\(\int \frac {(c+d x)^{3/2}}{(a+b x)^{7/2} (e+f x)^2} \, dx\) [1188]

Optimal result

Integrand size = 26, antiderivative size = 339 \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^{7/2} (e+f x)^2} \, dx=-\frac {(27 b d e-35 b c f+8 a d f) \sqrt {c+d x}}{15 (b e-a f)^3 (a+b x)^{3/2}}-\frac {\left (16 a^2 d^2 f^2+a b d f (83 d e-115 c f)+b^2 \left (6 d^2 e^2-95 c d e f+105 c^2 f^2\right )\right ) \sqrt {c+d x}}{15 (b c-a d) (b e-a f)^4 \sqrt {a+b x}}-\frac {2 (b c-a d) \sqrt {c+d x}}{5 b (b e-a f) (a+b x)^{5/2} (e+f x)}+\frac {(5 b d e-7 b c f+2 a d f) \sqrt {c+d x}}{5 b (b e-a f)^2 (a+b x)^{3/2} (e+f x)}-\frac {f \sqrt {d e-c f} (4 b d e-7 b c f+3 a d f) \text {arctanh}\left (\frac {\sqrt {d e-c f} \sqrt {a+b x}}{\sqrt {b e-a f} \sqrt {c+d x}}\right )}{(b e-a f)^{9/2}} \] Output:

-1/15*(8*a*d*f-35*b*c*f+27*b*d*e)*(d*x+c)^(1/2)/(-a*f+b*e)^3/(b*x+a)^(3/2) 
-1/15*(16*a^2*d^2*f^2+a*b*d*f*(-115*c*f+83*d*e)+b^2*(105*c^2*f^2-95*c*d*e* 
f+6*d^2*e^2))*(d*x+c)^(1/2)/(-a*d+b*c)/(-a*f+b*e)^4/(b*x+a)^(1/2)-2/5*(-a* 
d+b*c)*(d*x+c)^(1/2)/b/(-a*f+b*e)/(b*x+a)^(5/2)/(f*x+e)+1/5*(2*a*d*f-7*b*c 
*f+5*b*d*e)*(d*x+c)^(1/2)/b/(-a*f+b*e)^2/(b*x+a)^(3/2)/(f*x+e)-f*(-c*f+d*e 
)^(1/2)*(3*a*d*f-7*b*c*f+4*b*d*e)*arctanh((-c*f+d*e)^(1/2)*(b*x+a)^(1/2)/( 
-a*f+b*e)^(1/2)/(d*x+c)^(1/2))/(-a*f+b*e)^(9/2)
 

Mathematica [A] (verified)

Time = 1.16 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.27 \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^{7/2} (e+f x)^2} \, dx=-\frac {\sqrt {c+d x} \left (15 a^4 d f^2 (3 d e-c f+2 d f x)+5 a^3 b f \left (3 c^2 f^2-c d f (31 e+37 f x)+d^2 \left (12 e^2+29 e f x+8 f^2 x^2\right )\right )+a b^3 f \left (d^2 e x^2 (68 e+83 f x)-c d x \left (144 e^2+289 e f x+115 f^2 x^2\right )+c^2 \left (-32 e^2+168 e f x+245 f^2 x^2\right )\right )+a^2 b^2 f \left (c^2 f (116 e+161 f x)+d^2 x \left (140 e^2+201 e f x+16 f^2 x^2\right )-c d \left (40 e^2+313 e f x+273 f^2 x^2\right )\right )+b^4 \left (6 d^2 e^2 x^2 (e+f x)+c d e x \left (12 e^2-68 e f x-95 f^2 x^2\right )+c^2 \left (6 e^3-14 e^2 f x+70 e f^2 x^2+105 f^3 x^3\right )\right )\right )}{15 (b c-a d) (b e-a f)^4 (a+b x)^{5/2} (e+f x)}+\frac {f \sqrt {d e-c f} (4 b d e-7 b c f+3 a d f) \arctan \left (\frac {\sqrt {d e-c f} \sqrt {a+b x}}{\sqrt {-b e+a f} \sqrt {c+d x}}\right )}{(-b e+a f)^{9/2}} \] Input:

Integrate[(c + d*x)^(3/2)/((a + b*x)^(7/2)*(e + f*x)^2),x]
 

Output:

-1/15*(Sqrt[c + d*x]*(15*a^4*d*f^2*(3*d*e - c*f + 2*d*f*x) + 5*a^3*b*f*(3* 
c^2*f^2 - c*d*f*(31*e + 37*f*x) + d^2*(12*e^2 + 29*e*f*x + 8*f^2*x^2)) + a 
*b^3*f*(d^2*e*x^2*(68*e + 83*f*x) - c*d*x*(144*e^2 + 289*e*f*x + 115*f^2*x 
^2) + c^2*(-32*e^2 + 168*e*f*x + 245*f^2*x^2)) + a^2*b^2*f*(c^2*f*(116*e + 
 161*f*x) + d^2*x*(140*e^2 + 201*e*f*x + 16*f^2*x^2) - c*d*(40*e^2 + 313*e 
*f*x + 273*f^2*x^2)) + b^4*(6*d^2*e^2*x^2*(e + f*x) + c*d*e*x*(12*e^2 - 68 
*e*f*x - 95*f^2*x^2) + c^2*(6*e^3 - 14*e^2*f*x + 70*e*f^2*x^2 + 105*f^3*x^ 
3))))/((b*c - a*d)*(b*e - a*f)^4*(a + b*x)^(5/2)*(e + f*x)) + (f*Sqrt[d*e 
- c*f]*(4*b*d*e - 7*b*c*f + 3*a*d*f)*ArcTan[(Sqrt[d*e - c*f]*Sqrt[a + b*x] 
)/(Sqrt[-(b*e) + a*f]*Sqrt[c + d*x])])/(-(b*e) + a*f)^(9/2)
 

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 380, normalized size of antiderivative = 1.12, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {109, 27, 168, 27, 169, 27, 169, 27, 104, 221}

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.

\(\displaystyle \int \frac {(c+d x)^{3/2}}{(a+b x)^{7/2} (e+f x)^2} \, dx\)

\(\Big \downarrow \) 109

\(\displaystyle -\frac {2 \int -\frac {b c (6 d e-7 c f)-a d (d e-2 c f)+d (5 b d e-6 b c f+a d f) x}{2 (a+b x)^{5/2} \sqrt {c+d x} (e+f x)^2}dx}{5 b (b e-a f)}-\frac {2 \sqrt {c+d x} (b c-a d)}{5 b (a+b x)^{5/2} (e+f x) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {b c (6 d e-7 c f)-a d (d e-2 c f)+d (5 b d e-6 b c f+a d f) x}{(a+b x)^{5/2} \sqrt {c+d x} (e+f x)^2}dx}{5 b (b e-a f)}-\frac {2 \sqrt {c+d x} (b c-a d)}{5 b (a+b x)^{5/2} (e+f x) (b e-a f)}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {\frac {\sqrt {c+d x} (2 a d f-7 b c f+5 b d e)}{(a+b x)^{3/2} (e+f x) (b e-a f)}-\frac {\int -\frac {b (d e-c f) (b c (27 d e-35 c f)-a d (7 d e-15 c f)+4 d (5 b d e-7 b c f+2 a d f) x)}{2 (a+b x)^{5/2} \sqrt {c+d x} (e+f x)}dx}{(b e-a f) (d e-c f)}}{5 b (b e-a f)}-\frac {2 \sqrt {c+d x} (b c-a d)}{5 b (a+b x)^{5/2} (e+f x) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {b \int \frac {b c (27 d e-35 c f)-a d (7 d e-15 c f)+4 d (5 b d e-7 b c f+2 a d f) x}{(a+b x)^{5/2} \sqrt {c+d x} (e+f x)}dx}{2 (b e-a f)}+\frac {\sqrt {c+d x} (2 a d f-7 b c f+5 b d e)}{(a+b x)^{3/2} (e+f x) (b e-a f)}}{5 b (b e-a f)}-\frac {2 \sqrt {c+d x} (b c-a d)}{5 b (a+b x)^{5/2} (e+f x) (b e-a f)}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {\frac {b \left (-\frac {2 \int -\frac {(b c-a d) \left (a d f (29 d e-45 c f)+b \left (6 d^2 e^2-95 c d f e+105 c^2 f^2\right )-2 d f (27 b d e-35 b c f+8 a d f) x\right )}{2 (a+b x)^{3/2} \sqrt {c+d x} (e+f x)}dx}{3 (b c-a d) (b e-a f)}-\frac {2 \sqrt {c+d x} (8 a d f-35 b c f+27 b d e)}{3 (a+b x)^{3/2} (b e-a f)}\right )}{2 (b e-a f)}+\frac {\sqrt {c+d x} (2 a d f-7 b c f+5 b d e)}{(a+b x)^{3/2} (e+f x) (b e-a f)}}{5 b (b e-a f)}-\frac {2 \sqrt {c+d x} (b c-a d)}{5 b (a+b x)^{5/2} (e+f x) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {b \left (\frac {\int \frac {a d f (29 d e-45 c f)+b \left (6 d^2 e^2-95 c d f e+105 c^2 f^2\right )-2 d f (27 b d e-35 b c f+8 a d f) x}{(a+b x)^{3/2} \sqrt {c+d x} (e+f x)}dx}{3 (b e-a f)}-\frac {2 \sqrt {c+d x} (8 a d f-35 b c f+27 b d e)}{3 (a+b x)^{3/2} (b e-a f)}\right )}{2 (b e-a f)}+\frac {\sqrt {c+d x} (2 a d f-7 b c f+5 b d e)}{(a+b x)^{3/2} (e+f x) (b e-a f)}}{5 b (b e-a f)}-\frac {2 \sqrt {c+d x} (b c-a d)}{5 b (a+b x)^{5/2} (e+f x) (b e-a f)}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {\frac {b \left (\frac {-\frac {2 \int \frac {15 (b c-a d) f (d e-c f) (4 b d e-7 b c f+3 a d f)}{2 \sqrt {a+b x} \sqrt {c+d x} (e+f x)}dx}{(b c-a d) (b e-a f)}-\frac {2 \sqrt {c+d x} \left (16 a^2 d^2 f^2+a b d f (83 d e-115 c f)+b^2 \left (105 c^2 f^2-95 c d e f+6 d^2 e^2\right )\right )}{\sqrt {a+b x} (b c-a d) (b e-a f)}}{3 (b e-a f)}-\frac {2 \sqrt {c+d x} (8 a d f-35 b c f+27 b d e)}{3 (a+b x)^{3/2} (b e-a f)}\right )}{2 (b e-a f)}+\frac {\sqrt {c+d x} (2 a d f-7 b c f+5 b d e)}{(a+b x)^{3/2} (e+f x) (b e-a f)}}{5 b (b e-a f)}-\frac {2 \sqrt {c+d x} (b c-a d)}{5 b (a+b x)^{5/2} (e+f x) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {b \left (\frac {-\frac {15 f (d e-c f) (3 a d f-7 b c f+4 b d e) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} (e+f x)}dx}{b e-a f}-\frac {2 \sqrt {c+d x} \left (16 a^2 d^2 f^2+a b d f (83 d e-115 c f)+b^2 \left (105 c^2 f^2-95 c d e f+6 d^2 e^2\right )\right )}{\sqrt {a+b x} (b c-a d) (b e-a f)}}{3 (b e-a f)}-\frac {2 \sqrt {c+d x} (8 a d f-35 b c f+27 b d e)}{3 (a+b x)^{3/2} (b e-a f)}\right )}{2 (b e-a f)}+\frac {\sqrt {c+d x} (2 a d f-7 b c f+5 b d e)}{(a+b x)^{3/2} (e+f x) (b e-a f)}}{5 b (b e-a f)}-\frac {2 \sqrt {c+d x} (b c-a d)}{5 b (a+b x)^{5/2} (e+f x) (b e-a f)}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {\frac {b \left (\frac {-\frac {30 f (d e-c f) (3 a d f-7 b c f+4 b d e) \int \frac {1}{b e-a f-\frac {(d e-c f) (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{b e-a f}-\frac {2 \sqrt {c+d x} \left (16 a^2 d^2 f^2+a b d f (83 d e-115 c f)+b^2 \left (105 c^2 f^2-95 c d e f+6 d^2 e^2\right )\right )}{\sqrt {a+b x} (b c-a d) (b e-a f)}}{3 (b e-a f)}-\frac {2 \sqrt {c+d x} (8 a d f-35 b c f+27 b d e)}{3 (a+b x)^{3/2} (b e-a f)}\right )}{2 (b e-a f)}+\frac {\sqrt {c+d x} (2 a d f-7 b c f+5 b d e)}{(a+b x)^{3/2} (e+f x) (b e-a f)}}{5 b (b e-a f)}-\frac {2 \sqrt {c+d x} (b c-a d)}{5 b (a+b x)^{5/2} (e+f x) (b e-a f)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {b \left (\frac {-\frac {2 \sqrt {c+d x} \left (16 a^2 d^2 f^2+a b d f (83 d e-115 c f)+b^2 \left (105 c^2 f^2-95 c d e f+6 d^2 e^2\right )\right )}{\sqrt {a+b x} (b c-a d) (b e-a f)}-\frac {30 f \sqrt {d e-c f} (3 a d f-7 b c f+4 b d e) \text {arctanh}\left (\frac {\sqrt {a+b x} \sqrt {d e-c f}}{\sqrt {c+d x} \sqrt {b e-a f}}\right )}{(b e-a f)^{3/2}}}{3 (b e-a f)}-\frac {2 \sqrt {c+d x} (8 a d f-35 b c f+27 b d e)}{3 (a+b x)^{3/2} (b e-a f)}\right )}{2 (b e-a f)}+\frac {\sqrt {c+d x} (2 a d f-7 b c f+5 b d e)}{(a+b x)^{3/2} (e+f x) (b e-a f)}}{5 b (b e-a f)}-\frac {2 \sqrt {c+d x} (b c-a d)}{5 b (a+b x)^{5/2} (e+f x) (b e-a f)}\)

Input:

Int[(c + d*x)^(3/2)/((a + b*x)^(7/2)*(e + f*x)^2),x]
 

Output:

(-2*(b*c - a*d)*Sqrt[c + d*x])/(5*b*(b*e - a*f)*(a + b*x)^(5/2)*(e + f*x)) 
 + (((5*b*d*e - 7*b*c*f + 2*a*d*f)*Sqrt[c + d*x])/((b*e - a*f)*(a + b*x)^( 
3/2)*(e + f*x)) + (b*((-2*(27*b*d*e - 35*b*c*f + 8*a*d*f)*Sqrt[c + d*x])/( 
3*(b*e - a*f)*(a + b*x)^(3/2)) + ((-2*(16*a^2*d^2*f^2 + a*b*d*f*(83*d*e - 
115*c*f) + b^2*(6*d^2*e^2 - 95*c*d*e*f + 105*c^2*f^2))*Sqrt[c + d*x])/((b* 
c - a*d)*(b*e - a*f)*Sqrt[a + b*x]) - (30*f*Sqrt[d*e - c*f]*(4*b*d*e - 7*b 
*c*f + 3*a*d*f)*ArcTanh[(Sqrt[d*e - c*f]*Sqrt[a + b*x])/(Sqrt[b*e - a*f]*S 
qrt[c + d*x])])/(b*e - a*f)^(3/2))/(3*(b*e - a*f))))/(2*(b*e - a*f)))/(5*b 
*(b*e - a*f))
 

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 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 109

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f 
*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) 
 Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) 
+ c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) 
 + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, 
d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || 
IntegersQ[m, n + p] || IntegersQ[p, m + n])
 

rule 168

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
 

rule 169

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 
2*m, 2*n, 2*p]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(6685\) vs. \(2(303)=606\).

Time = 0.45 (sec) , antiderivative size = 6686, normalized size of antiderivative = 19.72

method	result	size
default	\(\text {Expression too large to display}\)	\(6686\)

Input:

int((d*x+c)^(3/2)/(b*x+a)^(7/2)/(f*x+e)^2,x,method=_RETURNVERBOSE)
 

Output:

result too large to display
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1652 vs. \(2 (303) = 606\).

Time = 33.55 (sec) , antiderivative size = 3486, normalized size of antiderivative = 10.28 \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^{7/2} (e+f x)^2} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^(3/2)/(b*x+a)^(7/2)/(f*x+e)^2,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {(c+d x)^{3/2}}{(a+b x)^{7/2} (e+f x)^2} \, dx=\int \frac {\left (c + d x\right )^{\frac {3}{2}}}{\left (a + b x\right )^{\frac {7}{2}} \left (e + f x\right )^{2}}\, dx \] Input:

integrate((d*x+c)**(3/2)/(b*x+a)**(7/2)/(f*x+e)**2,x)
 

Output:

Integral((c + d*x)**(3/2)/((a + b*x)**(7/2)*(e + f*x)**2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^{7/2} (e+f x)^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((d*x+c)^(3/2)/(b*x+a)^(7/2)/(f*x+e)^2,x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(((a*d)/f>0)', see `assume?` for 
more detai
 

Giac [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^{7/2} (e+f x)^2} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)^(3/2)/(b*x+a)^(7/2)/(f*x+e)^2,x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^{7/2} (e+f x)^2} \, dx=\int \frac {{\left (c+d\,x\right )}^{3/2}}{{\left (e+f\,x\right )}^2\,{\left (a+b\,x\right )}^{7/2}} \,d x \] Input:

int((c + d*x)^(3/2)/((e + f*x)^2*(a + b*x)^(7/2)),x)
 

Output:

int((c + d*x)^(3/2)/((e + f*x)^2*(a + b*x)^(7/2)), x)
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int \frac {(c+d x)^{3/2}}{(a+b x)^{7/2} (e+f x)^2} \, dx=\int \frac {\left (d x +c \right )^{\frac {3}{2}}}{\left (b x +a \right )^{\frac {7}{2}} \left (f x +e \right )^{2}}d x \] Input:

int((d*x+c)^(3/2)/(b*x+a)^(7/2)/(f*x+e)^2,x)
 

Output:

int((d*x+c)^(3/2)/(b*x+a)^(7/2)/(f*x+e)^2,x)