\(\int \sqrt {a+\frac {b}{x}} \sqrt {c+\frac {d}{x}} \, dx\) [46]

Optimal result

Integrand size = 23, antiderivative size = 123 \[ \int \sqrt {a+\frac {b}{x}} \sqrt {c+\frac {d}{x}} \, dx=\sqrt {a+\frac {b}{x}} \sqrt {c+\frac {d}{x}} x+\frac {(b c+a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+\frac {b}{x}}}{\sqrt {a} \sqrt {c+\frac {d}{x}}}\right )}{\sqrt {a} \sqrt {c}}-2 \sqrt {b} \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b} \sqrt {c+\frac {d}{x}}}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.55 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.43 \[ \int \sqrt {a+\frac {b}{x}} \sqrt {c+\frac {d}{x}} \, dx=\frac {\sqrt {a+\frac {b}{x}} \sqrt {c+\frac {d}{x}} x \left ((b c+a d) \sqrt {b+a x} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d+c x}}{\sqrt {c} \sqrt {b+a x}}\right )+\sqrt {a} \sqrt {c} \left ((b+a x) \sqrt {d+c x}-2 \sqrt {b} \sqrt {d} \sqrt {b+a x} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+c x}}{\sqrt {d} \sqrt {b+a x}}\right )\right )\right )}{\sqrt {a} \sqrt {c} (b+a x) \sqrt {d+c x}} \] Input:

Integrate[Sqrt[a + b/x]*Sqrt[c + d/x],x]
 

Output:

(Sqrt[a + b/x]*Sqrt[c + d/x]*x*((b*c + a*d)*Sqrt[b + a*x]*ArcTanh[(Sqrt[a] 
*Sqrt[d + c*x])/(Sqrt[c]*Sqrt[b + a*x])] + Sqrt[a]*Sqrt[c]*((b + a*x)*Sqrt 
[d + c*x] - 2*Sqrt[b]*Sqrt[d]*Sqrt[b + a*x]*ArcTanh[(Sqrt[b]*Sqrt[d + c*x] 
)/(Sqrt[d]*Sqrt[b + a*x])])))/(Sqrt[a]*Sqrt[c]*(b + a*x)*Sqrt[d + c*x])
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {899, 108, 27, 175, 66, 104, 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.

\(\displaystyle \int \sqrt {a+\frac {b}{x}} \sqrt {c+\frac {d}{x}} \, dx\)

\(\Big \downarrow \) 899

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

\(\Big \downarrow \) 108

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 175

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

\(\Big \downarrow \) 66

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

\(\Big \downarrow \) 104

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

\(\Big \downarrow \) 221

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

Input:

Int[Sqrt[a + b/x]*Sqrt[c + d/x],x]
 

Output:

Sqrt[a + b/x]*Sqrt[c + d/x]*x + ((2*(b*c + a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + 
b/x])/(Sqrt[a]*Sqrt[c + d/x])])/(Sqrt[a]*Sqrt[c]) - 4*Sqrt[b]*Sqrt[d]*ArcT 
anh[(Sqrt[d]*Sqrt[a + b/x])/(Sqrt[b]*Sqrt[c + d/x])])/2
 

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 66

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

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 108

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) 
, x] - Simp[1/(b*(m + 1))   Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* 
x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c 
, d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 
*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
 

rule 175

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ 
)))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b   Int[(c + d*x)^n*(e + f*x)^p, x] 
, x] + Simp[(b*g - a*h)/b   Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] 
 /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
 

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 899

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(219\) vs. \(2(95)=190\).

Time = 0.27 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.79

method	result	size
default	\(-\frac {\sqrt {\frac {a x +b}{x}}\, x \sqrt {\frac {c x +d}{x}}\, \left (2 b d \ln \left (\frac {a d x +b c x +2 \sqrt {b d}\, \sqrt {\left (a x +b \right ) \left (c x +d \right )}+2 b d}{x}\right ) \sqrt {a c}-\ln \left (\frac {2 a c x +2 \sqrt {\left (a x +b \right ) \left (c x +d \right )}\, \sqrt {a c}+a d +b c}{2 \sqrt {a c}}\right ) \sqrt {b d}\, a d -\ln \left (\frac {2 a c x +2 \sqrt {\left (a x +b \right ) \left (c x +d \right )}\, \sqrt {a c}+a d +b c}{2 \sqrt {a c}}\right ) \sqrt {b d}\, b c -2 \sqrt {\left (a x +b \right ) \left (c x +d \right )}\, \sqrt {a c}\, \sqrt {b d}\right )}{2 \sqrt {\left (a x +b \right ) \left (c x +d \right )}\, \sqrt {a c}\, \sqrt {b d}}\)	\(220\)

Input:

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

Output:

-1/2*((a*x+b)/x)^(1/2)*x*((c*x+d)/x)^(1/2)*(2*b*d*ln((a*d*x+b*c*x+2*(b*d)^ 
(1/2)*((a*x+b)*(c*x+d))^(1/2)+2*b*d)/x)*(a*c)^(1/2)-ln(1/2*(2*a*c*x+2*((a* 
x+b)*(c*x+d))^(1/2)*(a*c)^(1/2)+a*d+b*c)/(a*c)^(1/2))*(b*d)^(1/2)*a*d-ln(1 
/2*(2*a*c*x+2*((a*x+b)*(c*x+d))^(1/2)*(a*c)^(1/2)+a*d+b*c)/(a*c)^(1/2))*(b 
*d)^(1/2)*b*c-2*((a*x+b)*(c*x+d))^(1/2)*(a*c)^(1/2)*(b*d)^(1/2))/((a*x+b)* 
(c*x+d))^(1/2)/(a*c)^(1/2)/(b*d)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.74 (sec) , antiderivative size = 824, normalized size of antiderivative = 6.70 \[ \int \sqrt {a+\frac {b}{x}} \sqrt {c+\frac {d}{x}} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/4*(4*a*c*x*sqrt((a*x + b)/x)*sqrt((c*x + d)/x) + 2*sqrt(b*d)*a*c*log(-( 
8*b^2*d^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 - 4*(2*b*d*x + (b*c + a*d) 
*x^2)*sqrt(b*d)*sqrt((a*x + b)/x)*sqrt((c*x + d)/x) + 8*(b^2*c*d + a*b*d^2 
)*x)/x^2) + sqrt(a*c)*(b*c + a*d)*log(-8*a^2*c^2*x^2 - b^2*c^2 - 6*a*b*c*d 
 - a^2*d^2 - 4*(2*a*c*x^2 + (b*c + a*d)*x)*sqrt(a*c)*sqrt((a*x + b)/x)*sqr 
t((c*x + d)/x) - 8*(a*b*c^2 + a^2*c*d)*x))/(a*c), 1/4*(4*a*c*x*sqrt((a*x + 
 b)/x)*sqrt((c*x + d)/x) + 4*sqrt(-b*d)*a*c*arctan(2*sqrt(-b*d)*x*sqrt((a* 
x + b)/x)*sqrt((c*x + d)/x)/(2*b*d + (b*c + a*d)*x)) + sqrt(a*c)*(b*c + a* 
d)*log(-8*a^2*c^2*x^2 - b^2*c^2 - 6*a*b*c*d - a^2*d^2 - 4*(2*a*c*x^2 + (b* 
c + a*d)*x)*sqrt(a*c)*sqrt((a*x + b)/x)*sqrt((c*x + d)/x) - 8*(a*b*c^2 + a 
^2*c*d)*x))/(a*c), 1/2*(2*a*c*x*sqrt((a*x + b)/x)*sqrt((c*x + d)/x) + sqrt 
(b*d)*a*c*log(-(8*b^2*d^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 - 4*(2*b*d 
*x + (b*c + a*d)*x^2)*sqrt(b*d)*sqrt((a*x + b)/x)*sqrt((c*x + d)/x) + 8*(b 
^2*c*d + a*b*d^2)*x)/x^2) - sqrt(-a*c)*(b*c + a*d)*arctan(2*sqrt(-a*c)*x*s 
qrt((a*x + b)/x)*sqrt((c*x + d)/x)/(2*a*c*x + b*c + a*d)))/(a*c), 1/2*(2*a 
*c*x*sqrt((a*x + b)/x)*sqrt((c*x + d)/x) + 2*sqrt(-b*d)*a*c*arctan(2*sqrt( 
-b*d)*x*sqrt((a*x + b)/x)*sqrt((c*x + d)/x)/(2*b*d + (b*c + a*d)*x)) - sqr 
t(-a*c)*(b*c + a*d)*arctan(2*sqrt(-a*c)*x*sqrt((a*x + b)/x)*sqrt((c*x + d) 
/x)/(2*a*c*x + b*c + a*d)))/(a*c)]
 

Sympy [F]

\[ \int \sqrt {a+\frac {b}{x}} \sqrt {c+\frac {d}{x}} \, dx=\int \sqrt {a + \frac {b}{x}} \sqrt {c + \frac {d}{x}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \sqrt {a+\frac {b}{x}} \sqrt {c+\frac {d}{x}} \, dx=\int { \sqrt {a + \frac {b}{x}} \sqrt {c + \frac {d}{x}} \,d x } \] Input:

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

Output:

integrate(sqrt(a + b/x)*sqrt(c + d/x), x)
 

Giac [F]

\[ \int \sqrt {a+\frac {b}{x}} \sqrt {c+\frac {d}{x}} \, dx=\int { \sqrt {a + \frac {b}{x}} \sqrt {c + \frac {d}{x}} \,d x } \] Input:

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

Output:

integrate(sqrt(a + b/x)*sqrt(c + d/x), x)
 

Mupad [B] (verification not implemented)

Time = 22.30 (sec) , antiderivative size = 4674, normalized size of antiderivative = 38.00 \[ \int \sqrt {a+\frac {b}{x}} \sqrt {c+\frac {d}{x}} \, dx=\text {Too large to display} \] Input:

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

Output:

atan(((b*d)^(1/2)*(2*(b*d)^(1/2)*(2*(b*d)^(1/2)*(2*(b*d)^(1/2)*((2*(4*a^(9 
/2)*b^9*c^(19/2) - 4*a^(13/2)*b^7*c^(15/2)*d^2 - 4*a^(15/2)*b^6*c^(13/2)*d 
^3 + 4*a^(19/2)*b^4*c^(9/2)*d^5))/(a^7*c^7*d^9) - (((a + b/x)^(1/2) - a^(1 
/2))*(32*a^4*b^9*c^10 - 120*a^5*b^8*c^9*d + 288*a^6*b^7*c^8*d^2 - 400*a^7* 
b^6*c^7*d^3 + 288*a^8*b^5*c^6*d^4 - 120*a^9*b^4*c^5*d^5 + 32*a^10*b^3*c^4* 
d^6))/(2*a^7*c^7*d^9*((c + d/x)^(1/2) - c^(1/2)))) - (2*(8*a^5*b^9*c^9*d + 
 16*a^6*b^8*c^8*d^2 - 48*a^7*b^7*c^7*d^3 + 16*a^8*b^6*c^6*d^4 + 8*a^9*b^5* 
c^5*d^5))/(a^7*c^7*d^9) + (((a + b/x)^(1/2) - a^(1/2))*(16*a^(7/2)*b^10*c^ 
(21/2) - 76*a^(9/2)*b^9*c^(19/2)*d + 228*a^(11/2)*b^8*c^(17/2)*d^2 - 168*a 
^(13/2)*b^7*c^(15/2)*d^3 - 168*a^(15/2)*b^6*c^(13/2)*d^4 + 228*a^(17/2)*b^ 
5*c^(11/2)*d^5 - 76*a^(19/2)*b^4*c^(9/2)*d^6 + 16*a^(21/2)*b^3*c^(7/2)*d^7 
))/(2*a^7*c^7*d^9*((c + d/x)^(1/2) - c^(1/2)))) - (2*(a^(7/2)*b^11*c^(21/2 
) + 16*a^(9/2)*b^10*c^(19/2)*d - 42*a^(11/2)*b^9*c^(17/2)*d^2 + 25*a^(13/2 
)*b^8*c^(15/2)*d^3 + 25*a^(15/2)*b^7*c^(13/2)*d^4 - 42*a^(17/2)*b^6*c^(11/ 
2)*d^5 + 16*a^(19/2)*b^5*c^(9/2)*d^6 + a^(21/2)*b^4*c^(7/2)*d^7))/(a^7*c^7 
*d^9) + (((a + b/x)^(1/2) - a^(1/2))*(146*a^4*b^10*c^10*d - 556*a^5*b^9*c^ 
9*d^2 + 1006*a^6*b^8*c^8*d^3 - 1192*a^7*b^7*c^7*d^4 + 1006*a^8*b^6*c^6*d^5 
 - 556*a^9*b^5*c^5*d^6 + 146*a^10*b^4*c^4*d^7))/(2*a^7*c^7*d^9*((c + d/x)^ 
(1/2) - c^(1/2)))) + (2*(2*a^4*b^11*c^10*d + 8*a^5*b^10*c^9*d^2 - 2*a^6*b^ 
9*c^8*d^3 - 16*a^7*b^8*c^7*d^4 - 2*a^8*b^7*c^6*d^5 + 8*a^9*b^6*c^5*d^6 ...
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.92 \[ \int \sqrt {a+\frac {b}{x}} \sqrt {c+\frac {d}{x}} \, dx=\frac {\sqrt {c x +d}\, \sqrt {a x +b}\, a c +\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {c}\, \sqrt {a x +b}+\sqrt {a}\, \sqrt {c x +d}}{\sqrt {a d -b c}}\right ) a d +\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {c}\, \sqrt {a x +b}+\sqrt {a}\, \sqrt {c x +d}}{\sqrt {a d -b c}}\right ) b c +\sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\sqrt {c}\, \sqrt {a x +b}+\sqrt {a}\, \sqrt {c x +d}-\sqrt {2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+a d +b c}\right ) a c +\sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\sqrt {c}\, \sqrt {a x +b}+\sqrt {a}\, \sqrt {c x +d}+\sqrt {2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+a d +b c}\right ) a c -\sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (2 \sqrt {c}\, \sqrt {a}\, \sqrt {c x +d}\, \sqrt {a x +b}+2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+2 a c x \right ) a c}{a c} \] Input:

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

Output:

(sqrt(c*x + d)*sqrt(a*x + b)*a*c + sqrt(c)*sqrt(a)*log((sqrt(c)*sqrt(a*x + 
 b) + sqrt(a)*sqrt(c*x + d))/sqrt(a*d - b*c))*a*d + sqrt(c)*sqrt(a)*log((s 
qrt(c)*sqrt(a*x + b) + sqrt(a)*sqrt(c*x + d))/sqrt(a*d - b*c))*b*c + sqrt( 
d)*sqrt(b)*log(sqrt(c)*sqrt(a*x + b) + sqrt(a)*sqrt(c*x + d) - sqrt(2*sqrt 
(d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c))*a*c + sqrt(d)*sqrt(b)*log(sqrt(c 
)*sqrt(a*x + b) + sqrt(a)*sqrt(c*x + d) + sqrt(2*sqrt(d)*sqrt(c)*sqrt(b)*s 
qrt(a) + a*d + b*c))*a*c - sqrt(d)*sqrt(b)*log(2*sqrt(c)*sqrt(a)*sqrt(c*x 
+ d)*sqrt(a*x + b) + 2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + 2*a*c*x)*a*c)/(a* 
c)