\(\int \frac {\sqrt {c+d x}}{x^2 (a+b x)^2} \, dx\) [129]

Optimal result

Integrand size = 20, antiderivative size = 140 \[ \int \frac {\sqrt {c+d x}}{x^2 (a+b x)^2} \, dx=-\frac {2 b \sqrt {c+d x}}{a^2 (a+b x)}-\frac {\sqrt {c+d x}}{a x (a+b x)}+\frac {(4 b c-a d) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3 \sqrt {c}}-\frac {\sqrt {b} (4 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^3 \sqrt {b c-a d}} \] Output:

-2*b*(d*x+c)^(1/2)/a^2/(b*x+a)-(d*x+c)^(1/2)/a/x/(b*x+a)+(-a*d+4*b*c)*arct 
anh((d*x+c)^(1/2)/c^(1/2))/a^3/c^(1/2)-b^(1/2)*(-3*a*d+4*b*c)*arctanh(b^(1 
/2)*(d*x+c)^(1/2)/(-a*d+b*c)^(1/2))/a^3/(-a*d+b*c)^(1/2)
 

Mathematica [A] (verified)

Time = 0.52 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt {c+d x}}{x^2 (a+b x)^2} \, dx=\frac {-\frac {a (a+2 b x) \sqrt {c+d x}}{x (a+b x)}+\frac {\sqrt {b} (4 b c-3 a d) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{\sqrt {-b c+a d}}+\frac {(4 b c-a d) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{\sqrt {c}}}{a^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.10, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {110, 27, 168, 27, 174, 73, 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.

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

\(\Big \downarrow \) 110

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 174

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

-(Sqrt[c + d*x]/(a*x*(a + b*x))) - ((4*b*Sqrt[c + d*x])/(a*(a + b*x)) + (( 
-2*(4*b*c - a*d)*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/(a*Sqrt[c]) + (2*Sqrt[b]* 
(4*b*c - 3*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(a*Sqrt[ 
b*c - a*d]))/a)/(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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 110

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[1/((m + 1)*(b*e - a*f))   Int[(a + b*x)^(m + 1) 
*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + 
p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && Gt 
Q[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, 
 m + n])
 

rule 168

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(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] + S 
imp[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*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
 

rule 174

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

Time = 0.45 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.95

method	result	size
pseudoelliptic	\(\frac {4 \left (b x +a \right ) \sqrt {c}\, \left (b c -\frac {3 a d}{4}\right ) b x \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )-\left (x \left (b x +a \right ) \left (a d -4 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {x d +c}}{\sqrt {c}}\right )+a \sqrt {c}\, \left (2 b x +a \right ) \sqrt {x d +c}\right ) \sqrt {\left (a d -b c \right ) b}}{a^{3} \left (b x +a \right ) \sqrt {\left (a d -b c \right ) b}\, x \sqrt {c}}\)	\(133\)
derivativedivides	\(2 d^{3} \left (-\frac {b \left (\frac {\sqrt {x d +c}\, a d}{2 \left (x d +c \right ) b +2 a d -2 b c}+\frac {\left (3 a d -4 b c \right ) \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{a^{3} d^{3}}+\frac {-\frac {a \sqrt {x d +c}}{2 x}-\frac {\left (a d -4 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {x d +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}}{a^{3} d^{3}}\right )\)	\(136\)
default	\(2 d^{3} \left (-\frac {b \left (\frac {\sqrt {x d +c}\, a d}{2 \left (x d +c \right ) b +2 a d -2 b c}+\frac {\left (3 a d -4 b c \right ) \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{a^{3} d^{3}}+\frac {-\frac {a \sqrt {x d +c}}{2 x}-\frac {\left (a d -4 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {x d +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}}{a^{3} d^{3}}\right )\)	\(136\)
risch	\(-\frac {\sqrt {x d +c}}{a^{2} x}-\frac {d \left (\frac {2 b \left (\frac {\sqrt {x d +c}\, a d}{2 \left (x d +c \right ) b +2 a d -2 b c}+\frac {\left (3 a d -4 b c \right ) \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{a d}-\frac {\left (-a d +4 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {x d +c}}{\sqrt {c}}\right )}{a d \sqrt {c}}\right )}{a^{2}}\)	\(139\)

Input:

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

Output:

4/((a*d-b*c)*b)^(1/2)*((b*x+a)*c^(1/2)*(b*c-3/4*a*d)*b*x*arctan(b*(d*x+c)^ 
(1/2)/((a*d-b*c)*b)^(1/2))-1/4*(x*(b*x+a)*(a*d-4*b*c)*arctanh((d*x+c)^(1/2 
)/c^(1/2))+a*c^(1/2)*(2*b*x+a)*(d*x+c)^(1/2))*((a*d-b*c)*b)^(1/2))/c^(1/2) 
/a^3/(b*x+a)/x
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 756, normalized size of antiderivative = 5.40 \[ \int \frac {\sqrt {c+d x}}{x^2 (a+b x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/2*(((4*b^2*c^2 - 3*a*b*c*d)*x^2 + (4*a*b*c^2 - 3*a^2*c*d)*x)*sqrt(b/(b 
*c - a*d))*log((b*d*x + 2*b*c - a*d + 2*(b*c - a*d)*sqrt(d*x + c)*sqrt(b/( 
b*c - a*d)))/(b*x + a)) + ((4*b^2*c - a*b*d)*x^2 + (4*a*b*c - a^2*d)*x)*sq 
rt(c)*log((d*x - 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) + 2*(2*a*b*c*x + a^2*c) 
*sqrt(d*x + c))/(a^3*b*c*x^2 + a^4*c*x), 1/2*(2*((4*b^2*c^2 - 3*a*b*c*d)*x 
^2 + (4*a*b*c^2 - 3*a^2*c*d)*x)*sqrt(-b/(b*c - a*d))*arctan(sqrt(d*x + c)* 
sqrt(-b/(b*c - a*d))) - ((4*b^2*c - a*b*d)*x^2 + (4*a*b*c - a^2*d)*x)*sqrt 
(c)*log((d*x - 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) - 2*(2*a*b*c*x + a^2*c)*s 
qrt(d*x + c))/(a^3*b*c*x^2 + a^4*c*x), -1/2*(2*((4*b^2*c - a*b*d)*x^2 + (4 
*a*b*c - a^2*d)*x)*sqrt(-c)*arctan(sqrt(-c)/sqrt(d*x + c)) + ((4*b^2*c^2 - 
 3*a*b*c*d)*x^2 + (4*a*b*c^2 - 3*a^2*c*d)*x)*sqrt(b/(b*c - a*d))*log((b*d* 
x + 2*b*c - a*d + 2*(b*c - a*d)*sqrt(d*x + c)*sqrt(b/(b*c - a*d)))/(b*x + 
a)) + 2*(2*a*b*c*x + a^2*c)*sqrt(d*x + c))/(a^3*b*c*x^2 + a^4*c*x), (((4*b 
^2*c^2 - 3*a*b*c*d)*x^2 + (4*a*b*c^2 - 3*a^2*c*d)*x)*sqrt(-b/(b*c - a*d))* 
arctan(sqrt(d*x + c)*sqrt(-b/(b*c - a*d))) - ((4*b^2*c - a*b*d)*x^2 + (4*a 
*b*c - a^2*d)*x)*sqrt(-c)*arctan(sqrt(-c)/sqrt(d*x + c)) - (2*a*b*c*x + a^ 
2*c)*sqrt(d*x + c))/(a^3*b*c*x^2 + a^4*c*x)]
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((d*x+c)^(1/2)/x^2/(b*x+a)^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 [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.19 \[ \int \frac {\sqrt {c+d x}}{x^2 (a+b x)^2} \, dx=\frac {{\left (4 \, b^{2} c - 3 \, a b d\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} a^{3}} - \frac {{\left (4 \, b c - a d\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{a^{3} \sqrt {-c}} - \frac {2 \, {\left (d x + c\right )}^{\frac {3}{2}} b d - 2 \, \sqrt {d x + c} b c d + \sqrt {d x + c} a d^{2}}{{\left ({\left (d x + c\right )}^{2} b - 2 \, {\left (d x + c\right )} b c + b c^{2} + {\left (d x + c\right )} a d - a c d\right )} a^{2}} \] Input:

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

Output:

(4*b^2*c - 3*a*b*d)*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^ 
2*c + a*b*d)*a^3) - (4*b*c - a*d)*arctan(sqrt(d*x + c)/sqrt(-c))/(a^3*sqrt 
(-c)) - (2*(d*x + c)^(3/2)*b*d - 2*sqrt(d*x + c)*b*c*d + sqrt(d*x + c)*a*d 
^2)/(((d*x + c)^2*b - 2*(d*x + c)*b*c + b*c^2 + (d*x + c)*a*d - a*c*d)*a^2 
)
 

Mupad [B] (verification not implemented)

Time = 0.73 (sec) , antiderivative size = 1175, normalized size of antiderivative = 8.39 \[ \int \frac {\sqrt {c+d x}}{x^2 (a+b x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

(atan((((-b*(a*d - b*c))^(1/2)*(3*a*d - 4*b*c)*((4*(c + d*x)^(1/2)*(5*a^2* 
b^3*d^4 + 16*b^5*c^2*d^2 - 16*a*b^4*c*d^3))/a^4 - ((-b*(a*d - b*c))^(1/2)* 
(3*a*d - 4*b*c)*((2*(2*a^7*b^2*d^4 - 4*a^6*b^3*c*d^3))/a^6 - (2*(2*a^7*b^2 
*d^3 - 4*a^6*b^3*c*d^2)*(-b*(a*d - b*c))^(1/2)*(3*a*d - 4*b*c)*(c + d*x)^( 
1/2))/(a^4*(a^4*d - a^3*b*c))))/(2*(a^4*d - a^3*b*c)))*1i)/(2*(a^4*d - a^3 
*b*c)) + ((-b*(a*d - b*c))^(1/2)*(3*a*d - 4*b*c)*((4*(c + d*x)^(1/2)*(5*a^ 
2*b^3*d^4 + 16*b^5*c^2*d^2 - 16*a*b^4*c*d^3))/a^4 + ((-b*(a*d - b*c))^(1/2 
)*(3*a*d - 4*b*c)*((2*(2*a^7*b^2*d^4 - 4*a^6*b^3*c*d^3))/a^6 + (2*(2*a^7*b 
^2*d^3 - 4*a^6*b^3*c*d^2)*(-b*(a*d - b*c))^(1/2)*(3*a*d - 4*b*c)*(c + d*x) 
^(1/2))/(a^4*(a^4*d - a^3*b*c))))/(2*(a^4*d - a^3*b*c)))*1i)/(2*(a^4*d - a 
^3*b*c)))/((4*(3*a^2*b^3*d^5 + 16*b^5*c^2*d^3 - 16*a*b^4*c*d^4))/a^6 - ((- 
b*(a*d - b*c))^(1/2)*(3*a*d - 4*b*c)*((4*(c + d*x)^(1/2)*(5*a^2*b^3*d^4 + 
16*b^5*c^2*d^2 - 16*a*b^4*c*d^3))/a^4 - ((-b*(a*d - b*c))^(1/2)*(3*a*d - 4 
*b*c)*((2*(2*a^7*b^2*d^4 - 4*a^6*b^3*c*d^3))/a^6 - (2*(2*a^7*b^2*d^3 - 4*a 
^6*b^3*c*d^2)*(-b*(a*d - b*c))^(1/2)*(3*a*d - 4*b*c)*(c + d*x)^(1/2))/(a^4 
*(a^4*d - a^3*b*c))))/(2*(a^4*d - a^3*b*c))))/(2*(a^4*d - a^3*b*c)) + ((-b 
*(a*d - b*c))^(1/2)*(3*a*d - 4*b*c)*((4*(c + d*x)^(1/2)*(5*a^2*b^3*d^4 + 1 
6*b^5*c^2*d^2 - 16*a*b^4*c*d^3))/a^4 + ((-b*(a*d - b*c))^(1/2)*(3*a*d - 4* 
b*c)*((2*(2*a^7*b^2*d^4 - 4*a^6*b^3*c*d^3))/a^6 + (2*(2*a^7*b^2*d^3 - 4*a^ 
6*b^3*c*d^2)*(-b*(a*d - b*c))^(1/2)*(3*a*d - 4*b*c)*(c + d*x)^(1/2))/(a...
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 550, normalized size of antiderivative = 3.93 \[ \int \frac {\sqrt {c+d x}}{x^2 (a+b x)^2} \, dx=\frac {-6 \sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) a^{2} c d x +8 \sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) a b \,c^{2} x -6 \sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) a b c d \,x^{2}+8 \sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) b^{2} c^{2} x^{2}-2 \sqrt {d x +c}\, a^{3} c d +2 \sqrt {d x +c}\, a^{2} b \,c^{2}-4 \sqrt {d x +c}\, a^{2} b c d x +4 \sqrt {d x +c}\, a \,b^{2} c^{2} x +\sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}-\sqrt {c}\right ) a^{3} d^{2} x -5 \sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}-\sqrt {c}\right ) a^{2} b c d x +\sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}-\sqrt {c}\right ) a^{2} b \,d^{2} x^{2}+4 \sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}-\sqrt {c}\right ) a \,b^{2} c^{2} x -5 \sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}-\sqrt {c}\right ) a \,b^{2} c d \,x^{2}+4 \sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}-\sqrt {c}\right ) b^{3} c^{2} x^{2}-\sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}+\sqrt {c}\right ) a^{3} d^{2} x +5 \sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}+\sqrt {c}\right ) a^{2} b c d x -\sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}+\sqrt {c}\right ) a^{2} b \,d^{2} x^{2}-4 \sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}+\sqrt {c}\right ) a \,b^{2} c^{2} x +5 \sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}+\sqrt {c}\right ) a \,b^{2} c d \,x^{2}-4 \sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}+\sqrt {c}\right ) b^{3} c^{2} x^{2}}{2 a^{3} c x \left (a b d x -b^{2} c x +a^{2} d -a b c \right )} \] Input:

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

Output:

( - 6*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b 
*c)))*a**2*c*d*x + 8*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt( 
b)*sqrt(a*d - b*c)))*a*b*c**2*x - 6*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + 
 d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a*b*c*d*x**2 + 8*sqrt(b)*sqrt(a*d - b* 
c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*b**2*c**2*x**2 - 2*sq 
rt(c + d*x)*a**3*c*d + 2*sqrt(c + d*x)*a**2*b*c**2 - 4*sqrt(c + d*x)*a**2* 
b*c*d*x + 4*sqrt(c + d*x)*a*b**2*c**2*x + sqrt(c)*log(sqrt(c + d*x) - sqrt 
(c))*a**3*d**2*x - 5*sqrt(c)*log(sqrt(c + d*x) - sqrt(c))*a**2*b*c*d*x + s 
qrt(c)*log(sqrt(c + d*x) - sqrt(c))*a**2*b*d**2*x**2 + 4*sqrt(c)*log(sqrt( 
c + d*x) - sqrt(c))*a*b**2*c**2*x - 5*sqrt(c)*log(sqrt(c + d*x) - sqrt(c)) 
*a*b**2*c*d*x**2 + 4*sqrt(c)*log(sqrt(c + d*x) - sqrt(c))*b**3*c**2*x**2 - 
 sqrt(c)*log(sqrt(c + d*x) + sqrt(c))*a**3*d**2*x + 5*sqrt(c)*log(sqrt(c + 
 d*x) + sqrt(c))*a**2*b*c*d*x - sqrt(c)*log(sqrt(c + d*x) + sqrt(c))*a**2* 
b*d**2*x**2 - 4*sqrt(c)*log(sqrt(c + d*x) + sqrt(c))*a*b**2*c**2*x + 5*sqr 
t(c)*log(sqrt(c + d*x) + sqrt(c))*a*b**2*c*d*x**2 - 4*sqrt(c)*log(sqrt(c + 
 d*x) + sqrt(c))*b**3*c**2*x**2)/(2*a**3*c*x*(a**2*d - a*b*c + a*b*d*x - b 
**2*c*x))