\(\int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^{5/2}} \, dx\) [406]

Optimal result

Integrand size = 22, antiderivative size = 142 \[ \int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^{5/2}} \, dx=-\frac {5 c (b c-a d) \sqrt {c+d x}}{a^3 \sqrt {a+b x}}-\frac {5 (b c-a d) (c+d x)^{3/2}}{3 a^2 (a+b x)^{3/2}}-\frac {(c+d x)^{5/2}}{a x (a+b x)^{3/2}}+\frac {5 c^{3/2} (b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{7/2}} \] Output:

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

Mathematica [A] (verified)

Time = 10.11 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.88 \[ \int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^{5/2}} \, dx=-\frac {3 a^{5/2} (c+d x)^{5/2}+5 (b c-a d) x \left (\sqrt {a} \sqrt {c+d x} (4 a c+3 b c x+a d x)-3 c^{3/2} (a+b x)^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )\right )}{3 a^{7/2} x (a+b x)^{3/2}} \] Input:

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

Output:

-1/3*(3*a^(5/2)*(c + d*x)^(5/2) + 5*(b*c - a*d)*x*(Sqrt[a]*Sqrt[c + d*x]*( 
4*a*c + 3*b*c*x + a*d*x) - 3*c^(3/2)*(a + b*x)^(3/2)*ArcTanh[(Sqrt[c]*Sqrt 
[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])]))/(a^(7/2)*x*(a + b*x)^(3/2))
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.98, 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.

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 104

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

-((c + d*x)^(5/2)/(a*x*(a + b*x)^(3/2))) - (5*(b*c - a*d)*((2*(c + d*x)^(3 
/2))/(3*a*(a + b*x)^(3/2)) + (c*((2*Sqrt[c + d*x])/(a*Sqrt[a + b*x]) - (2* 
Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/a^(3/2)) 
)/a))/(2*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. \(501\) vs. \(2(116)=232\).

Time = 0.25 (sec) , antiderivative size = 502, normalized size of antiderivative = 3.54

method	result	size
default	\(-\frac {\sqrt {x d +c}\, \left (15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a \,b^{2} c^{2} d \,x^{3}-15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) b^{3} c^{3} x^{3}+30 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{2} b \,c^{2} d \,x^{2}-30 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a \,b^{2} c^{3} x^{2}+15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{3} c^{2} d x -15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{2} b \,c^{3} x -4 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a^{2} d^{2} x^{2}-20 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a b c d \,x^{2}+30 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, b^{2} c^{2} x^{2}-28 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a^{2} c d x +40 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a b \,c^{2} x +6 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, a^{2} c^{2} \sqrt {a c}\right )}{6 a^{3} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, x \sqrt {a c}\, \left (b x +a \right )^{\frac {3}{2}}}\)	\(502\)

Input:

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

Output:

-1/6*(d*x+c)^(1/2)*(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*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+30*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^2*d*x^2-30*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^3*x^2+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*c^2*d*x-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^2*b*c^3*x-4*((b*x+a)*(d* 
x+c))^(1/2)*(a*c)^(1/2)*a^2*d^2*x^2-20*((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+40*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1 
/2)*a*b*c^2*x+6*((b*x+a)*(d*x+c))^(1/2)*a^2*c^2*(a*c)^(1/2))/a^3/((b*x+a)* 
(d*x+c))^(1/2)/x/(a*c)^(1/2)/(b*x+a)^(3/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 237 vs. \(2 (116) = 232\).

Time = 0.34 (sec) , antiderivative size = 505, normalized size of antiderivative = 3.56 \[ \int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^{5/2}} \, dx=\left [-\frac {15 \, {\left ({\left (b^{3} c^{2} - a b^{2} c d\right )} x^{3} + 2 \, {\left (a b^{2} c^{2} - a^{2} b c d\right )} x^{2} + {\left (a^{2} b c^{2} - a^{3} c d\right )} x\right )} \sqrt {\frac {c}{a}} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a^{2} c + {\left (a b c + a^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {c}{a}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (3 \, a^{2} c^{2} + {\left (15 \, b^{2} c^{2} - 10 \, a b c d - 2 \, a^{2} d^{2}\right )} x^{2} + 2 \, {\left (10 \, a b c^{2} - 7 \, a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{12 \, {\left (a^{3} b^{2} x^{3} + 2 \, a^{4} b x^{2} + a^{5} x\right )}}, -\frac {15 \, {\left ({\left (b^{3} c^{2} - a b^{2} c d\right )} x^{3} + 2 \, {\left (a b^{2} c^{2} - a^{2} b c d\right )} x^{2} + {\left (a^{2} b c^{2} - a^{3} c d\right )} x\right )} \sqrt {-\frac {c}{a}} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {c}{a}}}{2 \, {\left (b c d x^{2} + a c^{2} + {\left (b c^{2} + a c d\right )} x\right )}}\right ) + 2 \, {\left (3 \, a^{2} c^{2} + {\left (15 \, b^{2} c^{2} - 10 \, a b c d - 2 \, a^{2} d^{2}\right )} x^{2} + 2 \, {\left (10 \, a b c^{2} - 7 \, a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6 \, {\left (a^{3} b^{2} x^{3} + 2 \, a^{4} b x^{2} + a^{5} x\right )}}\right ] \] Input:

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

Output:

[-1/12*(15*((b^3*c^2 - a*b^2*c*d)*x^3 + 2*(a*b^2*c^2 - a^2*b*c*d)*x^2 + (a 
^2*b*c^2 - a^3*c*d)*x)*sqrt(c/a)*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a 
^2*d^2)*x^2 - 4*(2*a^2*c + (a*b*c + a^2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)* 
sqrt(c/a) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) + 4*(3*a^2*c^2 + (15*b^2*c^2 - 1 
0*a*b*c*d - 2*a^2*d^2)*x^2 + 2*(10*a*b*c^2 - 7*a^2*c*d)*x)*sqrt(b*x + a)*s 
qrt(d*x + c))/(a^3*b^2*x^3 + 2*a^4*b*x^2 + a^5*x), -1/6*(15*((b^3*c^2 - a* 
b^2*c*d)*x^3 + 2*(a*b^2*c^2 - a^2*b*c*d)*x^2 + (a^2*b*c^2 - a^3*c*d)*x)*sq 
rt(-c/a)*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sq 
rt(-c/a)/(b*c*d*x^2 + a*c^2 + (b*c^2 + a*c*d)*x)) + 2*(3*a^2*c^2 + (15*b^2 
*c^2 - 10*a*b*c*d - 2*a^2*d^2)*x^2 + 2*(10*a*b*c^2 - 7*a^2*c*d)*x)*sqrt(b* 
x + a)*sqrt(d*x + c))/(a^3*b^2*x^3 + 2*a^4*b*x^2 + a^5*x)]
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((d*x+c)^(5/2)/x^2/(b*x+a)^(5/2),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. 990 vs. \(2 (116) = 232\).

Time = 1.32 (sec) , antiderivative size = 990, normalized size of antiderivative = 6.97 \[ \int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

5*(sqrt(b*d)*b*c^3*abs(b) - sqrt(b*d)*a*c^2*d*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) - 2*(sqrt(b*d)*b^3*c^4*abs(b) 
 - 2*sqrt(b*d)*a*b^2*c^3*d*abs(b) + sqrt(b*d)*a^2*b*c^2*d^2*abs(b) - sqrt( 
b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b*c 
^3*abs(b) - sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b* 
d - a*b*d))^2*a*c^2*d*abs(b))/((b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2 - 2*(s 
qrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^2*c - 2* 
(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b*d + 
(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4)*a^3) - 
4/3*(6*sqrt(b*d)*b^7*c^5*abs(b) - 23*sqrt(b*d)*a*b^6*c^4*d*abs(b) + 32*sqr 
t(b*d)*a^2*b^5*c^3*d^2*abs(b) - 18*sqrt(b*d)*a^3*b^4*c^2*d^3*abs(b) + 2*sq 
rt(b*d)*a^4*b^3*c*d^4*abs(b) + sqrt(b*d)*a^5*b^2*d^5*abs(b) - 12*sqrt(b*d) 
*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^5*c^4 
*abs(b) + 36*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b 
*d - a*b*d))^2*a*b^4*c^3*d*abs(b) - 36*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^3*c^2*d^2*abs(b) + 12*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^2*c*d^3*abs(b) + 6*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b 
*x + a)*b*d - a*b*d))^4*b^3*c^3*abs(b) - 9*sqrt(b*d)*(sqrt(b*d)*sqrt(b*...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.62 (sec) , antiderivative size = 1411, normalized size of antiderivative = 9.94 \[ \int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(15*sqrt(c)*sqrt(a)*sqrt(a + b*x)*log( - sqrt(2*sqrt(d)*sqrt(c)*sqrt(b)*sq 
rt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))*a**3*b 
**2*c*d**2*x - 90*sqrt(c)*sqrt(a)*sqrt(a + b*x)*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**3*c**2*d*x + 15*sqrt(c)*sqrt(a)*sqrt(a + b*x)*log( - sqrt( 
2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + s 
qrt(b)*sqrt(c + d*x))*a**2*b**3*c*d**2*x**2 + 75*sqrt(c)*sqrt(a)*sqrt(a + 
b*x)*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**4*c**3*x - 90*sqrt(c)*sqrt(a)* 
sqrt(a + b*x)*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**4*c**2*d*x**2 + 75*sq 
rt(c)*sqrt(a)*sqrt(a + b*x)*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**5*c**3*x* 
*2 + 15*sqrt(c)*sqrt(a)*sqrt(a + b*x)*log(sqrt(2*sqrt(d)*sqrt(c)*sqrt(b)*s 
qrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))*a**3* 
b**2*c*d**2*x - 90*sqrt(c)*sqrt(a)*sqrt(a + b*x)*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**3*c**2*d*x + 15*sqrt(c)*sqrt(a)*sqrt(a + b*x)*log(sqrt(2*sqr 
t(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**3*c*d**2*x**2 + 75*sqrt(c)*sqrt(a)*sqrt(a + b*...