\(\int \frac {a+b x+c x^2}{\sqrt {-1+x} \sqrt {1+x} (d+e x)^3} \, dx\) [59]

Optimal result

Integrand size = 32, antiderivative size = 210 \[ \int \frac {a+b x+c x^2}{\sqrt {-1+x} \sqrt {1+x} (d+e x)^3} \, dx=-\frac {\left (c d^2-e (b d-a e)\right ) \sqrt {-1+x} \sqrt {1+x}}{2 e \left (d^2-e^2\right ) (d+e x)^2}+\frac {\left (c \left (d^3-4 d e^2\right )-e \left (3 a d e-b \left (d^2+2 e^2\right )\right )\right ) \sqrt {-1+x} \sqrt {1+x}}{2 e \left (d^2-e^2\right )^2 (d+e x)}-\frac {\left (3 b d e-a \left (2 d^2+e^2\right )-c \left (d^2+2 e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {d+e} \sqrt {1+x}}{\sqrt {d-e} \sqrt {-1+x}}\right )}{(d-e)^{5/2} (d+e)^{5/2}} \] Output:

-1/2*(c*d^2-e*(-a*e+b*d))*(-1+x)^(1/2)*(1+x)^(1/2)/e/(d^2-e^2)/(e*x+d)^2+1 
/2*(c*(d^3-4*d*e^2)-e*(3*a*d*e-b*(d^2+2*e^2)))*(-1+x)^(1/2)*(1+x)^(1/2)/e/ 
(d^2-e^2)^2/(e*x+d)-(3*b*d*e-a*(2*d^2+e^2)-c*(d^2+2*e^2))*arctanh((d+e)^(1 
/2)*(1+x)^(1/2)/(d-e)^(1/2)/(-1+x)^(1/2))/(d-e)^(5/2)/(d+e)^(5/2)
 

Mathematica [A] (verified)

Time = 0.55 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.88 \[ \int \frac {a+b x+c x^2}{\sqrt {-1+x} \sqrt {1+x} (d+e x)^3} \, dx=\frac {\sqrt {-1+x} \sqrt {1+x} \left (a e \left (-4 d^2+e^2-3 d e x\right )+c d \left (-3 d e+d^2 x-4 e^2 x\right )+b \left (2 d^3+d e^2+d^2 e x+2 e^3 x\right )\right )}{2 (d-e)^2 (d+e)^2 (d+e x)^2}-\frac {\left (-3 b d e+a \left (2 d^2+e^2\right )+c \left (d^2+2 e^2\right )\right ) \arctan \left (\frac {\sqrt {d-e} \sqrt {\frac {-1+x}{1+x}}}{\sqrt {-d-e}}\right )}{(-d-e)^{5/2} (d-e)^{5/2}} \] Input:

Integrate[(a + b*x + c*x^2)/(Sqrt[-1 + x]*Sqrt[1 + x]*(d + e*x)^3),x]
 

Output:

(Sqrt[-1 + x]*Sqrt[1 + x]*(a*e*(-4*d^2 + e^2 - 3*d*e*x) + c*d*(-3*d*e + d^ 
2*x - 4*e^2*x) + b*(2*d^3 + d*e^2 + d^2*e*x + 2*e^3*x)))/(2*(d - e)^2*(d + 
 e)^2*(d + e*x)^2) - ((-3*b*d*e + a*(2*d^2 + e^2) + c*(d^2 + 2*e^2))*ArcTa 
n[(Sqrt[d - e]*Sqrt[(-1 + x)/(1 + x)])/Sqrt[-d - e]])/((-d - e)^(5/2)*(d - 
 e)^(5/2))
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.10, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2113, 2182, 25, 679, 488, 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.

\(\displaystyle \int \frac {a+b x+c x^2}{\sqrt {x-1} \sqrt {x+1} (d+e x)^3} \, dx\)

\(\Big \downarrow \) 2113

\(\displaystyle \frac {\sqrt {x^2-1} \int \frac {c x^2+b x+a}{(d+e x)^3 \sqrt {x^2-1}}dx}{\sqrt {x-1} \sqrt {x+1}}\)

\(\Big \downarrow \) 2182

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 679

\(\displaystyle \frac {\sqrt {x^2-1} \left (\frac {\frac {\sqrt {x^2-1} \left (-3 a d e^2+b d^2 e+2 b e^3+c d^3-4 c d e^2\right )}{e \left (d^2-e^2\right ) (d+e x)}-\frac {\left (-a \left (2 d^2+e^2\right )+3 b d e-c \left (d^2+2 e^2\right )\right ) \int \frac {1}{(d+e x) \sqrt {x^2-1}}dx}{d^2-e^2}}{2 \left (d^2-e^2\right )}-\frac {\sqrt {x^2-1} \left (a e^2-b d e+c d^2\right )}{2 e \left (d^2-e^2\right ) (d+e x)^2}\right )}{\sqrt {x-1} \sqrt {x+1}}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {\sqrt {x^2-1} \left (\frac {\frac {\left (-a \left (2 d^2+e^2\right )+3 b d e-c \left (d^2+2 e^2\right )\right ) \int \frac {1}{d^2-e^2-\frac {(-e-d x)^2}{x^2-1}}d\frac {-e-d x}{\sqrt {x^2-1}}}{d^2-e^2}+\frac {\sqrt {x^2-1} \left (-3 a d e^2+b d^2 e+2 b e^3+c d^3-4 c d e^2\right )}{e \left (d^2-e^2\right ) (d+e x)}}{2 \left (d^2-e^2\right )}-\frac {\sqrt {x^2-1} \left (a e^2-b d e+c d^2\right )}{2 e \left (d^2-e^2\right ) (d+e x)^2}\right )}{\sqrt {x-1} \sqrt {x+1}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\sqrt {x^2-1} \left (\frac {\frac {\text {arctanh}\left (\frac {-d x-e}{\sqrt {x^2-1} \sqrt {d^2-e^2}}\right ) \left (-a \left (2 d^2+e^2\right )+3 b d e-c \left (d^2+2 e^2\right )\right )}{\left (d^2-e^2\right )^{3/2}}+\frac {\sqrt {x^2-1} \left (-3 a d e^2+b d^2 e+2 b e^3+c d^3-4 c d e^2\right )}{e \left (d^2-e^2\right ) (d+e x)}}{2 \left (d^2-e^2\right )}-\frac {\sqrt {x^2-1} \left (a e^2-b d e+c d^2\right )}{2 e \left (d^2-e^2\right ) (d+e x)^2}\right )}{\sqrt {x-1} \sqrt {x+1}}\)

Input:

Int[(a + b*x + c*x^2)/(Sqrt[-1 + x]*Sqrt[1 + x]*(d + e*x)^3),x]
 

Output:

(Sqrt[-1 + x^2]*(-1/2*((c*d^2 - b*d*e + a*e^2)*Sqrt[-1 + x^2])/(e*(d^2 - e 
^2)*(d + e*x)^2) + (((c*d^3 + b*d^2*e - 3*a*d*e^2 - 4*c*d*e^2 + 2*b*e^3)*S 
qrt[-1 + x^2])/(e*(d^2 - e^2)*(d + e*x)) + ((3*b*d*e - a*(2*d^2 + e^2) - c 
*(d^2 + 2*e^2))*ArcTanh[(-e - d*x)/(Sqrt[d^2 - e^2]*Sqrt[-1 + x^2])])/(d^2 
 - e^2)^(3/2))/(2*(d^2 - e^2))))/(Sqrt[-1 + x]*Sqrt[1 + x])
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 219

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])
 

rule 488

Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ 
Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ 
[{a, b, c, d}, x]
 

rule 679

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1 
)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[(c*d*f + a*e*g)/(c*d^2 + a*e^2) 
 Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, 
 p}, x] && EqQ[Simplify[m + 2*p + 3], 0]
 

rule 2113

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_. 
)*(x_))^(p_.), x_Symbol] :> Simp[(a + b*x)^FracPart[m]*((c + d*x)^FracPart[ 
m]/(a*c + b*d*x^2)^FracPart[m])   Int[Px*(a*c + b*d*x^2)^m*(e + f*x)^p, x], 
 x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a 
*d, 0] && EqQ[m, n] &&  !IntegerQ[m]
 

rule 2182

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
 With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, 
 d + e*x, x]}, Simp[e*R*(d + e*x)^(m + 1)*((a + b*x^2)^(p + 1)/((m + 1)*(b* 
d^2 + a*e^2))), x] + Simp[1/((m + 1)*(b*d^2 + a*e^2))   Int[(d + e*x)^(m + 
1)*(a + b*x^2)^p*ExpandToSum[(m + 1)*(b*d^2 + a*e^2)*Qx + b*d*R*(m + 1) - b 
*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, d, e, p}, x] && PolyQ[Pq, 
 x] && NeQ[b*d^2 + a*e^2, 0] && LtQ[m, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1094\) vs. \(2(186)=372\).

Time = 0.94 (sec) , antiderivative size = 1095, normalized size of antiderivative = 5.21

method	result	size
default	\(\text {Expression too large to display}\)	\(1095\)

Input:

int((c*x^2+b*x+a)/(-1+x)^(1/2)/(1+x)^(1/2)/(e*x+d)^3,x,method=_RETURNVERBO 
SE)
 

Output:

-1/2*(3*a*d*e^3*x*(x^2-1)^(1/2)*((d^2-e^2)/e^2)^(1/2)-b*d^2*e^2*x*(x^2-1)^ 
(1/2)*((d^2-e^2)/e^2)^(1/2)-c*d^3*e*x*(x^2-1)^(1/2)*((d^2-e^2)/e^2)^(1/2)- 
a*e^4*(x^2-1)^(1/2)*((d^2-e^2)/e^2)^(1/2)+2*ln(-2*(-(x^2-1)^(1/2)*((d^2-e^ 
2)/e^2)^(1/2)*e+x*d+e)/(e*x+d))*c*e^4*x^2+ln(-2*(-(x^2-1)^(1/2)*((d^2-e^2) 
/e^2)^(1/2)*e+x*d+e)/(e*x+d))*a*d^2*e^2-3*ln(-2*(-(x^2-1)^(1/2)*((d^2-e^2) 
/e^2)^(1/2)*e+x*d+e)/(e*x+d))*b*d^3*e+2*ln(-2*(-(x^2-1)^(1/2)*((d^2-e^2)/e 
^2)^(1/2)*e+x*d+e)/(e*x+d))*c*d^2*e^2+ln(-2*(-(x^2-1)^(1/2)*((d^2-e^2)/e^2 
)^(1/2)*e+x*d+e)/(e*x+d))*a*e^4*x^2-2*b*e^4*x*(x^2-1)^(1/2)*((d^2-e^2)/e^2 
)^(1/2)+4*a*d^2*e^2*(x^2-1)^(1/2)*((d^2-e^2)/e^2)^(1/2)-2*b*d^3*e*(x^2-1)^ 
(1/2)*((d^2-e^2)/e^2)^(1/2)-b*d*e^3*(x^2-1)^(1/2)*((d^2-e^2)/e^2)^(1/2)+3* 
c*d^2*e^2*(x^2-1)^(1/2)*((d^2-e^2)/e^2)^(1/2)+2*ln(-2*(-(x^2-1)^(1/2)*((d^ 
2-e^2)/e^2)^(1/2)*e+x*d+e)/(e*x+d))*a*d^2*e^2*x^2-3*ln(-2*(-(x^2-1)^(1/2)* 
((d^2-e^2)/e^2)^(1/2)*e+x*d+e)/(e*x+d))*b*d*e^3*x^2+ln(-2*(-(x^2-1)^(1/2)* 
((d^2-e^2)/e^2)^(1/2)*e+x*d+e)/(e*x+d))*c*d^2*e^2*x^2+4*ln(-2*(-(x^2-1)^(1 
/2)*((d^2-e^2)/e^2)^(1/2)*e+x*d+e)/(e*x+d))*a*d^3*e*x+2*ln(-2*(-(x^2-1)^(1 
/2)*((d^2-e^2)/e^2)^(1/2)*e+x*d+e)/(e*x+d))*a*d*e^3*x-6*ln(-2*(-(x^2-1)^(1 
/2)*((d^2-e^2)/e^2)^(1/2)*e+x*d+e)/(e*x+d))*b*d^2*e^2*x+2*ln(-2*(-(x^2-1)^ 
(1/2)*((d^2-e^2)/e^2)^(1/2)*e+x*d+e)/(e*x+d))*c*d^3*e*x+4*ln(-2*(-(x^2-1)^ 
(1/2)*((d^2-e^2)/e^2)^(1/2)*e+x*d+e)/(e*x+d))*c*d*e^3*x+4*c*d*e^3*x*(x^2-1 
)^(1/2)*((d^2-e^2)/e^2)^(1/2)+ln(-2*(-(x^2-1)^(1/2)*((d^2-e^2)/e^2)^(1/...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 587 vs. \(2 (186) = 372\).

Time = 0.13 (sec) , antiderivative size = 1186, normalized size of antiderivative = 5.65 \[ \int \frac {a+b x+c x^2}{\sqrt {-1+x} \sqrt {1+x} (d+e x)^3} \, dx=\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)/(-1+x)^(1/2)/(1+x)^(1/2)/(e*x+d)^3,x, algorithm="f 
ricas")
 

Output:

[1/2*(c*d^7 + b*d^6*e - (3*a + 5*c)*d^5*e^2 + b*d^4*e^3 + (3*a + 4*c)*d^3* 
e^4 - 2*b*d^2*e^5 + (c*d^5*e^2 + b*d^4*e^3 - (3*a + 5*c)*d^3*e^4 + b*d^2*e 
^5 + (3*a + 4*c)*d*e^6 - 2*b*e^7)*x^2 + ((2*a + c)*d^4*e^2 - 3*b*d^3*e^3 + 
 (a + 2*c)*d^2*e^4 + ((2*a + c)*d^2*e^4 - 3*b*d*e^5 + (a + 2*c)*e^6)*x^2 + 
 2*((2*a + c)*d^3*e^3 - 3*b*d^2*e^4 + (a + 2*c)*d*e^5)*x)*sqrt(d^2 - e^2)* 
log((d^2*x + d*e + (d^2 - e^2 + sqrt(d^2 - e^2)*d)*sqrt(x + 1)*sqrt(x - 1) 
 + sqrt(d^2 - e^2)*(d*x + e))/(e*x + d)) + (2*b*d^5*e^2 - (4*a + 3*c)*d^4* 
e^3 - b*d^3*e^4 + (5*a + 3*c)*d^2*e^5 - b*d*e^6 - a*e^7 + (c*d^5*e^2 + b*d 
^4*e^3 - (3*a + 5*c)*d^3*e^4 + b*d^2*e^5 + (3*a + 4*c)*d*e^6 - 2*b*e^7)*x) 
*sqrt(x + 1)*sqrt(x - 1) + 2*(c*d^6*e + b*d^5*e^2 - (3*a + 5*c)*d^4*e^3 + 
b*d^3*e^4 + (3*a + 4*c)*d^2*e^5 - 2*b*d*e^6)*x)/(d^8*e^2 - 3*d^6*e^4 + 3*d 
^4*e^6 - d^2*e^8 + (d^6*e^4 - 3*d^4*e^6 + 3*d^2*e^8 - e^10)*x^2 + 2*(d^7*e 
^3 - 3*d^5*e^5 + 3*d^3*e^7 - d*e^9)*x), 1/2*(c*d^7 + b*d^6*e - (3*a + 5*c) 
*d^5*e^2 + b*d^4*e^3 + (3*a + 4*c)*d^3*e^4 - 2*b*d^2*e^5 + (c*d^5*e^2 + b* 
d^4*e^3 - (3*a + 5*c)*d^3*e^4 + b*d^2*e^5 + (3*a + 4*c)*d*e^6 - 2*b*e^7)*x 
^2 - 2*((2*a + c)*d^4*e^2 - 3*b*d^3*e^3 + (a + 2*c)*d^2*e^4 + ((2*a + c)*d 
^2*e^4 - 3*b*d*e^5 + (a + 2*c)*e^6)*x^2 + 2*((2*a + c)*d^3*e^3 - 3*b*d^2*e 
^4 + (a + 2*c)*d*e^5)*x)*sqrt(-d^2 + e^2)*arctan(-(sqrt(-d^2 + e^2)*e*sqrt 
(x + 1)*sqrt(x - 1) - sqrt(-d^2 + e^2)*(e*x + d))/(d^2 - e^2)) + (2*b*d^5* 
e^2 - (4*a + 3*c)*d^4*e^3 - b*d^3*e^4 + (5*a + 3*c)*d^2*e^5 - b*d*e^6 -...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {a+b x+c x^2}{\sqrt {-1+x} \sqrt {1+x} (d+e x)^3} \, dx=\text {Timed out} \] Input:

integrate((c*x**2+b*x+a)/(-1+x)**(1/2)/(1+x)**(1/2)/(e*x+d)**3,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b x+c x^2}{\sqrt {-1+x} \sqrt {1+x} (d+e x)^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((c*x^2+b*x+a)/(-1+x)^(1/2)/(1+x)^(1/2)/(e*x+d)^3,x, algorithm="m 
axima")
 

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((e-d)*(e+d)>0)', see `assume?` f 
or more de
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 627 vs. \(2 (186) = 372\).

Time = 0.36 (sec) , antiderivative size = 627, normalized size of antiderivative = 2.99 \[ \int \frac {a+b x+c x^2}{\sqrt {-1+x} \sqrt {1+x} (d+e x)^3} \, dx=-\frac {{\left (2 \, a d^{2} + c d^{2} - 3 \, b d e + a e^{2} + 2 \, c e^{2}\right )} \arctan \left (\frac {e {\left (\sqrt {x + 1} - \sqrt {x - 1}\right )}^{2} + 2 \, d}{2 \, \sqrt {-d^{2} + e^{2}}}\right )}{{\left (d^{4} - 2 \, d^{2} e^{2} + e^{4}\right )} \sqrt {-d^{2} + e^{2}}} + \frac {2 \, {\left (2 \, c d^{4} e {\left (\sqrt {x + 1} - \sqrt {x - 1}\right )}^{6} - 2 \, a d^{2} e^{3} {\left (\sqrt {x + 1} - \sqrt {x - 1}\right )}^{6} - 5 \, c d^{2} e^{3} {\left (\sqrt {x + 1} - \sqrt {x - 1}\right )}^{6} + 3 \, b d e^{4} {\left (\sqrt {x + 1} - \sqrt {x - 1}\right )}^{6} - a e^{5} {\left (\sqrt {x + 1} - \sqrt {x - 1}\right )}^{6} + 4 \, c d^{5} {\left (\sqrt {x + 1} - \sqrt {x - 1}\right )}^{4} + 4 \, b d^{4} e {\left (\sqrt {x + 1} - \sqrt {x - 1}\right )}^{4} - 12 \, a d^{3} e^{2} {\left (\sqrt {x + 1} - \sqrt {x - 1}\right )}^{4} - 14 \, c d^{3} e^{2} {\left (\sqrt {x + 1} - \sqrt {x - 1}\right )}^{4} + 10 \, b d^{2} e^{3} {\left (\sqrt {x + 1} - \sqrt {x - 1}\right )}^{4} - 6 \, a d e^{4} {\left (\sqrt {x + 1} - \sqrt {x - 1}\right )}^{4} - 8 \, c d e^{4} {\left (\sqrt {x + 1} - \sqrt {x - 1}\right )}^{4} + 4 \, b e^{5} {\left (\sqrt {x + 1} - \sqrt {x - 1}\right )}^{4} + 8 \, c d^{4} e {\left (\sqrt {x + 1} - \sqrt {x - 1}\right )}^{2} + 16 \, b d^{3} e^{2} {\left (\sqrt {x + 1} - \sqrt {x - 1}\right )}^{2} - 40 \, a d^{2} e^{3} {\left (\sqrt {x + 1} - \sqrt {x - 1}\right )}^{2} - 44 \, c d^{2} e^{3} {\left (\sqrt {x + 1} - \sqrt {x - 1}\right )}^{2} + 20 \, b d e^{4} {\left (\sqrt {x + 1} - \sqrt {x - 1}\right )}^{2} + 4 \, a e^{5} {\left (\sqrt {x + 1} - \sqrt {x - 1}\right )}^{2} + 8 \, c d^{3} e^{2} + 8 \, b d^{2} e^{3} - 24 \, a d e^{4} - 32 \, c d e^{4} + 16 \, b e^{5}\right )}}{{\left (d^{4} e^{2} - 2 \, d^{2} e^{4} + e^{6}\right )} {\left (e {\left (\sqrt {x + 1} - \sqrt {x - 1}\right )}^{4} + 4 \, d {\left (\sqrt {x + 1} - \sqrt {x - 1}\right )}^{2} + 4 \, e\right )}^{2}} \] Input:

integrate((c*x^2+b*x+a)/(-1+x)^(1/2)/(1+x)^(1/2)/(e*x+d)^3,x, algorithm="g 
iac")
 

Output:

-(2*a*d^2 + c*d^2 - 3*b*d*e + a*e^2 + 2*c*e^2)*arctan(1/2*(e*(sqrt(x + 1) 
- sqrt(x - 1))^2 + 2*d)/sqrt(-d^2 + e^2))/((d^4 - 2*d^2*e^2 + e^4)*sqrt(-d 
^2 + e^2)) + 2*(2*c*d^4*e*(sqrt(x + 1) - sqrt(x - 1))^6 - 2*a*d^2*e^3*(sqr 
t(x + 1) - sqrt(x - 1))^6 - 5*c*d^2*e^3*(sqrt(x + 1) - sqrt(x - 1))^6 + 3* 
b*d*e^4*(sqrt(x + 1) - sqrt(x - 1))^6 - a*e^5*(sqrt(x + 1) - sqrt(x - 1))^ 
6 + 4*c*d^5*(sqrt(x + 1) - sqrt(x - 1))^4 + 4*b*d^4*e*(sqrt(x + 1) - sqrt( 
x - 1))^4 - 12*a*d^3*e^2*(sqrt(x + 1) - sqrt(x - 1))^4 - 14*c*d^3*e^2*(sqr 
t(x + 1) - sqrt(x - 1))^4 + 10*b*d^2*e^3*(sqrt(x + 1) - sqrt(x - 1))^4 - 6 
*a*d*e^4*(sqrt(x + 1) - sqrt(x - 1))^4 - 8*c*d*e^4*(sqrt(x + 1) - sqrt(x - 
 1))^4 + 4*b*e^5*(sqrt(x + 1) - sqrt(x - 1))^4 + 8*c*d^4*e*(sqrt(x + 1) - 
sqrt(x - 1))^2 + 16*b*d^3*e^2*(sqrt(x + 1) - sqrt(x - 1))^2 - 40*a*d^2*e^3 
*(sqrt(x + 1) - sqrt(x - 1))^2 - 44*c*d^2*e^3*(sqrt(x + 1) - sqrt(x - 1))^ 
2 + 20*b*d*e^4*(sqrt(x + 1) - sqrt(x - 1))^2 + 4*a*e^5*(sqrt(x + 1) - sqrt 
(x - 1))^2 + 8*c*d^3*e^2 + 8*b*d^2*e^3 - 24*a*d*e^4 - 32*c*d*e^4 + 16*b*e^ 
5)/((d^4*e^2 - 2*d^2*e^4 + e^6)*(e*(sqrt(x + 1) - sqrt(x - 1))^4 + 4*d*(sq 
rt(x + 1) - sqrt(x - 1))^2 + 4*e)^2)
 

Mupad [B] (verification not implemented)

Time = 46.17 (sec) , antiderivative size = 7235, normalized size of antiderivative = 34.45 \[ \int \frac {a+b x+c x^2}{\sqrt {-1+x} \sqrt {1+x} (d+e x)^3} \, dx=\text {Too large to display} \] Input:

int((a + b*x + c*x^2)/((x - 1)^(1/2)*(x + 1)^(1/2)*(d + e*x)^3),x)
 

Output:

((((x - 1)^(1/2) - 1i)^2*(2*c*e^3 + c*d^2*e)*12i)/(d^2*((x + 1)^(1/2) - 1) 
^2*(d^4 + e^4 - 2*d^2*e^2)) - (2*(7*c*d^4 + 14*c*d^2*e^2)*((x - 1)^(1/2) - 
 1i))/(7*d^3*((x + 1)^(1/2) - 1)*(d^4 + e^4 - 2*d^2*e^2)) + (((x - 1)^(1/2 
) - 1i)^4*(2*c*e^3 - c*d^2*e)*24i)/(d^2*((x + 1)^(1/2) - 1)^4*(d^4 + e^4 - 
 2*d^2*e^2)) - (2*(21*c*d^4 - 102*c*d^2*e^2)*((x - 1)^(1/2) - 1i)^5)/(3*d^ 
3*((x + 1)^(1/2) - 1)^5*(d^4 + e^4 - 2*d^2*e^2)) - (2*(35*c*d^4 - 170*c*d^ 
2*e^2)*((x - 1)^(1/2) - 1i)^3)/(5*d^3*((x + 1)^(1/2) - 1)^3*(d^4 + e^4 - 2 
*d^2*e^2)) + (c*((x - 1)^(1/2) - 1i)^7*(d^2*1i + e^2*2i)*2i)/(d*((x + 1)^( 
1/2) - 1)^7*(d^4 + e^4 - 2*d^2*e^2)) + (12*c*e*((x - 1)^(1/2) - 1i)^6*(d^2 
*1i + e^2*2i))/(d^2*((x + 1)^(1/2) - 1)^6*(d^4 + e^4 - 2*d^2*e^2)))/(((x - 
 1)^(1/2) - 1i)^8/((x + 1)^(1/2) - 1)^8 - (e*((x - 1)^(1/2) - 1i)*8i)/(d*( 
(x + 1)^(1/2) - 1)) + (e*((x - 1)^(1/2) - 1i)^3*8i)/(d*((x + 1)^(1/2) - 1) 
^3) + (e*((x - 1)^(1/2) - 1i)^5*8i)/(d*((x + 1)^(1/2) - 1)^5) - (e*((x - 1 
)^(1/2) - 1i)^7*8i)/(d*((x + 1)^(1/2) - 1)^7) - (((x - 1)^(1/2) - 1i)^2*(4 
*d^2 + 16*e^2))/(d^2*((x + 1)^(1/2) - 1)^2) - (((x - 1)^(1/2) - 1i)^6*(4*d 
^2 + 16*e^2))/(d^2*((x + 1)^(1/2) - 1)^6) + (((x - 1)^(1/2) - 1i)^4*(6*d^2 
 - 32*e^2))/(d^2*((x + 1)^(1/2) - 1)^4) + 1) - ((2*((x - 1)^(1/2) - 1i)^3* 
(16*b*e^3 + 11*b*d^2*e))/(d^2*((x + 1)^(1/2) - 1)^3*(d^4 + e^4 - 2*d^2*e^2 
)) - (6*b*e*((x - 1)^(1/2) - 1i)^7)/(((x + 1)^(1/2) - 1)^7*(d^4 + e^4 - 2* 
d^2*e^2)) - (6*b*e*((x - 1)^(1/2) - 1i))/(((x + 1)^(1/2) - 1)*(d^4 + e^...
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 3120, normalized size of antiderivative = 14.86 \[ \int \frac {a+b x+c x^2}{\sqrt {-1+x} \sqrt {1+x} (d+e x)^3} \, dx =\text {Too large to display} \] Input:

int((c*x^2+b*x+a)/(-1+x)^(1/2)/(1+x)^(1/2)/(e*x+d)^3,x)
 

Output:

( - 4*sqrt(d + e)*sqrt(d - e)*log(sqrt(d + e)*sqrt(d - e) + sqrt(x + 1)*sq 
rt(x - 1)*e + d + e*x)*a*d**5*e - 8*sqrt(d + e)*sqrt(d - e)*log(sqrt(d + e 
)*sqrt(d - e) + sqrt(x + 1)*sqrt(x - 1)*e + d + e*x)*a*d**4*e**2*x - 4*sqr 
t(d + e)*sqrt(d - e)*log(sqrt(d + e)*sqrt(d - e) + sqrt(x + 1)*sqrt(x - 1) 
*e + d + e*x)*a*d**3*e**3*x**2 - 2*sqrt(d + e)*sqrt(d - e)*log(sqrt(d + e) 
*sqrt(d - e) + sqrt(x + 1)*sqrt(x - 1)*e + d + e*x)*a*d**3*e**3 - 4*sqrt(d 
 + e)*sqrt(d - e)*log(sqrt(d + e)*sqrt(d - e) + sqrt(x + 1)*sqrt(x - 1)*e 
+ d + e*x)*a*d**2*e**4*x - 2*sqrt(d + e)*sqrt(d - e)*log(sqrt(d + e)*sqrt( 
d - e) + sqrt(x + 1)*sqrt(x - 1)*e + d + e*x)*a*d*e**5*x**2 + 6*sqrt(d + e 
)*sqrt(d - e)*log(sqrt(d + e)*sqrt(d - e) + sqrt(x + 1)*sqrt(x - 1)*e + d 
+ e*x)*b*d**4*e**2 + 12*sqrt(d + e)*sqrt(d - e)*log(sqrt(d + e)*sqrt(d - e 
) + sqrt(x + 1)*sqrt(x - 1)*e + d + e*x)*b*d**3*e**3*x + 6*sqrt(d + e)*sqr 
t(d - e)*log(sqrt(d + e)*sqrt(d - e) + sqrt(x + 1)*sqrt(x - 1)*e + d + e*x 
)*b*d**2*e**4*x**2 - 2*sqrt(d + e)*sqrt(d - e)*log(sqrt(d + e)*sqrt(d - e) 
 + sqrt(x + 1)*sqrt(x - 1)*e + d + e*x)*c*d**5*e - 4*sqrt(d + e)*sqrt(d - 
e)*log(sqrt(d + e)*sqrt(d - e) + sqrt(x + 1)*sqrt(x - 1)*e + d + e*x)*c*d* 
*4*e**2*x - 2*sqrt(d + e)*sqrt(d - e)*log(sqrt(d + e)*sqrt(d - e) + sqrt(x 
 + 1)*sqrt(x - 1)*e + d + e*x)*c*d**3*e**3*x**2 - 4*sqrt(d + e)*sqrt(d - e 
)*log(sqrt(d + e)*sqrt(d - e) + sqrt(x + 1)*sqrt(x - 1)*e + d + e*x)*c*d** 
3*e**3 - 8*sqrt(d + e)*sqrt(d - e)*log(sqrt(d + e)*sqrt(d - e) + sqrt(x...