\(\int \frac {(c+d x)^2 \sqrt {a x+b x^2}}{x^5} \, dx\) [17]

Optimal result

Integrand size = 24, antiderivative size = 103 \[ \int \frac {(c+d x)^2 \sqrt {a x+b x^2}}{x^5} \, dx=\frac {8 c (b c-a d) \left (a x+b x^2\right )^{3/2}}{35 a^2 x^4}-\frac {8 (2 b c-5 a d) (b c-a d) \left (a x+b x^2\right )^{3/2}}{105 a^3 x^3}-\frac {2 (c+d x)^2 \left (a x+b x^2\right )^{3/2}}{7 a x^5} \] Output:

8/35*c*(-a*d+b*c)*(b*x^2+a*x)^(3/2)/a^2/x^4-8/105*(-5*a*d+2*b*c)*(-a*d+b*c 
)*(b*x^2+a*x)^(3/2)/a^3/x^3-2/7*(d*x+c)^2*(b*x^2+a*x)^(3/2)/a/x^5
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.68 \[ \int \frac {(c+d x)^2 \sqrt {a x+b x^2}}{x^5} \, dx=-\frac {2 (x (a+b x))^{3/2} \left (8 b^2 c^2 x^2-4 a b c x (3 c+7 d x)+a^2 \left (15 c^2+42 c d x+35 d^2 x^2\right )\right )}{105 a^3 x^5} \] Input:

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

Output:

(-2*(x*(a + b*x))^(3/2)*(8*b^2*c^2*x^2 - 4*a*b*c*x*(3*c + 7*d*x) + a^2*(15 
*c^2 + 42*c*d*x + 35*d^2*x^2)))/(105*a^3*x^5)
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.34, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1262, 27, 1220, 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.

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

\(\Big \downarrow \) 1262

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1220

\(\displaystyle \frac {-\frac {\left (35 a^2 d^2-28 a b c d+8 b^2 c^2\right ) \int \frac {\sqrt {b x^2+a x}}{x^4}dx}{7 a}-\frac {4 b c^2 \left (a x+b x^2\right )^{3/2}}{7 a x^5}}{2 b}-\frac {d^2 \left (a x+b x^2\right )^{3/2}}{b x^4}\)

\(\Big \downarrow \) 1129

\(\displaystyle \frac {-\frac {\left (35 a^2 d^2-28 a b c d+8 b^2 c^2\right ) \left (-\frac {2 b \int \frac {\sqrt {b x^2+a x}}{x^3}dx}{5 a}-\frac {2 \left (a x+b x^2\right )^{3/2}}{5 a x^4}\right )}{7 a}-\frac {4 b c^2 \left (a x+b x^2\right )^{3/2}}{7 a x^5}}{2 b}-\frac {d^2 \left (a x+b x^2\right )^{3/2}}{b x^4}\)

\(\Big \downarrow \) 1123

\(\displaystyle \frac {-\frac {\left (\frac {4 b \left (a x+b x^2\right )^{3/2}}{15 a^2 x^3}-\frac {2 \left (a x+b x^2\right )^{3/2}}{5 a x^4}\right ) \left (35 a^2 d^2-28 a b c d+8 b^2 c^2\right )}{7 a}-\frac {4 b c^2 \left (a x+b x^2\right )^{3/2}}{7 a x^5}}{2 b}-\frac {d^2 \left (a x+b x^2\right )^{3/2}}{b x^4}\)

Input:

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

Output:

-((d^2*(a*x + b*x^2)^(3/2))/(b*x^4)) + ((-4*b*c^2*(a*x + b*x^2)^(3/2))/(7* 
a*x^5) - ((8*b^2*c^2 - 28*a*b*c*d + 35*a^2*d^2)*((-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))/(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]
 

Maple [A] (verified)

Time = 0.51 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.66

method	result	size
pseudoelliptic	\(-\frac {2 \sqrt {x \left (b x +a \right )}\, \left (b x +a \right ) \left (\left (\frac {7}{3} d^{2} x^{2}+\frac {14}{5} c d x +c^{2}\right ) a^{2}-\frac {4 x b \left (\frac {7 d x}{3}+c \right ) c a}{5}+\frac {8 b^{2} c^{2} x^{2}}{15}\right )}{7 x^{4} a^{3}}\)	\(68\)
gosper	\(-\frac {2 \left (b x +a \right ) \left (35 a^{2} d^{2} x^{2}-28 a b c d \,x^{2}+8 b^{2} c^{2} x^{2}+42 a^{2} c d x -12 a b \,c^{2} x +15 a^{2} c^{2}\right ) \sqrt {b \,x^{2}+a x}}{105 x^{4} a^{3}}\)	\(81\)
orering	\(-\frac {2 \left (b x +a \right ) \left (35 a^{2} d^{2} x^{2}-28 a b c d \,x^{2}+8 b^{2} c^{2} x^{2}+42 a^{2} c d x -12 a b \,c^{2} x +15 a^{2} c^{2}\right ) \sqrt {b \,x^{2}+a x}}{105 x^{4} a^{3}}\)	\(81\)
trager	\(-\frac {2 \left (35 d^{2} x^{3} a^{2} b -28 a \,b^{2} c d \,x^{3}+8 b^{3} c^{2} x^{3}+35 a^{3} d^{2} x^{2}+14 x^{2} a^{2} b c d -4 a \,b^{2} c^{2} x^{2}+42 a^{3} c d x +3 a^{2} b \,c^{2} x +15 c^{2} a^{3}\right ) \sqrt {b \,x^{2}+a x}}{105 x^{4} a^{3}}\)	\(115\)
risch	\(-\frac {2 \left (b x +a \right ) \left (35 d^{2} x^{3} a^{2} b -28 a \,b^{2} c d \,x^{3}+8 b^{3} c^{2} x^{3}+35 a^{3} d^{2} x^{2}+14 x^{2} a^{2} b c d -4 a \,b^{2} c^{2} x^{2}+42 a^{3} c d x +3 a^{2} b \,c^{2} x +15 c^{2} a^{3}\right )}{105 x^{3} \sqrt {x \left (b x +a \right )}\, a^{3}}\)	\(118\)
default	\(c^{2} \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{7 a \,x^{5}}-\frac {4 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{5 a \,x^{4}}+\frac {4 b \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{15 a^{2} x^{3}}\right )}{7 a}\right )-\frac {2 d^{2} \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{3 a \,x^{3}}+2 c d \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{5 a \,x^{4}}+\frac {4 b \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{15 a^{2} x^{3}}\right )\)	\(138\)

Input:

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

Output:

-2/7*(x*(b*x+a))^(1/2)*(b*x+a)*((7/3*d^2*x^2+14/5*c*d*x+c^2)*a^2-4/5*x*b*( 
7/3*d*x+c)*c*a+8/15*b^2*c^2*x^2)/x^4/a^3
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.05 \[ \int \frac {(c+d x)^2 \sqrt {a x+b x^2}}{x^5} \, dx=-\frac {2 \, {\left (15 \, a^{3} c^{2} + {\left (8 \, b^{3} c^{2} - 28 \, a b^{2} c d + 35 \, a^{2} b d^{2}\right )} x^{3} - {\left (4 \, a b^{2} c^{2} - 14 \, a^{2} b c d - 35 \, a^{3} d^{2}\right )} x^{2} + 3 \, {\left (a^{2} b c^{2} + 14 \, a^{3} c d\right )} x\right )} \sqrt {b x^{2} + a x}}{105 \, a^{3} x^{4}} \] Input:

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

Output:

-2/105*(15*a^3*c^2 + (8*b^3*c^2 - 28*a*b^2*c*d + 35*a^2*b*d^2)*x^3 - (4*a* 
b^2*c^2 - 14*a^2*b*c*d - 35*a^3*d^2)*x^2 + 3*(a^2*b*c^2 + 14*a^3*c*d)*x)*s 
qrt(b*x^2 + a*x)/(a^3*x^4)
 

Sympy [F]

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

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

Output:

Integral(sqrt(x*(a + b*x))*(c + d*x)**2/x**5, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 199 vs. \(2 (91) = 182\).

Time = 0.04 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.93 \[ \int \frac {(c+d x)^2 \sqrt {a x+b x^2}}{x^5} \, dx=-\frac {16 \, \sqrt {b x^{2} + a x} b^{3} c^{2}}{105 \, a^{3} x} + \frac {8 \, \sqrt {b x^{2} + a x} b^{2} c d}{15 \, a^{2} x} - \frac {2 \, \sqrt {b x^{2} + a x} b d^{2}}{3 \, a x} + \frac {8 \, \sqrt {b x^{2} + a x} b^{2} c^{2}}{105 \, a^{2} x^{2}} - \frac {4 \, \sqrt {b x^{2} + a x} b c d}{15 \, a x^{2}} - \frac {2 \, \sqrt {b x^{2} + a x} d^{2}}{3 \, x^{2}} - \frac {2 \, \sqrt {b x^{2} + a x} b c^{2}}{35 \, a x^{3}} - \frac {4 \, \sqrt {b x^{2} + a x} c d}{5 \, x^{3}} - \frac {2 \, \sqrt {b x^{2} + a x} c^{2}}{7 \, x^{4}} \] Input:

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

Output:

-16/105*sqrt(b*x^2 + a*x)*b^3*c^2/(a^3*x) + 8/15*sqrt(b*x^2 + a*x)*b^2*c*d 
/(a^2*x) - 2/3*sqrt(b*x^2 + a*x)*b*d^2/(a*x) + 8/105*sqrt(b*x^2 + a*x)*b^2 
*c^2/(a^2*x^2) - 4/15*sqrt(b*x^2 + a*x)*b*c*d/(a*x^2) - 2/3*sqrt(b*x^2 + a 
*x)*d^2/x^2 - 2/35*sqrt(b*x^2 + a*x)*b*c^2/(a*x^3) - 4/5*sqrt(b*x^2 + a*x) 
*c*d/x^3 - 2/7*sqrt(b*x^2 + a*x)*c^2/x^4
                                                                                    
                                                                                    
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 351 vs. \(2 (91) = 182\).

Time = 0.17 (sec) , antiderivative size = 351, normalized size of antiderivative = 3.41 \[ \int \frac {(c+d x)^2 \sqrt {a x+b x^2}}{x^5} \, dx=\frac {2 \, {\left (105 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{6} b d^{2} + 210 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{5} b^{\frac {3}{2}} c d + 105 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{5} a \sqrt {b} d^{2} + 140 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{4} b^{2} c^{2} + 350 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{4} a b c d + 35 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{4} a^{2} d^{2} + 315 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{3} a b^{\frac {3}{2}} c^{2} + 210 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{3} a^{2} \sqrt {b} c d + 273 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} a^{2} b c^{2} + 42 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} a^{3} c d + 105 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} a^{3} \sqrt {b} c^{2} + 15 \, a^{4} c^{2}\right )}}{105 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{7}} \] Input:

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

Output:

2/105*(105*(sqrt(b)*x - sqrt(b*x^2 + a*x))^6*b*d^2 + 210*(sqrt(b)*x - sqrt 
(b*x^2 + a*x))^5*b^(3/2)*c*d + 105*(sqrt(b)*x - sqrt(b*x^2 + a*x))^5*a*sqr 
t(b)*d^2 + 140*(sqrt(b)*x - sqrt(b*x^2 + a*x))^4*b^2*c^2 + 350*(sqrt(b)*x 
- sqrt(b*x^2 + a*x))^4*a*b*c*d + 35*(sqrt(b)*x - sqrt(b*x^2 + a*x))^4*a^2* 
d^2 + 315*(sqrt(b)*x - sqrt(b*x^2 + a*x))^3*a*b^(3/2)*c^2 + 210*(sqrt(b)*x 
 - sqrt(b*x^2 + a*x))^3*a^2*sqrt(b)*c*d + 273*(sqrt(b)*x - sqrt(b*x^2 + a* 
x))^2*a^2*b*c^2 + 42*(sqrt(b)*x - sqrt(b*x^2 + a*x))^2*a^3*c*d + 105*(sqrt 
(b)*x - sqrt(b*x^2 + a*x))*a^3*sqrt(b)*c^2 + 15*a^4*c^2)/(sqrt(b)*x - sqrt 
(b*x^2 + a*x))^7
 

Mupad [B] (verification not implemented)

Time = 9.95 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.93 \[ \int \frac {(c+d x)^2 \sqrt {a x+b x^2}}{x^5} \, dx=\frac {8\,b^2\,c^2\,\sqrt {b\,x^2+a\,x}}{105\,a^2\,x^2}-\frac {2\,d^2\,\sqrt {b\,x^2+a\,x}}{3\,x^2}-\frac {4\,c\,d\,\sqrt {b\,x^2+a\,x}}{5\,x^3}-\frac {2\,c^2\,\sqrt {b\,x^2+a\,x}}{7\,x^4}-\frac {16\,b^3\,c^2\,\sqrt {b\,x^2+a\,x}}{105\,a^3\,x}-\frac {2\,b\,c^2\,\sqrt {b\,x^2+a\,x}}{35\,a\,x^3}-\frac {2\,b\,d^2\,\sqrt {b\,x^2+a\,x}}{3\,a\,x}+\frac {8\,b^2\,c\,d\,\sqrt {b\,x^2+a\,x}}{15\,a^2\,x}-\frac {4\,b\,c\,d\,\sqrt {b\,x^2+a\,x}}{15\,a\,x^2} \] Input:

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

Output:

(8*b^2*c^2*(a*x + b*x^2)^(1/2))/(105*a^2*x^2) - (2*d^2*(a*x + b*x^2)^(1/2) 
)/(3*x^2) - (4*c*d*(a*x + b*x^2)^(1/2))/(5*x^3) - (2*c^2*(a*x + b*x^2)^(1/ 
2))/(7*x^4) - (16*b^3*c^2*(a*x + b*x^2)^(1/2))/(105*a^3*x) - (2*b*c^2*(a*x 
 + b*x^2)^(1/2))/(35*a*x^3) - (2*b*d^2*(a*x + b*x^2)^(1/2))/(3*a*x) + (8*b 
^2*c*d*(a*x + b*x^2)^(1/2))/(15*a^2*x) - (4*b*c*d*(a*x + b*x^2)^(1/2))/(15 
*a*x^2)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 215, normalized size of antiderivative = 2.09 \[ \int \frac {(c+d x)^2 \sqrt {a x+b x^2}}{x^5} \, dx=\frac {-\frac {2 \sqrt {x}\, \sqrt {b x +a}\, a^{3} c^{2}}{7}-\frac {4 \sqrt {x}\, \sqrt {b x +a}\, a^{3} c d x}{5}-\frac {2 \sqrt {x}\, \sqrt {b x +a}\, a^{3} d^{2} x^{2}}{3}-\frac {2 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b \,c^{2} x}{35}-\frac {4 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b c d \,x^{2}}{15}-\frac {2 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b \,d^{2} x^{3}}{3}+\frac {8 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{2} c^{2} x^{2}}{105}+\frac {8 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{2} c d \,x^{3}}{15}-\frac {16 \sqrt {x}\, \sqrt {b x +a}\, b^{3} c^{2} x^{3}}{105}+\frac {2 \sqrt {b}\, a^{2} b \,d^{2} x^{4}}{21}-\frac {8 \sqrt {b}\, a \,b^{2} c d \,x^{4}}{15}+\frac {16 \sqrt {b}\, b^{3} c^{2} x^{4}}{105}}{a^{3} x^{4}} \] Input:

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

Output:

(2*( - 15*sqrt(x)*sqrt(a + b*x)*a**3*c**2 - 42*sqrt(x)*sqrt(a + b*x)*a**3* 
c*d*x - 35*sqrt(x)*sqrt(a + b*x)*a**3*d**2*x**2 - 3*sqrt(x)*sqrt(a + b*x)* 
a**2*b*c**2*x - 14*sqrt(x)*sqrt(a + b*x)*a**2*b*c*d*x**2 - 35*sqrt(x)*sqrt 
(a + b*x)*a**2*b*d**2*x**3 + 4*sqrt(x)*sqrt(a + b*x)*a*b**2*c**2*x**2 + 28 
*sqrt(x)*sqrt(a + b*x)*a*b**2*c*d*x**3 - 8*sqrt(x)*sqrt(a + b*x)*b**3*c**2 
*x**3 + 5*sqrt(b)*a**2*b*d**2*x**4 - 28*sqrt(b)*a*b**2*c*d*x**4 + 8*sqrt(b 
)*b**3*c**2*x**4))/(105*a**3*x**4)