\(\int \frac {\sqrt {a x+b x^2}}{x^3 (c+d x)} \, dx\) [35]

Optimal result

Integrand size = 24, antiderivative size = 111 \[ \int \frac {\sqrt {a x+b x^2}}{x^3 (c+d x)} \, dx=-\frac {2 \sqrt {a x+b x^2}}{3 c x^2}-\frac {2 (b c-3 a d) \sqrt {a x+b x^2}}{3 a c^2 x}-\frac {2 d \sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a x+b x^2}}\right )}{c^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.20 \[ \int \frac {\sqrt {a x+b x^2}}{x^3 (c+d x)} \, dx=-\frac {2 \sqrt {x (a+b x)} \left (\sqrt {c} \sqrt {a+b x} (b c x+a (c-3 d x))+3 a d \sqrt {-b c+a d} x^{3/2} \arctan \left (\frac {-d \sqrt {x} \sqrt {a+b x}+\sqrt {b} (c+d x)}{\sqrt {c} \sqrt {-b c+a d}}\right )\right )}{3 a c^{5/2} x^2 \sqrt {a+b x}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.18, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1261, 107, 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.

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

\(\Big \downarrow \) 1261

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

\(\Big \downarrow \) 107

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 104

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(Sqrt[a*x + b*x^2]*((-2*(a + b*x)^(3/2))/(3*a*c*x^(3/2)) - (d*((-2*Sqrt[a 
+ b*x])/(c*Sqrt[x]) + (2*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b*c - a*d]*Sqrt[x]) 
/(Sqrt[c]*Sqrt[a + b*x])])/c^(3/2)))/c))/(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 107

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[(a*d*f*(m + 1) + b*c*f*(n + 
 1) + b*d*e*(p + 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, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x 
] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[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 = 0.62 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.79

method	result	size
pseudoelliptic	\(\frac {-\frac {2 \sqrt {x \left (b x +a \right )}\, \left (-3 a d x +c b x +a c \right )}{3 x^{2}}-\frac {2 \left (a d -b c \right ) a d \arctan \left (\frac {\sqrt {x \left (b x +a \right )}\, c}{x \sqrt {c \left (a d -b c \right )}}\right )}{\sqrt {c \left (a d -b c \right )}}}{a \,c^{2}}\)	\(88\)
risch	\(-\frac {2 \left (b x +a \right ) \left (-3 a d x +c b x +a c \right )}{3 a \,c^{2} \sqrt {x \left (b x +a \right )}\, x}-\frac {\left (a d -b c \right ) \ln \left (\frac {-\frac {2 c \left (a d -b c \right )}{d^{2}}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{x +\frac {c}{d}}\right )}{c^{2} \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}\)	\(179\)
default	\(-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{3 c a \,x^{3}}+\frac {d^{2} \left (\sqrt {b \,x^{2}+a x}+\frac {a \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{2 \sqrt {b}}\right )}{c^{3}}-\frac {d \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{a \,x^{2}}+\frac {2 b \left (\sqrt {b \,x^{2}+a x}+\frac {a \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{2 \sqrt {b}}\right )}{a}\right )}{c^{2}}-\frac {d^{2} \left (\sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}+\frac {\left (a d -2 b c \right ) \ln \left (\frac {\frac {a d -2 b c}{2 d}+b \left (x +\frac {c}{d}\right )}{\sqrt {b}}+\sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}\right )}{2 d \sqrt {b}}+\frac {c \left (a d -b c \right ) \ln \left (\frac {-\frac {2 c \left (a d -b c \right )}{d^{2}}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}\right )}{c^{3}}\)	\(434\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.95 \[ \int \frac {\sqrt {a x+b x^2}}{x^3 (c+d x)} \, dx=\left [\frac {3 \, a d x^{2} \sqrt {\frac {b c - a d}{c}} \log \left (\frac {a c + {\left (2 \, b c - a d\right )} x - 2 \, \sqrt {b x^{2} + a x} c \sqrt {\frac {b c - a d}{c}}}{d x + c}\right ) - 2 \, \sqrt {b x^{2} + a x} {\left (a c + {\left (b c - 3 \, a d\right )} x\right )}}{3 \, a c^{2} x^{2}}, -\frac {2 \, {\left (3 \, a d x^{2} \sqrt {-\frac {b c - a d}{c}} \arctan \left (-\frac {\sqrt {b x^{2} + a x} c \sqrt {-\frac {b c - a d}{c}}}{{\left (b c - a d\right )} x}\right ) + \sqrt {b x^{2} + a x} {\left (a c + {\left (b c - 3 \, a d\right )} x\right )}\right )}}{3 \, a c^{2} x^{2}}\right ] \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.63 \[ \int \frac {\sqrt {a x+b x^2}}{x^3 (c+d x)} \, dx=-\frac {2 \, {\left (b c d - a d^{2}\right )} \arctan \left (-\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} d + \sqrt {b} c}{\sqrt {-b c^{2} + a c d}}\right )}{\sqrt {-b c^{2} + a c d} c^{2}} + \frac {2 \, {\left (3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} b c - 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} a d + 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} a \sqrt {b} c + a^{2} c\right )}}{3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{3} c^{2}} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.68 \[ \int \frac {\sqrt {a x+b x^2}}{x^3 (c+d x)} \, dx=\frac {-2 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b x +a}-\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a d \,x^{2}-2 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a d \,x^{2}-\frac {2 \sqrt {x}\, \sqrt {b x +a}\, a \,c^{2}}{3}+2 \sqrt {x}\, \sqrt {b x +a}\, a c d x -\frac {2 \sqrt {x}\, \sqrt {b x +a}\, b \,c^{2} x}{3}-\frac {2 \sqrt {b}\, a c d \,x^{2}}{3}-\frac {2 \sqrt {b}\, b \,c^{2} x^{2}}{3}}{a \,c^{3} x^{2}} \] Input:

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