.15
method | result | size |
default | \(\frac {g \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}\right )}{e \sqrt {c \,e^{2}}}+\frac {2 \left (d g -e f \right ) \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )}}{e^{2} \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )}\) | \(134\) |
Input:
int((g*x+f)/(e*x+d)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x,method=_RETUR NVERBOSE)
Output:
g/e/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d *e+c*d^2)^(1/2))+2*(d*g-e*f)/e^2/(-b*e^2+2*c*d*e)/(x+d/e)*(-c*e^2*(x+d/e)^ 2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)
Time = 0.34 (sec) , antiderivative size = 405, normalized size of antiderivative = 3.46 \[ \int \frac {f+g x}{(d+e x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\left [-\frac {{\left ({\left (2 \, c d e - b e^{2}\right )} g x + {\left (2 \, c d^{2} - b d e\right )} g\right )} \sqrt {-c} \log \left (8 \, c^{2} e^{2} x^{2} + 8 \, b c e^{2} x - 4 \, c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2} - 4 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, c e x + b e\right )} \sqrt {-c}\right ) + 4 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (c e f - c d g\right )}}{2 \, {\left (2 \, c^{2} d^{2} e^{2} - b c d e^{3} + {\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} x\right )}}, -\frac {{\left ({\left (2 \, c d e - b e^{2}\right )} g x + {\left (2 \, c d^{2} - b d e\right )} g\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, c e x + b e\right )} \sqrt {c}}{2 \, {\left (c^{2} e^{2} x^{2} + b c e^{2} x - c^{2} d^{2} + b c d e\right )}}\right ) + 2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (c e f - c d g\right )}}{2 \, c^{2} d^{2} e^{2} - b c d e^{3} + {\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} x}\right ] \] Input:
integrate((g*x+f)/(e*x+d)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algori thm="fricas")
Output:
[-1/2*(((2*c*d*e - b*e^2)*g*x + (2*c*d^2 - b*d*e)*g)*sqrt(-c)*log(8*c^2*e^ 2*x^2 + 8*b*c*e^2*x - 4*c^2*d^2 + 4*b*c*d*e + b^2*e^2 - 4*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*e*x + b*e)*sqrt(-c)) + 4*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(c*e*f - c*d*g))/(2*c^2*d^2*e^2 - b*c*d*e^3 + (2 *c^2*d*e^3 - b*c*e^4)*x), -(((2*c*d*e - b*e^2)*g*x + (2*c*d^2 - b*d*e)*g)* sqrt(c)*arctan(1/2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*e*x + b *e)*sqrt(c)/(c^2*e^2*x^2 + b*c*e^2*x - c^2*d^2 + b*c*d*e)) + 2*sqrt(-c*e^2 *x^2 - b*e^2*x + c*d^2 - b*d*e)*(c*e*f - c*d*g))/(2*c^2*d^2*e^2 - b*c*d*e^ 3 + (2*c^2*d*e^3 - b*c*e^4)*x)]
\[ \int \frac {f+g x}{(d+e x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\int \frac {f + g x}{\sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (d + e x\right )}\, dx \] Input:
integrate((g*x+f)/(e*x+d)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2),x)
Output:
Integral((f + g*x)/(sqrt(-(d + e*x)*(b*e - c*d + c*e*x))*(d + e*x)), x)
Exception generated. \[ \int \frac {f+g x}{(d+e x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((g*x+f)/(e*x+d)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algori thm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-2*c*d>0)', see `assume?` for more deta
Leaf count of result is larger than twice the leaf count of optimal. 288 vs. \(2 (109) = 218\).
Time = 0.39 (sec) , antiderivative size = 288, normalized size of antiderivative = 2.46 \[ \int \frac {f+g x}{(d+e x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=-\frac {g \log \left ({\left | b c d^{2} e^{2} - 2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )} \sqrt {-c} c d^{2} {\left | e \right |} - 2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )} b \sqrt {-c} d e {\left | e \right |} - 4 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{2} c d e - {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{2} b e^{2} + 2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{3} \sqrt {-c} {\left | e \right |} \right |}\right )}{3 \, \sqrt {-c} e {\left | e \right |}} \] Input:
integrate((g*x+f)/(e*x+d)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algori thm="giac")
Output:
-1/3*g*log(abs(b*c*d^2*e^2 - 2*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))*sqrt(-c)*c*d^2*abs(e) - 2*(sqrt(-c*e^2)*x - sqrt(-c*e^2 *x^2 - b*e^2*x + c*d^2 - b*d*e))*b*sqrt(-c)*d*e*abs(e) - 4*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))^2*c*d*e - (sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))^2*b*e^2 + 2*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))^3*sqrt(-c)*abs(e)))/(sqrt(-c) *e*abs(e))
Timed out. \[ \int \frac {f+g x}{(d+e x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\int \frac {f+g\,x}{\left (d+e\,x\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}} \,d x \] Input:
int((f + g*x)/((d + e*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2)),x)
Output:
int((f + g*x)/((d + e*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2)), x)
Time = 0.25 (sec) , antiderivative size = 486, normalized size of antiderivative = 4.15 \[ \int \frac {f+g x}{(d+e x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\frac {2 i \left (\sqrt {c}\, \mathit {asinh} \left (\frac {\sqrt {-c e x -b e +c d}\, i}{\sqrt {-b e +2 c d}}\right ) b^{2} d \,e^{2} g +\sqrt {c}\, \mathit {asinh} \left (\frac {\sqrt {-c e x -b e +c d}\, i}{\sqrt {-b e +2 c d}}\right ) b^{2} e^{3} g x -4 \sqrt {c}\, \mathit {asinh} \left (\frac {\sqrt {-c e x -b e +c d}\, i}{\sqrt {-b e +2 c d}}\right ) b c \,d^{2} e g -4 \sqrt {c}\, \mathit {asinh} \left (\frac {\sqrt {-c e x -b e +c d}\, i}{\sqrt {-b e +2 c d}}\right ) b c d \,e^{2} g x +4 \sqrt {c}\, \mathit {asinh} \left (\frac {\sqrt {-c e x -b e +c d}\, i}{\sqrt {-b e +2 c d}}\right ) c^{2} d^{3} g +4 \sqrt {c}\, \mathit {asinh} \left (\frac {\sqrt {-c e x -b e +c d}\, i}{\sqrt {-b e +2 c d}}\right ) c^{2} d^{2} e g x -\sqrt {e x +d}\, \sqrt {b e -2 c d}\, \sqrt {-b e +2 c d}\, \sqrt {-c e x -b e +c d}\, c d g +\sqrt {e x +d}\, \sqrt {b e -2 c d}\, \sqrt {-b e +2 c d}\, \sqrt {-c e x -b e +c d}\, c e f +\sqrt {c}\, b c \,d^{2} e g -\sqrt {c}\, b c d \,e^{2} f +\sqrt {c}\, b c d \,e^{2} g x -\sqrt {c}\, b c \,e^{3} f x -2 \sqrt {c}\, c^{2} d^{3} g +2 \sqrt {c}\, c^{2} d^{2} e f -2 \sqrt {c}\, c^{2} d^{2} e g x +2 \sqrt {c}\, c^{2} d \,e^{2} f x \right )}{c \,e^{2} \left (b^{2} e^{3} x -4 b c d \,e^{2} x +4 c^{2} d^{2} e x +b^{2} d \,e^{2}-4 b c \,d^{2} e +4 c^{2} d^{3}\right )} \] Input:
int((g*x+f)/(e*x+d)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x)
Output:
(2*i*(sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b **2*d*e**2*g + sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**2*e**3*g*x - 4*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqr t( - b*e + 2*c*d))*b*c*d**2*e*g - 4*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e *x)*i)/sqrt( - b*e + 2*c*d))*b*c*d*e**2*g*x + 4*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*c**2*d**3*g + 4*sqrt(c)*asinh((sq rt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*c**2*d**2*e*g*x - sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)* c*d*g + sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*c*e*f + sqrt(c)*b*c*d**2*e*g - sqrt(c)*b*c*d*e**2*f + sqrt(c )*b*c*d*e**2*g*x - sqrt(c)*b*c*e**3*f*x - 2*sqrt(c)*c**2*d**3*g + 2*sqrt(c )*c**2*d**2*e*f - 2*sqrt(c)*c**2*d**2*e*g*x + 2*sqrt(c)*c**2*d*e**2*f*x))/ (c*e**2*(b**2*d*e**2 + b**2*e**3*x - 4*b*c*d**2*e - 4*b*c*d*e**2*x + 4*c** 2*d**3 + 4*c**2*d**2*e*x))