\(\int \frac {\sqrt {a+b x^2} (c+d x^2)}{(e+f x^2)^2} \, dx\) [264]

Optimal result

Integrand size = 28, antiderivative size = 143 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )}{\left (e+f x^2\right )^2} \, dx=-\frac {(d e-c f) x \sqrt {a+b x^2}}{2 e f \left (e+f x^2\right )}+\frac {\sqrt {b} d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f^2}-\frac {\left (2 b d e^2-a f (d e+c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{2 e^{3/2} f^2 \sqrt {b e-a f}} \] Output:

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

Mathematica [A] (verified)

Time = 0.89 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )}{\left (e+f x^2\right )^2} \, dx=\frac {\frac {f (-d e+c f) x \sqrt {a+b x^2}}{e \left (e+f x^2\right )}+\frac {\left (2 b d e^2-a f (d e+c f)\right ) \arctan \left (\frac {-f x \sqrt {a+b x^2}+\sqrt {b} \left (e+f x^2\right )}{\sqrt {e} \sqrt {-b e+a f}}\right )}{e^{3/2} \sqrt {-b e+a f}}-2 \sqrt {b} d \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{2 f^2} \] Input:

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

Output:

((f*(-(d*e) + c*f)*x*Sqrt[a + b*x^2])/(e*(e + f*x^2)) + ((2*b*d*e^2 - a*f* 
(d*e + c*f))*ArcTan[(-(f*x*Sqrt[a + b*x^2]) + Sqrt[b]*(e + f*x^2))/(Sqrt[e 
]*Sqrt[-(b*e) + a*f])])/(e^(3/2)*Sqrt[-(b*e) + a*f]) - 2*Sqrt[b]*d*Log[-(S 
qrt[b]*x) + Sqrt[a + b*x^2]])/(2*f^2)
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.08, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {401, 25, 398, 224, 219, 291, 221}

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.

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

\(\Big \downarrow \) 401

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 398

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

-1/2*((d*e - c*f)*x*Sqrt[a + b*x^2])/(e*f*(e + f*x^2)) + ((2*Sqrt[b]*d*e*A 
rcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/f - ((2*b*d*e^2 - a*f*(d*e + c*f))*Ar 
cTanh[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(Sqrt[e]*f*Sqrt[b*e 
- a*f]))/(2*e*f)
 

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 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 291

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst 
[Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, 
d}, x] && NeQ[b*c - a*d, 0]
 

rule 398

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

rule 401

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x 
_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ 
q/(a*b*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1))   Int[(a + b*x^2)^(p + 1)*( 
c + d*x^2)^(q - 1)*Simp[c*(b*e*2*(p + 1) + b*e - a*f) + d*(b*e*2*(p + 1) + 
(b*e - a*f)*(2*q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && L 
tQ[p, -1] && GtQ[q, 0]
 

Maple [A] (verified)

Time = 0.80 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.06

method	result	size
pseudoelliptic	\(\frac {-\left (f \,x^{2}+e \right ) \left (a c \,f^{2}+a d e f -2 b d \,e^{2}\right ) \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )+\left (2 d e \sqrt {b}\, \left (f \,x^{2}+e \right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )+\sqrt {b \,x^{2}+a}\, f x \left (c f -d e \right )\right ) \sqrt {\left (a f -b e \right ) e}}{2 \sqrt {\left (a f -b e \right ) e}\, f^{2} e \left (f \,x^{2}+e \right )}\)	\(151\)
default	\(\text {Expression too large to display}\)	\(1981\)

Input:

int((b*x^2+a)^(1/2)*(d*x^2+c)/(f*x^2+e)^2,x,method=_RETURNVERBOSE)
 

Output:

1/2*(-(f*x^2+e)*(a*c*f^2+a*d*e*f-2*b*d*e^2)*arctan(e*(b*x^2+a)^(1/2)/x/((a 
*f-b*e)*e)^(1/2))+(2*d*e*b^(1/2)*(f*x^2+e)*arctanh((b*x^2+a)^(1/2)/x/b^(1/ 
2))+(b*x^2+a)^(1/2)*f*x*(c*f-d*e))*((a*f-b*e)*e)^(1/2))/((a*f-b*e)*e)^(1/2 
)/f^2/e/(f*x^2+e)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 294 vs. \(2 (121) = 242\).

Time = 1.03 (sec) , antiderivative size = 1267, normalized size of antiderivative = 8.86 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )}{\left (e+f x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/8*(4*(b*d*e^3*f + a*c*e*f^3 - (b*c + a*d)*e^2*f^2)*sqrt(b*x^2 + a)*x - 
 4*(b*d*e^4 - a*d*e^3*f + (b*d*e^3*f - a*d*e^2*f^2)*x^2)*sqrt(b)*log(-2*b* 
x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + (2*b*d*e^3 - a*d*e^2*f - a*c*e*f^ 
2 + (2*b*d*e^2*f - a*d*e*f^2 - a*c*f^3)*x^2)*sqrt(b*e^2 - a*e*f)*log(((8*b 
^2*e^2 - 8*a*b*e*f + a^2*f^2)*x^4 + a^2*e^2 + 2*(4*a*b*e^2 - 3*a^2*e*f)*x^ 
2 + 4*((2*b*e - a*f)*x^3 + a*e*x)*sqrt(b*e^2 - a*e*f)*sqrt(b*x^2 + a))/(f^ 
2*x^4 + 2*e*f*x^2 + e^2)))/(b*e^4*f^2 - a*e^3*f^3 + (b*e^3*f^3 - a*e^2*f^4 
)*x^2), -1/8*(4*(b*d*e^3*f + a*c*e*f^3 - (b*c + a*d)*e^2*f^2)*sqrt(b*x^2 + 
 a)*x + 8*(b*d*e^4 - a*d*e^3*f + (b*d*e^3*f - a*d*e^2*f^2)*x^2)*sqrt(-b)*a 
rctan(sqrt(-b)*x/sqrt(b*x^2 + a)) + (2*b*d*e^3 - a*d*e^2*f - a*c*e*f^2 + ( 
2*b*d*e^2*f - a*d*e*f^2 - a*c*f^3)*x^2)*sqrt(b*e^2 - a*e*f)*log(((8*b^2*e^ 
2 - 8*a*b*e*f + a^2*f^2)*x^4 + a^2*e^2 + 2*(4*a*b*e^2 - 3*a^2*e*f)*x^2 + 4 
*((2*b*e - a*f)*x^3 + a*e*x)*sqrt(b*e^2 - a*e*f)*sqrt(b*x^2 + a))/(f^2*x^4 
 + 2*e*f*x^2 + e^2)))/(b*e^4*f^2 - a*e^3*f^3 + (b*e^3*f^3 - a*e^2*f^4)*x^2 
), -1/4*(2*(b*d*e^3*f + a*c*e*f^3 - (b*c + a*d)*e^2*f^2)*sqrt(b*x^2 + a)*x 
 - (2*b*d*e^3 - a*d*e^2*f - a*c*e*f^2 + (2*b*d*e^2*f - a*d*e*f^2 - a*c*f^3 
)*x^2)*sqrt(-b*e^2 + a*e*f)*arctan(1/2*sqrt(-b*e^2 + a*e*f)*((2*b*e - a*f) 
*x^2 + a*e)*sqrt(b*x^2 + a)/((b^2*e^2 - a*b*e*f)*x^3 + (a*b*e^2 - a^2*e*f) 
*x)) - 2*(b*d*e^4 - a*d*e^3*f + (b*d*e^3*f - a*d*e^2*f^2)*x^2)*sqrt(b)*log 
(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a))/(b*e^4*f^2 - a*e^3*f^3 + ...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

integrate(sqrt(b*x^2 + a)*(d*x^2 + c)/(f*x^2 + e)^2, x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 349 vs. \(2 (121) = 242\).

Time = 0.15 (sec) , antiderivative size = 349, normalized size of antiderivative = 2.44 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )}{\left (e+f x^2\right )^2} \, dx=-\frac {\sqrt {b} d \log \left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2}\right )}{2 \, f^{2}} + \frac {{\left (2 \, b^{\frac {3}{2}} d e^{2} - a \sqrt {b} d e f - a \sqrt {b} c f^{2}\right )} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} f + 2 \, b e - a f}{2 \, \sqrt {-b^{2} e^{2} + a b e f}}\right )}{2 \, \sqrt {-b^{2} e^{2} + a b e f} e f^{2}} - \frac {2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b^{\frac {3}{2}} d e^{2} - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b^{\frac {3}{2}} c e f - {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a \sqrt {b} d e f + {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a \sqrt {b} c f^{2} + a^{2} \sqrt {b} d e f - a^{2} \sqrt {b} c f^{2}}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} f + 4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b e - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a f + a^{2} f\right )} e f^{2}} \] Input:

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

Output:

-1/2*sqrt(b)*d*log((sqrt(b)*x - sqrt(b*x^2 + a))^2)/f^2 + 1/2*(2*b^(3/2)*d 
*e^2 - a*sqrt(b)*d*e*f - a*sqrt(b)*c*f^2)*arctan(1/2*((sqrt(b)*x - sqrt(b* 
x^2 + a))^2*f + 2*b*e - a*f)/sqrt(-b^2*e^2 + a*b*e*f))/(sqrt(-b^2*e^2 + a* 
b*e*f)*e*f^2) - (2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b^(3/2)*d*e^2 - 2*(sqrt 
(b)*x - sqrt(b*x^2 + a))^2*b^(3/2)*c*e*f - (sqrt(b)*x - sqrt(b*x^2 + a))^2 
*a*sqrt(b)*d*e*f + (sqrt(b)*x - sqrt(b*x^2 + a))^2*a*sqrt(b)*c*f^2 + a^2*s 
qrt(b)*d*e*f - a^2*sqrt(b)*c*f^2)/(((sqrt(b)*x - sqrt(b*x^2 + a))^4*f + 4* 
(sqrt(b)*x - sqrt(b*x^2 + a))^2*b*e - 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a* 
f + a^2*f)*e*f^2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 929, normalized size of antiderivative = 6.50 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )}{\left (e+f x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x** 
2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a*c*e*f**2 - sqrt(e)*sqrt(a*f - 
 b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x 
)/(sqrt(e)*sqrt(b)))*a*c*f**3*x**2 - sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a* 
f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)) 
)*a*d*e**2*f - sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqr 
t(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a*d*e*f**2*x**2 + 2* 
sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - 
 sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*b*d*e**3 + 2*sqrt(e)*sqrt(a*f - b*e 
)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(s 
qrt(e)*sqrt(b)))*b*d*e**2*f*x**2 - sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f 
- b*e) + sqrt(f)*sqrt(a + b*x**2) + sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))* 
a*c*e*f**2 - sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) + sqrt(f)*sqrt( 
a + b*x**2) + sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a*c*f**3*x**2 - sqrt(e 
)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) + sqrt(f)*sqrt(a + b*x**2) + sqrt( 
f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a*d*e**2*f - sqrt(e)*sqrt(a*f - b*e)*atan 
((sqrt(a*f - b*e) + sqrt(f)*sqrt(a + b*x**2) + sqrt(f)*sqrt(b)*x)/(sqrt(e) 
*sqrt(b)))*a*d*e*f**2*x**2 + 2*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b* 
e) + sqrt(f)*sqrt(a + b*x**2) + sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*b*d* 
e**3 + 2*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) + sqrt(f)*sqrt(a...