\(\int \frac {(c+d x)^{3/2}}{x^2 \sqrt {a+b x}} \, dx\) [328]

Optimal result

Integrand size = 22, antiderivative size = 119 \[ \int \frac {(c+d x)^{3/2}}{x^2 \sqrt {a+b x}} \, dx=-\frac {c \sqrt {a+b x} \sqrt {c+d x}}{a x}+\frac {\sqrt {c} (b c-3 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2}}+\frac {2 d^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b}} \] Output:

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

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00 \[ \int \frac {(c+d x)^{3/2}}{x^2 \sqrt {a+b x}} \, dx=-\frac {c \sqrt {a+b x} \sqrt {c+d x}}{a x}+\frac {\sqrt {c} (b c-3 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2}}+\frac {2 d^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b}} \] Input:

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

Output:

-((c*Sqrt[a + b*x]*Sqrt[c + d*x])/(a*x)) + (Sqrt[c]*(b*c - 3*a*d)*ArcTanh[ 
(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/a^(3/2) + (2*d^(3/2)*Arc 
Tanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/Sqrt[b]
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {109, 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.

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

\(\Big \downarrow \) 109

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 175

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

\(\Big \downarrow \) 66

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

\(\Big \downarrow \) 104

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

-((c*Sqrt[a + b*x]*Sqrt[c + d*x])/(a*x)) - ((-2*Sqrt[c]*(b*c - 3*a*d)*ArcT 
anh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/Sqrt[a] - (4*a*d^(3/ 
2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/Sqrt[b])/(2*a 
)
 

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 109

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(222\) vs. \(2(91)=182\).

Time = 0.23 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.87

method	result	size
default	\(\frac {\sqrt {x d +c}\, \sqrt {b x +a}\, \left (2 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a \,d^{2} x \sqrt {a c}-3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a c d x \sqrt {d b}+\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) b \,c^{2} x \sqrt {d b}-2 c \sqrt {d b}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\right )}{2 a \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, x \sqrt {d b}\, \sqrt {a c}}\)	\(223\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 187 vs. \(2 (91) = 182\).

Time = 0.36 (sec) , antiderivative size = 858, normalized size of antiderivative = 7.21 \[ \int \frac {(c+d x)^{3/2}}{x^2 \sqrt {a+b x}} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/4*(2*a*d*x*sqrt(d/b)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 
+ 4*(2*b^2*d*x + b^2*c + a*b*d)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(d/b) + 8* 
(b^2*c*d + a*b*d^2)*x) - (b*c - 3*a*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*sqrt(b*x + 
 a)*sqrt(d*x + c)*c)/(a*x), -1/4*(4*a*d*x*sqrt(-d/b)*arctan(1/2*(2*b*d*x + 
 b*c + a*d)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(-d/b)/(b*d^2*x^2 + a*c*d + (b 
*c*d + a*d^2)*x)) + (b*c - 3*a*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)*s 
qrt(d*x + c)*sqrt(c/a) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) + 4*sqrt(b*x + a)*s 
qrt(d*x + c)*c)/(a*x), 1/2*(a*d*x*sqrt(d/b)*log(8*b^2*d^2*x^2 + b^2*c^2 + 
6*a*b*c*d + a^2*d^2 + 4*(2*b^2*d*x + b^2*c + a*b*d)*sqrt(b*x + a)*sqrt(d*x 
 + c)*sqrt(d/b) + 8*(b^2*c*d + a*b*d^2)*x) - (b*c - 3*a*d)*x*sqrt(-c/a)*ar 
ctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(-c/a)/(b 
*c*d*x^2 + a*c^2 + (b*c^2 + a*c*d)*x)) - 2*sqrt(b*x + a)*sqrt(d*x + c)*c)/ 
(a*x), -1/2*(2*a*d*x*sqrt(-d/b)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(b*x 
+ a)*sqrt(d*x + c)*sqrt(-d/b)/(b*d^2*x^2 + a*c*d + (b*c*d + a*d^2)*x)) + ( 
b*c - 3*a*d)*x*sqrt(-c/a)*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(b*x + a) 
*sqrt(d*x + c)*sqrt(-c/a)/(b*c*d*x^2 + a*c^2 + (b*c^2 + a*c*d)*x)) + 2*sqr 
t(b*x + a)*sqrt(d*x + c)*c)/(a*x)]
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((d*x+c)^(3/2)/x^2/(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. 477 vs. \(2 (91) = 182\).

Time = 0.27 (sec) , antiderivative size = 477, normalized size of antiderivative = 4.01 \[ \int \frac {(c+d x)^{3/2}}{x^2 \sqrt {a+b x}} \, dx=-b {\left (\frac {\sqrt {b d} d {\left | b \right |} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{b^{3}} - \frac {{\left (\sqrt {b d} b c^{2} {\left | b \right |} - 3 \, \sqrt {b d} a c d {\left | b \right |}\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} a b^{2}} + \frac {2 \, {\left (\sqrt {b d} b^{3} c^{3} {\left | b \right |} - 2 \, \sqrt {b d} a b^{2} c^{2} d {\left | b \right |} + \sqrt {b d} a^{2} b c d^{2} {\left | b \right |} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b c^{2} {\left | b \right |} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a c d {\left | b \right |}\right )}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4}\right )} a b}\right )} \] Input:

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

Output:

-b*(sqrt(b*d)*d*abs(b)*log((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + 
a)*b*d - a*b*d))^2)/b^3 - (sqrt(b*d)*b*c^2*abs(b) - 3*sqrt(b*d)*a*c*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*b^2) + 2*( 
sqrt(b*d)*b^3*c^3*abs(b) - 2*sqrt(b*d)*a*b^2*c^2*d*abs(b) + sqrt(b*d)*a^2* 
b*c*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^2*abs(b) - sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqr 
t(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*c*d*abs(b))/((b^4*c^2 - 2*a*b^3*c*d 
+ a^2*b^2*d^2 - 2*(sqrt(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*b))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 360, normalized size of antiderivative = 3.03 \[ \int \frac {(c+d x)^{3/2}}{x^2 \sqrt {a+b x}} \, dx=\frac {-2 \sqrt {d x +c}\, \sqrt {b x +a}\, a b c +3 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+a d +b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}\right ) a b d x -\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+a d +b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}\right ) b^{2} c x +3 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+a d +b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}\right ) a b d x -\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+a d +b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}\right ) b^{2} c x -3 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (2 \sqrt {d}\, \sqrt {b}\, \sqrt {d x +c}\, \sqrt {b x +a}+2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+2 b d x \right ) a b d x +\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (2 \sqrt {d}\, \sqrt {b}\, \sqrt {d x +c}\, \sqrt {b x +a}+2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+2 b d x \right ) b^{2} c x +4 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{2} d x}{2 a^{2} b x} \] Input:

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

Output:

( - 2*sqrt(c + d*x)*sqrt(a + b*x)*a*b*c + 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) + sqr 
t(b)*sqrt(c + d*x))*a*b*d*x - 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**2*c*x + 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*b*d*x 
- 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**2*c*x - 3*sqrt(c)*sqrt 
(a)*log(2*sqrt(d)*sqrt(b)*sqrt(c + d*x)*sqrt(a + b*x) + 2*sqrt(d)*sqrt(c)* 
sqrt(b)*sqrt(a) + 2*b*d*x)*a*b*d*x + sqrt(c)*sqrt(a)*log(2*sqrt(d)*sqrt(b) 
*sqrt(c + d*x)*sqrt(a + b*x) + 2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + 2*b*d*x 
)*b**2*c*x + 4*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c 
 + d*x))/sqrt(a*d - b*c))*a**2*d*x)/(2*a**2*b*x)