\(\int \frac {(c+d x)^{3/2}}{(a+b x)^{7/2} (e+f x)} \, dx\) [1187]

Optimal result

Integrand size = 26, antiderivative size = 266 \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^{7/2} (e+f x)} \, dx=-\frac {2 (b c-a d) \sqrt {c+d x}}{5 b (b e-a f) (a+b x)^{5/2}}-\frac {2 (6 b d e-5 b c f-a d f) \sqrt {c+d x}}{15 b (b e-a f)^2 (a+b x)^{3/2}}+\frac {2 \left (2 a^2 d^2 f^2-2 a b d f (7 d e-5 c f)-b^2 \left (3 d^2 e^2-20 c d e f+15 c^2 f^2\right )\right ) \sqrt {c+d x}}{15 b (b c-a d) (b e-a f)^3 \sqrt {a+b x}}-\frac {2 f (d e-c f)^{3/2} \text {arctanh}\left (\frac {\sqrt {d e-c f} \sqrt {a+b x}}{\sqrt {b e-a f} \sqrt {c+d x}}\right )}{(b e-a f)^{7/2}} \] Output:

-2/5*(-a*d+b*c)*(d*x+c)^(1/2)/b/(-a*f+b*e)/(b*x+a)^(5/2)-2/15*(-a*d*f-5*b* 
c*f+6*b*d*e)*(d*x+c)^(1/2)/b/(-a*f+b*e)^2/(b*x+a)^(3/2)+2/15*(2*a^2*d^2*f^ 
2-2*a*b*d*f*(-5*c*f+7*d*e)-b^2*(15*c^2*f^2-20*c*d*e*f+3*d^2*e^2))*(d*x+c)^ 
(1/2)/b/(-a*d+b*c)/(-a*f+b*e)^3/(b*x+a)^(1/2)-2*f*(-c*f+d*e)^(3/2)*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) 
^(7/2)
 

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.00 \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^{7/2} (e+f x)} \, dx=-\frac {2 \sqrt {c+d x} \left (-5 a^3 d f (-3 d e+4 c f+d f x)+a^2 b f \left (23 c^2 f+d^2 x (35 e-2 f x)-2 c d (5 e+12 f x)\right )+a b^2 f \left (14 d^2 e x^2-2 c d x (21 e+5 f x)+c^2 (-11 e+35 f x)\right )+b^3 \left (3 d^2 e^2 x^2+2 c d e x (3 e-10 f x)+c^2 \left (3 e^2-5 e f x+15 f^2 x^2\right )\right )\right )}{15 (b c-a d) (b e-a f)^3 (a+b x)^{5/2}}-\frac {2 f (d e-c f)^{3/2} \arctan \left (\frac {\sqrt {d e-c f} \sqrt {a+b x}}{\sqrt {-b e+a f} \sqrt {c+d x}}\right )}{(-b e+a f)^{7/2}} \] Input:

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

Output:

(-2*Sqrt[c + d*x]*(-5*a^3*d*f*(-3*d*e + 4*c*f + d*f*x) + a^2*b*f*(23*c^2*f 
 + d^2*x*(35*e - 2*f*x) - 2*c*d*(5*e + 12*f*x)) + a*b^2*f*(14*d^2*e*x^2 - 
2*c*d*x*(21*e + 5*f*x) + c^2*(-11*e + 35*f*x)) + b^3*(3*d^2*e^2*x^2 + 2*c* 
d*e*x*(3*e - 10*f*x) + c^2*(3*e^2 - 5*e*f*x + 15*f^2*x^2))))/(15*(b*c - a* 
d)*(b*e - a*f)^3*(a + b*x)^(5/2)) - (2*f*(d*e - c*f)^(3/2)*ArcTan[(Sqrt[d* 
e - c*f]*Sqrt[a + b*x])/(Sqrt[-(b*e) + a*f]*Sqrt[c + d*x])])/(-(b*e) + a*f 
)^(7/2)
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.79, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {107, 105, 105, 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.

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

\(\Big \downarrow \) 107

\(\displaystyle -\frac {f \int \frac {(c+d x)^{3/2}}{(a+b x)^{5/2} (e+f x)}dx}{b e-a f}-\frac {2 b (c+d x)^{5/2}}{5 (a+b x)^{5/2} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 105

\(\displaystyle -\frac {f \left (\frac {(d e-c f) \left (\frac {(d e-c f) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} (e+f x)}dx}{b e-a f}-\frac {2 \sqrt {c+d x}}{\sqrt {a+b x} (b e-a f)}\right )}{b e-a f}-\frac {2 (c+d x)^{3/2}}{3 (a+b x)^{3/2} (b e-a f)}\right )}{b e-a f}-\frac {2 b (c+d x)^{5/2}}{5 (a+b x)^{5/2} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 104

\(\displaystyle -\frac {f \left (\frac {(d e-c f) \left (\frac {2 (d e-c f) \int \frac {1}{b e-a f-\frac {(d e-c f) (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{b e-a f}-\frac {2 \sqrt {c+d x}}{\sqrt {a+b x} (b e-a f)}\right )}{b e-a f}-\frac {2 (c+d x)^{3/2}}{3 (a+b x)^{3/2} (b e-a f)}\right )}{b e-a f}-\frac {2 b (c+d x)^{5/2}}{5 (a+b x)^{5/2} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {f \left (\frac {(d e-c f) \left (\frac {2 \sqrt {d e-c f} \text {arctanh}\left (\frac {\sqrt {a+b x} \sqrt {d e-c f}}{\sqrt {c+d x} \sqrt {b e-a f}}\right )}{(b e-a f)^{3/2}}-\frac {2 \sqrt {c+d x}}{\sqrt {a+b x} (b e-a f)}\right )}{b e-a f}-\frac {2 (c+d x)^{3/2}}{3 (a+b x)^{3/2} (b e-a f)}\right )}{b e-a f}-\frac {2 b (c+d x)^{5/2}}{5 (a+b x)^{5/2} (b c-a d) (b e-a f)}\)

Input:

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

Output:

(-2*b*(c + d*x)^(5/2))/(5*(b*c - a*d)*(b*e - a*f)*(a + b*x)^(5/2)) - (f*(( 
-2*(c + d*x)^(3/2))/(3*(b*e - a*f)*(a + b*x)^(3/2)) + ((d*e - c*f)*((-2*Sq 
rt[c + d*x])/((b*e - a*f)*Sqrt[a + b*x]) + (2*Sqrt[d*e - c*f]*ArcTanh[(Sqr 
t[d*e - c*f]*Sqrt[a + b*x])/(Sqrt[b*e - a*f]*Sqrt[c + d*x])])/(b*e - a*f)^ 
(3/2)))/(b*e - a*f)))/(b*e - a*f)
 

Defintions of rubi rules used

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 105

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

rule 107

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(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] + Simp[(a*d*f*(m + 1) + b*c*f*(n + 
 1) + b*d*e*(p + 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, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x 
] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[m, 1])
 

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. \(3206\) vs. \(2(236)=472\).

Time = 0.41 (sec) , antiderivative size = 3207, normalized size of antiderivative = 12.06

method	result	size
default	\(\text {Expression too large to display}\)	\(3207\)

Input:

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

Output:

-1/15*(-40*a^3*c*d*f^2*((b*x+a)*(d*x+c))^(1/2)*((c*f-d*e)*(a*f-b*e)/f^2)^( 
1/2)+30*a^3*d^2*e*f*((b*x+a)*(d*x+c))^(1/2)*((c*f-d*e)*(a*f-b*e)/f^2)^(1/2 
)+46*a^2*b*c^2*f^2*((b*x+a)*(d*x+c))^(1/2)*((c*f-d*e)*(a*f-b*e)/f^2)^(1/2) 
+15*ln((a*d*f*x+b*c*f*x-2*b*d*e*x+2*((b*x+a)*(d*x+c))^(1/2)*((c*f-d*e)*(a* 
f-b*e)/f^2)^(1/2)*f+2*a*c*f-a*d*e-b*c*e)/(f*x+e))*a*b^3*d^3*e^2*x^3-15*ln( 
(a*d*f*x+b*c*f*x-2*b*d*e*x+2*((b*x+a)*(d*x+c))^(1/2)*((c*f-d*e)*(a*f-b*e)/ 
f^2)^(1/2)*f+2*a*c*f-a*d*e-b*c*e)/(f*x+e))*b^4*c*d^2*e^2*x^3+45*ln((a*d*f* 
x+b*c*f*x-2*b*d*e*x+2*((b*x+a)*(d*x+c))^(1/2)*((c*f-d*e)*(a*f-b*e)/f^2)^(1 
/2)*f+2*a*c*f-a*d*e-b*c*e)/(f*x+e))*a^2*b^2*d^3*e^2*x^2-45*ln((a*d*f*x+b*c 
*f*x-2*b*d*e*x+2*((b*x+a)*(d*x+c))^(1/2)*((c*f-d*e)*(a*f-b*e)/f^2)^(1/2)*f 
+2*a*c*f-a*d*e-b*c*e)/(f*x+e))*a*b^3*c^3*f^2*x^2+45*ln((a*d*f*x+b*c*f*x-2* 
b*d*e*x+2*((b*x+a)*(d*x+c))^(1/2)*((c*f-d*e)*(a*f-b*e)/f^2)^(1/2)*f+2*a*c* 
f-a*d*e-b*c*e)/(f*x+e))*a^3*b*d^3*e^2*x-45*ln((a*d*f*x+b*c*f*x-2*b*d*e*x+2 
*((b*x+a)*(d*x+c))^(1/2)*((c*f-d*e)*(a*f-b*e)/f^2)^(1/2)*f+2*a*c*f-a*d*e-b 
*c*e)/(f*x+e))*a^2*b^2*c^3*f^2*x-45*ln((a*d*f*x+b*c*f*x-2*b*d*e*x+2*((b*x+ 
a)*(d*x+c))^(1/2)*((c*f-d*e)*(a*f-b*e)/f^2)^(1/2)*f+2*a*c*f-a*d*e-b*c*e)/( 
f*x+e))*a^2*b^2*c*d^2*e^2*x+30*ln((a*d*f*x+b*c*f*x-2*b*d*e*x+2*((b*x+a)*(d 
*x+c))^(1/2)*((c*f-d*e)*(a*f-b*e)/f^2)^(1/2)*f+2*a*c*f-a*d*e-b*c*e)/(f*x+e 
))*a^3*b*c^2*d*e*f+12*b^3*c*d*e^2*x*((b*x+a)*(d*x+c))^(1/2)*((c*f-d*e)*(a* 
f-b*e)/f^2)^(1/2)+45*ln((a*d*f*x+b*c*f*x-2*b*d*e*x+2*((b*x+a)*(d*x+c))^...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 940 vs. \(2 (236) = 472\).

Time = 5.99 (sec) , antiderivative size = 2062, normalized size of antiderivative = 7.75 \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^{7/2} (e+f x)} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/30*(15*(((b^4*c*d - a*b^3*d^2)*e*f - (b^4*c^2 - a*b^3*c*d)*f^2)*x^3 + ( 
a^3*b*c*d - a^4*d^2)*e*f - (a^3*b*c^2 - a^4*c*d)*f^2 + 3*((a*b^3*c*d - a^2 
*b^2*d^2)*e*f - (a*b^3*c^2 - a^2*b^2*c*d)*f^2)*x^2 + 3*((a^2*b^2*c*d - a^3 
*b*d^2)*e*f - (a^2*b^2*c^2 - a^3*b*c*d)*f^2)*x)*sqrt((d*e - c*f)/(b*e - a* 
f))*log((8*a^2*c^2*f^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*e^2 - 8*(a*b*c^2 
+ a^2*c*d)*e*f + (8*b^2*d^2*e^2 - 8*(b^2*c*d + a*b*d^2)*e*f + (b^2*c^2 + 6 
*a*b*c*d + a^2*d^2)*f^2)*x^2 - 4*(2*a^2*c*f^2 + (b^2*c + a*b*d)*e^2 - (3*a 
*b*c + a^2*d)*e*f + (2*b^2*d*e^2 - (b^2*c + 3*a*b*d)*e*f + (a*b*c + a^2*d) 
*f^2)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt((d*e - c*f)/(b*e - a*f)) + 2*(4* 
(b^2*c*d + a*b*d^2)*e^2 - (3*b^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*e*f + 4*(a* 
b*c^2 + a^2*c*d)*f^2)*x)/(f^2*x^2 + 2*e*f*x + e^2)) - 4*(3*b^3*c^2*e^2 - ( 
11*a*b^2*c^2 + 10*a^2*b*c*d - 15*a^3*d^2)*e*f + (23*a^2*b*c^2 - 20*a^3*c*d 
)*f^2 + (3*b^3*d^2*e^2 - 2*(10*b^3*c*d - 7*a*b^2*d^2)*e*f + (15*b^3*c^2 - 
10*a*b^2*c*d - 2*a^2*b*d^2)*f^2)*x^2 + (6*b^3*c*d*e^2 - (5*b^3*c^2 + 42*a* 
b^2*c*d - 35*a^2*b*d^2)*e*f + (35*a*b^2*c^2 - 24*a^2*b*c*d - 5*a^3*d^2)*f^ 
2)*x)*sqrt(b*x + a)*sqrt(d*x + c))/((a^3*b^4*c - a^4*b^3*d)*e^3 - 3*(a^4*b 
^3*c - a^5*b^2*d)*e^2*f + 3*(a^5*b^2*c - a^6*b*d)*e*f^2 - (a^6*b*c - a^7*d 
)*f^3 + ((b^7*c - a*b^6*d)*e^3 - 3*(a*b^6*c - a^2*b^5*d)*e^2*f + 3*(a^2*b^ 
5*c - a^3*b^4*d)*e*f^2 - (a^3*b^4*c - a^4*b^3*d)*f^3)*x^3 + 3*((a*b^6*c - 
a^2*b^5*d)*e^3 - 3*(a^2*b^5*c - a^3*b^4*d)*e^2*f + 3*(a^3*b^4*c - a^4*b...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((d*x+c)^(3/2)/(b*x+a)^(7/2)/(f*x+e),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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2604 vs. \(2 (236) = 472\).

Time = 21.35 (sec) , antiderivative size = 2604, normalized size of antiderivative = 9.79 \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^{7/2} (e+f x)} \, dx=\text {Too large to display} \] Input:

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

Output:

2*(sqrt(b*d)*d^2*e^2*f*abs(b) - 2*sqrt(b*d)*c*d*e*f^2*abs(b) + sqrt(b*d)*c 
^2*f^3*abs(b))*arctan(1/2*(2*b^2*d*e - b^2*c*f - a*b*d*f + (sqrt(b*d)*sqrt 
(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*f)/(sqrt(-b^2*d^2*e^2 + 
 b^2*c*d*e*f + a*b*d^2*e*f - a*b*c*d*f^2)*b))/(sqrt(-b^2*d^2*e^2 + b^2*c*d 
*e*f + a*b*d^2*e*f - a*b*c*d*f^2)*(b^3*e^3 - 3*a*b^2*e^2*f + 3*a^2*b*e*f^2 
 - a^3*f^3)*b) - 4/15*(3*sqrt(b*d)*b^9*c^4*d^2*e^2*abs(b) - 12*sqrt(b*d)*a 
*b^8*c^3*d^3*e^2*abs(b) + 18*sqrt(b*d)*a^2*b^7*c^2*d^4*e^2*abs(b) - 12*sqr 
t(b*d)*a^3*b^6*c*d^5*e^2*abs(b) + 3*sqrt(b*d)*a^4*b^5*d^6*e^2*abs(b) - 20* 
sqrt(b*d)*b^9*c^5*d*e*f*abs(b) + 94*sqrt(b*d)*a*b^8*c^4*d^2*e*f*abs(b) - 1 
76*sqrt(b*d)*a^2*b^7*c^3*d^3*e*f*abs(b) + 164*sqrt(b*d)*a^3*b^6*c^2*d^4*e* 
f*abs(b) - 76*sqrt(b*d)*a^4*b^5*c*d^5*e*f*abs(b) + 14*sqrt(b*d)*a^5*b^4*d^ 
6*e*f*abs(b) + 15*sqrt(b*d)*b^9*c^6*f^2*abs(b) - 70*sqrt(b*d)*a*b^8*c^5*d* 
f^2*abs(b) + 128*sqrt(b*d)*a^2*b^7*c^4*d^2*f^2*abs(b) - 112*sqrt(b*d)*a^3* 
b^6*c^3*d^3*f^2*abs(b) + 43*sqrt(b*d)*a^4*b^5*c^2*d^4*f^2*abs(b) - 2*sqrt( 
b*d)*a^5*b^4*c*d^5*f^2*abs(b) - 2*sqrt(b*d)*a^6*b^3*d^6*f^2*abs(b) + 70*sq 
rt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2* 
b^7*c^4*d*e*f*abs(b) - 280*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c 
 + (b*x + a)*b*d - a*b*d))^2*a*b^6*c^3*d^2*e*f*abs(b) + 420*sqrt(b*d)*(sqr 
t(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2*b^5*c^2* 
d^3*e*f*abs(b) - 280*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + ...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 4.48 (sec) , antiderivative size = 5205, normalized size of antiderivative = 19.57 \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^{7/2} (e+f x)} \, dx =\text {Too large to display} \] Input:

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

Output:

(15*sqrt(a + b*x)*sqrt(c*f - d*e)*sqrt(a*f - b*e)*log( - sqrt(2*sqrt(d)*sq 
rt(b)*sqrt(c*f - d*e)*sqrt(a*f - b*e) + a*d*f + b*c*f - 2*b*d*e) + sqrt(f) 
*sqrt(d)*sqrt(a + b*x) + sqrt(f)*sqrt(b)*sqrt(c + d*x))*a**3*b**2*c*d*f**2 
 - 15*sqrt(a + b*x)*sqrt(c*f - d*e)*sqrt(a*f - b*e)*log( - sqrt(2*sqrt(d)* 
sqrt(b)*sqrt(c*f - d*e)*sqrt(a*f - b*e) + a*d*f + b*c*f - 2*b*d*e) + sqrt( 
f)*sqrt(d)*sqrt(a + b*x) + sqrt(f)*sqrt(b)*sqrt(c + d*x))*a**3*b**2*d**2*e 
*f - 15*sqrt(a + b*x)*sqrt(c*f - d*e)*sqrt(a*f - b*e)*log( - sqrt(2*sqrt(d 
)*sqrt(b)*sqrt(c*f - d*e)*sqrt(a*f - b*e) + a*d*f + b*c*f - 2*b*d*e) + sqr 
t(f)*sqrt(d)*sqrt(a + b*x) + sqrt(f)*sqrt(b)*sqrt(c + d*x))*a**2*b**3*c**2 
*f**2 + 15*sqrt(a + b*x)*sqrt(c*f - d*e)*sqrt(a*f - b*e)*log( - sqrt(2*sqr 
t(d)*sqrt(b)*sqrt(c*f - d*e)*sqrt(a*f - b*e) + a*d*f + b*c*f - 2*b*d*e) + 
sqrt(f)*sqrt(d)*sqrt(a + b*x) + sqrt(f)*sqrt(b)*sqrt(c + d*x))*a**2*b**3*c 
*d*e*f + 30*sqrt(a + b*x)*sqrt(c*f - d*e)*sqrt(a*f - b*e)*log( - sqrt(2*sq 
rt(d)*sqrt(b)*sqrt(c*f - d*e)*sqrt(a*f - b*e) + a*d*f + b*c*f - 2*b*d*e) + 
 sqrt(f)*sqrt(d)*sqrt(a + b*x) + sqrt(f)*sqrt(b)*sqrt(c + d*x))*a**2*b**3* 
c*d*f**2*x - 30*sqrt(a + b*x)*sqrt(c*f - d*e)*sqrt(a*f - b*e)*log( - sqrt( 
2*sqrt(d)*sqrt(b)*sqrt(c*f - d*e)*sqrt(a*f - b*e) + a*d*f + b*c*f - 2*b*d* 
e) + sqrt(f)*sqrt(d)*sqrt(a + b*x) + sqrt(f)*sqrt(b)*sqrt(c + d*x))*a**2*b 
**3*d**2*e*f*x - 30*sqrt(a + b*x)*sqrt(c*f - d*e)*sqrt(a*f - b*e)*log( - s 
qrt(2*sqrt(d)*sqrt(b)*sqrt(c*f - d*e)*sqrt(a*f - b*e) + a*d*f + b*c*f -...