\(\int \frac {A+B x}{x^6 \sqrt {b x+c x^2}} \, dx\) [152]

Optimal result

Integrand size = 22, antiderivative size = 195 \[ \int \frac {A+B x}{x^6 \sqrt {b x+c x^2}} \, dx=-\frac {2 A \sqrt {b x+c x^2}}{11 b x^6}-\frac {2 (11 b B-10 A c) \sqrt {b x+c x^2}}{99 b^2 x^5}+\frac {16 c (11 b B-10 A c) \sqrt {b x+c x^2}}{693 b^3 x^4}-\frac {32 c^2 (11 b B-10 A c) \sqrt {b x+c x^2}}{1155 b^4 x^3}+\frac {128 c^3 (11 b B-10 A c) \sqrt {b x+c x^2}}{3465 b^5 x^2}-\frac {256 c^4 (11 b B-10 A c) \sqrt {b x+c x^2}}{3465 b^6 x} \] Output:

-2/11*A*(c*x^2+b*x)^(1/2)/b/x^6-2/99*(-10*A*c+11*B*b)*(c*x^2+b*x)^(1/2)/b^ 
2/x^5+16/693*c*(-10*A*c+11*B*b)*(c*x^2+b*x)^(1/2)/b^3/x^4-32/1155*c^2*(-10 
*A*c+11*B*b)*(c*x^2+b*x)^(1/2)/b^4/x^3+128/3465*c^3*(-10*A*c+11*B*b)*(c*x^ 
2+b*x)^(1/2)/b^5/x^2-256/3465*c^4*(-10*A*c+11*B*b)*(c*x^2+b*x)^(1/2)/b^6/x
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.63 \[ \int \frac {A+B x}{x^6 \sqrt {b x+c x^2}} \, dx=-\frac {2 \sqrt {x (b+c x)} \left (11 b B x \left (35 b^4-40 b^3 c x+48 b^2 c^2 x^2-64 b c^3 x^3+128 c^4 x^4\right )+5 A \left (63 b^5-70 b^4 c x+80 b^3 c^2 x^2-96 b^2 c^3 x^3+128 b c^4 x^4-256 c^5 x^5\right )\right )}{3465 b^6 x^6} \] Input:

Integrate[(A + B*x)/(x^6*Sqrt[b*x + c*x^2]),x]
 

Output:

(-2*Sqrt[x*(b + c*x)]*(11*b*B*x*(35*b^4 - 40*b^3*c*x + 48*b^2*c^2*x^2 - 64 
*b*c^3*x^3 + 128*c^4*x^4) + 5*A*(63*b^5 - 70*b^4*c*x + 80*b^3*c^2*x^2 - 96 
*b^2*c^3*x^3 + 128*b*c^4*x^4 - 256*c^5*x^5)))/(3465*b^6*x^6)
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1220, 1129, 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.

\(\displaystyle \int \frac {A+B x}{x^6 \sqrt {b x+c x^2}} \, dx\)

\(\Big \downarrow \) 1220

\(\displaystyle \frac {(11 b B-10 A c) \int \frac {1}{x^5 \sqrt {c x^2+b x}}dx}{11 b}-\frac {2 A \sqrt {b x+c x^2}}{11 b x^6}\)

\(\Big \downarrow \) 1129

\(\displaystyle \frac {(11 b B-10 A c) \left (-\frac {8 c \int \frac {1}{x^4 \sqrt {c x^2+b x}}dx}{9 b}-\frac {2 \sqrt {b x+c x^2}}{9 b x^5}\right )}{11 b}-\frac {2 A \sqrt {b x+c x^2}}{11 b x^6}\)

\(\Big \downarrow \) 1129

\(\displaystyle \frac {(11 b B-10 A c) \left (-\frac {8 c \left (-\frac {6 c \int \frac {1}{x^3 \sqrt {c x^2+b x}}dx}{7 b}-\frac {2 \sqrt {b x+c x^2}}{7 b x^4}\right )}{9 b}-\frac {2 \sqrt {b x+c x^2}}{9 b x^5}\right )}{11 b}-\frac {2 A \sqrt {b x+c x^2}}{11 b x^6}\)

\(\Big \downarrow \) 1129

\(\displaystyle \frac {(11 b B-10 A c) \left (-\frac {8 c \left (-\frac {6 c \left (-\frac {4 c \int \frac {1}{x^2 \sqrt {c x^2+b x}}dx}{5 b}-\frac {2 \sqrt {b x+c x^2}}{5 b x^3}\right )}{7 b}-\frac {2 \sqrt {b x+c x^2}}{7 b x^4}\right )}{9 b}-\frac {2 \sqrt {b x+c x^2}}{9 b x^5}\right )}{11 b}-\frac {2 A \sqrt {b x+c x^2}}{11 b x^6}\)

\(\Big \downarrow \) 1129

\(\displaystyle \frac {(11 b B-10 A c) \left (-\frac {8 c \left (-\frac {6 c \left (-\frac {4 c \left (-\frac {2 c \int \frac {1}{x \sqrt {c x^2+b x}}dx}{3 b}-\frac {2 \sqrt {b x+c x^2}}{3 b x^2}\right )}{5 b}-\frac {2 \sqrt {b x+c x^2}}{5 b x^3}\right )}{7 b}-\frac {2 \sqrt {b x+c x^2}}{7 b x^4}\right )}{9 b}-\frac {2 \sqrt {b x+c x^2}}{9 b x^5}\right )}{11 b}-\frac {2 A \sqrt {b x+c x^2}}{11 b x^6}\)

\(\Big \downarrow \) 1123

\(\displaystyle \frac {\left (-\frac {8 c \left (-\frac {6 c \left (-\frac {4 c \left (\frac {4 c \sqrt {b x+c x^2}}{3 b^2 x}-\frac {2 \sqrt {b x+c x^2}}{3 b x^2}\right )}{5 b}-\frac {2 \sqrt {b x+c x^2}}{5 b x^3}\right )}{7 b}-\frac {2 \sqrt {b x+c x^2}}{7 b x^4}\right )}{9 b}-\frac {2 \sqrt {b x+c x^2}}{9 b x^5}\right ) (11 b B-10 A c)}{11 b}-\frac {2 A \sqrt {b x+c x^2}}{11 b x^6}\)

Input:

Int[(A + B*x)/(x^6*Sqrt[b*x + c*x^2]),x]
 

Output:

(-2*A*Sqrt[b*x + c*x^2])/(11*b*x^6) + ((11*b*B - 10*A*c)*((-2*Sqrt[b*x + c 
*x^2])/(9*b*x^5) - (8*c*((-2*Sqrt[b*x + c*x^2])/(7*b*x^4) - (6*c*((-2*Sqrt 
[b*x + c*x^2])/(5*b*x^3) - (4*c*((-2*Sqrt[b*x + c*x^2])/(3*b*x^2) + (4*c*S 
qrt[b*x + c*x^2])/(3*b^2*x)))/(5*b)))/(7*b)))/(9*b)))/(11*b)
 

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 = 1.01 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.51

method	result	size
pseudoelliptic	\(-\frac {2 \sqrt {x \left (c x +b \right )}\, \left (\left (\frac {11 B x}{9}+A \right ) b^{5}-\frac {10 \left (\frac {44 B x}{35}+A \right ) c x \,b^{4}}{9}+\frac {80 c^{2} \left (\frac {33 B x}{25}+A \right ) x^{2} b^{3}}{63}-\frac {32 \left (\frac {22 B x}{15}+A \right ) c^{3} x^{3} b^{2}}{21}+\frac {128 c^{4} x^{4} \left (\frac {11 B x}{5}+A \right ) b}{63}-\frac {256 A \,c^{5} x^{5}}{63}\right )}{11 b^{6} x^{6}}\)	\(100\)
trager	\(-\frac {2 \left (-1280 A \,c^{5} x^{5}+1408 B b \,c^{4} x^{5}+640 A b \,c^{4} x^{4}-704 B \,b^{2} c^{3} x^{4}-480 A \,b^{2} c^{3} x^{3}+528 B \,b^{3} c^{2} x^{3}+400 A \,b^{3} c^{2} x^{2}-440 B \,b^{4} c \,x^{2}-350 A \,b^{4} c x +385 B \,b^{5} x +315 A \,b^{5}\right ) \sqrt {c \,x^{2}+b x}}{3465 b^{6} x^{6}}\)	\(129\)
risch	\(-\frac {2 \left (c x +b \right ) \left (-1280 A \,c^{5} x^{5}+1408 B b \,c^{4} x^{5}+640 A b \,c^{4} x^{4}-704 B \,b^{2} c^{3} x^{4}-480 A \,b^{2} c^{3} x^{3}+528 B \,b^{3} c^{2} x^{3}+400 A \,b^{3} c^{2} x^{2}-440 B \,b^{4} c \,x^{2}-350 A \,b^{4} c x +385 B \,b^{5} x +315 A \,b^{5}\right )}{3465 b^{6} x^{5} \sqrt {x \left (c x +b \right )}}\)	\(132\)
gosper	\(-\frac {2 \left (c x +b \right ) \left (-1280 A \,c^{5} x^{5}+1408 B b \,c^{4} x^{5}+640 A b \,c^{4} x^{4}-704 B \,b^{2} c^{3} x^{4}-480 A \,b^{2} c^{3} x^{3}+528 B \,b^{3} c^{2} x^{3}+400 A \,b^{3} c^{2} x^{2}-440 B \,b^{4} c \,x^{2}-350 A \,b^{4} c x +385 B \,b^{5} x +315 A \,b^{5}\right )}{3465 x^{5} b^{6} \sqrt {c \,x^{2}+b x}}\)	\(134\)
orering	\(-\frac {2 \left (c x +b \right ) \left (-1280 A \,c^{5} x^{5}+1408 B b \,c^{4} x^{5}+640 A b \,c^{4} x^{4}-704 B \,b^{2} c^{3} x^{4}-480 A \,b^{2} c^{3} x^{3}+528 B \,b^{3} c^{2} x^{3}+400 A \,b^{3} c^{2} x^{2}-440 B \,b^{4} c \,x^{2}-350 A \,b^{4} c x +385 B \,b^{5} x +315 A \,b^{5}\right )}{3465 x^{5} b^{6} \sqrt {c \,x^{2}+b x}}\)	\(134\)
default	\(A \left (-\frac {2 \sqrt {c \,x^{2}+b x}}{11 b \,x^{6}}-\frac {10 c \left (-\frac {2 \sqrt {c \,x^{2}+b x}}{9 b \,x^{5}}-\frac {8 c \left (-\frac {2 \sqrt {c \,x^{2}+b x}}{7 b \,x^{4}}-\frac {6 c \left (-\frac {2 \sqrt {c \,x^{2}+b x}}{5 b \,x^{3}}-\frac {4 c \left (-\frac {2 \sqrt {c \,x^{2}+b x}}{3 b \,x^{2}}+\frac {4 c \sqrt {c \,x^{2}+b x}}{3 b^{2} x}\right )}{5 b}\right )}{7 b}\right )}{9 b}\right )}{11 b}\right )+B \left (-\frac {2 \sqrt {c \,x^{2}+b x}}{9 b \,x^{5}}-\frac {8 c \left (-\frac {2 \sqrt {c \,x^{2}+b x}}{7 b \,x^{4}}-\frac {6 c \left (-\frac {2 \sqrt {c \,x^{2}+b x}}{5 b \,x^{3}}-\frac {4 c \left (-\frac {2 \sqrt {c \,x^{2}+b x}}{3 b \,x^{2}}+\frac {4 c \sqrt {c \,x^{2}+b x}}{3 b^{2} x}\right )}{5 b}\right )}{7 b}\right )}{9 b}\right )\)	\(268\)

Input:

int((B*x+A)/x^6/(c*x^2+b*x)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-2/11*(x*(c*x+b))^(1/2)*((11/9*B*x+A)*b^5-10/9*(44/35*B*x+A)*c*x*b^4+80/63 
*c^2*(33/25*B*x+A)*x^2*b^3-32/21*(22/15*B*x+A)*c^3*x^3*b^2+128/63*c^4*x^4* 
(11/5*B*x+A)*b-256/63*A*c^5*x^5)/b^6/x^6
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.67 \[ \int \frac {A+B x}{x^6 \sqrt {b x+c x^2}} \, dx=-\frac {2 \, {\left (315 \, A b^{5} + 128 \, {\left (11 \, B b c^{4} - 10 \, A c^{5}\right )} x^{5} - 64 \, {\left (11 \, B b^{2} c^{3} - 10 \, A b c^{4}\right )} x^{4} + 48 \, {\left (11 \, B b^{3} c^{2} - 10 \, A b^{2} c^{3}\right )} x^{3} - 40 \, {\left (11 \, B b^{4} c - 10 \, A b^{3} c^{2}\right )} x^{2} + 35 \, {\left (11 \, B b^{5} - 10 \, A b^{4} c\right )} x\right )} \sqrt {c x^{2} + b x}}{3465 \, b^{6} x^{6}} \] Input:

integrate((B*x+A)/x^6/(c*x^2+b*x)^(1/2),x, algorithm="fricas")
 

Output:

-2/3465*(315*A*b^5 + 128*(11*B*b*c^4 - 10*A*c^5)*x^5 - 64*(11*B*b^2*c^3 - 
10*A*b*c^4)*x^4 + 48*(11*B*b^3*c^2 - 10*A*b^2*c^3)*x^3 - 40*(11*B*b^4*c - 
10*A*b^3*c^2)*x^2 + 35*(11*B*b^5 - 10*A*b^4*c)*x)*sqrt(c*x^2 + b*x)/(b^6*x 
^6)
 

Sympy [F]

\[ \int \frac {A+B x}{x^6 \sqrt {b x+c x^2}} \, dx=\int \frac {A + B x}{x^{6} \sqrt {x \left (b + c x\right )}}\, dx \] Input:

integrate((B*x+A)/x**6/(c*x**2+b*x)**(1/2),x)
 

Output:

Integral((A + B*x)/(x**6*sqrt(x*(b + c*x))), x)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.25 \[ \int \frac {A+B x}{x^6 \sqrt {b x+c x^2}} \, dx=-\frac {256 \, \sqrt {c x^{2} + b x} B c^{4}}{315 \, b^{5} x} + \frac {512 \, \sqrt {c x^{2} + b x} A c^{5}}{693 \, b^{6} x} + \frac {128 \, \sqrt {c x^{2} + b x} B c^{3}}{315 \, b^{4} x^{2}} - \frac {256 \, \sqrt {c x^{2} + b x} A c^{4}}{693 \, b^{5} x^{2}} - \frac {32 \, \sqrt {c x^{2} + b x} B c^{2}}{105 \, b^{3} x^{3}} + \frac {64 \, \sqrt {c x^{2} + b x} A c^{3}}{231 \, b^{4} x^{3}} + \frac {16 \, \sqrt {c x^{2} + b x} B c}{63 \, b^{2} x^{4}} - \frac {160 \, \sqrt {c x^{2} + b x} A c^{2}}{693 \, b^{3} x^{4}} - \frac {2 \, \sqrt {c x^{2} + b x} B}{9 \, b x^{5}} + \frac {20 \, \sqrt {c x^{2} + b x} A c}{99 \, b^{2} x^{5}} - \frac {2 \, \sqrt {c x^{2} + b x} A}{11 \, b x^{6}} \] Input:

integrate((B*x+A)/x^6/(c*x^2+b*x)^(1/2),x, algorithm="maxima")
 

Output:

-256/315*sqrt(c*x^2 + b*x)*B*c^4/(b^5*x) + 512/693*sqrt(c*x^2 + b*x)*A*c^5 
/(b^6*x) + 128/315*sqrt(c*x^2 + b*x)*B*c^3/(b^4*x^2) - 256/693*sqrt(c*x^2 
+ b*x)*A*c^4/(b^5*x^2) - 32/105*sqrt(c*x^2 + b*x)*B*c^2/(b^3*x^3) + 64/231 
*sqrt(c*x^2 + b*x)*A*c^3/(b^4*x^3) + 16/63*sqrt(c*x^2 + b*x)*B*c/(b^2*x^4) 
 - 160/693*sqrt(c*x^2 + b*x)*A*c^2/(b^3*x^4) - 2/9*sqrt(c*x^2 + b*x)*B/(b* 
x^5) + 20/99*sqrt(c*x^2 + b*x)*A*c/(b^2*x^5) - 2/11*sqrt(c*x^2 + b*x)*A/(b 
*x^6)
 

Giac [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.59 \[ \int \frac {A+B x}{x^6 \sqrt {b x+c x^2}} \, dx=\frac {2 \, {\left (11088 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} B c^{2} + 18480 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} B b c^{\frac {3}{2}} + 18480 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} A c^{\frac {5}{2}} + 11880 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} B b^{2} c + 39600 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} A b c^{2} + 3465 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b^{3} \sqrt {c} + 34650 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b^{2} c^{\frac {3}{2}} + 385 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b^{4} + 15400 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b^{3} c + 3465 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{4} \sqrt {c} + 315 \, A b^{5}\right )}}{3465 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{11}} \] Input:

integrate((B*x+A)/x^6/(c*x^2+b*x)^(1/2),x, algorithm="giac")
 

Output:

2/3465*(11088*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*c^2 + 18480*(sqrt(c)*x - 
 sqrt(c*x^2 + b*x))^5*B*b*c^(3/2) + 18480*(sqrt(c)*x - sqrt(c*x^2 + b*x))^ 
5*A*c^(5/2) + 11880*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^2*c + 39600*(sqr 
t(c)*x - sqrt(c*x^2 + b*x))^4*A*b*c^2 + 3465*(sqrt(c)*x - sqrt(c*x^2 + b*x 
))^3*B*b^3*sqrt(c) + 34650*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^2*c^(3/2) 
 + 385*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^4 + 15400*(sqrt(c)*x - sqrt(c 
*x^2 + b*x))^2*A*b^3*c + 3465*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^4*sqrt(c 
) + 315*A*b^5)/(sqrt(c)*x - sqrt(c*x^2 + b*x))^11
 

Mupad [B] (verification not implemented)

Time = 5.26 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.91 \[ \int \frac {A+B x}{x^6 \sqrt {b x+c x^2}} \, dx=\frac {\sqrt {c\,x^2+b\,x}\,\left (320\,A\,c^3-352\,B\,b\,c^2\right )}{1155\,b^4\,x^3}-\frac {\sqrt {c\,x^2+b\,x}\,\left (1280\,A\,c^4-1408\,B\,b\,c^3\right )}{3465\,b^5\,x^2}-\frac {\left (160\,A\,c^2-176\,B\,b\,c\right )\,\sqrt {c\,x^2+b\,x}}{693\,b^3\,x^4}-\frac {2\,A\,\sqrt {c\,x^2+b\,x}}{11\,b\,x^6}+\frac {\sqrt {c\,x^2+b\,x}\,\left (20\,A\,c-22\,B\,b\right )}{99\,b^2\,x^5}+\frac {256\,c^4\,\sqrt {c\,x^2+b\,x}\,\left (10\,A\,c-11\,B\,b\right )}{3465\,b^6\,x} \] Input:

int((A + B*x)/(x^6*(b*x + c*x^2)^(1/2)),x)
 

Output:

((b*x + c*x^2)^(1/2)*(320*A*c^3 - 352*B*b*c^2))/(1155*b^4*x^3) - ((b*x + c 
*x^2)^(1/2)*(1280*A*c^4 - 1408*B*b*c^3))/(3465*b^5*x^2) - ((160*A*c^2 - 17 
6*B*b*c)*(b*x + c*x^2)^(1/2))/(693*b^3*x^4) - (2*A*(b*x + c*x^2)^(1/2))/(1 
1*b*x^6) + ((b*x + c*x^2)^(1/2)*(20*A*c - 22*B*b))/(99*b^2*x^5) + (256*c^4 
*(b*x + c*x^2)^(1/2)*(10*A*c - 11*B*b))/(3465*b^6*x)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.16 \[ \int \frac {A+B x}{x^6 \sqrt {b x+c x^2}} \, dx=\frac {-\frac {2 \sqrt {x}\, \sqrt {c x +b}\, a \,b^{5}}{11}+\frac {20 \sqrt {x}\, \sqrt {c x +b}\, a \,b^{4} c x}{99}-\frac {160 \sqrt {x}\, \sqrt {c x +b}\, a \,b^{3} c^{2} x^{2}}{693}+\frac {64 \sqrt {x}\, \sqrt {c x +b}\, a \,b^{2} c^{3} x^{3}}{231}-\frac {256 \sqrt {x}\, \sqrt {c x +b}\, a b \,c^{4} x^{4}}{693}+\frac {512 \sqrt {x}\, \sqrt {c x +b}\, a \,c^{5} x^{5}}{693}-\frac {2 \sqrt {x}\, \sqrt {c x +b}\, b^{6} x}{9}+\frac {16 \sqrt {x}\, \sqrt {c x +b}\, b^{5} c \,x^{2}}{63}-\frac {32 \sqrt {x}\, \sqrt {c x +b}\, b^{4} c^{2} x^{3}}{105}+\frac {128 \sqrt {x}\, \sqrt {c x +b}\, b^{3} c^{3} x^{4}}{315}-\frac {256 \sqrt {x}\, \sqrt {c x +b}\, b^{2} c^{4} x^{5}}{315}-\frac {512 \sqrt {c}\, a \,c^{5} x^{6}}{693}+\frac {256 \sqrt {c}\, b^{2} c^{4} x^{6}}{315}}{b^{6} x^{6}} \] Input:

int((B*x+A)/x^6/(c*x^2+b*x)^(1/2),x)
 

Output:

(2*( - 315*sqrt(x)*sqrt(b + c*x)*a*b**5 + 350*sqrt(x)*sqrt(b + c*x)*a*b**4 
*c*x - 400*sqrt(x)*sqrt(b + c*x)*a*b**3*c**2*x**2 + 480*sqrt(x)*sqrt(b + c 
*x)*a*b**2*c**3*x**3 - 640*sqrt(x)*sqrt(b + c*x)*a*b*c**4*x**4 + 1280*sqrt 
(x)*sqrt(b + c*x)*a*c**5*x**5 - 385*sqrt(x)*sqrt(b + c*x)*b**6*x + 440*sqr 
t(x)*sqrt(b + c*x)*b**5*c*x**2 - 528*sqrt(x)*sqrt(b + c*x)*b**4*c**2*x**3 
+ 704*sqrt(x)*sqrt(b + c*x)*b**3*c**3*x**4 - 1408*sqrt(x)*sqrt(b + c*x)*b* 
*2*c**4*x**5 - 1280*sqrt(c)*a*c**5*x**6 + 1408*sqrt(c)*b**2*c**4*x**6))/(3 
465*b**6*x**6)