\(\int \frac {1}{x (c+d x)^2 \sqrt {a x+b x^2}} \, dx\) [154]

Optimal result

Integrand size = 24, antiderivative size = 145 \[ \int \frac {1}{x (c+d x)^2 \sqrt {a x+b x^2}} \, dx=-\frac {(2 b c-3 a d) \sqrt {a x+b x^2}}{a c^2 (b c-a d) x}-\frac {d \sqrt {a x+b x^2}}{c (b c-a d) x (c+d x)}-\frac {d (4 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a x+b x^2}}\right )}{c^{5/2} (b c-a d)^{3/2}} \] Output:

-(-3*a*d+2*b*c)*(b*x^2+a*x)^(1/2)/a/c^2/(-a*d+b*c)/x-d*(b*x^2+a*x)^(1/2)/c 
/(-a*d+b*c)/x/(d*x+c)-d*(-3*a*d+4*b*c)*arctanh((-a*d+b*c)^(1/2)*x/c^(1/2)/ 
(b*x^2+a*x)^(1/2))/c^(5/2)/(-a*d+b*c)^(3/2)
 

Mathematica [A] (verified)

Time = 0.80 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x (c+d x)^2 \sqrt {a x+b x^2}} \, dx=\frac {\frac {\sqrt {c} (a+b x) (2 b c (c+d x)-a d (2 c+3 d x))}{a (-b c+a d) (c+d x)}-\frac {d (4 b c-3 a d) \sqrt {x} \sqrt {a+b x} \arctan \left (\frac {-d \sqrt {x} \sqrt {a+b x}+\sqrt {b} (c+d x)}{\sqrt {c} \sqrt {-b c+a d}}\right )}{(-b c+a d)^{3/2}}}{c^{5/2} \sqrt {x (a+b x)}} \] Input:

Integrate[1/(x*(c + d*x)^2*Sqrt[a*x + b*x^2]),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.22, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1261, 114, 27, 169, 27, 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.

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

\(\Big \downarrow \) 1261

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

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 169

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 104

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

\(\Big \downarrow \) 221

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

Input:

Int[1/(x*(c + d*x)^2*Sqrt[a*x + b*x^2]),x]
 

Output:

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

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 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 114

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[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*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || 
 IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
 

rule 169

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] && LtQ[m, -1] && IntegersQ[ 
2*m, 2*n, 2*p]
 

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]
 

rule 1261

Int[((e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((b_.)*(x_) + (c_.)*(x_)^2) 
^(p_), x_Symbol] :> Simp[(e*x)^m*((b*x + c*x^2)^p/(x^(m + p)*(b + c*x)^p)) 
  Int[x^(m + p)*(f + g*x)^n*(b + c*x)^p, x], x] /; FreeQ[{b, c, e, f, g, m, 
 n}, x] &&  !IGtQ[n, 0]
 

Maple [A] (verified)

Time = 0.67 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.91

method	result	size
pseudoelliptic	\(\frac {-2 \sqrt {x \left (b x +a \right )}\, \left (-b \,c^{2}+d \left (-b x +a \right ) c +\frac {3 a \,d^{2} x}{2}\right ) \sqrt {c \left (a d -b c \right )}+d x a \arctan \left (\frac {\sqrt {x \left (b x +a \right )}\, c}{x \sqrt {c \left (a d -b c \right )}}\right ) \left (d x +c \right ) \left (3 a d -4 b c \right )}{\sqrt {c \left (a d -b c \right )}\, c^{2} x \left (a d -b c \right ) \left (d x +c \right ) a}\)	\(132\)
default	\(-\frac {2 \sqrt {b \,x^{2}+a x}}{c^{2} a x}+\frac {\ln \left (\frac {-\frac {2 c \left (a d -b c \right )}{d^{2}}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{x +\frac {c}{d}}\right )}{c^{2} \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}-\frac {\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{c \left (a d -b c \right ) \left (x +\frac {c}{d}\right )}-\frac {\left (a d -2 b c \right ) d \ln \left (\frac {-\frac {2 c \left (a d -b c \right )}{d^{2}}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{x +\frac {c}{d}}\right )}{2 c \left (a d -b c \right ) \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}}{d c}\)	\(386\)
risch	\(-\frac {2 \left (b x +a \right )}{a \,c^{2} \sqrt {x \left (b x +a \right )}}-\frac {d \left (-\frac {\ln \left (\frac {-\frac {2 c \left (a d -b c \right )}{d^{2}}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{x +\frac {c}{d}}\right )}{d \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}+\frac {c \left (\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{c \left (a d -b c \right ) \left (x +\frac {c}{d}\right )}-\frac {\left (a d -2 b c \right ) d \ln \left (\frac {-\frac {2 c \left (a d -b c \right )}{d^{2}}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{x +\frac {c}{d}}\right )}{2 c \left (a d -b c \right ) \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}\right )}{d^{2}}\right )}{c^{2}}\)	\(391\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 488, normalized size of antiderivative = 3.37 \[ \int \frac {1}{x (c+d x)^2 \sqrt {a x+b x^2}} \, dx=\left [\frac {\sqrt {b c^{2} - a c d} {\left ({\left (4 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{2} + {\left (4 \, a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x\right )} \log \left (\frac {a c + {\left (2 \, b c - a d\right )} x - 2 \, \sqrt {b c^{2} - a c d} \sqrt {b x^{2} + a x}}{d x + c}\right ) - 2 \, {\left (2 \, b^{2} c^{4} - 4 \, a b c^{3} d + 2 \, a^{2} c^{2} d^{2} + {\left (2 \, b^{2} c^{3} d - 5 \, a b c^{2} d^{2} + 3 \, a^{2} c d^{3}\right )} x\right )} \sqrt {b x^{2} + a x}}{2 \, {\left ({\left (a b^{2} c^{5} d - 2 \, a^{2} b c^{4} d^{2} + a^{3} c^{3} d^{3}\right )} x^{2} + {\left (a b^{2} c^{6} - 2 \, a^{2} b c^{5} d + a^{3} c^{4} d^{2}\right )} x\right )}}, \frac {\sqrt {-b c^{2} + a c d} {\left ({\left (4 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{2} + {\left (4 \, a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x\right )} \arctan \left (\frac {\sqrt {-b c^{2} + a c d} \sqrt {b x^{2} + a x}}{b c x + a c}\right ) - {\left (2 \, b^{2} c^{4} - 4 \, a b c^{3} d + 2 \, a^{2} c^{2} d^{2} + {\left (2 \, b^{2} c^{3} d - 5 \, a b c^{2} d^{2} + 3 \, a^{2} c d^{3}\right )} x\right )} \sqrt {b x^{2} + a x}}{{\left (a b^{2} c^{5} d - 2 \, a^{2} b c^{4} d^{2} + a^{3} c^{3} d^{3}\right )} x^{2} + {\left (a b^{2} c^{6} - 2 \, a^{2} b c^{5} d + a^{3} c^{4} d^{2}\right )} x}\right ] \] Input:

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

Output:

[1/2*(sqrt(b*c^2 - a*c*d)*((4*a*b*c*d^2 - 3*a^2*d^3)*x^2 + (4*a*b*c^2*d - 
3*a^2*c*d^2)*x)*log((a*c + (2*b*c - a*d)*x - 2*sqrt(b*c^2 - a*c*d)*sqrt(b* 
x^2 + a*x))/(d*x + c)) - 2*(2*b^2*c^4 - 4*a*b*c^3*d + 2*a^2*c^2*d^2 + (2*b 
^2*c^3*d - 5*a*b*c^2*d^2 + 3*a^2*c*d^3)*x)*sqrt(b*x^2 + a*x))/((a*b^2*c^5* 
d - 2*a^2*b*c^4*d^2 + a^3*c^3*d^3)*x^2 + (a*b^2*c^6 - 2*a^2*b*c^5*d + a^3* 
c^4*d^2)*x), (sqrt(-b*c^2 + a*c*d)*((4*a*b*c*d^2 - 3*a^2*d^3)*x^2 + (4*a*b 
*c^2*d - 3*a^2*c*d^2)*x)*arctan(sqrt(-b*c^2 + a*c*d)*sqrt(b*x^2 + a*x)/(b* 
c*x + a*c)) - (2*b^2*c^4 - 4*a*b*c^3*d + 2*a^2*c^2*d^2 + (2*b^2*c^3*d - 5* 
a*b*c^2*d^2 + 3*a^2*c*d^3)*x)*sqrt(b*x^2 + a*x))/((a*b^2*c^5*d - 2*a^2*b*c 
^4*d^2 + a^3*c^3*d^3)*x^2 + (a*b^2*c^6 - 2*a^2*b*c^5*d + a^3*c^4*d^2)*x)]
 

Sympy [F]

\[ \int \frac {1}{x (c+d x)^2 \sqrt {a x+b x^2}} \, dx=\int \frac {1}{x \sqrt {x \left (a + b x\right )} \left (c + d x\right )^{2}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {1}{x (c+d x)^2 \sqrt {a x+b x^2}} \, dx=\int { \frac {1}{\sqrt {b x^{2} + a x} {\left (d x + c\right )}^{2} x} \,d x } \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 739 vs. \(2 (131) = 262\).

Time = 0.48 (sec) , antiderivative size = 739, normalized size of antiderivative = 5.10 \[ \int \frac {1}{x (c+d x)^2 \sqrt {a x+b x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

1/6*(6*sqrt(b - 2*b*c/(d*x + c) + b*c^2/(d*x + c)^2 + a*d/(d*x + c) - a*c* 
d/(d*x + c)^2)*c^2*d^2*sgn(1/(d*x + c))*sgn(d)/(b*c^5*sgn(1/(d*x + c))^2*s 
gn(d)^2 - a*c^4*d*sgn(1/(d*x + c))^2*sgn(d)^2) - (4*b*c*d^3*log(abs(8*b^2* 
c^2*d - 8*a*b*c*d^2 + a^2*d^3 - 8*sqrt(b*c^2 - a*c*d)*b^(3/2)*c*abs(d) + 4 
*sqrt(b*c^2 - a*c*d)*a*sqrt(b)*d*abs(d))) - 3*a*d^4*log(abs(8*b^2*c^2*d - 
8*a*b*c*d^2 + a^2*d^3 - 8*sqrt(b*c^2 - a*c*d)*b^(3/2)*c*abs(d) + 4*sqrt(b* 
c^2 - a*c*d)*a*sqrt(b)*d*abs(d))) + 6*sqrt(b*c^2 - a*c*d)*sqrt(b)*d^2*abs( 
d))*sgn(1/(d*x + c))*sgn(d)/(sqrt(b*c^2 - a*c*d)*b*c^3*abs(d) - sqrt(b*c^2 
 - a*c*d)*a*c^2*d*abs(d)) + (4*b*c*d^3 - 3*a*d^4)*log(abs(-2*sqrt(b*c^2 - 
a*c*d)*c*(sqrt(b - 2*b*c/(d*x + c) + b*c^2/(d*x + c)^2 + a*d/(d*x + c) - a 
*c*d/(d*x + c)^2) + sqrt(b*c^2*d^2 - a*c*d^3)/((d*x + c)*d))^3*abs(d) + 2* 
b^2*c^2*d - 3*a*b*c*d^2 + a^2*d^3 - 2*sqrt(b*c^2 - a*c*d)*(3*b*c - 2*a*d)* 
(sqrt(b - 2*b*c/(d*x + c) + b*c^2/(d*x + c)^2 + a*d/(d*x + c) - a*c*d/(d*x 
 + c)^2) + sqrt(b*c^2*d^2 - a*c*d^3)/((d*x + c)*d))*abs(d) + (6*b*c^2*d - 
5*a*c*d^2)*(sqrt(b - 2*b*c/(d*x + c) + b*c^2/(d*x + c)^2 + a*d/(d*x + c) - 
 a*c*d/(d*x + c)^2) + sqrt(b*c^2*d^2 - a*c*d^3)/((d*x + c)*d))^2))/((b*c^3 
 - a*c^2*d)*sqrt(b*c^2 - a*c*d)*abs(d)*sgn(1/(d*x + c))*sgn(d)))/d
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x (c+d x)^2 \sqrt {a x+b x^2}} \, dx=\int \frac {1}{x\,\sqrt {b\,x^2+a\,x}\,{\left (c+d\,x\right )}^2} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 712, normalized size of antiderivative = 4.91 \[ \int \frac {1}{x (c+d x)^2 \sqrt {a x+b x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(3*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x) - 
 sqrt(x)*sqrt(d)*sqrt(b))/(sqrt(c)*sqrt(b)))*a**2*c*d**2*x + 3*sqrt(c)*sqr 
t(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x) - sqrt(x)*sqrt( 
d)*sqrt(b))/(sqrt(c)*sqrt(b)))*a**2*d**3*x**2 - 4*sqrt(c)*sqrt(a*d - b*c)* 
atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x) - sqrt(x)*sqrt(d)*sqrt(b))/( 
sqrt(c)*sqrt(b)))*a*b*c**2*d*x - 4*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d 
- b*c) - sqrt(d)*sqrt(a + b*x) - sqrt(x)*sqrt(d)*sqrt(b))/(sqrt(c)*sqrt(b) 
))*a*b*c*d**2*x**2 + 3*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) + sqr 
t(d)*sqrt(a + b*x) + sqrt(x)*sqrt(d)*sqrt(b))/(sqrt(c)*sqrt(b)))*a**2*c*d* 
*2*x + 3*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) + sqrt(d)*sqrt(a + 
b*x) + sqrt(x)*sqrt(d)*sqrt(b))/(sqrt(c)*sqrt(b)))*a**2*d**3*x**2 - 4*sqrt 
(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(x 
)*sqrt(d)*sqrt(b))/(sqrt(c)*sqrt(b)))*a*b*c**2*d*x - 4*sqrt(c)*sqrt(a*d - 
b*c)*atan((sqrt(a*d - b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(x)*sqrt(d)*sqrt( 
b))/(sqrt(c)*sqrt(b)))*a*b*c*d**2*x**2 - 2*sqrt(x)*sqrt(a + b*x)*a**2*c**2 
*d**2 - 3*sqrt(x)*sqrt(a + b*x)*a**2*c*d**3*x + 4*sqrt(x)*sqrt(a + b*x)*a* 
b*c**3*d + 5*sqrt(x)*sqrt(a + b*x)*a*b*c**2*d**2*x - 2*sqrt(x)*sqrt(a + b* 
x)*b**2*c**4 - 2*sqrt(x)*sqrt(a + b*x)*b**2*c**3*d*x + sqrt(b)*a**2*c**2*d 
**2*x + sqrt(b)*a**2*c*d**3*x**2 - 3*sqrt(b)*a*b*c**3*d*x - 3*sqrt(b)*a*b* 
c**2*d**2*x**2 + 2*sqrt(b)*b**2*c**4*x + 2*sqrt(b)*b**2*c**3*d*x**2)/(a...