\(\int \frac {1}{x^2 (c+d x) \sqrt {a x+b x^2}} \, dx\) [148]

Optimal result

Integrand size = 24, antiderivative size = 117 \[ \int \frac {1}{x^2 (c+d x) \sqrt {a x+b x^2}} \, dx=-\frac {2 \sqrt {a x+b x^2}}{3 a c x^2}+\frac {2 (2 b c+3 a d) \sqrt {a x+b x^2}}{3 a^2 c^2 x}+\frac {2 d^2 \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a x+b x^2}}\right )}{c^{5/2} \sqrt {b c-a d}} \] Output:

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

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.13 \[ \int \frac {1}{x^2 (c+d x) \sqrt {a x+b x^2}} \, dx=\frac {2 \left (\frac {\sqrt {c} (a+b x) (-a c+2 b c x+3 a d x)}{a^2}-\frac {3 d^2 x^{3/2} \sqrt {a+b x} \arctan \left (\frac {-d \sqrt {x} \sqrt {a+b x}+\sqrt {b} (c+d x)}{\sqrt {c} \sqrt {-b c+a d}}\right )}{\sqrt {-b c+a d}}\right )}{3 c^{5/2} x \sqrt {x (a+b x)}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.29, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1261, 115, 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.

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

\(\Big \downarrow \) 1261

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

\(\Big \downarrow \) 115

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 169

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 104

\(\displaystyle \frac {\sqrt {x} \sqrt {a+b x} \left (-\frac {-\frac {6 a d^2 \int \frac {1}{c-\frac {(b c-a d) x}{a+b x}}d\frac {\sqrt {x}}{\sqrt {a+b x}}}{c}-\frac {2 \sqrt {a+b x} (3 a d+2 b c)}{a c \sqrt {x}}}{3 a c}-\frac {2 \sqrt {a+b x}}{3 a c x^{3/2}}\right )}{\sqrt {a x+b x^2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\sqrt {x} \sqrt {a+b x} \left (-\frac {-\frac {6 a d^2 \text {arctanh}\left (\frac {\sqrt {x} \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x}}\right )}{c^{3/2} \sqrt {b c-a d}}-\frac {2 \sqrt {a+b x} (3 a d+2 b c)}{a c \sqrt {x}}}{3 a c}-\frac {2 \sqrt {a+b x}}{3 a c x^{3/2}}\right )}{\sqrt {a x+b x^2}}\)

Input:

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

Output:

(Sqrt[x]*Sqrt[a + b*x]*((-2*Sqrt[a + b*x])/(3*a*c*x^(3/2)) - ((-2*(2*b*c + 
 3*a*d)*Sqrt[a + b*x])/(a*c*Sqrt[x]) - (6*a*d^2*ArcTanh[(Sqrt[b*c - a*d]*S 
qrt[x])/(Sqrt[c]*Sqrt[a + b*x])])/(c^(3/2)*Sqrt[b*c - a*d]))/(3*a*c)))/Sqr 
t[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 115

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

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 = 0.63 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.93

method	result	size
pseudoelliptic	\(\frac {-2 d^{2} \arctan \left (\frac {\sqrt {x \left (b x +a \right )}\, c}{x \sqrt {c \left (a d -b c \right )}}\right ) a^{2} x^{3}+2 \left (-\frac {\left (x \left (b x +a \right )\right )^{\frac {3}{2}} c}{3}+\sqrt {x \left (b x +a \right )}\, x^{2} \left (a d +b c \right )\right ) \sqrt {c \left (a d -b c \right )}}{a^{2} c^{2} x^{3} \sqrt {c \left (a d -b c \right )}}\)	\(109\)
risch	\(-\frac {2 \left (b x +a \right ) \left (-3 a d x -2 c b x +a c \right )}{3 a^{2} c^{2} \sqrt {x \left (b x +a \right )}\, x}-\frac {d \ln \left (\frac {-\frac {2 c \left (a d -b c \right )}{d^{2}}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{x +\frac {c}{d}}\right )}{c^{2} \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}\)	\(173\)
default	\(\frac {-\frac {2 \sqrt {b \,x^{2}+a x}}{3 a \,x^{2}}+\frac {4 b \sqrt {b \,x^{2}+a x}}{3 a^{2} x}}{c}-\frac {d \ln \left (\frac {-\frac {2 c \left (a d -b c \right )}{d^{2}}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{x +\frac {c}{d}}\right )}{c^{2} \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}+\frac {2 d \sqrt {b \,x^{2}+a x}}{c^{2} a x}\)	\(201\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 300, normalized size of antiderivative = 2.56 \[ \int \frac {1}{x^2 (c+d x) \sqrt {a x+b x^2}} \, dx=\left [\frac {3 \, \sqrt {b c^{2} - a c d} a^{2} d^{2} x^{2} \log \left (\frac {a c + {\left (2 \, b c - a d\right )} x + 2 \, \sqrt {b c^{2} - a c d} \sqrt {b x^{2} + a x}}{d x + c}\right ) - 2 \, {\left (a b c^{3} - a^{2} c^{2} d - {\left (2 \, b^{2} c^{3} + a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x\right )} \sqrt {b x^{2} + a x}}{3 \, {\left (a^{2} b c^{4} - a^{3} c^{3} d\right )} x^{2}}, -\frac {2 \, {\left (3 \, \sqrt {-b c^{2} + a c d} a^{2} d^{2} x^{2} \arctan \left (\frac {\sqrt {-b c^{2} + a c d} \sqrt {b x^{2} + a x}}{b c x + a c}\right ) + {\left (a b c^{3} - a^{2} c^{2} d - {\left (2 \, b^{2} c^{3} + a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x\right )} \sqrt {b x^{2} + a x}\right )}}{3 \, {\left (a^{2} b c^{4} - a^{3} c^{3} d\right )} x^{2}}\right ] \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.24 \[ \int \frac {1}{x^2 (c+d x) \sqrt {a x+b x^2}} \, dx=\frac {2 \, d^{2} \arctan \left (-\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} d + \sqrt {b} c}{\sqrt {-b c^{2} + a c d}}\right )}{\sqrt {-b c^{2} + a c d} c^{2}} - \frac {2 \, {\left (3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} d - 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} c - a c\right )}}{3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{3} c^{2}} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 263, normalized size of antiderivative = 2.25 \[ \int \frac {1}{x^2 (c+d x) \sqrt {a x+b x^2}} \, dx=\frac {-2 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b x +a}-\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a^{2} d^{2} x^{2}-2 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a^{2} d^{2} x^{2}-\frac {2 \sqrt {x}\, \sqrt {b x +a}\, a^{2} c^{2} d}{3}+2 \sqrt {x}\, \sqrt {b x +a}\, a^{2} c \,d^{2} x +\frac {2 \sqrt {x}\, \sqrt {b x +a}\, a b \,c^{3}}{3}-\frac {2 \sqrt {x}\, \sqrt {b x +a}\, a b \,c^{2} d x}{3}-\frac {4 \sqrt {x}\, \sqrt {b x +a}\, b^{2} c^{3} x}{3}-\frac {2 \sqrt {b}\, a^{2} c \,d^{2} x^{2}}{3}-\frac {2 \sqrt {b}\, a b \,c^{2} d \,x^{2}}{3}+\frac {4 \sqrt {b}\, b^{2} c^{3} x^{2}}{3}}{a^{2} c^{3} x^{2} \left (a d -b c \right )} \] Input:

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

Output:

(2*( - 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**2*x**2 - 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**2*x**2 - sqrt(x)*sqrt(a + b* 
x)*a**2*c**2*d + 3*sqrt(x)*sqrt(a + b*x)*a**2*c*d**2*x + sqrt(x)*sqrt(a + 
b*x)*a*b*c**3 - sqrt(x)*sqrt(a + b*x)*a*b*c**2*d*x - 2*sqrt(x)*sqrt(a + b* 
x)*b**2*c**3*x - sqrt(b)*a**2*c*d**2*x**2 - sqrt(b)*a*b*c**2*d*x**2 + 2*sq 
rt(b)*b**2*c**3*x**2))/(3*a**2*c**3*x**2*(a*d - b*c))