\(\int \frac {(d+e x)^2}{(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx\) [278]

Optimal result

Integrand size = 46, antiderivative size = 211 \[ \int \frac {(d+e x)^2}{(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c g}-\frac {(2 c e f-4 c d g+b e g) \arctan \left (\frac {\sqrt {c} (d+e x)}{\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{c^{3/2} g^2}+\frac {2 (e f-d g)^{3/2} \arctan \left (\frac {\sqrt {c e f+c d g-b e g} (d+e x)}{\sqrt {e f-d g} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{g^2 \sqrt {c e f+c d g-b e g}} \] Output:

-(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/c/g-(b*e*g-4*c*d*g+2*c*e*f)*arctan 
(c^(1/2)*(e*x+d)/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2))/c^(3/2)/g^2+2*(-d 
*g+e*f)^(3/2)*arctan((-b*e*g+c*d*g+c*e*f)^(1/2)*(e*x+d)/(-d*g+e*f)^(1/2)/( 
d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2))/g^2/(-b*e*g+c*d*g+c*e*f)^(1/2)
 

Mathematica [A] (verified)

Time = 0.99 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.31 \[ \int \frac {(d+e x)^2}{(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\frac {\sqrt {c d-b e-c e x} \left (\sqrt {c e f+c d g-b e g} (2 c e f-4 c d g+b e g) \sqrt {d+e x} \arctan \left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {c} \sqrt {d+e x}}\right )-\sqrt {c} \left (g \sqrt {c e f+c d g-b e g} (d+e x) \sqrt {-b e+c (d-e x)}+2 c (e f-d g)^{3/2} \sqrt {d+e x} \arctan \left (\frac {\sqrt {e f-d g} \sqrt {c d-b e-c e x}}{\sqrt {c e f+c d g-b e g} \sqrt {d+e x}}\right )\right )\right )}{c^{3/2} g^2 \sqrt {c e f+c d g-b e g} \sqrt {(d+e x) (-b e+c (d-e x))}} \] Input:

Integrate[(d + e*x)^2/((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2] 
),x]
 

Output:

(Sqrt[c*d - b*e - c*e*x]*(Sqrt[c*e*f + c*d*g - b*e*g]*(2*c*e*f - 4*c*d*g + 
 b*e*g)*Sqrt[d + e*x]*ArcTan[Sqrt[c*d - b*e - c*e*x]/(Sqrt[c]*Sqrt[d + e*x 
])] - Sqrt[c]*(g*Sqrt[c*e*f + c*d*g - b*e*g]*(d + e*x)*Sqrt[-(b*e) + c*(d 
- e*x)] + 2*c*(e*f - d*g)^(3/2)*Sqrt[d + e*x]*ArcTan[(Sqrt[e*f - d*g]*Sqrt 
[c*d - b*e - c*e*x])/(Sqrt[c*e*f + c*d*g - b*e*g]*Sqrt[d + e*x])])))/(c^(3 
/2)*g^2*Sqrt[c*e*f + c*d*g - b*e*g]*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))] 
)
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.22, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {1267, 27, 1269, 1092, 217, 1154, 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.

\(\displaystyle \int \frac {(d+e x)^2}{(f+g x) \sqrt {-b d e-b e^2 x+c d^2-c e^2 x^2}} \, dx\)

\(\Big \downarrow \) 1267

\(\displaystyle -\frac {\int \frac {e^2 g \left (-2 c g d^2+b e^2 f+e (2 c e f-4 c d g+b e g) x\right )}{2 (f+g x) \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{c e^2 g^2}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c g}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {-2 c g d^2+b e^2 f+e (2 c e f-4 c d g+b e g) x}{(f+g x) \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{2 c g}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c g}\)

\(\Big \downarrow \) 1269

\(\displaystyle -\frac {\frac {e (b e g-4 c d g+2 c e f) \int \frac {1}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{g}-\frac {2 c (e f-d g)^2 \int \frac {1}{(f+g x) \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{g}}{2 c g}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c g}\)

\(\Big \downarrow \) 1092

\(\displaystyle -\frac {\frac {2 e (b e g-4 c d g+2 c e f) \int \frac {1}{-\frac {(b+2 c x)^2 e^4}{-c x^2 e^2-b x e^2+d (c d-b e)}-4 c e^2}d\left (-\frac {e^2 (b+2 c x)}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}\right )}{g}-\frac {2 c (e f-d g)^2 \int \frac {1}{(f+g x) \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{g}}{2 c g}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c g}\)

\(\Big \downarrow \) 217

\(\displaystyle -\frac {\frac {\arctan \left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right ) (b e g-4 c d g+2 c e f)}{\sqrt {c} g}-\frac {2 c (e f-d g)^2 \int \frac {1}{(f+g x) \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{g}}{2 c g}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c g}\)

\(\Big \downarrow \) 1154

\(\displaystyle -\frac {\frac {4 c (e f-d g)^2 \int \frac {1}{-\frac {\left (2 c g d^2+b e (e f-2 d g)+e^2 (2 c f-b g) x\right )^2}{-c x^2 e^2-b x e^2+d (c d-b e)}-4 (e f-d g) (c e f+c d g-b e g)}d\frac {2 c g d^2+b e (e f-2 d g)+e^2 (2 c f-b g) x}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}}{g}+\frac {\arctan \left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right ) (b e g-4 c d g+2 c e f)}{\sqrt {c} g}}{2 c g}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c g}\)

\(\Big \downarrow \) 217

\(\displaystyle -\frac {\frac {\arctan \left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right ) (b e g-4 c d g+2 c e f)}{\sqrt {c} g}-\frac {2 c (e f-d g)^{3/2} \arctan \left (\frac {e^2 x (2 c f-b g)+b e (e f-2 d g)+2 c d^2 g}{2 \sqrt {e f-d g} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} \sqrt {-b e g+c d g+c e f}}\right )}{g \sqrt {-b e g+c d g+c e f}}}{2 c g}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c g}\)

Input:

Int[(d + e*x)^2/((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2]),x]
 

Output:

-(Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]/(c*g)) - (((2*c*e*f - 4*c*d*g 
+ b*e*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]*g) - (2*c*(e*f - d*g)^(3/2)*ArcTan[(2*c*d^2*g + b*e 
*(e*f - 2*d*g) + e^2*(2*c*f - b*g)*x)/(2*Sqrt[e*f - d*g]*Sqrt[c*e*f + c*d* 
g - b*e*g]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])])/(g*Sqrt[c*e*f + c* 
d*g - b*e*g]))/(2*c*g)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

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])
 

rule 1092

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]
 

rule 1154

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]
 

rule 1267

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[g^n*(d + e*x)^(m + n - 1)*((a + b 
*x + c*x^2)^(p + 1)/(c*e^(n - 1)*(m + n + 2*p + 1))), x] + Simp[1/(c*e^n*(m 
 + n + 2*p + 1))   Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^n*(m 
 + n + 2*p + 1)*(f + g*x)^n - c*g^n*(m + n + 2*p + 1)*(d + e*x)^n - g^n*(d 
+ e*x)^(n - 2)*(b*d*e*(p + 1) + a*e^2*(m + n - 1) - c*d^2*(m + n + 2*p + 1) 
 - e*(2*c*d - b*e)*(m + n + p)*x), x], x], x] /; FreeQ[{a, b, c, d, e, f, g 
, m, p}, x] && IGtQ[n, 1] && IntegerQ[m] && NeQ[m + n + 2*p + 1, 0]
 

rule 1269

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(481\) vs. \(2(193)=386\).

Time = 2.14 (sec) , antiderivative size = 482, normalized size of antiderivative = 2.28

method	result	size
default	\(\frac {e \left (e g \left (-\frac {\sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{c \,e^{2}}-\frac {b \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}\right )}{2 c \sqrt {c \,e^{2}}}\right )+\frac {2 d g \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}\right )}{\sqrt {c \,e^{2}}}-\frac {e f \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}\right )}{\sqrt {c \,e^{2}}}\right )}{g^{2}}-\frac {\left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right ) \ln \left (\frac {-\frac {2 \left (b d e \,g^{2}-b \,e^{2} f g -c \,d^{2} g^{2}+c \,e^{2} f^{2}\right )}{g^{2}}-\frac {e^{2} \left (b g -2 c f \right ) \left (x +\frac {f}{g}\right )}{g}+2 \sqrt {-\frac {b d e \,g^{2}-b \,e^{2} f g -c \,d^{2} g^{2}+c \,e^{2} f^{2}}{g^{2}}}\, \sqrt {-\left (x +\frac {f}{g}\right )^{2} c \,e^{2}-\frac {e^{2} \left (b g -2 c f \right ) \left (x +\frac {f}{g}\right )}{g}-\frac {b d e \,g^{2}-b \,e^{2} f g -c \,d^{2} g^{2}+c \,e^{2} f^{2}}{g^{2}}}}{x +\frac {f}{g}}\right )}{g^{3} \sqrt {-\frac {b d e \,g^{2}-b \,e^{2} f g -c \,d^{2} g^{2}+c \,e^{2} f^{2}}{g^{2}}}}\)	\(482\)

Input:

int((e*x+d)^2/(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x,method=_RET 
URNVERBOSE)
 

Output:

e/g^2*(e*g*(-1/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-1/2*b/c/(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/(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))-e*f/(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)))-(d^2*g^2-2*d*e*f*g+e^2*f^2)/g 
^3/(-(b*d*e*g^2-b*e^2*f*g-c*d^2*g^2+c*e^2*f^2)/g^2)^(1/2)*ln((-2*(b*d*e*g^ 
2-b*e^2*f*g-c*d^2*g^2+c*e^2*f^2)/g^2-e^2*(b*g-2*c*f)/g*(x+f/g)+2*(-(b*d*e* 
g^2-b*e^2*f*g-c*d^2*g^2+c*e^2*f^2)/g^2)^(1/2)*(-(x+f/g)^2*c*e^2-e^2*(b*g-2 
*c*f)/g*(x+f/g)-(b*d*e*g^2-b*e^2*f*g-c*d^2*g^2+c*e^2*f^2)/g^2)^(1/2))/(x+f 
/g))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {(d+e x)^2}{(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\text {Timed out} \] Input:

integrate((e*x+d)^2/(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algo 
rithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {(d+e x)^2}{(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\int \frac {\left (d + e x\right )^{2}}{\sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (f + g x\right )}\, dx \] Input:

integrate((e*x+d)**2/(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2),x 
)
 

Output:

Integral((d + e*x)**2/(sqrt(-(d + e*x)*(b*e - c*d + c*e*x))*(f + g*x)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^2}{(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*x+d)^2/(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algo 
rithm="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((4*c*e^2>0)', see `assume?` for 
more detai
 

Giac [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^2}{(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((e*x+d)^2/(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algo 
rithm="giac")
 

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^2}{(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\int \frac {{\left (d+e\,x\right )}^2}{\left (f+g\,x\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}} \,d x \] Input:

int((d + e*x)^2/((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2)),x)
 

Output:

int((d + e*x)^2/((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2)), x 
)
 

Reduce [B] (verification not implemented)

Time = 0.47 (sec) , antiderivative size = 2483, normalized size of antiderivative = 11.77 \[ \int \frac {(d+e x)^2}{(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx =\text {Too large to display} \] Input:

int((e*x+d)^2/(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x)
 

Output:

(i*( - sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))* 
b**3*e**3*g**2 + 7*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b* 
e + 2*c*d))*b**2*c*d*e**2*g**2 - sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x) 
*i)/sqrt( - b*e + 2*c*d))*b**2*c*e**3*f*g - 14*sqrt(c)*asinh((sqrt( - b*e 
+ c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b*c**2*d**2*e*g**2 + 2*sqrt(c)*asi 
nh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b*c**2*e**3*f**2 + 
 8*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*c**3 
*d**3*g**2 + 4*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 
2*c*d))*c**3*d**2*e*f*g - 4*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/s 
qrt( - b*e + 2*c*d))*c**3*d*e**2*f**2 - sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sq 
rt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b*c*e*g**2 + sqrt(d + e*x)*s 
qrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*c**2*d*g* 
*2 + 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**2*e*f*g - sqrt(d*g - e*f)*sqrt(b*e*g - c*d*g - c*e*f)*log((s 
qrt(g)*sqrt(b*e - 2*c*d)*sqrt( - b*e + c*d - c*e*x)*i - sqrt(2*sqrt(c)*sqr 
t(d*g - e*f)*sqrt(b*e*g - c*d*g - c*e*f) - b*e*g + 2*c*e*f)*sqrt( - b*e + 
2*c*d) + sqrt(g)*sqrt(c)*sqrt(d + e*x)*sqrt( - b*e + 2*c*d)*i)/sqrt( - b*e 
 + 2*c*d))*b*c**2*d*e*g + sqrt(d*g - e*f)*sqrt(b*e*g - c*d*g - c*e*f)*log( 
(sqrt(g)*sqrt(b*e - 2*c*d)*sqrt( - b*e + c*d - c*e*x)*i - sqrt(2*sqrt(c)*s 
qrt(d*g - e*f)*sqrt(b*e*g - c*d*g - c*e*f) - b*e*g + 2*c*e*f)*sqrt( - b...