\(\int \frac {(c+d x)^{5/2}}{x^4 \sqrt {a+b x}} \, dx\) [338]

Optimal result

Integrand size = 22, antiderivative size = 157 \[ \int \frac {(c+d x)^{5/2}}{x^4 \sqrt {a+b x}} \, dx=-\frac {5 (b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}{8 a^3 x}+\frac {5 (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}{12 a^2 x^2}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 a x^3}+\frac {5 (b c-a d)^3 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{7/2} \sqrt {c}} \] Output:

-5/8*(-a*d+b*c)^2*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^3/x+5/12*(-a*d+b*c)*(b*x+a 
)^(1/2)*(d*x+c)^(3/2)/a^2/x^2-1/3*(b*x+a)^(1/2)*(d*x+c)^(5/2)/a/x^3+5/8*(- 
a*d+b*c)^3*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(7/2)/c^ 
(1/2)
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.90 \[ \int \frac {(c+d x)^{5/2}}{x^4 \sqrt {a+b x}} \, dx=\frac {(-b c+a d)^3 \left (\frac {\sqrt {a} \sqrt {a+b x} \sqrt {c+d x} \left (15 b^2 c^2 x^2-10 a b c x (c+4 d x)+a^2 \left (8 c^2+26 c d x+33 d^2 x^2\right )\right )}{(b c-a d)^3 x^3}-\frac {15 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {c}}\right )}{24 a^{7/2}} \] Input:

Integrate[(c + d*x)^(5/2)/(x^4*Sqrt[a + b*x]),x]
 

Output:

((-(b*c) + a*d)^3*((Sqrt[a]*Sqrt[a + b*x]*Sqrt[c + d*x]*(15*b^2*c^2*x^2 - 
10*a*b*c*x*(c + 4*d*x) + a^2*(8*c^2 + 26*c*d*x + 33*d^2*x^2)))/((b*c - a*d 
)^3*x^3) - (15*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/S 
qrt[c]))/(24*a^(7/2))
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {105, 105, 105, 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.

\(\displaystyle \int \frac {(c+d x)^{5/2}}{x^4 \sqrt {a+b x}} \, dx\)

\(\Big \downarrow \) 105

\(\displaystyle -\frac {5 (b c-a d) \int \frac {(c+d x)^{3/2}}{x^3 \sqrt {a+b x}}dx}{6 a}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 a x^3}\)

\(\Big \downarrow \) 105

\(\displaystyle -\frac {5 (b c-a d) \left (-\frac {3 (b c-a d) \int \frac {\sqrt {c+d x}}{x^2 \sqrt {a+b x}}dx}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}\right )}{6 a}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 a x^3}\)

\(\Big \downarrow \) 105

\(\displaystyle -\frac {5 (b c-a d) \left (-\frac {3 (b c-a d) \left (-\frac {(b c-a d) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x}\right )}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}\right )}{6 a}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 a x^3}\)

\(\Big \downarrow \) 104

\(\displaystyle -\frac {5 (b c-a d) \left (-\frac {3 (b c-a d) \left (-\frac {(b c-a d) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{a}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x}\right )}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}\right )}{6 a}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 a x^3}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {5 (b c-a d) \left (-\frac {3 (b c-a d) \left (\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} \sqrt {c}}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x}\right )}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}\right )}{6 a}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 a x^3}\)

Input:

Int[(c + d*x)^(5/2)/(x^4*Sqrt[a + b*x]),x]
 

Output:

-1/3*(Sqrt[a + b*x]*(c + d*x)^(5/2))/(a*x^3) - (5*(b*c - a*d)*(-1/2*(Sqrt[ 
a + b*x]*(c + d*x)^(3/2))/(a*x^2) - (3*(b*c - a*d)*(-((Sqrt[a + b*x]*Sqrt[ 
c + d*x])/(a*x)) + ((b*c - a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*S 
qrt[c + d*x])])/(a^(3/2)*Sqrt[c])))/(4*a)))/(6*a)
 

Defintions of rubi rules used

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 105

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 + 1)/((m + 
1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f)))   Int[(a 
+ b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, 
e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] 
 ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(404\) vs. \(2(125)=250\).

Time = 0.23 (sec) , antiderivative size = 405, normalized size of antiderivative = 2.58

method	result	size
default	\(-\frac {\sqrt {x d +c}\, \sqrt {b x +a}\, \left (15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{3} d^{3} x^{3}-45 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{2} b c \,d^{2} x^{3}+45 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a \,b^{2} c^{2} d \,x^{3}-15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) b^{3} c^{3} x^{3}+66 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a^{2} d^{2} x^{2}-80 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a b c d \,x^{2}+30 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, b^{2} c^{2} x^{2}+52 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a^{2} c d x -20 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a b \,c^{2} x +16 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, a^{2} c^{2} \sqrt {a c}\right )}{48 a^{3} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, x^{3} \sqrt {a c}}\)	\(405\)

Input:

int((d*x+c)^(5/2)/x^4/(b*x+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/48*(d*x+c)^(1/2)*(b*x+a)^(1/2)/a^3*(15*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(( 
b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^3*d^3*x^3-45*ln((a*d*x+b*c*x+2*(a*c)^(1/ 
2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^2*b*c*d^2*x^3+45*ln((a*d*x+b*c*x+2* 
(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a*b^2*c^2*d*x^3-15*ln((a*d*x 
+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*b^3*c^3*x^3+66*((b* 
x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^2*d^2*x^2-80*((b*x+a)*(d*x+c))^(1/2)*(a* 
c)^(1/2)*a*b*c*d*x^2+30*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*b^2*c^2*x^2+52 
*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^2*c*d*x-20*((b*x+a)*(d*x+c))^(1/2)* 
(a*c)^(1/2)*a*b*c^2*x+16*((b*x+a)*(d*x+c))^(1/2)*a^2*c^2*(a*c)^(1/2))/((b* 
x+a)*(d*x+c))^(1/2)/x^3/(a*c)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 438, normalized size of antiderivative = 2.79 \[ \int \frac {(c+d x)^{5/2}}{x^4 \sqrt {a+b x}} \, dx=\left [-\frac {15 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {a c} x^{3} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (8 \, a^{3} c^{3} + {\left (15 \, a b^{2} c^{3} - 40 \, a^{2} b c^{2} d + 33 \, a^{3} c d^{2}\right )} x^{2} - 2 \, {\left (5 \, a^{2} b c^{3} - 13 \, a^{3} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, a^{4} c x^{3}}, -\frac {15 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-a c} x^{3} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left (8 \, a^{3} c^{3} + {\left (15 \, a b^{2} c^{3} - 40 \, a^{2} b c^{2} d + 33 \, a^{3} c d^{2}\right )} x^{2} - 2 \, {\left (5 \, a^{2} b c^{3} - 13 \, a^{3} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, a^{4} c x^{3}}\right ] \] Input:

integrate((d*x+c)^(5/2)/x^4/(b*x+a)^(1/2),x, algorithm="fricas")
 

Output:

[-1/96*(15*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(a*c)*x 
^3*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 - 4*(2*a*c + (b*c 
+ a*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(a*b*c^2 + a^2*c*d)*x) 
/x^2) + 4*(8*a^3*c^3 + (15*a*b^2*c^3 - 40*a^2*b*c^2*d + 33*a^3*c*d^2)*x^2 
- 2*(5*a^2*b*c^3 - 13*a^3*c^2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^4*c*x^ 
3), -1/48*(15*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(-a* 
c)*x^3*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d* 
x + c)/(a*b*c*d*x^2 + a^2*c^2 + (a*b*c^2 + a^2*c*d)*x)) + 2*(8*a^3*c^3 + ( 
15*a*b^2*c^3 - 40*a^2*b*c^2*d + 33*a^3*c*d^2)*x^2 - 2*(5*a^2*b*c^3 - 13*a^ 
3*c^2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^4*c*x^3)]
 

Sympy [F]

\[ \int \frac {(c+d x)^{5/2}}{x^4 \sqrt {a+b x}} \, dx=\int \frac {\left (c + d x\right )^{\frac {5}{2}}}{x^{4} \sqrt {a + b x}}\, dx \] Input:

integrate((d*x+c)**(5/2)/x**4/(b*x+a)**(1/2),x)
 

Output:

Integral((c + d*x)**(5/2)/(x**4*sqrt(a + b*x)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(c+d x)^{5/2}}{x^4 \sqrt {a+b x}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((d*x+c)^(5/2)/x^4/(b*x+a)^(1/2),x, algorithm="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(a*d-b*c>0)', see `assume?` for m 
ore detail
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2148 vs. \(2 (125) = 250\).

Time = 1.65 (sec) , antiderivative size = 2148, normalized size of antiderivative = 13.68 \[ \int \frac {(c+d x)^{5/2}}{x^4 \sqrt {a+b x}} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^(5/2)/x^4/(b*x+a)^(1/2),x, algorithm="giac")
 

Output:

1/24*b^3*(15*(sqrt(b*d)*b^3*c^3*abs(b) - 3*sqrt(b*d)*a*b^2*c^2*d*abs(b) + 
3*sqrt(b*d)*a^2*b*c*d^2*abs(b) - sqrt(b*d)*a^3*d^3*abs(b))*arctan(-1/2*(b^ 
2*c + a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b* 
d))^2)/(sqrt(-a*b*c*d)*b))/(sqrt(-a*b*c*d)*a^3*b^4) - 2*(15*sqrt(b*d)*b^13 
*c^8*abs(b) - 130*sqrt(b*d)*a*b^12*c^7*d*abs(b) + 498*sqrt(b*d)*a^2*b^11*c 
^6*d^2*abs(b) - 1098*sqrt(b*d)*a^3*b^10*c^5*d^3*abs(b) + 1520*sqrt(b*d)*a^ 
4*b^9*c^4*d^4*abs(b) - 1350*sqrt(b*d)*a^5*b^8*c^3*d^5*abs(b) + 750*sqrt(b* 
d)*a^6*b^7*c^2*d^6*abs(b) - 238*sqrt(b*d)*a^7*b^6*c*d^7*abs(b) + 33*sqrt(b 
*d)*a^8*b^5*d^8*abs(b) - 75*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2* 
c + (b*x + a)*b*d - a*b*d))^2*b^11*c^7*abs(b) + 405*sqrt(b*d)*(sqrt(b*d)*s 
qrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^10*c^6*d*abs(b) 
- 783*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a* 
b*d))^2*a^2*b^9*c^5*d^2*abs(b) + 417*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - 
sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^3*b^8*c^4*d^3*abs(b) + 687*sqrt(b 
*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^4* 
b^7*c^3*d^4*abs(b) - 1233*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c 
+ (b*x + a)*b*d - a*b*d))^2*a^5*b^6*c^2*d^5*abs(b) + 747*sqrt(b*d)*(sqrt(b 
*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^6*b^5*c*d^6*a 
bs(b) - 165*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b* 
d - a*b*d))^2*a^7*b^4*d^7*abs(b) + 150*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + ...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{5/2}}{x^4 \sqrt {a+b x}} \, dx=\int \frac {{\left (c+d\,x\right )}^{5/2}}{x^4\,\sqrt {a+b\,x}} \,d x \] Input:

int((c + d*x)^(5/2)/(x^4*(a + b*x)^(1/2)),x)
 

Output:

int((c + d*x)^(5/2)/(x^4*(a + b*x)^(1/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.58 (sec) , antiderivative size = 950, normalized size of antiderivative = 6.05 \[ \int \frac {(c+d x)^{5/2}}{x^4 \sqrt {a+b x}} \, dx =\text {Too large to display} \] Input:

int((d*x+c)^(5/2)/x^4/(b*x+a)^(1/2),x)
 

Output:

( - 16*sqrt(c + d*x)*sqrt(a + b*x)*a**4*c**3*d - 52*sqrt(c + d*x)*sqrt(a + 
 b*x)*a**4*c**2*d**2*x - 66*sqrt(c + d*x)*sqrt(a + b*x)*a**4*c*d**3*x**2 - 
 16*sqrt(c + d*x)*sqrt(a + b*x)*a**3*b*c**4 - 32*sqrt(c + d*x)*sqrt(a + b* 
x)*a**3*b*c**3*d*x + 14*sqrt(c + d*x)*sqrt(a + b*x)*a**3*b*c**2*d**2*x**2 
+ 20*sqrt(c + d*x)*sqrt(a + b*x)*a**2*b**2*c**4*x + 50*sqrt(c + d*x)*sqrt( 
a + b*x)*a**2*b**2*c**3*d*x**2 - 30*sqrt(c + d*x)*sqrt(a + b*x)*a*b**3*c** 
4*x**2 + 15*sqrt(c)*sqrt(a)*log( - sqrt(2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) 
+ a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))*a**4*d**4*x* 
*3 - 30*sqrt(c)*sqrt(a)*log( - sqrt(2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + a* 
d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))*a**3*b*c*d**3*x* 
*3 + 30*sqrt(c)*sqrt(a)*log( - sqrt(2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + a* 
d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))*a*b**3*c**3*d*x* 
*3 - 15*sqrt(c)*sqrt(a)*log( - sqrt(2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + a* 
d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))*b**4*c**4*x**3 + 
 15*sqrt(c)*sqrt(a)*log(sqrt(2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c 
) + sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))*a**4*d**4*x**3 - 30*sqr 
t(c)*sqrt(a)*log(sqrt(2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sqr 
t(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))*a**3*b*c*d**3*x**3 + 30*sqrt(c 
)*sqrt(a)*log(sqrt(2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt(d 
)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))*a*b**3*c**3*d*x**3 - 15*sqrt(c...