\(\int \frac {\sqrt {a x+b x^2}}{x (c+d x)^2} \, dx\) [41]

Optimal result

Integrand size = 24, antiderivative size = 77 \[ \int \frac {\sqrt {a x+b x^2}}{x (c+d x)^2} \, dx=\frac {\sqrt {a x+b x^2}}{c (c+d x)}+\frac {a \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a x+b x^2}}\right )}{c^{3/2} \sqrt {b c-a d}} \] Output:

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

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.40 \[ \int \frac {\sqrt {a x+b x^2}}{x (c+d x)^2} \, dx=\frac {\sqrt {x (a+b x)} \left (\frac {\sqrt {c}}{c+d x}-\frac {a \arctan \left (\frac {-d \sqrt {x} \sqrt {a+b x}+\sqrt {b} (c+d x)}{\sqrt {c} \sqrt {-b c+a d}}\right )}{\sqrt {-b c+a d} \sqrt {x} \sqrt {a+b x}}\right )}{c^{3/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.38, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1261, 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.

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

\(\Big \downarrow \) 1261

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 104

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

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]
 

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 = 0.60 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.84

method	result	size
pseudoelliptic	\(\frac {\frac {\sqrt {x \left (b x +a \right )}}{d x +c}-\frac {a \arctan \left (\frac {\sqrt {x \left (b x +a \right )}\, c}{x \sqrt {c \left (a d -b c \right )}}\right )}{\sqrt {c \left (a d -b c \right )}}}{c}\)	\(65\)
default	\(\frac {\sqrt {b \,x^{2}+a x}+\frac {a \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{2 \sqrt {b}}}{c^{2}}-\frac {\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}}}+\frac {\left (a d -2 b c \right ) \ln \left (\frac {\frac {a d -2 b c}{2 d}+b \left (x +\frac {c}{d}\right )}{\sqrt {b}}+\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}}}\right )}{2 d \sqrt {b}}+\frac {c \left (a d -b c \right ) \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^{2} \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}}{c^{2}}-\frac {\frac {d^{2} \left (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}}\right )^{\frac {3}{2}}}{c \left (a d -b c \right ) \left (x +\frac {c}{d}\right )}-\frac {\left (a d -2 b c \right ) d \left (\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}}}+\frac {\left (a d -2 b c \right ) \ln \left (\frac {\frac {a d -2 b c}{2 d}+b \left (x +\frac {c}{d}\right )}{\sqrt {b}}+\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}}}\right )}{2 d \sqrt {b}}+\frac {c \left (a d -b c \right ) \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^{2} \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}\right )}{2 c \left (a d -b c \right )}-\frac {2 b \,d^{2} \left (\frac {\left (2 b \left (x +\frac {c}{d}\right )+\frac {a d -2 b c}{d}\right ) \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}}}}{4 b}+\frac {\left (-\frac {4 b c \left (a d -b c \right )}{d^{2}}-\frac {\left (a d -2 b c \right )^{2}}{d^{2}}\right ) \ln \left (\frac {\frac {a d -2 b c}{2 d}+b \left (x +\frac {c}{d}\right )}{\sqrt {b}}+\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}}}\right )}{8 b^{\frac {3}{2}}}\right )}{c \left (a d -b c \right )}}{d c}\)	\(924\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 251, normalized size of antiderivative = 3.26 \[ \int \frac {\sqrt {a x+b x^2}}{x (c+d x)^2} \, dx=\left [\frac {\sqrt {b c^{2} - a c d} {\left (a d x + a c\right )} \log \left (\frac {a c + {\left (2 \, b c - a d\right )} x + 2 \, \sqrt {b c^{2} - a c d} \sqrt {b x^{2} + a x}}{d x + c}\right ) + 2 \, {\left (b c^{2} - a c d\right )} \sqrt {b x^{2} + a x}}{2 \, {\left (b c^{4} - a c^{3} d + {\left (b c^{3} d - a c^{2} d^{2}\right )} x\right )}}, -\frac {\sqrt {-b c^{2} + a c d} {\left (a d x + a c\right )} \arctan \left (\frac {\sqrt {-b c^{2} + a c d} \sqrt {b x^{2} + a x}}{b c x + a c}\right ) - {\left (b c^{2} - a c d\right )} \sqrt {b x^{2} + a x}}{b c^{4} - a c^{3} d + {\left (b c^{3} d - a c^{2} d^{2}\right )} x}\right ] \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

integrate(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. 313 vs. \(2 (65) = 130\).

Time = 0.42 (sec) , antiderivative size = 313, normalized size of antiderivative = 4.06 \[ \int \frac {\sqrt {a x+b x^2}}{x (c+d x)^2} \, dx=-\frac {1}{2} \, d^{3} {\left (\frac {a \log \left ({\left | 2 \, b c d - a d^{2} - 2 \, \sqrt {b c^{2} - a c d} {\left (\sqrt {b - \frac {2 \, b c}{d x + c} + \frac {b c^{2}}{{\left (d x + c\right )}^{2}} + \frac {a d}{d x + c} - \frac {a c d}{{\left (d x + c\right )}^{2}}} + \frac {\sqrt {b c^{2} d^{2} - a c d^{3}}}{{\left (d x + c\right )} d}\right )} {\left | d \right |} \right |}\right ) \mathrm {sgn}\left (\frac {1}{d x + c}\right ) \mathrm {sgn}\left (d\right )}{\sqrt {b c^{2} - a c d} c d^{2} {\left | d \right |}} - \frac {2 \, \sqrt {b - \frac {2 \, b c}{d x + c} + \frac {b c^{2}}{{\left (d x + c\right )}^{2}} + \frac {a d}{d x + c} - \frac {a c d}{{\left (d x + c\right )}^{2}}} \mathrm {sgn}\left (\frac {1}{d x + c}\right ) \mathrm {sgn}\left (d\right )}{c d^{4}} - \frac {{\left (a d^{2} \log \left ({\left | 2 \, b c d - a d^{2} - 2 \, \sqrt {b c^{2} - a c d} \sqrt {b} {\left | d \right |} \right |}\right ) - 2 \, \sqrt {b c^{2} - a c d} \sqrt {b} {\left | d \right |}\right )} \mathrm {sgn}\left (\frac {1}{d x + c}\right ) \mathrm {sgn}\left (d\right )}{\sqrt {b c^{2} - a c d} c d^{4} {\left | d \right |}}\right )} \] Input:

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

Output:

-1/2*d^3*(a*log(abs(2*b*c*d - a*d^2 - 2*sqrt(b*c^2 - a*c*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) + sqr 
t(b*c^2*d^2 - a*c*d^3)/((d*x + c)*d))*abs(d)))*sgn(1/(d*x + c))*sgn(d)/(sq 
rt(b*c^2 - a*c*d)*c*d^2*abs(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)*sgn(1/(d*x + c))*sgn(d)/(c*d^4 
) - (a*d^2*log(abs(2*b*c*d - a*d^2 - 2*sqrt(b*c^2 - a*c*d)*sqrt(b)*abs(d)) 
) - 2*sqrt(b*c^2 - a*c*d)*sqrt(b)*abs(d))*sgn(1/(d*x + c))*sgn(d)/(sqrt(b* 
c^2 - a*c*d)*c*d^4*abs(d)))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 266, normalized size of antiderivative = 3.45 \[ \int \frac {\sqrt {a x+b x^2}}{x (c+d x)^2} \, dx=\frac {-\sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b x +a}-\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a c -\sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b x +a}-\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a d x -\sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a c -\sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a d x +\sqrt {x}\, \sqrt {b x +a}\, a c d -\sqrt {x}\, \sqrt {b x +a}\, b \,c^{2}}{c^{2} \left (a \,d^{2} x -b c d x +a c d -b \,c^{2}\right )} \] Input:

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

Output:

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