\(\int \frac {(c+d x)^3 \sqrt {a x+b x^2}}{x^8} \, dx\) [29]

Optimal result

Integrand size = 24, antiderivative size = 290 \[ \int \frac {(c+d x)^3 \sqrt {a x+b x^2}}{x^8} \, dx=-\frac {2 c^3 \left (a x+b x^2\right )^{3/2}}{13 a x^8}+\frac {2 c^2 (10 b c-39 a d) \left (a x+b x^2\right )^{3/2}}{143 a^2 x^7}-\frac {2 c \left (80 b^2 c^2-312 a b c d+429 a^2 d^2\right ) \left (a x+b x^2\right )^{3/2}}{1287 a^3 x^6}-\frac {2 \left (429 d^3-\frac {2 b c \left (80 b^2 c^2-312 a b c d+429 a^2 d^2\right )}{a^3}\right ) \left (a x+b x^2\right )^{3/2}}{3003 a x^5}+\frac {8 b \left (429 a^3 d^3-2 b c \left (80 b^2 c^2-312 a b c d+429 a^2 d^2\right )\right ) \left (a x+b x^2\right )^{3/2}}{15015 a^5 x^4}-\frac {16 b^2 \left (429 a^3 d^3-2 b c \left (80 b^2 c^2-312 a b c d+429 a^2 d^2\right )\right ) \left (a x+b x^2\right )^{3/2}}{45045 a^6 x^3} \] Output:

-2/13*c^3*(b*x^2+a*x)^(3/2)/a/x^8+2/143*c^2*(-39*a*d+10*b*c)*(b*x^2+a*x)^( 
3/2)/a^2/x^7-2/1287*c*(429*a^2*d^2-312*a*b*c*d+80*b^2*c^2)*(b*x^2+a*x)^(3/ 
2)/a^3/x^6-2/3003*(429*d^3-2*b*c*(429*a^2*d^2-312*a*b*c*d+80*b^2*c^2)/a^3) 
*(b*x^2+a*x)^(3/2)/a/x^5+8/15015*b*(429*a^3*d^3-2*b*c*(429*a^2*d^2-312*a*b 
*c*d+80*b^2*c^2))*(b*x^2+a*x)^(3/2)/a^5/x^4-16/45045*b^2*(429*a^3*d^3-2*b* 
c*(429*a^2*d^2-312*a*b*c*d+80*b^2*c^2))*(b*x^2+a*x)^(3/2)/a^6/x^3
 

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.68 \[ \int \frac {(c+d x)^3 \sqrt {a x+b x^2}}{x^8} \, dx=-\frac {2 (x (a+b x))^{3/2} \left (-1280 b^5 c^3 x^5+384 a b^4 c^2 x^4 (5 c+13 d x)-48 a^2 b^3 c x^3 \left (50 c^2+156 c d x+143 d^2 x^2\right )+15 a^5 \left (231 c^3+819 c^2 d x+1001 c d^2 x^2+429 d^3 x^3\right )+8 a^3 b^2 x^2 \left (350 c^3+1170 c^2 d x+1287 c d^2 x^2+429 d^3 x^3\right )-6 a^4 b x \left (525 c^3+1820 c^2 d x+2145 c d^2 x^2+858 d^3 x^3\right )\right )}{45045 a^6 x^8} \] Input:

Integrate[((c + d*x)^3*Sqrt[a*x + b*x^2])/x^8,x]
 

Output:

(-2*(x*(a + b*x))^(3/2)*(-1280*b^5*c^3*x^5 + 384*a*b^4*c^2*x^4*(5*c + 13*d 
*x) - 48*a^2*b^3*c*x^3*(50*c^2 + 156*c*d*x + 143*d^2*x^2) + 15*a^5*(231*c^ 
3 + 819*c^2*d*x + 1001*c*d^2*x^2 + 429*d^3*x^3) + 8*a^3*b^2*x^2*(350*c^3 + 
 1170*c^2*d*x + 1287*c*d^2*x^2 + 429*d^3*x^3) - 6*a^4*b*x*(525*c^3 + 1820* 
c^2*d*x + 2145*c*d^2*x^2 + 858*d^3*x^3)))/(45045*a^6*x^8)
 

Rubi [A] (verified)

Time = 1.07 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.02, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {1262, 27, 2169, 27, 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.

\(\displaystyle \int \frac {\sqrt {a x+b x^2} (c+d x)^3}{x^8} \, dx\)

\(\Big \downarrow \) 1262

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2169

\(\displaystyle \frac {-\frac {\int -\frac {\left (16 b^2 c^3+3 d \left (16 b^2 c^2-22 a b d c+11 a^2 d^2\right ) x\right ) \sqrt {b x^2+a x}}{2 x^8}dx}{4 b}-\frac {3 d^2 \left (a x+b x^2\right )^{3/2} (2 b c-a d)}{4 b x^7}}{2 b}-\frac {d^3 \left (a x+b x^2\right )^{3/2}}{3 b x^6}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {\left (16 b^2 c^3+3 d \left (16 b^2 c^2-22 a b d c+11 a^2 d^2\right ) x\right ) \sqrt {b x^2+a x}}{x^8}dx}{8 b}-\frac {3 d^2 \left (a x+b x^2\right )^{3/2} (2 b c-a d)}{4 b x^7}}{2 b}-\frac {d^3 \left (a x+b x^2\right )^{3/2}}{3 b x^6}\)

\(\Big \downarrow \) 1220

\(\displaystyle \frac {\frac {-\frac {\left (-429 a^3 d^3+858 a^2 b c d^2-624 a b^2 c^2 d+160 b^3 c^3\right ) \int \frac {\sqrt {b x^2+a x}}{x^7}dx}{13 a}-\frac {32 b^2 c^3 \left (a x+b x^2\right )^{3/2}}{13 a x^8}}{8 b}-\frac {3 d^2 \left (a x+b x^2\right )^{3/2} (2 b c-a d)}{4 b x^7}}{2 b}-\frac {d^3 \left (a x+b x^2\right )^{3/2}}{3 b x^6}\)

\(\Big \downarrow \) 1129

\(\displaystyle \frac {\frac {-\frac {\left (-429 a^3 d^3+858 a^2 b c d^2-624 a b^2 c^2 d+160 b^3 c^3\right ) \left (-\frac {8 b \int \frac {\sqrt {b x^2+a x}}{x^6}dx}{11 a}-\frac {2 \left (a x+b x^2\right )^{3/2}}{11 a x^7}\right )}{13 a}-\frac {32 b^2 c^3 \left (a x+b x^2\right )^{3/2}}{13 a x^8}}{8 b}-\frac {3 d^2 \left (a x+b x^2\right )^{3/2} (2 b c-a d)}{4 b x^7}}{2 b}-\frac {d^3 \left (a x+b x^2\right )^{3/2}}{3 b x^6}\)

\(\Big \downarrow \) 1129

\(\displaystyle \frac {\frac {-\frac {\left (-429 a^3 d^3+858 a^2 b c d^2-624 a b^2 c^2 d+160 b^3 c^3\right ) \left (-\frac {8 b \left (-\frac {2 b \int \frac {\sqrt {b x^2+a x}}{x^5}dx}{3 a}-\frac {2 \left (a x+b x^2\right )^{3/2}}{9 a x^6}\right )}{11 a}-\frac {2 \left (a x+b x^2\right )^{3/2}}{11 a x^7}\right )}{13 a}-\frac {32 b^2 c^3 \left (a x+b x^2\right )^{3/2}}{13 a x^8}}{8 b}-\frac {3 d^2 \left (a x+b x^2\right )^{3/2} (2 b c-a d)}{4 b x^7}}{2 b}-\frac {d^3 \left (a x+b x^2\right )^{3/2}}{3 b x^6}\)

\(\Big \downarrow \) 1129

\(\displaystyle \frac {\frac {-\frac {\left (-429 a^3 d^3+858 a^2 b c d^2-624 a b^2 c^2 d+160 b^3 c^3\right ) \left (-\frac {8 b \left (-\frac {2 b \left (-\frac {4 b \int \frac {\sqrt {b x^2+a x}}{x^4}dx}{7 a}-\frac {2 \left (a x+b x^2\right )^{3/2}}{7 a x^5}\right )}{3 a}-\frac {2 \left (a x+b x^2\right )^{3/2}}{9 a x^6}\right )}{11 a}-\frac {2 \left (a x+b x^2\right )^{3/2}}{11 a x^7}\right )}{13 a}-\frac {32 b^2 c^3 \left (a x+b x^2\right )^{3/2}}{13 a x^8}}{8 b}-\frac {3 d^2 \left (a x+b x^2\right )^{3/2} (2 b c-a d)}{4 b x^7}}{2 b}-\frac {d^3 \left (a x+b x^2\right )^{3/2}}{3 b x^6}\)

\(\Big \downarrow \) 1129

\(\displaystyle \frac {\frac {-\frac {\left (-429 a^3 d^3+858 a^2 b c d^2-624 a b^2 c^2 d+160 b^3 c^3\right ) \left (-\frac {8 b \left (-\frac {2 b \left (-\frac {4 b \left (-\frac {2 b \int \frac {\sqrt {b x^2+a x}}{x^3}dx}{5 a}-\frac {2 \left (a x+b x^2\right )^{3/2}}{5 a x^4}\right )}{7 a}-\frac {2 \left (a x+b x^2\right )^{3/2}}{7 a x^5}\right )}{3 a}-\frac {2 \left (a x+b x^2\right )^{3/2}}{9 a x^6}\right )}{11 a}-\frac {2 \left (a x+b x^2\right )^{3/2}}{11 a x^7}\right )}{13 a}-\frac {32 b^2 c^3 \left (a x+b x^2\right )^{3/2}}{13 a x^8}}{8 b}-\frac {3 d^2 \left (a x+b x^2\right )^{3/2} (2 b c-a d)}{4 b x^7}}{2 b}-\frac {d^3 \left (a x+b x^2\right )^{3/2}}{3 b x^6}\)

\(\Big \downarrow \) 1123

\(\displaystyle \frac {\frac {-\frac {\left (-\frac {8 b \left (-\frac {2 b \left (-\frac {4 b \left (\frac {4 b \left (a x+b x^2\right )^{3/2}}{15 a^2 x^3}-\frac {2 \left (a x+b x^2\right )^{3/2}}{5 a x^4}\right )}{7 a}-\frac {2 \left (a x+b x^2\right )^{3/2}}{7 a x^5}\right )}{3 a}-\frac {2 \left (a x+b x^2\right )^{3/2}}{9 a x^6}\right )}{11 a}-\frac {2 \left (a x+b x^2\right )^{3/2}}{11 a x^7}\right ) \left (-429 a^3 d^3+858 a^2 b c d^2-624 a b^2 c^2 d+160 b^3 c^3\right )}{13 a}-\frac {32 b^2 c^3 \left (a x+b x^2\right )^{3/2}}{13 a x^8}}{8 b}-\frac {3 d^2 \left (a x+b x^2\right )^{3/2} (2 b c-a d)}{4 b x^7}}{2 b}-\frac {d^3 \left (a x+b x^2\right )^{3/2}}{3 b x^6}\)

Input:

Int[((c + d*x)^3*Sqrt[a*x + b*x^2])/x^8,x]
 

Output:

-1/3*(d^3*(a*x + b*x^2)^(3/2))/(b*x^6) + ((-3*d^2*(2*b*c - a*d)*(a*x + b*x 
^2)^(3/2))/(4*b*x^7) + ((-32*b^2*c^3*(a*x + b*x^2)^(3/2))/(13*a*x^8) - ((1 
60*b^3*c^3 - 624*a*b^2*c^2*d + 858*a^2*b*c*d^2 - 429*a^3*d^3)*((-2*(a*x + 
b*x^2)^(3/2))/(11*a*x^7) - (8*b*((-2*(a*x + b*x^2)^(3/2))/(9*a*x^6) - (2*b 
*((-2*(a*x + b*x^2)^(3/2))/(7*a*x^5) - (4*b*((-2*(a*x + b*x^2)^(3/2))/(5*a 
*x^4) + (4*b*(a*x + b*x^2)^(3/2))/(15*a^2*x^3)))/(7*a)))/(3*a)))/(11*a)))/ 
(13*a))/(8*b))/(2*b)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 
]
 

rule 1262

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

rule 2169

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, S 
imp[f*(d + e*x)^(m + q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q 
+ 2*p + 1))), x] + Simp[1/(c*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + 
b*x + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 
1)*(d + e*x)^q + e*f*(m + p + q)*(d + e*x)^(q - 2)*(b*d - 2*a*e + (2*c*d - 
b*e)*x), x], x], x] /; NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, 
 p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2 
, 0]
 

Maple [A] (verified)

Time = 0.68 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.65

method	result	size
pseudoelliptic	\(-\frac {2 \left (\left (\frac {13}{7} d^{3} x^{3}+\frac {13}{3} c \,d^{2} x^{2}+\frac {39}{11} c^{2} d x +c^{3}\right ) a^{5}-\frac {10 x b \left (\frac {286}{175} d^{3} x^{3}+\frac {143}{35} c \,d^{2} x^{2}+\frac {52}{15} c^{2} d x +c^{3}\right ) a^{4}}{11}+\frac {80 x^{2} b^{2} \left (\frac {429}{350} d^{3} x^{3}+\frac {1287}{350} c \,d^{2} x^{2}+\frac {117}{35} c^{2} d x +c^{3}\right ) a^{3}}{99}-\frac {160 x^{3} b^{3} \left (\frac {143}{50} d^{2} x^{2}+\frac {78}{25} c d x +c^{2}\right ) c \,a^{2}}{231}+\frac {128 \left (\frac {13 d x}{5}+c \right ) x^{4} b^{4} c^{2} a}{231}-\frac {256 b^{5} c^{3} x^{5}}{693}\right ) \sqrt {x \left (b x +a \right )}\, \left (b x +a \right )}{13 x^{7} a^{6}}\)	\(188\)
gosper	\(-\frac {2 \left (b x +a \right ) \left (3432 a^{3} b^{2} d^{3} x^{5}-6864 a^{2} b^{3} c \,d^{2} x^{5}+4992 a \,b^{4} c^{2} d \,x^{5}-1280 b^{5} c^{3} x^{5}-5148 a^{4} b \,d^{3} x^{4}+10296 a^{3} b^{2} c \,d^{2} x^{4}-7488 a^{2} b^{3} c^{2} d \,x^{4}+1920 a \,b^{4} c^{3} x^{4}+6435 a^{5} d^{3} x^{3}-12870 a^{4} b c \,d^{2} x^{3}+9360 a^{3} b^{2} c^{2} d \,x^{3}-2400 a^{2} b^{3} c^{3} x^{3}+15015 a^{5} c \,d^{2} x^{2}-10920 a^{4} b \,c^{2} d \,x^{2}+2800 a^{3} b^{2} c^{3} x^{2}+12285 a^{5} c^{2} d x -3150 a^{4} b \,c^{3} x +3465 a^{5} c^{3}\right ) \sqrt {b \,x^{2}+a x}}{45045 x^{7} a^{6}}\)	\(253\)
orering	\(-\frac {2 \left (b x +a \right ) \left (3432 a^{3} b^{2} d^{3} x^{5}-6864 a^{2} b^{3} c \,d^{2} x^{5}+4992 a \,b^{4} c^{2} d \,x^{5}-1280 b^{5} c^{3} x^{5}-5148 a^{4} b \,d^{3} x^{4}+10296 a^{3} b^{2} c \,d^{2} x^{4}-7488 a^{2} b^{3} c^{2} d \,x^{4}+1920 a \,b^{4} c^{3} x^{4}+6435 a^{5} d^{3} x^{3}-12870 a^{4} b c \,d^{2} x^{3}+9360 a^{3} b^{2} c^{2} d \,x^{3}-2400 a^{2} b^{3} c^{3} x^{3}+15015 a^{5} c \,d^{2} x^{2}-10920 a^{4} b \,c^{2} d \,x^{2}+2800 a^{3} b^{2} c^{3} x^{2}+12285 a^{5} c^{2} d x -3150 a^{4} b \,c^{3} x +3465 a^{5} c^{3}\right ) \sqrt {b \,x^{2}+a x}}{45045 x^{7} a^{6}}\)	\(253\)
trager	\(-\frac {2 \left (3432 a^{3} b^{3} d^{3} x^{6}-6864 a^{2} b^{4} c \,d^{2} x^{6}+4992 a \,b^{5} c^{2} d \,x^{6}-1280 b^{6} c^{3} x^{6}-1716 a^{4} b^{2} d^{3} x^{5}+3432 a^{3} b^{3} c \,d^{2} x^{5}-2496 a^{2} b^{4} c^{2} d \,x^{5}+640 a \,b^{5} c^{3} x^{5}+1287 a^{5} b \,d^{3} x^{4}-2574 a^{4} b^{2} c \,d^{2} x^{4}+1872 a^{3} b^{3} c^{2} d \,x^{4}-480 a^{2} b^{4} c^{3} x^{4}+6435 a^{6} d^{3} x^{3}+2145 a^{5} b c \,d^{2} x^{3}-1560 a^{4} b^{2} c^{2} d \,x^{3}+400 a^{3} b^{3} c^{3} x^{3}+15015 a^{6} c \,d^{2} x^{2}+1365 a^{5} b \,c^{2} d \,x^{2}-350 a^{4} b^{2} c^{3} x^{2}+12285 a^{6} c^{2} d x +315 a^{5} b \,c^{3} x +3465 a^{6} c^{3}\right ) \sqrt {b \,x^{2}+a x}}{45045 x^{7} a^{6}}\)	\(306\)
risch	\(-\frac {2 \left (b x +a \right ) \left (3432 a^{3} b^{3} d^{3} x^{6}-6864 a^{2} b^{4} c \,d^{2} x^{6}+4992 a \,b^{5} c^{2} d \,x^{6}-1280 b^{6} c^{3} x^{6}-1716 a^{4} b^{2} d^{3} x^{5}+3432 a^{3} b^{3} c \,d^{2} x^{5}-2496 a^{2} b^{4} c^{2} d \,x^{5}+640 a \,b^{5} c^{3} x^{5}+1287 a^{5} b \,d^{3} x^{4}-2574 a^{4} b^{2} c \,d^{2} x^{4}+1872 a^{3} b^{3} c^{2} d \,x^{4}-480 a^{2} b^{4} c^{3} x^{4}+6435 a^{6} d^{3} x^{3}+2145 a^{5} b c \,d^{2} x^{3}-1560 a^{4} b^{2} c^{2} d \,x^{3}+400 a^{3} b^{3} c^{3} x^{3}+15015 a^{6} c \,d^{2} x^{2}+1365 a^{5} b \,c^{2} d \,x^{2}-350 a^{4} b^{2} c^{3} x^{2}+12285 a^{6} c^{2} d x +315 a^{5} b \,c^{3} x +3465 a^{6} c^{3}\right )}{45045 x^{6} \sqrt {x \left (b x +a \right )}\, a^{6}}\)	\(309\)
default	\(c^{3} \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{13 a \,x^{8}}-\frac {10 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{11 a \,x^{7}}-\frac {8 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{9 a \,x^{6}}-\frac {2 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{7 a \,x^{5}}-\frac {4 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{5 a \,x^{4}}+\frac {4 b \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{15 a^{2} x^{3}}\right )}{7 a}\right )}{3 a}\right )}{11 a}\right )}{13 a}\right )+d^{3} \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{7 a \,x^{5}}-\frac {4 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{5 a \,x^{4}}+\frac {4 b \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{15 a^{2} x^{3}}\right )}{7 a}\right )+3 c \,d^{2} \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{9 a \,x^{6}}-\frac {2 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{7 a \,x^{5}}-\frac {4 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{5 a \,x^{4}}+\frac {4 b \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{15 a^{2} x^{3}}\right )}{7 a}\right )}{3 a}\right )+3 c^{2} d \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{11 a \,x^{7}}-\frac {8 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{9 a \,x^{6}}-\frac {2 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{7 a \,x^{5}}-\frac {4 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{5 a \,x^{4}}+\frac {4 b \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{15 a^{2} x^{3}}\right )}{7 a}\right )}{3 a}\right )}{11 a}\right )\)	\(442\)

Input:

int((d*x+c)^3*(b*x^2+a*x)^(1/2)/x^8,x,method=_RETURNVERBOSE)
 

Output:

-2/13*((13/7*d^3*x^3+13/3*c*d^2*x^2+39/11*c^2*d*x+c^3)*a^5-10/11*x*b*(286/ 
175*d^3*x^3+143/35*c*d^2*x^2+52/15*c^2*d*x+c^3)*a^4+80/99*x^2*b^2*(429/350 
*d^3*x^3+1287/350*c*d^2*x^2+117/35*c^2*d*x+c^3)*a^3-160/231*x^3*b^3*(143/5 
0*d^2*x^2+78/25*c*d*x+c^2)*c*a^2+128/231*(13/5*d*x+c)*x^4*b^4*c^2*a-256/69 
3*b^5*c^3*x^5)*(x*(b*x+a))^(1/2)*(b*x+a)/x^7/a^6
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.96 \[ \int \frac {(c+d x)^3 \sqrt {a x+b x^2}}{x^8} \, dx=-\frac {2 \, {\left (3465 \, a^{6} c^{3} - 8 \, {\left (160 \, b^{6} c^{3} - 624 \, a b^{5} c^{2} d + 858 \, a^{2} b^{4} c d^{2} - 429 \, a^{3} b^{3} d^{3}\right )} x^{6} + 4 \, {\left (160 \, a b^{5} c^{3} - 624 \, a^{2} b^{4} c^{2} d + 858 \, a^{3} b^{3} c d^{2} - 429 \, a^{4} b^{2} d^{3}\right )} x^{5} - 3 \, {\left (160 \, a^{2} b^{4} c^{3} - 624 \, a^{3} b^{3} c^{2} d + 858 \, a^{4} b^{2} c d^{2} - 429 \, a^{5} b d^{3}\right )} x^{4} + 5 \, {\left (80 \, a^{3} b^{3} c^{3} - 312 \, a^{4} b^{2} c^{2} d + 429 \, a^{5} b c d^{2} + 1287 \, a^{6} d^{3}\right )} x^{3} - 35 \, {\left (10 \, a^{4} b^{2} c^{3} - 39 \, a^{5} b c^{2} d - 429 \, a^{6} c d^{2}\right )} x^{2} + 315 \, {\left (a^{5} b c^{3} + 39 \, a^{6} c^{2} d\right )} x\right )} \sqrt {b x^{2} + a x}}{45045 \, a^{6} x^{7}} \] Input:

integrate((d*x+c)^3*(b*x^2+a*x)^(1/2)/x^8,x, algorithm="fricas")
 

Output:

-2/45045*(3465*a^6*c^3 - 8*(160*b^6*c^3 - 624*a*b^5*c^2*d + 858*a^2*b^4*c* 
d^2 - 429*a^3*b^3*d^3)*x^6 + 4*(160*a*b^5*c^3 - 624*a^2*b^4*c^2*d + 858*a^ 
3*b^3*c*d^2 - 429*a^4*b^2*d^3)*x^5 - 3*(160*a^2*b^4*c^3 - 624*a^3*b^3*c^2* 
d + 858*a^4*b^2*c*d^2 - 429*a^5*b*d^3)*x^4 + 5*(80*a^3*b^3*c^3 - 312*a^4*b 
^2*c^2*d + 429*a^5*b*c*d^2 + 1287*a^6*d^3)*x^3 - 35*(10*a^4*b^2*c^3 - 39*a 
^5*b*c^2*d - 429*a^6*c*d^2)*x^2 + 315*(a^5*b*c^3 + 39*a^6*c^2*d)*x)*sqrt(b 
*x^2 + a*x)/(a^6*x^7)
 

Sympy [F]

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

integrate((d*x+c)**3*(b*x**2+a*x)**(1/2)/x**8,x)
 

Output:

Integral(sqrt(x*(a + b*x))*(c + d*x)**3/x**8, x)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 530, normalized size of antiderivative = 1.83 \[ \int \frac {(c+d x)^3 \sqrt {a x+b x^2}}{x^8} \, dx=\frac {512 \, \sqrt {b x^{2} + a x} b^{6} c^{3}}{9009 \, a^{6} x} - \frac {256 \, \sqrt {b x^{2} + a x} b^{5} c^{2} d}{1155 \, a^{5} x} + \frac {32 \, \sqrt {b x^{2} + a x} b^{4} c d^{2}}{105 \, a^{4} x} - \frac {16 \, \sqrt {b x^{2} + a x} b^{3} d^{3}}{105 \, a^{3} x} - \frac {256 \, \sqrt {b x^{2} + a x} b^{5} c^{3}}{9009 \, a^{5} x^{2}} + \frac {128 \, \sqrt {b x^{2} + a x} b^{4} c^{2} d}{1155 \, a^{4} x^{2}} - \frac {16 \, \sqrt {b x^{2} + a x} b^{3} c d^{2}}{105 \, a^{3} x^{2}} + \frac {8 \, \sqrt {b x^{2} + a x} b^{2} d^{3}}{105 \, a^{2} x^{2}} + \frac {64 \, \sqrt {b x^{2} + a x} b^{4} c^{3}}{3003 \, a^{4} x^{3}} - \frac {32 \, \sqrt {b x^{2} + a x} b^{3} c^{2} d}{385 \, a^{3} x^{3}} + \frac {4 \, \sqrt {b x^{2} + a x} b^{2} c d^{2}}{35 \, a^{2} x^{3}} - \frac {2 \, \sqrt {b x^{2} + a x} b d^{3}}{35 \, a x^{3}} - \frac {160 \, \sqrt {b x^{2} + a x} b^{3} c^{3}}{9009 \, a^{3} x^{4}} + \frac {16 \, \sqrt {b x^{2} + a x} b^{2} c^{2} d}{231 \, a^{2} x^{4}} - \frac {2 \, \sqrt {b x^{2} + a x} b c d^{2}}{21 \, a x^{4}} - \frac {2 \, \sqrt {b x^{2} + a x} d^{3}}{7 \, x^{4}} + \frac {20 \, \sqrt {b x^{2} + a x} b^{2} c^{3}}{1287 \, a^{2} x^{5}} - \frac {2 \, \sqrt {b x^{2} + a x} b c^{2} d}{33 \, a x^{5}} - \frac {2 \, \sqrt {b x^{2} + a x} c d^{2}}{3 \, x^{5}} - \frac {2 \, \sqrt {b x^{2} + a x} b c^{3}}{143 \, a x^{6}} - \frac {6 \, \sqrt {b x^{2} + a x} c^{2} d}{11 \, x^{6}} - \frac {2 \, \sqrt {b x^{2} + a x} c^{3}}{13 \, x^{7}} \] Input:

integrate((d*x+c)^3*(b*x^2+a*x)^(1/2)/x^8,x, algorithm="maxima")
 

Output:

512/9009*sqrt(b*x^2 + a*x)*b^6*c^3/(a^6*x) - 256/1155*sqrt(b*x^2 + a*x)*b^ 
5*c^2*d/(a^5*x) + 32/105*sqrt(b*x^2 + a*x)*b^4*c*d^2/(a^4*x) - 16/105*sqrt 
(b*x^2 + a*x)*b^3*d^3/(a^3*x) - 256/9009*sqrt(b*x^2 + a*x)*b^5*c^3/(a^5*x^ 
2) + 128/1155*sqrt(b*x^2 + a*x)*b^4*c^2*d/(a^4*x^2) - 16/105*sqrt(b*x^2 + 
a*x)*b^3*c*d^2/(a^3*x^2) + 8/105*sqrt(b*x^2 + a*x)*b^2*d^3/(a^2*x^2) + 64/ 
3003*sqrt(b*x^2 + a*x)*b^4*c^3/(a^4*x^3) - 32/385*sqrt(b*x^2 + a*x)*b^3*c^ 
2*d/(a^3*x^3) + 4/35*sqrt(b*x^2 + a*x)*b^2*c*d^2/(a^2*x^3) - 2/35*sqrt(b*x 
^2 + a*x)*b*d^3/(a*x^3) - 160/9009*sqrt(b*x^2 + a*x)*b^3*c^3/(a^3*x^4) + 1 
6/231*sqrt(b*x^2 + a*x)*b^2*c^2*d/(a^2*x^4) - 2/21*sqrt(b*x^2 + a*x)*b*c*d 
^2/(a*x^4) - 2/7*sqrt(b*x^2 + a*x)*d^3/x^4 + 20/1287*sqrt(b*x^2 + a*x)*b^2 
*c^3/(a^2*x^5) - 2/33*sqrt(b*x^2 + a*x)*b*c^2*d/(a*x^5) - 2/3*sqrt(b*x^2 + 
 a*x)*c*d^2/x^5 - 2/143*sqrt(b*x^2 + a*x)*b*c^3/(a*x^6) - 6/11*sqrt(b*x^2 
+ a*x)*c^2*d/x^6 - 2/13*sqrt(b*x^2 + a*x)*c^3/x^7
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 806 vs. \(2 (266) = 532\).

Time = 0.15 (sec) , antiderivative size = 806, normalized size of antiderivative = 2.78 \[ \int \frac {(c+d x)^3 \sqrt {a x+b x^2}}{x^8} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^3*(b*x^2+a*x)^(1/2)/x^8,x, algorithm="giac")
 

Output:

2/45045*(60060*(sqrt(b)*x - sqrt(b*x^2 + a*x))^10*b^2*d^3 + 270270*(sqrt(b 
)*x - sqrt(b*x^2 + a*x))^9*b^(5/2)*c*d^2 + 135135*(sqrt(b)*x - sqrt(b*x^2 
+ a*x))^9*a*b^(3/2)*d^3 + 432432*(sqrt(b)*x - sqrt(b*x^2 + a*x))^8*b^3*c^2 
*d + 756756*(sqrt(b)*x - sqrt(b*x^2 + a*x))^8*a*b^2*c*d^2 + 117117*(sqrt(b 
)*x - sqrt(b*x^2 + a*x))^8*a^2*b*d^3 + 240240*(sqrt(b)*x - sqrt(b*x^2 + a* 
x))^7*b^(7/2)*c^3 + 1441440*(sqrt(b)*x - sqrt(b*x^2 + a*x))^7*a*b^(5/2)*c^ 
2*d + 855855*(sqrt(b)*x - sqrt(b*x^2 + a*x))^7*a^2*b^(3/2)*c*d^2 + 45045*( 
sqrt(b)*x - sqrt(b*x^2 + a*x))^7*a^3*sqrt(b)*d^3 + 926640*(sqrt(b)*x - sqr 
t(b*x^2 + a*x))^6*a*b^3*c^3 + 2007720*(sqrt(b)*x - sqrt(b*x^2 + a*x))^6*a^ 
2*b^2*c^2*d + 482625*(sqrt(b)*x - sqrt(b*x^2 + a*x))^6*a^3*b*c*d^2 + 6435* 
(sqrt(b)*x - sqrt(b*x^2 + a*x))^6*a^4*d^3 + 1531530*(sqrt(b)*x - sqrt(b*x^ 
2 + a*x))^5*a^2*b^(5/2)*c^3 + 1486485*(sqrt(b)*x - sqrt(b*x^2 + a*x))^5*a^ 
3*b^(3/2)*c^2*d + 135135*(sqrt(b)*x - sqrt(b*x^2 + a*x))^5*a^4*sqrt(b)*c*d 
^2 + 1401400*(sqrt(b)*x - sqrt(b*x^2 + a*x))^4*a^3*b^2*c^3 + 615615*(sqrt( 
b)*x - sqrt(b*x^2 + a*x))^4*a^4*b*c^2*d + 15015*(sqrt(b)*x - sqrt(b*x^2 + 
a*x))^4*a^5*c*d^2 + 765765*(sqrt(b)*x - sqrt(b*x^2 + a*x))^3*a^4*b^(3/2)*c 
^3 + 135135*(sqrt(b)*x - sqrt(b*x^2 + a*x))^3*a^5*sqrt(b)*c^2*d + 249795*( 
sqrt(b)*x - sqrt(b*x^2 + a*x))^2*a^5*b*c^3 + 12285*(sqrt(b)*x - sqrt(b*x^2 
 + a*x))^2*a^6*c^2*d + 45045*(sqrt(b)*x - sqrt(b*x^2 + a*x))*a^6*sqrt(b)*c 
^3 + 3465*a^7*c^3)/(sqrt(b)*x - sqrt(b*x^2 + a*x))^13
 

Mupad [B] (verification not implemented)

Time = 13.46 (sec) , antiderivative size = 530, normalized size of antiderivative = 1.83 \[ \int \frac {(c+d x)^3 \sqrt {a x+b x^2}}{x^8} \, dx=\frac {20\,b^2\,c^3\,\sqrt {b\,x^2+a\,x}}{1287\,a^2\,x^5}-\frac {2\,d^3\,\sqrt {b\,x^2+a\,x}}{7\,x^4}-\frac {2\,c\,d^2\,\sqrt {b\,x^2+a\,x}}{3\,x^5}-\frac {6\,c^2\,d\,\sqrt {b\,x^2+a\,x}}{11\,x^6}-\frac {2\,c^3\,\sqrt {b\,x^2+a\,x}}{13\,x^7}-\frac {160\,b^3\,c^3\,\sqrt {b\,x^2+a\,x}}{9009\,a^3\,x^4}+\frac {64\,b^4\,c^3\,\sqrt {b\,x^2+a\,x}}{3003\,a^4\,x^3}-\frac {256\,b^5\,c^3\,\sqrt {b\,x^2+a\,x}}{9009\,a^5\,x^2}+\frac {512\,b^6\,c^3\,\sqrt {b\,x^2+a\,x}}{9009\,a^6\,x}+\frac {8\,b^2\,d^3\,\sqrt {b\,x^2+a\,x}}{105\,a^2\,x^2}-\frac {16\,b^3\,d^3\,\sqrt {b\,x^2+a\,x}}{105\,a^3\,x}-\frac {2\,b\,c^3\,\sqrt {b\,x^2+a\,x}}{143\,a\,x^6}-\frac {2\,b\,d^3\,\sqrt {b\,x^2+a\,x}}{35\,a\,x^3}-\frac {2\,b\,c\,d^2\,\sqrt {b\,x^2+a\,x}}{21\,a\,x^4}-\frac {2\,b\,c^2\,d\,\sqrt {b\,x^2+a\,x}}{33\,a\,x^5}+\frac {4\,b^2\,c\,d^2\,\sqrt {b\,x^2+a\,x}}{35\,a^2\,x^3}+\frac {16\,b^2\,c^2\,d\,\sqrt {b\,x^2+a\,x}}{231\,a^2\,x^4}-\frac {16\,b^3\,c\,d^2\,\sqrt {b\,x^2+a\,x}}{105\,a^3\,x^2}-\frac {32\,b^3\,c^2\,d\,\sqrt {b\,x^2+a\,x}}{385\,a^3\,x^3}+\frac {32\,b^4\,c\,d^2\,\sqrt {b\,x^2+a\,x}}{105\,a^4\,x}+\frac {128\,b^4\,c^2\,d\,\sqrt {b\,x^2+a\,x}}{1155\,a^4\,x^2}-\frac {256\,b^5\,c^2\,d\,\sqrt {b\,x^2+a\,x}}{1155\,a^5\,x} \] Input:

int(((a*x + b*x^2)^(1/2)*(c + d*x)^3)/x^8,x)
 

Output:

(20*b^2*c^3*(a*x + b*x^2)^(1/2))/(1287*a^2*x^5) - (2*d^3*(a*x + b*x^2)^(1/ 
2))/(7*x^4) - (2*c*d^2*(a*x + b*x^2)^(1/2))/(3*x^5) - (6*c^2*d*(a*x + b*x^ 
2)^(1/2))/(11*x^6) - (2*c^3*(a*x + b*x^2)^(1/2))/(13*x^7) - (160*b^3*c^3*( 
a*x + b*x^2)^(1/2))/(9009*a^3*x^4) + (64*b^4*c^3*(a*x + b*x^2)^(1/2))/(300 
3*a^4*x^3) - (256*b^5*c^3*(a*x + b*x^2)^(1/2))/(9009*a^5*x^2) + (512*b^6*c 
^3*(a*x + b*x^2)^(1/2))/(9009*a^6*x) + (8*b^2*d^3*(a*x + b*x^2)^(1/2))/(10 
5*a^2*x^2) - (16*b^3*d^3*(a*x + b*x^2)^(1/2))/(105*a^3*x) - (2*b*c^3*(a*x 
+ b*x^2)^(1/2))/(143*a*x^6) - (2*b*d^3*(a*x + b*x^2)^(1/2))/(35*a*x^3) - ( 
2*b*c*d^2*(a*x + b*x^2)^(1/2))/(21*a*x^4) - (2*b*c^2*d*(a*x + b*x^2)^(1/2) 
)/(33*a*x^5) + (4*b^2*c*d^2*(a*x + b*x^2)^(1/2))/(35*a^2*x^3) + (16*b^2*c^ 
2*d*(a*x + b*x^2)^(1/2))/(231*a^2*x^4) - (16*b^3*c*d^2*(a*x + b*x^2)^(1/2) 
)/(105*a^3*x^2) - (32*b^3*c^2*d*(a*x + b*x^2)^(1/2))/(385*a^3*x^3) + (32*b 
^4*c*d^2*(a*x + b*x^2)^(1/2))/(105*a^4*x) + (128*b^4*c^2*d*(a*x + b*x^2)^( 
1/2))/(1155*a^4*x^2) - (256*b^5*c^2*d*(a*x + b*x^2)^(1/2))/(1155*a^5*x)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 531, normalized size of antiderivative = 1.83 \[ \int \frac {(c+d x)^3 \sqrt {a x+b x^2}}{x^8} \, dx=\frac {-\frac {2 \sqrt {x}\, \sqrt {b x +a}\, a^{6} d^{3} x^{3}}{7}+\frac {512 \sqrt {x}\, \sqrt {b x +a}\, b^{6} c^{3} x^{6}}{9009}+\frac {16 \sqrt {b}\, a^{3} b^{3} d^{3} x^{7}}{105}-\frac {2 \sqrt {x}\, \sqrt {b x +a}\, a^{6} c^{3}}{13}-\frac {512 \sqrt {b}\, b^{6} c^{3} x^{7}}{9009}-\frac {2 \sqrt {x}\, \sqrt {b x +a}\, a^{5} b \,c^{2} d \,x^{2}}{33}-\frac {2 \sqrt {x}\, \sqrt {b x +a}\, a^{5} b c \,d^{2} x^{3}}{21}+\frac {16 \sqrt {x}\, \sqrt {b x +a}\, a^{4} b^{2} c^{2} d \,x^{3}}{231}+\frac {4 \sqrt {x}\, \sqrt {b x +a}\, a^{4} b^{2} c \,d^{2} x^{4}}{35}-\frac {32 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{3} c^{2} d \,x^{4}}{385}-\frac {16 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{3} c \,d^{2} x^{5}}{105}+\frac {128 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{4} c^{2} d \,x^{5}}{1155}+\frac {32 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{4} c \,d^{2} x^{6}}{105}-\frac {256 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{5} c^{2} d \,x^{6}}{1155}-\frac {6 \sqrt {x}\, \sqrt {b x +a}\, a^{6} c^{2} d x}{11}-\frac {2 \sqrt {x}\, \sqrt {b x +a}\, a^{6} c \,d^{2} x^{2}}{3}-\frac {2 \sqrt {x}\, \sqrt {b x +a}\, a^{5} b \,c^{3} x}{143}-\frac {2 \sqrt {x}\, \sqrt {b x +a}\, a^{5} b \,d^{3} x^{4}}{35}+\frac {20 \sqrt {x}\, \sqrt {b x +a}\, a^{4} b^{2} c^{3} x^{2}}{1287}+\frac {8 \sqrt {x}\, \sqrt {b x +a}\, a^{4} b^{2} d^{3} x^{5}}{105}-\frac {160 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{3} c^{3} x^{3}}{9009}-\frac {16 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{3} d^{3} x^{6}}{105}+\frac {64 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{4} c^{3} x^{4}}{3003}-\frac {256 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{5} c^{3} x^{5}}{9009}-\frac {32 \sqrt {b}\, a^{2} b^{4} c \,d^{2} x^{7}}{105}+\frac {256 \sqrt {b}\, a \,b^{5} c^{2} d \,x^{7}}{1155}}{a^{6} x^{7}} \] Input:

int((d*x+c)^3*(b*x^2+a*x)^(1/2)/x^8,x)
 

Output:

(2*( - 3465*sqrt(x)*sqrt(a + b*x)*a**6*c**3 - 12285*sqrt(x)*sqrt(a + b*x)* 
a**6*c**2*d*x - 15015*sqrt(x)*sqrt(a + b*x)*a**6*c*d**2*x**2 - 6435*sqrt(x 
)*sqrt(a + b*x)*a**6*d**3*x**3 - 315*sqrt(x)*sqrt(a + b*x)*a**5*b*c**3*x - 
 1365*sqrt(x)*sqrt(a + b*x)*a**5*b*c**2*d*x**2 - 2145*sqrt(x)*sqrt(a + b*x 
)*a**5*b*c*d**2*x**3 - 1287*sqrt(x)*sqrt(a + b*x)*a**5*b*d**3*x**4 + 350*s 
qrt(x)*sqrt(a + b*x)*a**4*b**2*c**3*x**2 + 1560*sqrt(x)*sqrt(a + b*x)*a**4 
*b**2*c**2*d*x**3 + 2574*sqrt(x)*sqrt(a + b*x)*a**4*b**2*c*d**2*x**4 + 171 
6*sqrt(x)*sqrt(a + b*x)*a**4*b**2*d**3*x**5 - 400*sqrt(x)*sqrt(a + b*x)*a* 
*3*b**3*c**3*x**3 - 1872*sqrt(x)*sqrt(a + b*x)*a**3*b**3*c**2*d*x**4 - 343 
2*sqrt(x)*sqrt(a + b*x)*a**3*b**3*c*d**2*x**5 - 3432*sqrt(x)*sqrt(a + b*x) 
*a**3*b**3*d**3*x**6 + 480*sqrt(x)*sqrt(a + b*x)*a**2*b**4*c**3*x**4 + 249 
6*sqrt(x)*sqrt(a + b*x)*a**2*b**4*c**2*d*x**5 + 6864*sqrt(x)*sqrt(a + b*x) 
*a**2*b**4*c*d**2*x**6 - 640*sqrt(x)*sqrt(a + b*x)*a*b**5*c**3*x**5 - 4992 
*sqrt(x)*sqrt(a + b*x)*a*b**5*c**2*d*x**6 + 1280*sqrt(x)*sqrt(a + b*x)*b** 
6*c**3*x**6 + 3432*sqrt(b)*a**3*b**3*d**3*x**7 - 6864*sqrt(b)*a**2*b**4*c* 
d**2*x**7 + 4992*sqrt(b)*a*b**5*c**2*d*x**7 - 1280*sqrt(b)*b**6*c**3*x**7) 
)/(45045*a**6*x**7)