\(\int \frac {x}{(c+d x)^2 \sqrt {a x+b x^2}} \, dx\) [152]

Optimal result

Integrand size = 22, antiderivative size = 85 \[ \int \frac {x}{(c+d x)^2 \sqrt {a x+b x^2}} \, dx=\frac {\sqrt {a x+b x^2}}{(b c-a d) (c+d x)}-\frac {a \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a x+b x^2}}\right )}{\sqrt {c} (b c-a d)^{3/2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 4.57 (sec) , antiderivative size = 590, normalized size of antiderivative = 6.94 \[ \int \frac {x}{(c+d x)^2 \sqrt {a x+b x^2}} \, dx=\frac {\sqrt {x} \sqrt {a+b x} \left (\frac {a \sqrt {x}}{(b c-a d) (c+d x) \left (-\sqrt {a}+\sqrt {a+b x}\right )}+\frac {b x^{3/2}}{(b c-a d) (c+d x) \left (-\sqrt {a}+\sqrt {a+b x}\right )}+\frac {\sqrt {a} \sqrt {x} \sqrt {a+b x}}{(-b c+a d) (c+d x) \left (-\sqrt {a}+\sqrt {a+b x}\right )}+\frac {a \left (i \sqrt {a} \sqrt {d}+\sqrt {b c-a d}\right ) \arctan \left (\frac {\sqrt {-b c+2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {x}}{\sqrt {c} \left (\sqrt {a}-\sqrt {a+b x}\right )}\right )}{\sqrt {c} (b c-a d)^{3/2} \sqrt {-b c+2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}}}+\frac {a \arctan \left (\frac {\sqrt {-b c+2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {x}}{\sqrt {c} \left (\sqrt {a}-\sqrt {a+b x}\right )}\right )}{\sqrt {c} (b c-a d) \sqrt {-b c+2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}}}+\frac {i a^{3/2} \sqrt {d} \arctan \left (\frac {\sqrt {-b c+2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {x}}{\sqrt {c} \left (-\sqrt {a}+\sqrt {a+b x}\right )}\right )}{\sqrt {c} (b c-a d)^{3/2} \sqrt {-b c+2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}}}\right )}{\sqrt {x (a+b x)}} \] Input:

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

Output:

(Sqrt[x]*Sqrt[a + b*x]*((a*Sqrt[x])/((b*c - a*d)*(c + d*x)*(-Sqrt[a] + Sqr 
t[a + b*x])) + (b*x^(3/2))/((b*c - a*d)*(c + d*x)*(-Sqrt[a] + Sqrt[a + b*x 
])) + (Sqrt[a]*Sqrt[x]*Sqrt[a + b*x])/((-(b*c) + a*d)*(c + d*x)*(-Sqrt[a] 
+ Sqrt[a + b*x])) + (a*(I*Sqrt[a]*Sqrt[d] + Sqrt[b*c - a*d])*ArcTan[(Sqrt[ 
-(b*c) + 2*a*d - (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*Sqrt[x])/(Sqrt[c]* 
(Sqrt[a] - Sqrt[a + b*x]))])/(Sqrt[c]*(b*c - a*d)^(3/2)*Sqrt[-(b*c) + 2*a* 
d - (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]) + (a*ArcTan[(Sqrt[-(b*c) + 2*a 
*d + (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*Sqrt[x])/(Sqrt[c]*(Sqrt[a] - S 
qrt[a + b*x]))])/(Sqrt[c]*(b*c - a*d)*Sqrt[-(b*c) + 2*a*d + (2*I)*Sqrt[a]* 
Sqrt[d]*Sqrt[b*c - a*d]]) + (I*a^(3/2)*Sqrt[d]*ArcTan[(Sqrt[-(b*c) + 2*a*d 
 + (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*Sqrt[x])/(Sqrt[c]*(-Sqrt[a] + Sq 
rt[a + b*x]))])/(Sqrt[c]*(b*c - a*d)^(3/2)*Sqrt[-(b*c) + 2*a*d + (2*I)*Sqr 
t[a]*Sqrt[d]*Sqrt[b*c - a*d]])))/Sqrt[x*(a + b*x)]
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.22, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1228, 1154, 219}

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.

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

\(\Big \downarrow \) 1228

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1228

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + 
 b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e 
*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^ 
(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x 
] && EqQ[Simplify[m + 2*p + 3], 0]
 

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.86

method	result	size
pseudoelliptic	\(\frac {-\frac {\sqrt {x \left (b x +a \right )}}{d x +c}-\frac {a \arctan \left (\frac {\sqrt {x \left (b x +a \right )}\, c}{x \sqrt {c \left (a d -b c \right )}}\right )}{\sqrt {c \left (a d -b c \right )}}}{a d -b c}\)	\(73\)
default	\(-\frac {\ln \left (\frac {-\frac {2 c \left (a d -b c \right )}{d^{2}}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}-\frac {c \left (\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{c \left (a d -b c \right ) \left (x +\frac {c}{d}\right )}-\frac {\left (a d -2 b c \right ) d \ln \left (\frac {-\frac {2 c \left (a d -b c \right )}{d^{2}}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{x +\frac {c}{d}}\right )}{2 c \left (a d -b c \right ) \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}\right )}{d^{3}}\)	\(363\)

Input:

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

Output:

1/(a*d-b*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 = 297, normalized size of antiderivative = 3.49 \[ \int \frac {x}{(c+d x)^2 \sqrt {a x+b x^2}} \, dx=\left [-\frac {\sqrt {b c^{2} - a c d} {\left (a d x + a c\right )} \log \left (\frac {a c + {\left (2 \, b c - a d\right )} x + 2 \, \sqrt {b c^{2} - a c d} \sqrt {b x^{2} + a x}}{d x + c}\right ) - 2 \, {\left (b c^{2} - a c d\right )} \sqrt {b x^{2} + a x}}{2 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} + {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x\right )}}, \frac {\sqrt {-b c^{2} + a c d} {\left (a d x + a c\right )} \arctan \left (\frac {\sqrt {-b c^{2} + a c d} \sqrt {b x^{2} + a x}}{b c x + a c}\right ) + {\left (b c^{2} - a c d\right )} \sqrt {b x^{2} + a x}}{b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} + {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x}\right ] \] Input:

integrate(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)*(a*d*x + a*c)*log((a*c + (2*b*c - a*d)*x + 2*sq 
rt(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^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2 + (b^2*c^3*d - 2*a*b*c^2 
*d^2 + a^2*c*d^3)*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^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2 + (b^2*c^3*d - 2*a*b*c^2*d^2 
+ a^2*c*d^3)*x)]
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

Leaf count of result is larger than twice the leaf count of optimal. 379 vs. \(2 (73) = 146\).

Time = 0.29 (sec) , antiderivative size = 379, normalized size of antiderivative = 4.46 \[ \int \frac {x}{(c+d x)^2 \sqrt {a x+b x^2}} \, dx=\frac {\frac {2 \, \sqrt {b - \frac {2 \, b c}{d x + c} + \frac {b c^{2}}{{\left (d x + c\right )}^{2}} + \frac {a d}{d x + c} - \frac {a c d}{{\left (d x + c\right )}^{2}}} d^{2} \mathrm {sgn}\left (\frac {1}{d x + c}\right ) \mathrm {sgn}\left (d\right )}{b c \mathrm {sgn}\left (\frac {1}{d x + c}\right )^{2} \mathrm {sgn}\left (d\right )^{2} - a d \mathrm {sgn}\left (\frac {1}{d x + c}\right )^{2} \mathrm {sgn}\left (d\right )^{2}} - \frac {{\left (a d^{4} \log \left ({\left | 2 \, b c d - a d^{2} - 2 \, \sqrt {b c^{2} - a c d} \sqrt {b} {\left | d \right |} \right |}\right ) + 2 \, \sqrt {b c^{2} - a c d} \sqrt {b} d^{2} {\left | d \right |}\right )} \mathrm {sgn}\left (\frac {1}{d x + c}\right ) \mathrm {sgn}\left (d\right )}{\sqrt {b c^{2} - a c d} b c {\left | d \right |} - \sqrt {b c^{2} - a c d} a d {\left | d \right |}} + \frac {a d^{4} \log \left ({\left | 2 \, b c d - a d^{2} - 2 \, \sqrt {b c^{2} - a c d} {\left (\sqrt {b - \frac {2 \, b c}{d x + c} + \frac {b c^{2}}{{\left (d x + c\right )}^{2}} + \frac {a d}{d x + c} - \frac {a c d}{{\left (d x + c\right )}^{2}}} + \frac {\sqrt {b c^{2} d^{2} - a c d^{3}}}{{\left (d x + c\right )} d}\right )} {\left | d \right |} \right |}\right )}{\sqrt {b c^{2} - a c d} {\left (b c - a d\right )} {\left | d \right |} \mathrm {sgn}\left (\frac {1}{d x + c}\right ) \mathrm {sgn}\left (d\right )}}{2 \, d^{3}} \] Input:

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

Output:

1/2*(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)*d^2*sgn(1/(d*x + c))*sgn(d)/(b*c*sgn(1/(d*x + c))^2*sgn(d)^ 
2 - a*d*sgn(1/(d*x + c))^2*sgn(d)^2) - (a*d^4*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)*d^2*a 
bs(d))*sgn(1/(d*x + c))*sgn(d)/(sqrt(b*c^2 - a*c*d)*b*c*abs(d) - sqrt(b*c^ 
2 - a*c*d)*a*d*abs(d)) + a*d^4*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) + sqrt(b*c^2*d^2 - a*c*d^3)/((d*x + c)*d))*abs(d)))/(sqrt(b* 
c^2 - a*c*d)*(b*c - a*d)*abs(d)*sgn(1/(d*x + c))*sgn(d)))/d^3
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 293, normalized size of antiderivative = 3.45 \[ \int \frac {x}{(c+d x)^2 \sqrt {a x+b x^2}} \, dx=\frac {-\sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b x +a}-\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a c -\sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b x +a}-\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a d x -\sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a c -\sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a d x -\sqrt {x}\, \sqrt {b x +a}\, a c d +\sqrt {x}\, \sqrt {b x +a}\, b \,c^{2}}{c \left (a^{2} d^{3} x -2 a b c \,d^{2} x +b^{2} c^{2} d x +a^{2} c \,d^{2}-2 a b \,c^{2} d +b^{2} c^{3}\right )} \] Input:

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