\(\int \frac {(A+B x) (b x+c x^2)^{3/2}}{x^8} \, dx\) [124]

Optimal result

Integrand size = 22, antiderivative size = 125 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{x^8} \, dx=-\frac {2 A \left (b x+c x^2\right )^{5/2}}{11 b x^8}-\frac {2 (11 b B-6 A c) \left (b x+c x^2\right )^{5/2}}{99 b^2 x^7}+\frac {8 c (11 b B-6 A c) \left (b x+c x^2\right )^{5/2}}{693 b^3 x^6}-\frac {16 c^2 (11 b B-6 A c) \left (b x+c x^2\right )^{5/2}}{3465 b^4 x^5} \] Output:

-2/11*A*(c*x^2+b*x)^(5/2)/b/x^8-2/99*(-6*A*c+11*B*b)*(c*x^2+b*x)^(5/2)/b^2 
/x^7+8/693*c*(-6*A*c+11*B*b)*(c*x^2+b*x)^(5/2)/b^3/x^6-16/3465*c^2*(-6*A*c 
+11*B*b)*(c*x^2+b*x)^(5/2)/b^4/x^5
 

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.63 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{x^8} \, dx=-\frac {2 (x (b+c x))^{5/2} \left (11 b B x \left (35 b^2-20 b c x+8 c^2 x^2\right )+3 A \left (105 b^3-70 b^2 c x+40 b c^2 x^2-16 c^3 x^3\right )\right )}{3465 b^4 x^8} \] Input:

Integrate[((A + B*x)*(b*x + c*x^2)^(3/2))/x^8,x]
 

Output:

(-2*(x*(b + c*x))^(5/2)*(11*b*B*x*(35*b^2 - 20*b*c*x + 8*c^2*x^2) + 3*A*(1 
05*b^3 - 70*b^2*c*x + 40*b*c^2*x^2 - 16*c^3*x^3)))/(3465*b^4*x^8)
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1220, 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.

\(\displaystyle \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{x^8} \, dx\)

\(\Big \downarrow \) 1220

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

\(\Big \downarrow \) 1129

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

\(\Big \downarrow \) 1129

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

\(\Big \downarrow \) 1123

\(\displaystyle \frac {\left (-\frac {4 c \left (\frac {4 c \left (b x+c x^2\right )^{5/2}}{35 b^2 x^5}-\frac {2 \left (b x+c x^2\right )^{5/2}}{7 b x^6}\right )}{9 b}-\frac {2 \left (b x+c x^2\right )^{5/2}}{9 b x^7}\right ) (11 b B-6 A c)}{11 b}-\frac {2 A \left (b x+c x^2\right )^{5/2}}{11 b x^8}\)

Input:

Int[((A + B*x)*(b*x + c*x^2)^(3/2))/x^8,x]
 

Output:

(-2*A*(b*x + c*x^2)^(5/2))/(11*b*x^8) + ((11*b*B - 6*A*c)*((-2*(b*x + c*x^ 
2)^(5/2))/(9*b*x^7) - (4*c*((-2*(b*x + c*x^2)^(5/2))/(7*b*x^6) + (4*c*(b*x 
 + c*x^2)^(5/2))/(35*b^2*x^5)))/(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 = 73, normalized size of antiderivative = 0.58

method	result	size
pseudoelliptic	\(-\frac {2 \left (c x +b \right )^{2} \sqrt {x \left (c x +b \right )}\, \left (\left (\frac {11 B x}{9}+A \right ) b^{3}-\frac {2 c x \left (\frac {22 B x}{21}+A \right ) b^{2}}{3}+\frac {8 c^{2} \left (\frac {11 B x}{15}+A \right ) x^{2} b}{21}-\frac {16 A \,c^{3} x^{3}}{105}\right )}{11 x^{6} b^{4}}\)	\(73\)
gosper	\(-\frac {2 \left (c x +b \right ) \left (-48 A \,c^{3} x^{3}+88 x^{3} B b \,c^{2}+120 A b \,c^{2} x^{2}-220 x^{2} B \,b^{2} c -210 A \,b^{2} c x +385 x B \,b^{3}+315 A \,b^{3}\right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{3465 x^{7} b^{4}}\)	\(86\)
orering	\(-\frac {2 \left (c x +b \right ) \left (-48 A \,c^{3} x^{3}+88 x^{3} B b \,c^{2}+120 A b \,c^{2} x^{2}-220 x^{2} B \,b^{2} c -210 A \,b^{2} c x +385 x B \,b^{3}+315 A \,b^{3}\right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{3465 x^{7} b^{4}}\)	\(86\)
trager	\(-\frac {2 \left (-48 A \,c^{5} x^{5}+88 B b \,c^{4} x^{5}+24 A b \,c^{4} x^{4}-44 B \,b^{2} c^{3} x^{4}-18 A \,b^{2} c^{3} x^{3}+33 B \,b^{3} c^{2} x^{3}+15 A \,b^{3} c^{2} x^{2}+550 B \,b^{4} c \,x^{2}+420 A \,b^{4} c x +385 B \,b^{5} x +315 A \,b^{5}\right ) \sqrt {c \,x^{2}+b x}}{3465 b^{4} x^{6}}\)	\(129\)
risch	\(-\frac {2 \left (c x +b \right ) \left (-48 A \,c^{5} x^{5}+88 B b \,c^{4} x^{5}+24 A b \,c^{4} x^{4}-44 B \,b^{2} c^{3} x^{4}-18 A \,b^{2} c^{3} x^{3}+33 B \,b^{3} c^{2} x^{3}+15 A \,b^{3} c^{2} x^{2}+550 B \,b^{4} c \,x^{2}+420 A \,b^{4} c x +385 B \,b^{5} x +315 A \,b^{5}\right )}{3465 x^{5} \sqrt {x \left (c x +b \right )}\, b^{4}}\)	\(132\)
default	\(A \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{11 b \,x^{8}}-\frac {6 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{9 b \,x^{7}}-\frac {4 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{7 b \,x^{6}}+\frac {4 c \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{35 b^{2} x^{5}}\right )}{9 b}\right )}{11 b}\right )+B \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{9 b \,x^{7}}-\frac {4 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{7 b \,x^{6}}+\frac {4 c \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{35 b^{2} x^{5}}\right )}{9 b}\right )\)	\(164\)

Input:

int((B*x+A)*(c*x^2+b*x)^(3/2)/x^8,x,method=_RETURNVERBOSE)
 

Output:

-2/11*(c*x+b)^2*(x*(c*x+b))^(1/2)*((11/9*B*x+A)*b^3-2/3*c*x*(22/21*B*x+A)* 
b^2+8/21*c^2*(11/15*B*x+A)*x^2*b-16/105*A*c^3*x^3)/x^6/b^4
 

Fricas [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.04 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{x^8} \, dx=-\frac {2 \, {\left (315 \, A b^{5} + 8 \, {\left (11 \, B b c^{4} - 6 \, A c^{5}\right )} x^{5} - 4 \, {\left (11 \, B b^{2} c^{3} - 6 \, A b c^{4}\right )} x^{4} + 3 \, {\left (11 \, B b^{3} c^{2} - 6 \, A b^{2} c^{3}\right )} x^{3} + 5 \, {\left (110 \, B b^{4} c + 3 \, A b^{3} c^{2}\right )} x^{2} + 35 \, {\left (11 \, B b^{5} + 12 \, A b^{4} c\right )} x\right )} \sqrt {c x^{2} + b x}}{3465 \, b^{4} x^{6}} \] Input:

integrate((B*x+A)*(c*x^2+b*x)^(3/2)/x^8,x, algorithm="fricas")
 

Output:

-2/3465*(315*A*b^5 + 8*(11*B*b*c^4 - 6*A*c^5)*x^5 - 4*(11*B*b^2*c^3 - 6*A* 
b*c^4)*x^4 + 3*(11*B*b^3*c^2 - 6*A*b^2*c^3)*x^3 + 5*(110*B*b^4*c + 3*A*b^3 
*c^2)*x^2 + 35*(11*B*b^5 + 12*A*b^4*c)*x)*sqrt(c*x^2 + b*x)/(b^4*x^6)
 

Sympy [F]

\[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{x^8} \, dx=\int \frac {\left (x \left (b + c x\right )\right )^{\frac {3}{2}} \left (A + B x\right )}{x^{8}}\, dx \] Input:

integrate((B*x+A)*(c*x**2+b*x)**(3/2)/x**8,x)
 

Output:

Integral((x*(b + c*x))**(3/2)*(A + B*x)/x**8, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 268 vs. \(2 (109) = 218\).

Time = 0.04 (sec) , antiderivative size = 268, normalized size of antiderivative = 2.14 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{x^8} \, dx=-\frac {16 \, \sqrt {c x^{2} + b x} B c^{4}}{315 \, b^{3} x} + \frac {32 \, \sqrt {c x^{2} + b x} A c^{5}}{1155 \, b^{4} x} + \frac {8 \, \sqrt {c x^{2} + b x} B c^{3}}{315 \, b^{2} x^{2}} - \frac {16 \, \sqrt {c x^{2} + b x} A c^{4}}{1155 \, b^{3} x^{2}} - \frac {2 \, \sqrt {c x^{2} + b x} B c^{2}}{105 \, b x^{3}} + \frac {4 \, \sqrt {c x^{2} + b x} A c^{3}}{385 \, b^{2} x^{3}} + \frac {\sqrt {c x^{2} + b x} B c}{63 \, x^{4}} - \frac {2 \, \sqrt {c x^{2} + b x} A c^{2}}{231 \, b x^{4}} + \frac {\sqrt {c x^{2} + b x} B b}{9 \, x^{5}} + \frac {\sqrt {c x^{2} + b x} A c}{132 \, x^{5}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} B}{3 \, x^{6}} + \frac {3 \, \sqrt {c x^{2} + b x} A b}{44 \, x^{6}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} A}{4 \, x^{7}} \] Input:

integrate((B*x+A)*(c*x^2+b*x)^(3/2)/x^8,x, algorithm="maxima")
 

Output:

-16/315*sqrt(c*x^2 + b*x)*B*c^4/(b^3*x) + 32/1155*sqrt(c*x^2 + b*x)*A*c^5/ 
(b^4*x) + 8/315*sqrt(c*x^2 + b*x)*B*c^3/(b^2*x^2) - 16/1155*sqrt(c*x^2 + b 
*x)*A*c^4/(b^3*x^2) - 2/105*sqrt(c*x^2 + b*x)*B*c^2/(b*x^3) + 4/385*sqrt(c 
*x^2 + b*x)*A*c^3/(b^2*x^3) + 1/63*sqrt(c*x^2 + b*x)*B*c/x^4 - 2/231*sqrt( 
c*x^2 + b*x)*A*c^2/(b*x^4) + 1/9*sqrt(c*x^2 + b*x)*B*b/x^5 + 1/132*sqrt(c* 
x^2 + b*x)*A*c/x^5 - 1/3*(c*x^2 + b*x)^(3/2)*B/x^6 + 3/44*sqrt(c*x^2 + b*x 
)*A*b/x^6 - 1/4*(c*x^2 + b*x)^(3/2)*A/x^7
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 431 vs. \(2 (109) = 218\).

Time = 0.15 (sec) , antiderivative size = 431, normalized size of antiderivative = 3.45 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{x^8} \, dx=\frac {2 \, {\left (4620 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{8} B c^{3} + 17325 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{7} B b c^{\frac {5}{2}} + 6930 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{7} A c^{\frac {7}{2}} + 28413 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} B b^{2} c^{2} + 30492 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} A b c^{3} + 25410 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} B b^{3} c^{\frac {3}{2}} + 58905 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} A b^{2} c^{\frac {5}{2}} + 12870 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} B b^{4} c + 63855 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} A b^{3} c^{2} + 3465 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b^{5} \sqrt {c} + 41580 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b^{4} c^{\frac {3}{2}} + 385 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b^{6} + 16170 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b^{5} c + 3465 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{6} \sqrt {c} + 315 \, A b^{7}\right )}}{3465 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{11}} \] Input:

integrate((B*x+A)*(c*x^2+b*x)^(3/2)/x^8,x, algorithm="giac")
 

Output:

2/3465*(4620*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*B*c^3 + 17325*(sqrt(c)*x - 
sqrt(c*x^2 + b*x))^7*B*b*c^(5/2) + 6930*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7* 
A*c^(7/2) + 28413*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b^2*c^2 + 30492*(sqr 
t(c)*x - sqrt(c*x^2 + b*x))^6*A*b*c^3 + 25410*(sqrt(c)*x - sqrt(c*x^2 + b* 
x))^5*B*b^3*c^(3/2) + 58905*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^2*c^(5/2 
) + 12870*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^4*c + 63855*(sqrt(c)*x - s 
qrt(c*x^2 + b*x))^4*A*b^3*c^2 + 3465*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b 
^5*sqrt(c) + 41580*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^4*c^(3/2) + 385*( 
sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^6 + 16170*(sqrt(c)*x - sqrt(c*x^2 + b 
*x))^2*A*b^5*c + 3465*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^6*sqrt(c) + 315* 
A*b^7)/(sqrt(c)*x - sqrt(c*x^2 + b*x))^11
 

Mupad [B] (verification not implemented)

Time = 6.21 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.87 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{x^8} \, dx=\frac {4\,A\,c^3\,\sqrt {c\,x^2+b\,x}}{385\,b^2\,x^3}-\frac {8\,A\,c\,\sqrt {c\,x^2+b\,x}}{33\,x^5}-\frac {2\,B\,b\,\sqrt {c\,x^2+b\,x}}{9\,x^5}-\frac {20\,B\,c\,\sqrt {c\,x^2+b\,x}}{63\,x^4}-\frac {2\,A\,c^2\,\sqrt {c\,x^2+b\,x}}{231\,b\,x^4}-\frac {2\,A\,b\,\sqrt {c\,x^2+b\,x}}{11\,x^6}-\frac {16\,A\,c^4\,\sqrt {c\,x^2+b\,x}}{1155\,b^3\,x^2}+\frac {32\,A\,c^5\,\sqrt {c\,x^2+b\,x}}{1155\,b^4\,x}-\frac {2\,B\,c^2\,\sqrt {c\,x^2+b\,x}}{105\,b\,x^3}+\frac {8\,B\,c^3\,\sqrt {c\,x^2+b\,x}}{315\,b^2\,x^2}-\frac {16\,B\,c^4\,\sqrt {c\,x^2+b\,x}}{315\,b^3\,x} \] Input:

int(((b*x + c*x^2)^(3/2)*(A + B*x))/x^8,x)
 

Output:

(4*A*c^3*(b*x + c*x^2)^(1/2))/(385*b^2*x^3) - (8*A*c*(b*x + c*x^2)^(1/2))/ 
(33*x^5) - (2*B*b*(b*x + c*x^2)^(1/2))/(9*x^5) - (20*B*c*(b*x + c*x^2)^(1/ 
2))/(63*x^4) - (2*A*c^2*(b*x + c*x^2)^(1/2))/(231*b*x^4) - (2*A*b*(b*x + c 
*x^2)^(1/2))/(11*x^6) - (16*A*c^4*(b*x + c*x^2)^(1/2))/(1155*b^3*x^2) + (3 
2*A*c^5*(b*x + c*x^2)^(1/2))/(1155*b^4*x) - (2*B*c^2*(b*x + c*x^2)^(1/2))/ 
(105*b*x^3) + (8*B*c^3*(b*x + c*x^2)^(1/2))/(315*b^2*x^2) - (16*B*c^4*(b*x 
 + c*x^2)^(1/2))/(315*b^3*x)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.81 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{x^8} \, dx=\frac {-\frac {2 \sqrt {x}\, \sqrt {c x +b}\, a \,b^{5}}{11}-\frac {8 \sqrt {x}\, \sqrt {c x +b}\, a \,b^{4} c x}{33}-\frac {2 \sqrt {x}\, \sqrt {c x +b}\, a \,b^{3} c^{2} x^{2}}{231}+\frac {4 \sqrt {x}\, \sqrt {c x +b}\, a \,b^{2} c^{3} x^{3}}{385}-\frac {16 \sqrt {x}\, \sqrt {c x +b}\, a b \,c^{4} x^{4}}{1155}+\frac {32 \sqrt {x}\, \sqrt {c x +b}\, a \,c^{5} x^{5}}{1155}-\frac {2 \sqrt {x}\, \sqrt {c x +b}\, b^{6} x}{9}-\frac {20 \sqrt {x}\, \sqrt {c x +b}\, b^{5} c \,x^{2}}{63}-\frac {2 \sqrt {x}\, \sqrt {c x +b}\, b^{4} c^{2} x^{3}}{105}+\frac {8 \sqrt {x}\, \sqrt {c x +b}\, b^{3} c^{3} x^{4}}{315}-\frac {16 \sqrt {x}\, \sqrt {c x +b}\, b^{2} c^{4} x^{5}}{315}-\frac {32 \sqrt {c}\, a \,c^{5} x^{6}}{1155}+\frac {16 \sqrt {c}\, b^{2} c^{4} x^{6}}{315}}{b^{4} x^{6}} \] Input:

int((B*x+A)*(c*x^2+b*x)^(3/2)/x^8,x)
 

Output:

(2*( - 315*sqrt(x)*sqrt(b + c*x)*a*b**5 - 420*sqrt(x)*sqrt(b + c*x)*a*b**4 
*c*x - 15*sqrt(x)*sqrt(b + c*x)*a*b**3*c**2*x**2 + 18*sqrt(x)*sqrt(b + c*x 
)*a*b**2*c**3*x**3 - 24*sqrt(x)*sqrt(b + c*x)*a*b*c**4*x**4 + 48*sqrt(x)*s 
qrt(b + c*x)*a*c**5*x**5 - 385*sqrt(x)*sqrt(b + c*x)*b**6*x - 550*sqrt(x)* 
sqrt(b + c*x)*b**5*c*x**2 - 33*sqrt(x)*sqrt(b + c*x)*b**4*c**2*x**3 + 44*s 
qrt(x)*sqrt(b + c*x)*b**3*c**3*x**4 - 88*sqrt(x)*sqrt(b + c*x)*b**2*c**4*x 
**5 - 48*sqrt(c)*a*c**5*x**6 + 88*sqrt(c)*b**2*c**4*x**6))/(3465*b**4*x**6 
)