\(\int \frac {(A+B x) \sqrt {a x+b x^2}}{x^3 (c+d x)} \, dx\) [6]

Optimal result

Integrand size = 29, antiderivative size = 126 \[ \int \frac {(A+B x) \sqrt {a x+b x^2}}{x^3 (c+d x)} \, dx=-\frac {2 A \sqrt {a x+b x^2}}{3 c x^2}-\frac {2 (A b c+3 a B c-3 a A d) \sqrt {a x+b x^2}}{3 a c^2 x}+\frac {2 \sqrt {b c-a d} (B c-A d) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a x+b x^2}}\right )}{c^{5/2}} \] Output:

-2/3*A*(b*x^2+a*x)^(1/2)/c/x^2-2/3*(-3*A*a*d+A*b*c+3*B*a*c)*(b*x^2+a*x)^(1 
/2)/a/c^2/x+2*(-a*d+b*c)^(1/2)*(-A*d+B*c)*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 = 0.50 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.10 \[ \int \frac {(A+B x) \sqrt {a x+b x^2}}{x^3 (c+d x)} \, dx=\frac {2 \sqrt {x (a+b x)} \left (-\frac {\sqrt {c} (A b c x+3 a B c x+a A (c-3 d x))}{a}+\frac {3 \sqrt {-b c+a d} (B c-A d) x^{3/2} \arctan \left (\frac {-d \sqrt {x} \sqrt {a+b x}+\sqrt {b} (c+d x)}{\sqrt {c} \sqrt {-b c+a d}}\right )}{\sqrt {a+b x}}\right )}{3 c^{5/2} x^2} \] Input:

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

Output:

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(261\) vs. \(2(126)=252\).

Time = 1.06 (sec) , antiderivative size = 261, normalized size of antiderivative = 2.07, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2153, 2009}

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.

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

\(\Big \downarrow \) 2153

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {b c-a d} (B c-A d) \text {arctanh}\left (\frac {x (2 b c-a d)+a c}{2 \sqrt {c} \sqrt {a x+b x^2} \sqrt {b c-a d}}\right )}{c^{5/2}}-\frac {a d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right ) (B c-A d)}{\sqrt {b} c^3}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right ) (2 b c-a d) (B c-A d)}{\sqrt {b} c^3}+\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right ) (B c-A d)}{c^2}-\frac {2 \sqrt {a x+b x^2} (B c-A d)}{c^2 x}-\frac {2 A \left (a x+b x^2\right )^{3/2}}{3 a c x^3}\)

Input:

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

Output:

(-2*(B*c - A*d)*Sqrt[a*x + b*x^2])/(c^2*x) - (2*A*(a*x + b*x^2)^(3/2))/(3* 
a*c*x^3) + (2*Sqrt[b]*(B*c - A*d)*ArcTanh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/ 
c^2 - (a*d*(B*c - A*d)*ArcTanh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/(Sqrt[b]*c^ 
3) - ((2*b*c - a*d)*(B*c - A*d)*ArcTanh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/(S 
qrt[b]*c^3) + (Sqrt[b*c - a*d]*(B*c - A*d)*ArcTanh[(a*c + (2*b*c - a*d)*x) 
/(2*Sqrt[c]*Sqrt[b*c - a*d]*Sqrt[a*x + b*x^2])])/c^(5/2)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2153

Int[(Px_)*((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b 
_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Px*(d + e* 
x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m, n, p}, x] && PolyQ[Px, x] && (IntegerQ[p] || (IntegerQ[2*p] && IntegerQ 
[m] && ILtQ[n, 0])) &&  !(IGtQ[m, 0] && IGtQ[n, 0])
 

Maple [A] (verified)

Time = 0.68 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.94

method	result	size
pseudoelliptic	\(-\frac {2 \left (3 a \,x^{2} \left (a d -b c \right ) \left (A d -B c \right ) \arctan \left (\frac {\sqrt {x \left (b x +a \right )}\, c}{x \sqrt {c \left (a d -b c \right )}}\right )+\sqrt {c \left (a d -b c \right )}\, \sqrt {x \left (b x +a \right )}\, \left (\left (\left (3 B x +A \right ) a +A b x \right ) c -3 A a d x \right )\right )}{3 \sqrt {c \left (a d -b c \right )}\, c^{2} x^{2} a}\)	\(118\)
risch	\(-\frac {2 \left (b x +a \right ) \left (-3 A a d x +A b c x +3 B a c x +A a c \right )}{3 a \,c^{2} \sqrt {x \left (b x +a \right )}\, x}-\frac {\left (A a \,d^{2}-A b c d -B a c d +B b \,c^{2}\right ) \ln \left (\frac {-\frac {2 c \left (a d -b c \right )}{d^{2}}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{x +\frac {c}{d}}\right )}{d \,c^{2} \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}\)	\(208\)
default	\(-\frac {2 A \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{3 c a \,x^{3}}-\frac {\left (A d -B c \right ) \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{a \,x^{2}}+\frac {2 b \left (\sqrt {b \,x^{2}+a x}+\frac {a \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{2 \sqrt {b}}\right )}{a}\right )}{c^{2}}+\frac {\left (A d -B c \right ) d \left (\sqrt {b \,x^{2}+a x}+\frac {a \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{2 \sqrt {b}}\right )}{c^{3}}-\frac {d \left (A d -B c \right ) \left (\sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}+\frac {\left (a d -2 b c \right ) \ln \left (\frac {\frac {a d -2 b c}{2 d}+b \left (x +\frac {c}{d}\right )}{\sqrt {b}}+\sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}\right )}{2 d \sqrt {b}}+\frac {c \left (a d -b c \right ) \ln \left (\frac {-\frac {2 c \left (a d -b c \right )}{d^{2}}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}\right )}{c^{3}}\)	\(454\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 255, normalized size of antiderivative = 2.02 \[ \int \frac {(A+B x) \sqrt {a x+b x^2}}{x^3 (c+d x)} \, dx=\left [-\frac {3 \, {\left (B a c - A a d\right )} x^{2} \sqrt {\frac {b c - a d}{c}} \log \left (\frac {a c + {\left (2 \, b c - a d\right )} x - 2 \, \sqrt {b x^{2} + a x} c \sqrt {\frac {b c - a d}{c}}}{d x + c}\right ) + 2 \, {\left (A a c - {\left (3 \, A a d - {\left (3 \, B a + A b\right )} c\right )} x\right )} \sqrt {b x^{2} + a x}}{3 \, a c^{2} x^{2}}, \frac {2 \, {\left (3 \, {\left (B a c - A a d\right )} x^{2} \sqrt {-\frac {b c - a d}{c}} \arctan \left (-\frac {\sqrt {b x^{2} + a x} c \sqrt {-\frac {b c - a d}{c}}}{{\left (b c - a d\right )} x}\right ) - {\left (A a c - {\left (3 \, A a d - {\left (3 \, B a + A b\right )} c\right )} x\right )} \sqrt {b x^{2} + a x}\right )}}{3 \, a c^{2} x^{2}}\right ] \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 225 vs. \(2 (108) = 216\).

Time = 0.35 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.79 \[ \int \frac {(A+B x) \sqrt {a x+b x^2}}{x^3 (c+d x)} \, dx=\frac {2 \, {\left (B b c^{2} - B a c d - A b c d + A a d^{2}\right )} \arctan \left (-\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} d + \sqrt {b} c}{\sqrt {-b c^{2} + a c d}}\right )}{\sqrt {-b c^{2} + a c d} c^{2}} + \frac {2 \, {\left (3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} B a c + 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} A b c - 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} A a d + 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} A a \sqrt {b} c + A a^{2} c\right )}}{3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{3} c^{2}} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 282, normalized size of antiderivative = 2.24 \[ \int \frac {(A+B x) \sqrt {a x+b x^2}}{x^3 (c+d x)} \, dx=\frac {-2 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b x +a}-\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a d \,x^{2}+2 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b x +a}-\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) b c \,x^{2}-2 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a d \,x^{2}+2 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) b c \,x^{2}-\frac {2 \sqrt {x}\, \sqrt {b x +a}\, a \,c^{2}}{3}+2 \sqrt {x}\, \sqrt {b x +a}\, a c d x -\frac {8 \sqrt {x}\, \sqrt {b x +a}\, b \,c^{2} x}{3}-\frac {2 \sqrt {b}\, a c d \,x^{2}}{3}}{c^{3} x^{2}} \] Input:

int((B*x+A)*(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)))*b*c*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*d*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)))*b*c*x* 
*2 - sqrt(x)*sqrt(a + b*x)*a*c**2 + 3*sqrt(x)*sqrt(a + b*x)*a*c*d*x - 4*sq 
rt(x)*sqrt(a + b*x)*b*c**2*x - sqrt(b)*a*c*d*x**2))/(3*c**3*x**2)