\(\int \frac {(c+d x)^{3/2}}{x^4 \sqrt {a+b x}} \, dx\) [330]

Optimal result

Integrand size = 22, antiderivative size = 180 \[ \int \frac {(c+d x)^{3/2}}{x^4 \sqrt {a+b x}} \, dx=-\frac {(b c-a d) (5 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{8 a^3 c x}+\frac {(5 b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{12 a^2 c x^2}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 a c x^3}+\frac {(b c-a d)^2 (5 b c+a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{7/2} c^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.79 \[ \int \frac {(c+d x)^{3/2}}{x^4 \sqrt {a+b x}} \, dx=-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (15 b^2 c^2 x^2-2 a b c x (5 c+11 d x)+a^2 \left (8 c^2+14 c d x+3 d^2 x^2\right )\right )}{24 a^3 c x^3}+\frac {(b c-a d)^2 (5 b c+a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{7/2} c^{3/2}} \] Input:

Integrate[(c + d*x)^(3/2)/(x^4*Sqrt[a + b*x]),x]
 

Output:

-1/24*(Sqrt[a + b*x]*Sqrt[c + d*x]*(15*b^2*c^2*x^2 - 2*a*b*c*x*(5*c + 11*d 
*x) + a^2*(8*c^2 + 14*c*d*x + 3*d^2*x^2)))/(a^3*c*x^3) + ((b*c - a*d)^2*(5 
*b*c + a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(8*a 
^(7/2)*c^(3/2))
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.94, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {107, 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.

\(\displaystyle \int \frac {(c+d x)^{3/2}}{x^4 \sqrt {a+b x}} \, dx\)

\(\Big \downarrow \) 107

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 104

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

\(\Big \downarrow \) 221

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

Input:

Int[(c + d*x)^(3/2)/(x^4*Sqrt[a + b*x]),x]
 

Output:

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

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 107

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[(a*d*f*(m + 1) + b*c*f*(n + 
 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 
 1)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x 
] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[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. \(407\) vs. \(2(148)=296\).

Time = 0.23 (sec) , antiderivative size = 408, normalized size of antiderivative = 2.27

method	result	size
default	\(\frac {\sqrt {x d +c}\, \sqrt {b x +a}\, \left (3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{3} d^{3} x^{3}+9 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{2} b c \,d^{2} x^{3}-27 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a \,b^{2} c^{2} d \,x^{3}+15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) b^{3} c^{3} x^{3}-6 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a^{2} d^{2} x^{2}+44 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a b c d \,x^{2}-30 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, b^{2} c^{2} x^{2}-28 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a^{2} c d x +20 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a b \,c^{2} x -16 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, a^{2} c^{2} \sqrt {a c}\right )}{48 a^{3} c \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, x^{3} \sqrt {a c}}\)	\(408\)

Input:

int((d*x+c)^(3/2)/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/c*(3*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+9*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^3-27*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^3-6*((b*x+ 
a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^2*d^2*x^2+44*((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^2-28*( 
(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 = 0.28 (sec) , antiderivative size = 438, normalized size of antiderivative = 2.43 \[ \int \frac {(c+d x)^{3/2}}{x^4 \sqrt {a+b x}} \, dx=\left [\frac {3 \, {\left (5 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \sqrt {a c} x^{3} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (8 \, a^{3} c^{3} + {\left (15 \, a b^{2} c^{3} - 22 \, a^{2} b c^{2} d + 3 \, a^{3} c d^{2}\right )} x^{2} - 2 \, {\left (5 \, a^{2} b c^{3} - 7 \, a^{3} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, a^{4} c^{2} x^{3}}, -\frac {3 \, {\left (5 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \sqrt {-a c} x^{3} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left (8 \, a^{3} c^{3} + {\left (15 \, a b^{2} c^{3} - 22 \, a^{2} b c^{2} d + 3 \, a^{3} c d^{2}\right )} x^{2} - 2 \, {\left (5 \, a^{2} b c^{3} - 7 \, a^{3} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, a^{4} c^{2} x^{3}}\right ] \] Input:

integrate((d*x+c)^(3/2)/x^4/(b*x+a)^(1/2),x, algorithm="fricas")
 

Output:

[1/96*(3*(5*b^3*c^3 - 9*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 - 22*a^2*b*c^2*d + 3*a^3*c*d^2)*x^2 - 
 2*(5*a^2*b*c^3 - 7*a^3*c^2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^4*c^2*x^ 
3), -1/48*(3*(5*b^3*c^3 - 9*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 - 22*a^2*b*c^2*d + 3*a^3*c*d^2)*x^2 - 2*(5*a^2*b*c^3 - 7*a^3 
*c^2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^4*c^2*x^3)]
 

Sympy [F]

\[ \int \frac {(c+d x)^{3/2}}{x^4 \sqrt {a+b x}} \, dx=\int \frac {\left (c + d x\right )^{\frac {3}{2}}}{x^{4} \sqrt {a + b x}}\, dx \] Input:

integrate((d*x+c)**(3/2)/x**4/(b*x+a)**(1/2),x)
 

Output:

Integral((c + d*x)**(3/2)/(x**4*sqrt(a + b*x)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(c+d x)^{3/2}}{x^4 \sqrt {a+b x}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((d*x+c)^(3/2)/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. 2214 vs. \(2 (148) = 296\).

Time = 1.55 (sec) , antiderivative size = 2214, normalized size of antiderivative = 12.30 \[ \int \frac {(c+d x)^{3/2}}{x^4 \sqrt {a+b x}} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^(3/2)/x^4/(b*x+a)^(1/2),x, algorithm="giac")
 

Output:

1/24*b^3*(3*(5*sqrt(b*d)*b^3*c^3*abs(b) - 9*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*c) - 2*(15*sqrt(b*d)*b 
^13*c^8*abs(b) - 112*sqrt(b*d)*a*b^12*c^7*d*abs(b) + 360*sqrt(b*d)*a^2*b^1 
1*c^6*d^2*abs(b) - 648*sqrt(b*d)*a^3*b^10*c^5*d^3*abs(b) + 710*sqrt(b*d)*a 
^4*b^9*c^4*d^4*abs(b) - 480*sqrt(b*d)*a^5*b^8*c^3*d^5*abs(b) + 192*sqrt(b* 
d)*a^6*b^7*c^2*d^6*abs(b) - 40*sqrt(b*d)*a^7*b^6*c*d^7*abs(b) + 3*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) + 315*sqrt(b*d)*(sqrt(b*d)*sqr 
t(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^10*c^6*d*abs(b) - 
387*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) - 117*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sq 
rt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^3*b^8*c^4*d^3*abs(b) + 663*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) - 567*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) + 183*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*abs( 
b) - 15*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 + a) -...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{3/2}}{x^4 \sqrt {a+b x}} \, dx=\int \frac {{\left (c+d\,x\right )}^{3/2}}{x^4\,\sqrt {a+b\,x}} \,d x \] Input:

int((c + d*x)^(3/2)/(x^4*(a + b*x)^(1/2)),x)
 

Output:

int((c + d*x)^(3/2)/(x^4*(a + b*x)^(1/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.66 (sec) , antiderivative size = 1126, normalized size of antiderivative = 6.26 \[ \int \frac {(c+d x)^{3/2}}{x^4 \sqrt {a+b x}} \, dx =\text {Too large to display} \] Input:

int((d*x+c)^(3/2)/x^4/(b*x+a)^(1/2),x)
 

Output:

( - 16*sqrt(c + d*x)*sqrt(a + b*x)*a**4*c**3*d - 28*sqrt(c + d*x)*sqrt(a + 
 b*x)*a**4*c**2*d**2*x - 6*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 - 8*sqrt(c + d*x)*sqrt(a + b*x) 
*a**3*b*c**3*d*x + 38*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 + 14*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 - 3*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 
- 12*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 
+ 18*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**2*b**2*c**2*d**2 
*x**3 + 12*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 - 3*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 - 12*s 
qrt(c)*sqrt(a)*log(sqrt(2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + s 
qrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))*a**3*b*c*d**3*x**3 + 18*s...