\(\int \frac {c+d x}{x^5 \sqrt {a x+b x^2}} \, dx\) [133]

Optimal result

Integrand size = 22, antiderivative size = 160 \[ \int \frac {c+d x}{x^5 \sqrt {a x+b x^2}} \, dx=-\frac {2 c \sqrt {a x+b x^2}}{9 a x^5}+\frac {2 (8 b c-9 a d) \sqrt {a x+b x^2}}{63 a^2 x^4}-\frac {4 b (8 b c-9 a d) \sqrt {a x+b x^2}}{105 a^3 x^3}+\frac {16 b^2 (8 b c-9 a d) \sqrt {a x+b x^2}}{315 a^4 x^2}-\frac {32 b^3 (8 b c-9 a d) \sqrt {a x+b x^2}}{315 a^5 x} \] Output:

-2/9*c*(b*x^2+a*x)^(1/2)/a/x^5+2/63*(-9*a*d+8*b*c)*(b*x^2+a*x)^(1/2)/a^2/x 
^4-4/105*b*(-9*a*d+8*b*c)*(b*x^2+a*x)^(1/2)/a^3/x^3+16/315*b^2*(-9*a*d+8*b 
*c)*(b*x^2+a*x)^(1/2)/a^4/x^2-32/315*b^3*(-9*a*d+8*b*c)*(b*x^2+a*x)^(1/2)/ 
a^5/x
 

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.59 \[ \int \frac {c+d x}{x^5 \sqrt {a x+b x^2}} \, dx=-\frac {2 \sqrt {x (a+b x)} \left (128 b^4 c x^4+24 a^2 b^2 x^2 (2 c+3 d x)-16 a b^3 x^3 (4 c+9 d x)+5 a^4 (7 c+9 d x)-2 a^3 b x (20 c+27 d x)\right )}{315 a^5 x^5} \] Input:

Integrate[(c + d*x)/(x^5*Sqrt[a*x + b*x^2]),x]
 

Output:

(-2*Sqrt[x*(a + b*x)]*(128*b^4*c*x^4 + 24*a^2*b^2*x^2*(2*c + 3*d*x) - 16*a 
*b^3*x^3*(4*c + 9*d*x) + 5*a^4*(7*c + 9*d*x) - 2*a^3*b*x*(20*c + 27*d*x))) 
/(315*a^5*x^5)
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {1220, 1129, 1129, 1129, 1123}

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.

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

\(\Big \downarrow \) 1220

\(\displaystyle -\frac {(8 b c-9 a d) \int \frac {1}{x^4 \sqrt {b x^2+a x}}dx}{9 a}-\frac {2 c \sqrt {a x+b x^2}}{9 a x^5}\)

\(\Big \downarrow \) 1129

\(\displaystyle -\frac {(8 b c-9 a d) \left (-\frac {6 b \int \frac {1}{x^3 \sqrt {b x^2+a x}}dx}{7 a}-\frac {2 \sqrt {a x+b x^2}}{7 a x^4}\right )}{9 a}-\frac {2 c \sqrt {a x+b x^2}}{9 a x^5}\)

\(\Big \downarrow \) 1129

\(\displaystyle -\frac {(8 b c-9 a d) \left (-\frac {6 b \left (-\frac {4 b \int \frac {1}{x^2 \sqrt {b x^2+a x}}dx}{5 a}-\frac {2 \sqrt {a x+b x^2}}{5 a x^3}\right )}{7 a}-\frac {2 \sqrt {a x+b x^2}}{7 a x^4}\right )}{9 a}-\frac {2 c \sqrt {a x+b x^2}}{9 a x^5}\)

\(\Big \downarrow \) 1129

\(\displaystyle -\frac {(8 b c-9 a d) \left (-\frac {6 b \left (-\frac {4 b \left (-\frac {2 b \int \frac {1}{x \sqrt {b x^2+a x}}dx}{3 a}-\frac {2 \sqrt {a x+b x^2}}{3 a x^2}\right )}{5 a}-\frac {2 \sqrt {a x+b x^2}}{5 a x^3}\right )}{7 a}-\frac {2 \sqrt {a x+b x^2}}{7 a x^4}\right )}{9 a}-\frac {2 c \sqrt {a x+b x^2}}{9 a x^5}\)

\(\Big \downarrow \) 1123

\(\displaystyle -\frac {\left (-\frac {6 b \left (-\frac {4 b \left (\frac {4 b \sqrt {a x+b x^2}}{3 a^2 x}-\frac {2 \sqrt {a x+b x^2}}{3 a x^2}\right )}{5 a}-\frac {2 \sqrt {a x+b x^2}}{5 a x^3}\right )}{7 a}-\frac {2 \sqrt {a x+b x^2}}{7 a x^4}\right ) (8 b c-9 a d)}{9 a}-\frac {2 c \sqrt {a x+b x^2}}{9 a x^5}\)

Input:

Int[(c + d*x)/(x^5*Sqrt[a*x + b*x^2]),x]
 

Output:

(-2*c*Sqrt[a*x + b*x^2])/(9*a*x^5) - ((8*b*c - 9*a*d)*((-2*Sqrt[a*x + b*x^ 
2])/(7*a*x^4) - (6*b*((-2*Sqrt[a*x + b*x^2])/(5*a*x^3) - (4*b*((-2*Sqrt[a* 
x + b*x^2])/(3*a*x^2) + (4*b*Sqrt[a*x + b*x^2])/(3*a^2*x)))/(5*a)))/(7*a)) 
)/(9*a)
 

Defintions of rubi rules used

rule 1123

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[e*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(2*c*d - b 
*e))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 
0] && EqQ[m + 2*p + 2, 0]
 

rule 1129

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(-e)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((m + p + 1)*(2* 
c*d - b*e))), x] + Simp[c*(Simplify[m + 2*p + 2]/((m + p + 1)*(2*c*d - b*e) 
))   Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d 
, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[Simplify[m + 2*p + 
2], 0]
 

rule 1220

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d*g - e*f)*(d + e*x)^m*((a + b*x + c*x 
^2)^(p + 1)/((2*c*d - b*e)*(m + p + 1))), x] + Simp[(m*(g*(c*d - b*e) + c*e 
*f) + e*(p + 1)*(2*c*f - b*g))/(e*(2*c*d - b*e)*(m + p + 1))   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[c*d^2 - b*d*e + a*e^2, 0] && ((LtQ[m, -1] &&  !IGtQ[m + p + 1, 0 
]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0 
]
 

Maple [A] (verified)

Time = 0.47 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.52

method	result	size
pseudoelliptic	\(-\frac {2 \sqrt {x \left (b x +a \right )}\, \left (\left (\frac {9 d x}{7}+c \right ) a^{4}-\frac {8 \left (\frac {27 d x}{20}+c \right ) x b \,a^{3}}{7}+\frac {48 x^{2} b^{2} \left (\frac {3 d x}{2}+c \right ) a^{2}}{35}-\frac {64 \left (\frac {9 d x}{4}+c \right ) x^{3} b^{3} a}{35}+\frac {128 x^{4} b^{4} c}{35}\right )}{9 a^{5} x^{5}}\)	\(83\)
trager	\(-\frac {2 \left (-144 x^{4} a \,b^{3} d +128 x^{4} b^{4} c +72 a^{2} b^{2} d \,x^{3}-64 a \,b^{3} c \,x^{3}-54 a^{3} b d \,x^{2}+48 a^{2} b^{2} c \,x^{2}+45 a^{4} d x -40 a^{3} b c x +35 c \,a^{4}\right ) \sqrt {b \,x^{2}+a x}}{315 a^{5} x^{5}}\)	\(105\)
risch	\(-\frac {2 \left (b x +a \right ) \left (-144 x^{4} a \,b^{3} d +128 x^{4} b^{4} c +72 a^{2} b^{2} d \,x^{3}-64 a \,b^{3} c \,x^{3}-54 a^{3} b d \,x^{2}+48 a^{2} b^{2} c \,x^{2}+45 a^{4} d x -40 a^{3} b c x +35 c \,a^{4}\right )}{315 a^{5} x^{4} \sqrt {x \left (b x +a \right )}}\)	\(108\)
gosper	\(-\frac {2 \left (b x +a \right ) \left (-144 x^{4} a \,b^{3} d +128 x^{4} b^{4} c +72 a^{2} b^{2} d \,x^{3}-64 a \,b^{3} c \,x^{3}-54 a^{3} b d \,x^{2}+48 a^{2} b^{2} c \,x^{2}+45 a^{4} d x -40 a^{3} b c x +35 c \,a^{4}\right )}{315 x^{4} a^{5} \sqrt {b \,x^{2}+a x}}\)	\(110\)
orering	\(-\frac {2 \left (b x +a \right ) \left (-144 x^{4} a \,b^{3} d +128 x^{4} b^{4} c +72 a^{2} b^{2} d \,x^{3}-64 a \,b^{3} c \,x^{3}-54 a^{3} b d \,x^{2}+48 a^{2} b^{2} c \,x^{2}+45 a^{4} d x -40 a^{3} b c x +35 c \,a^{4}\right )}{315 x^{4} a^{5} \sqrt {b \,x^{2}+a x}}\)	\(110\)
default	\(c \left (-\frac {2 \sqrt {b \,x^{2}+a x}}{9 a \,x^{5}}-\frac {8 b \left (-\frac {2 \sqrt {b \,x^{2}+a x}}{7 a \,x^{4}}-\frac {6 b \left (-\frac {2 \sqrt {b \,x^{2}+a x}}{5 a \,x^{3}}-\frac {4 b \left (-\frac {2 \sqrt {b \,x^{2}+a x}}{3 a \,x^{2}}+\frac {4 b \sqrt {b \,x^{2}+a x}}{3 a^{2} x}\right )}{5 a}\right )}{7 a}\right )}{9 a}\right )+d \left (-\frac {2 \sqrt {b \,x^{2}+a x}}{7 a \,x^{4}}-\frac {6 b \left (-\frac {2 \sqrt {b \,x^{2}+a x}}{5 a \,x^{3}}-\frac {4 b \left (-\frac {2 \sqrt {b \,x^{2}+a x}}{3 a \,x^{2}}+\frac {4 b \sqrt {b \,x^{2}+a x}}{3 a^{2} x}\right )}{5 a}\right )}{7 a}\right )\)	\(216\)

Input:

int((d*x+c)/x^5/(b*x^2+a*x)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-2/9*(x*(b*x+a))^(1/2)*((9/7*d*x+c)*a^4-8/7*(27/20*d*x+c)*x*b*a^3+48/35*x^ 
2*b^2*(3/2*d*x+c)*a^2-64/35*(9/4*d*x+c)*x^3*b^3*a+128/35*x^4*b^4*c)/a^5/x^ 
5
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.66 \[ \int \frac {c+d x}{x^5 \sqrt {a x+b x^2}} \, dx=-\frac {2 \, {\left (35 \, a^{4} c + 16 \, {\left (8 \, b^{4} c - 9 \, a b^{3} d\right )} x^{4} - 8 \, {\left (8 \, a b^{3} c - 9 \, a^{2} b^{2} d\right )} x^{3} + 6 \, {\left (8 \, a^{2} b^{2} c - 9 \, a^{3} b d\right )} x^{2} - 5 \, {\left (8 \, a^{3} b c - 9 \, a^{4} d\right )} x\right )} \sqrt {b x^{2} + a x}}{315 \, a^{5} x^{5}} \] Input:

integrate((d*x+c)/x^5/(b*x^2+a*x)^(1/2),x, algorithm="fricas")
 

Output:

-2/315*(35*a^4*c + 16*(8*b^4*c - 9*a*b^3*d)*x^4 - 8*(8*a*b^3*c - 9*a^2*b^2 
*d)*x^3 + 6*(8*a^2*b^2*c - 9*a^3*b*d)*x^2 - 5*(8*a^3*b*c - 9*a^4*d)*x)*sqr 
t(b*x^2 + a*x)/(a^5*x^5)
 

Sympy [F]

\[ \int \frac {c+d x}{x^5 \sqrt {a x+b x^2}} \, dx=\int \frac {c + d x}{x^{5} \sqrt {x \left (a + b x\right )}}\, dx \] Input:

integrate((d*x+c)/x**5/(b*x**2+a*x)**(1/2),x)
 

Output:

Integral((c + d*x)/(x**5*sqrt(x*(a + b*x))), x)
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.24 \[ \int \frac {c+d x}{x^5 \sqrt {a x+b x^2}} \, dx=-\frac {256 \, \sqrt {b x^{2} + a x} b^{4} c}{315 \, a^{5} x} + \frac {32 \, \sqrt {b x^{2} + a x} b^{3} d}{35 \, a^{4} x} + \frac {128 \, \sqrt {b x^{2} + a x} b^{3} c}{315 \, a^{4} x^{2}} - \frac {16 \, \sqrt {b x^{2} + a x} b^{2} d}{35 \, a^{3} x^{2}} - \frac {32 \, \sqrt {b x^{2} + a x} b^{2} c}{105 \, a^{3} x^{3}} + \frac {12 \, \sqrt {b x^{2} + a x} b d}{35 \, a^{2} x^{3}} + \frac {16 \, \sqrt {b x^{2} + a x} b c}{63 \, a^{2} x^{4}} - \frac {2 \, \sqrt {b x^{2} + a x} d}{7 \, a x^{4}} - \frac {2 \, \sqrt {b x^{2} + a x} c}{9 \, a x^{5}} \] Input:

integrate((d*x+c)/x^5/(b*x^2+a*x)^(1/2),x, algorithm="maxima")
 

Output:

-256/315*sqrt(b*x^2 + a*x)*b^4*c/(a^5*x) + 32/35*sqrt(b*x^2 + a*x)*b^3*d/( 
a^4*x) + 128/315*sqrt(b*x^2 + a*x)*b^3*c/(a^4*x^2) - 16/35*sqrt(b*x^2 + a* 
x)*b^2*d/(a^3*x^2) - 32/105*sqrt(b*x^2 + a*x)*b^2*c/(a^3*x^3) + 12/35*sqrt 
(b*x^2 + a*x)*b*d/(a^2*x^3) + 16/63*sqrt(b*x^2 + a*x)*b*c/(a^2*x^4) - 2/7* 
sqrt(b*x^2 + a*x)*d/(a*x^4) - 2/9*sqrt(b*x^2 + a*x)*c/(a*x^5)
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.57 \[ \int \frac {c+d x}{x^5 \sqrt {a x+b x^2}} \, dx=\frac {2 \, {\left (630 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{5} b^{\frac {3}{2}} d + 1008 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{4} b^{2} c + 756 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{4} a b d + 1680 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{3} a b^{\frac {3}{2}} c + 315 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{3} a^{2} \sqrt {b} d + 1080 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} a^{2} b c + 45 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} a^{3} d + 315 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} a^{3} \sqrt {b} c + 35 \, a^{4} c\right )}}{315 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{9}} \] Input:

integrate((d*x+c)/x^5/(b*x^2+a*x)^(1/2),x, algorithm="giac")
 

Output:

2/315*(630*(sqrt(b)*x - sqrt(b*x^2 + a*x))^5*b^(3/2)*d + 1008*(sqrt(b)*x - 
 sqrt(b*x^2 + a*x))^4*b^2*c + 756*(sqrt(b)*x - sqrt(b*x^2 + a*x))^4*a*b*d 
+ 1680*(sqrt(b)*x - sqrt(b*x^2 + a*x))^3*a*b^(3/2)*c + 315*(sqrt(b)*x - sq 
rt(b*x^2 + a*x))^3*a^2*sqrt(b)*d + 1080*(sqrt(b)*x - sqrt(b*x^2 + a*x))^2* 
a^2*b*c + 45*(sqrt(b)*x - sqrt(b*x^2 + a*x))^2*a^3*d + 315*(sqrt(b)*x - sq 
rt(b*x^2 + a*x))*a^3*sqrt(b)*c + 35*a^4*c)/(sqrt(b)*x - sqrt(b*x^2 + a*x)) 
^9
 

Mupad [B] (verification not implemented)

Time = 9.18 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.91 \[ \int \frac {c+d x}{x^5 \sqrt {a x+b x^2}} \, dx=\frac {\sqrt {b\,x^2+a\,x}\,\left (128\,b^3\,c-144\,a\,b^2\,d\right )}{315\,a^4\,x^2}-\frac {\sqrt {b\,x^2+a\,x}\,\left (18\,a\,d-16\,b\,c\right )}{63\,a^2\,x^4}-\frac {\sqrt {b\,x^2+a\,x}\,\left (256\,b^4\,c-288\,a\,b^3\,d\right )}{315\,a^5\,x}-\frac {2\,c\,\sqrt {b\,x^2+a\,x}}{9\,a\,x^5}-\frac {\sqrt {b\,x^2+a\,x}\,\left (32\,b^2\,c-36\,a\,b\,d\right )}{105\,a^3\,x^3} \] Input:

int((c + d*x)/(x^5*(a*x + b*x^2)^(1/2)),x)
 

Output:

((a*x + b*x^2)^(1/2)*(128*b^3*c - 144*a*b^2*d))/(315*a^4*x^2) - ((a*x + b* 
x^2)^(1/2)*(18*a*d - 16*b*c))/(63*a^2*x^4) - ((a*x + b*x^2)^(1/2)*(256*b^4 
*c - 288*a*b^3*d))/(315*a^5*x) - (2*c*(a*x + b*x^2)^(1/2))/(9*a*x^5) - ((a 
*x + b*x^2)^(1/2)*(32*b^2*c - 36*a*b*d))/(105*a^3*x^3)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.18 \[ \int \frac {c+d x}{x^5 \sqrt {a x+b x^2}} \, dx=\frac {-\frac {2 \sqrt {x}\, \sqrt {b x +a}\, a^{4} c}{9}-\frac {2 \sqrt {x}\, \sqrt {b x +a}\, a^{4} d x}{7}+\frac {16 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b c x}{63}+\frac {12 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b d \,x^{2}}{35}-\frac {32 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{2} c \,x^{2}}{105}-\frac {16 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{2} d \,x^{3}}{35}+\frac {128 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{3} c \,x^{3}}{315}+\frac {32 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{3} d \,x^{4}}{35}-\frac {256 \sqrt {x}\, \sqrt {b x +a}\, b^{4} c \,x^{4}}{315}-\frac {32 \sqrt {b}\, a \,b^{3} d \,x^{5}}{35}+\frac {256 \sqrt {b}\, b^{4} c \,x^{5}}{315}}{a^{5} x^{5}} \] Input:

int((d*x+c)/x^5/(b*x^2+a*x)^(1/2),x)
 

Output:

(2*( - 35*sqrt(x)*sqrt(a + b*x)*a**4*c - 45*sqrt(x)*sqrt(a + b*x)*a**4*d*x 
 + 40*sqrt(x)*sqrt(a + b*x)*a**3*b*c*x + 54*sqrt(x)*sqrt(a + b*x)*a**3*b*d 
*x**2 - 48*sqrt(x)*sqrt(a + b*x)*a**2*b**2*c*x**2 - 72*sqrt(x)*sqrt(a + b* 
x)*a**2*b**2*d*x**3 + 64*sqrt(x)*sqrt(a + b*x)*a*b**3*c*x**3 + 144*sqrt(x) 
*sqrt(a + b*x)*a*b**3*d*x**4 - 128*sqrt(x)*sqrt(a + b*x)*b**4*c*x**4 - 144 
*sqrt(b)*a*b**3*d*x**5 + 128*sqrt(b)*b**4*c*x**5))/(315*a**5*x**5)