\(\int \frac {(a+b x)^{5/2} (a c-b c x)^{5/2}}{(e+f x)^7} \, dx\) [854]

Optimal result

Integrand size = 30, antiderivative size = 412 \[ \int \frac {(a+b x)^{5/2} (a c-b c x)^{5/2}}{(e+f x)^7} \, dx=-\frac {(a+b x)^{5/2} (a c-b c x)^{7/2}}{6 c (b e+a f) (e+f x)^6}-\frac {a b (a+b x)^{3/2} (a c-b c x)^{7/2}}{6 c (b e+a f)^2 (e+f x)^5}-\frac {a^2 b^2 \sqrt {a+b x} (a c-b c x)^{7/2}}{8 c (b e+a f)^3 (e+f x)^4}+\frac {a^3 b^3 \sqrt {a+b x} (a c-b c x)^{5/2}}{24 (b e-a f) (b e+a f)^3 (e+f x)^3}+\frac {5 a^4 b^4 c \sqrt {a+b x} (a c-b c x)^{3/2}}{48 (b e-a f)^2 (b e+a f)^3 (e+f x)^2}+\frac {5 a^5 b^5 c^2 \sqrt {a+b x} \sqrt {a c-b c x}}{16 (b e-a f)^3 (b e+a f)^3 (e+f x)}+\frac {5 a^6 b^6 c^{5/2} \arctan \left (\frac {\sqrt {c} \sqrt {b e+a f} \sqrt {a+b x}}{\sqrt {b e-a f} \sqrt {a c-b c x}}\right )}{8 (b e-a f)^{7/2} (b e+a f)^{7/2}} \] Output:

-1/6*(b*x+a)^(5/2)*(-b*c*x+a*c)^(7/2)/c/(a*f+b*e)/(f*x+e)^6-1/6*a*b*(b*x+a 
)^(3/2)*(-b*c*x+a*c)^(7/2)/c/(a*f+b*e)^2/(f*x+e)^5-1/8*a^2*b^2*(b*x+a)^(1/ 
2)*(-b*c*x+a*c)^(7/2)/c/(a*f+b*e)^3/(f*x+e)^4+1/24*a^3*b^3*(b*x+a)^(1/2)*( 
-b*c*x+a*c)^(5/2)/(-a*f+b*e)/(a*f+b*e)^3/(f*x+e)^3+5/48*a^4*b^4*c*(b*x+a)^ 
(1/2)*(-b*c*x+a*c)^(3/2)/(-a*f+b*e)^2/(a*f+b*e)^3/(f*x+e)^2+5/16*a^5*b^5*c 
^2*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/(-a*f+b*e)^3/(a*f+b*e)^3/(f*x+e)+5/8*a 
^6*b^6*c^(5/2)*arctan(c^(1/2)*(a*f+b*e)^(1/2)*(b*x+a)^(1/2)/(-a*f+b*e)^(1/ 
2)/(-b*c*x+a*c)^(1/2))/(-a*f+b*e)^(7/2)/(a*f+b*e)^(7/2)
 

Mathematica [A] (verified)

Time = 2.38 (sec) , antiderivative size = 323, normalized size of antiderivative = 0.78 \[ \int \frac {(a+b x)^{5/2} (a c-b c x)^{5/2}}{(e+f x)^7} \, dx=\frac {1}{48} c^2 \sqrt {c (a-b x)} \left (\frac {\sqrt {a+b x} \left (8 a^{10} f^5+8 b^{10} e^5 x^5-2 a^8 b^2 f^3 \left (13 e^2+6 e f x+13 f^2 x^2\right )-2 a^2 b^8 e^3 x^3 \left (13 e^2+6 e f x+13 f^2 x^2\right )+a^6 b^4 f \left (33 e^4+54 e^3 f x+122 e^2 f^2 x^2+54 e f^3 x^3+33 f^4 x^4\right )+a^4 b^6 e x \left (33 e^4+54 e^3 f x+122 e^2 f^2 x^2+54 e f^3 x^3+33 f^4 x^4\right )\right )}{(b e-a f)^3 (b e+a f)^3 (e+f x)^6}+\frac {30 a^6 b^6 \arctan \left (\frac {\sqrt {b e+a f} \sqrt {a+b x}}{\sqrt {b e-a f} \sqrt {a-b x}}\right )}{(b e-a f)^{7/2} (b e+a f)^{7/2} \sqrt {a-b x}}\right ) \] Input:

Integrate[((a + b*x)^(5/2)*(a*c - b*c*x)^(5/2))/(e + f*x)^7,x]
 

Output:

(c^2*Sqrt[c*(a - b*x)]*((Sqrt[a + b*x]*(8*a^10*f^5 + 8*b^10*e^5*x^5 - 2*a^ 
8*b^2*f^3*(13*e^2 + 6*e*f*x + 13*f^2*x^2) - 2*a^2*b^8*e^3*x^3*(13*e^2 + 6* 
e*f*x + 13*f^2*x^2) + a^6*b^4*f*(33*e^4 + 54*e^3*f*x + 122*e^2*f^2*x^2 + 5 
4*e*f^3*x^3 + 33*f^4*x^4) + a^4*b^6*e*x*(33*e^4 + 54*e^3*f*x + 122*e^2*f^2 
*x^2 + 54*e*f^3*x^3 + 33*f^4*x^4)))/((b*e - a*f)^3*(b*e + a*f)^3*(e + f*x) 
^6) + (30*a^6*b^6*ArcTan[(Sqrt[b*e + a*f]*Sqrt[a + b*x])/(Sqrt[b*e - a*f]* 
Sqrt[a - b*x])])/((b*e - a*f)^(7/2)*(b*e + a*f)^(7/2)*Sqrt[a - b*x])))/48
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {105, 105, 105, 105, 105, 105, 104, 218}

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.

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

\(\Big \downarrow \) 105

\(\displaystyle \frac {5 a b \int \frac {(a+b x)^{3/2} (a c-b c x)^{5/2}}{(e+f x)^6}dx}{6 (a f+b e)}-\frac {(a+b x)^{5/2} (a c-b c x)^{7/2}}{6 c (e+f x)^6 (a f+b e)}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {5 a b \left (\frac {3 a b \int \frac {\sqrt {a+b x} (a c-b c x)^{5/2}}{(e+f x)^5}dx}{5 (a f+b e)}-\frac {(a+b x)^{3/2} (a c-b c x)^{7/2}}{5 c (e+f x)^5 (a f+b e)}\right )}{6 (a f+b e)}-\frac {(a+b x)^{5/2} (a c-b c x)^{7/2}}{6 c (e+f x)^6 (a f+b e)}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {5 a b \left (\frac {3 a b \left (\frac {a b \int \frac {(a c-b c x)^{5/2}}{\sqrt {a+b x} (e+f x)^4}dx}{4 (a f+b e)}-\frac {\sqrt {a+b x} (a c-b c x)^{7/2}}{4 c (e+f x)^4 (a f+b e)}\right )}{5 (a f+b e)}-\frac {(a+b x)^{3/2} (a c-b c x)^{7/2}}{5 c (e+f x)^5 (a f+b e)}\right )}{6 (a f+b e)}-\frac {(a+b x)^{5/2} (a c-b c x)^{7/2}}{6 c (e+f x)^6 (a f+b e)}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {5 a b \left (\frac {3 a b \left (\frac {a b \left (\frac {5 a b c \int \frac {(a c-b c x)^{3/2}}{\sqrt {a+b x} (e+f x)^3}dx}{3 (b e-a f)}+\frac {\sqrt {a+b x} (a c-b c x)^{5/2}}{3 (e+f x)^3 (b e-a f)}\right )}{4 (a f+b e)}-\frac {\sqrt {a+b x} (a c-b c x)^{7/2}}{4 c (e+f x)^4 (a f+b e)}\right )}{5 (a f+b e)}-\frac {(a+b x)^{3/2} (a c-b c x)^{7/2}}{5 c (e+f x)^5 (a f+b e)}\right )}{6 (a f+b e)}-\frac {(a+b x)^{5/2} (a c-b c x)^{7/2}}{6 c (e+f x)^6 (a f+b e)}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {5 a b \left (\frac {3 a b \left (\frac {a b \left (\frac {5 a b c \left (\frac {3 a b c \int \frac {\sqrt {a c-b c x}}{\sqrt {a+b x} (e+f x)^2}dx}{2 (b e-a f)}+\frac {\sqrt {a+b x} (a c-b c x)^{3/2}}{2 (e+f x)^2 (b e-a f)}\right )}{3 (b e-a f)}+\frac {\sqrt {a+b x} (a c-b c x)^{5/2}}{3 (e+f x)^3 (b e-a f)}\right )}{4 (a f+b e)}-\frac {\sqrt {a+b x} (a c-b c x)^{7/2}}{4 c (e+f x)^4 (a f+b e)}\right )}{5 (a f+b e)}-\frac {(a+b x)^{3/2} (a c-b c x)^{7/2}}{5 c (e+f x)^5 (a f+b e)}\right )}{6 (a f+b e)}-\frac {(a+b x)^{5/2} (a c-b c x)^{7/2}}{6 c (e+f x)^6 (a f+b e)}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {5 a b \left (\frac {3 a b \left (\frac {a b \left (\frac {5 a b c \left (\frac {3 a b c \left (\frac {a b c \int \frac {1}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)}dx}{b e-a f}+\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{(e+f x) (b e-a f)}\right )}{2 (b e-a f)}+\frac {\sqrt {a+b x} (a c-b c x)^{3/2}}{2 (e+f x)^2 (b e-a f)}\right )}{3 (b e-a f)}+\frac {\sqrt {a+b x} (a c-b c x)^{5/2}}{3 (e+f x)^3 (b e-a f)}\right )}{4 (a f+b e)}-\frac {\sqrt {a+b x} (a c-b c x)^{7/2}}{4 c (e+f x)^4 (a f+b e)}\right )}{5 (a f+b e)}-\frac {(a+b x)^{3/2} (a c-b c x)^{7/2}}{5 c (e+f x)^5 (a f+b e)}\right )}{6 (a f+b e)}-\frac {(a+b x)^{5/2} (a c-b c x)^{7/2}}{6 c (e+f x)^6 (a f+b e)}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {5 a b \left (\frac {3 a b \left (\frac {a b \left (\frac {5 a b c \left (\frac {3 a b c \left (\frac {2 a b c \int \frac {1}{b e-a f+\frac {c (b e+a f) (a+b x)}{a c-b c x}}d\frac {\sqrt {a+b x}}{\sqrt {a c-b c x}}}{b e-a f}+\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{(e+f x) (b e-a f)}\right )}{2 (b e-a f)}+\frac {\sqrt {a+b x} (a c-b c x)^{3/2}}{2 (e+f x)^2 (b e-a f)}\right )}{3 (b e-a f)}+\frac {\sqrt {a+b x} (a c-b c x)^{5/2}}{3 (e+f x)^3 (b e-a f)}\right )}{4 (a f+b e)}-\frac {\sqrt {a+b x} (a c-b c x)^{7/2}}{4 c (e+f x)^4 (a f+b e)}\right )}{5 (a f+b e)}-\frac {(a+b x)^{3/2} (a c-b c x)^{7/2}}{5 c (e+f x)^5 (a f+b e)}\right )}{6 (a f+b e)}-\frac {(a+b x)^{5/2} (a c-b c x)^{7/2}}{6 c (e+f x)^6 (a f+b e)}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {5 a b \left (\frac {3 a b \left (\frac {a b \left (\frac {5 a b c \left (\frac {3 a b c \left (\frac {2 a b \sqrt {c} \arctan \left (\frac {\sqrt {c} \sqrt {a+b x} \sqrt {a f+b e}}{\sqrt {a c-b c x} \sqrt {b e-a f}}\right )}{(b e-a f)^{3/2} \sqrt {a f+b e}}+\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{(e+f x) (b e-a f)}\right )}{2 (b e-a f)}+\frac {\sqrt {a+b x} (a c-b c x)^{3/2}}{2 (e+f x)^2 (b e-a f)}\right )}{3 (b e-a f)}+\frac {\sqrt {a+b x} (a c-b c x)^{5/2}}{3 (e+f x)^3 (b e-a f)}\right )}{4 (a f+b e)}-\frac {\sqrt {a+b x} (a c-b c x)^{7/2}}{4 c (e+f x)^4 (a f+b e)}\right )}{5 (a f+b e)}-\frac {(a+b x)^{3/2} (a c-b c x)^{7/2}}{5 c (e+f x)^5 (a f+b e)}\right )}{6 (a f+b e)}-\frac {(a+b x)^{5/2} (a c-b c x)^{7/2}}{6 c (e+f x)^6 (a f+b e)}\)

Input:

Int[((a + b*x)^(5/2)*(a*c - b*c*x)^(5/2))/(e + f*x)^7,x]
 

Output:

-1/6*((a + b*x)^(5/2)*(a*c - b*c*x)^(7/2))/(c*(b*e + a*f)*(e + f*x)^6) + ( 
5*a*b*(-1/5*((a + b*x)^(3/2)*(a*c - b*c*x)^(7/2))/(c*(b*e + a*f)*(e + f*x) 
^5) + (3*a*b*(-1/4*(Sqrt[a + b*x]*(a*c - b*c*x)^(7/2))/(c*(b*e + a*f)*(e + 
 f*x)^4) + (a*b*((Sqrt[a + b*x]*(a*c - b*c*x)^(5/2))/(3*(b*e - a*f)*(e + f 
*x)^3) + (5*a*b*c*((Sqrt[a + b*x]*(a*c - b*c*x)^(3/2))/(2*(b*e - a*f)*(e + 
 f*x)^2) + (3*a*b*c*((Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/((b*e - a*f)*(e + f 
*x)) + (2*a*b*Sqrt[c]*ArcTan[(Sqrt[c]*Sqrt[b*e + a*f]*Sqrt[a + b*x])/(Sqrt 
[b*e - a*f]*Sqrt[a*c - b*c*x])])/((b*e - a*f)^(3/2)*Sqrt[b*e + a*f])))/(2* 
(b*e - a*f))))/(3*(b*e - a*f))))/(4*(b*e + a*f))))/(5*(b*e + a*f))))/(6*(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 218

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1622\) vs. \(2(358)=716\).

Time = 238.42 (sec) , antiderivative size = 1623, normalized size of antiderivative = 3.94

method	result	size
default	\(\text {Expression too large to display}\)	\(1623\)

Input:

int((b*x+a)^(5/2)*(-b*c*x+a*c)^(5/2)/(f*x+e)^7,x,method=_RETURNVERBOSE)
 

Output:

1/48*(b*x+a)^(1/2)*(c*(-b*x+a))^(1/2)*(15*ln(2*(b^2*c*e*x+a^2*c*f+(c*(a^2* 
f^2-b^2*e^2)/f^2)^(1/2)*(c*(-b^2*x^2+a^2))^(1/2)*f)/(f*x+e))*a^6*b^6*c*f^6 
*x^6-8*b^10*e^5*f*x^5*(c*(-b^2*x^2+a^2))^(1/2)*(c*(a^2*f^2-b^2*e^2)/f^2)^( 
1/2)-33*a^6*b^4*f^6*x^4*(c*(-b^2*x^2+a^2))^(1/2)*(c*(a^2*f^2-b^2*e^2)/f^2) 
^(1/2)+26*a^8*b^2*f^6*x^2*(c*(-b^2*x^2+a^2))^(1/2)*(c*(a^2*f^2-b^2*e^2)/f^ 
2)^(1/2)+26*a^8*b^2*e^2*f^4*(c*(-b^2*x^2+a^2))^(1/2)*(c*(a^2*f^2-b^2*e^2)/ 
f^2)^(1/2)-33*a^6*b^4*e^4*f^2*(c*(-b^2*x^2+a^2))^(1/2)*(c*(a^2*f^2-b^2*e^2 
)/f^2)^(1/2)+15*ln(2*(b^2*c*e*x+a^2*c*f+(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)*(c 
*(-b^2*x^2+a^2))^(1/2)*f)/(f*x+e))*a^6*b^6*c*e^6-54*a^6*b^4*e^3*f^3*x*(c*( 
-b^2*x^2+a^2))^(1/2)*(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)-33*a^4*b^6*e^5*f*x*(c 
*(-b^2*x^2+a^2))^(1/2)*(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)+225*ln(2*(b^2*c*e*x 
+a^2*c*f+(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)*(c*(-b^2*x^2+a^2))^(1/2)*f)/(f*x+ 
e))*a^6*b^6*c*e^4*f^2*x^2+12*a^2*b^8*e^4*f^2*x^4*(c*(-b^2*x^2+a^2))^(1/2)* 
(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)-33*a^4*b^6*e*f^5*x^5*(c*(-b^2*x^2+a^2))^(1 
/2)*(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)+26*a^2*b^8*e^3*f^3*x^5*(c*(-b^2*x^2+a^ 
2))^(1/2)*(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)-54*a^4*b^6*e^2*f^4*x^4*(c*(-b^2* 
x^2+a^2))^(1/2)*(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)+90*ln(2*(b^2*c*e*x+a^2*c*f 
+(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)*(c*(-b^2*x^2+a^2))^(1/2)*f)/(f*x+e))*a^6* 
b^6*c*e^5*f*x+90*ln(2*(b^2*c*e*x+a^2*c*f+(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)*( 
c*(-b^2*x^2+a^2))^(1/2)*f)/(f*x+e))*a^6*b^6*c*e*f^5*x^5+225*ln(2*(b^2*c...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{5/2} (a c-b c x)^{5/2}}{(e+f x)^7} \, dx=\text {Timed out} \] Input:

integrate((b*x+a)^(5/2)*(-b*c*x+a*c)^(5/2)/(f*x+e)^7,x, algorithm="fricas" 
)
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{5/2} (a c-b c x)^{5/2}}{(e+f x)^7} \, dx=\text {Timed out} \] Input:

integrate((b*x+a)**(5/2)*(-b*c*x+a*c)**(5/2)/(f*x+e)**7,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b x)^{5/2} (a c-b c x)^{5/2}}{(e+f x)^7} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((b*x+a)^(5/2)*(-b*c*x+a*c)^(5/2)/(f*x+e)^7,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*f-b*e)>0)', see `assume?` for 
 more deta
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3271 vs. \(2 (358) = 716\).

Time = 2.86 (sec) , antiderivative size = 3271, normalized size of antiderivative = 7.94 \[ \int \frac {(a+b x)^{5/2} (a c-b c x)^{5/2}}{(e+f x)^7} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^(5/2)*(-b*c*x+a*c)^(5/2)/(f*x+e)^7,x, algorithm="giac")
 

Output:

-1/24*(15*a^6*b^7*sqrt(-c)*c^2*arctan(-1/2*(2*b*c*e - (sqrt(b*x + a)*sqrt( 
-c) - sqrt(-(b*x + a)*c + 2*a*c))^2*f)/(sqrt(-b^2*e^2 + a^2*f^2)*c))/((b^6 
*e^6 - 3*a^2*b^4*e^4*f^2 + 3*a^4*b^2*e^2*f^4 - a^6*f^6)*sqrt(-b^2*e^2 + a^ 
2*f^2)) - 2*(8192*b^18*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c 
))^12*sqrt(-c)*c^8*e^11 - 12288*b^17*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x 
+ a)*c + 2*a*c))^14*sqrt(-c)*c^7*e^10*f - 49152*a^2*b^17*(sqrt(b*x + a)*sq 
rt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^10*sqrt(-c)*c^9*e^10*f + 7680*b^16*(s 
qrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^16*sqrt(-c)*c^6*e^9*f^ 
2 + 34816*a^2*b^16*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^1 
2*sqrt(-c)*c^8*e^9*f^2 + 122880*a^4*b^16*(sqrt(b*x + a)*sqrt(-c) - sqrt(-( 
b*x + a)*c + 2*a*c))^8*sqrt(-c)*c^10*e^9*f^2 - 2560*b^15*(sqrt(b*x + a)*sq 
rt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^18*sqrt(-c)*c^5*e^8*f^3 + 9216*a^2*b^ 
15*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^14*sqrt(-c)*c^7*e 
^8*f^3 + 36864*a^4*b^15*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a* 
c))^10*sqrt(-c)*c^9*e^8*f^3 - 163840*a^6*b^15*(sqrt(b*x + a)*sqrt(-c) - sq 
rt(-(b*x + a)*c + 2*a*c))^6*sqrt(-c)*c^11*e^8*f^3 + 480*b^14*(sqrt(b*x + a 
)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^20*sqrt(-c)*c^4*e^7*f^4 - 17280*a 
^2*b^14*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^16*sqrt(-c)* 
c^6*e^7*f^4 - 119808*a^4*b^14*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c 
+ 2*a*c))^12*sqrt(-c)*c^8*e^7*f^4 - 276480*a^6*b^14*(sqrt(b*x + a)*sqrt...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{5/2} (a c-b c x)^{5/2}}{(e+f x)^7} \, dx=\int \frac {{\left (a\,c-b\,c\,x\right )}^{5/2}\,{\left (a+b\,x\right )}^{5/2}}{{\left (e+f\,x\right )}^7} \,d x \] Input:

int(((a*c - b*c*x)^(5/2)*(a + b*x)^(5/2))/(e + f*x)^7,x)
 

Output:

int(((a*c - b*c*x)^(5/2)*(a + b*x)^(5/2))/(e + f*x)^7, x)
 

Reduce [B] (verification not implemented)

Time = 28.29 (sec) , antiderivative size = 10172, normalized size of antiderivative = 24.69 \[ \int \frac {(a+b x)^{5/2} (a c-b c x)^{5/2}}{(e+f x)^7} \, dx =\text {Too large to display} \] Input:

int((b*x+a)^(5/2)*(-b*c*x+a*c)^(5/2)/(f*x+e)^7,x)
 

Output:

(sqrt(c)*c**2*(30*sqrt(f)*sqrt(a)*sqrt(2*sqrt(f)*sqrt(a)*sqrt(a*f - b*e)*s 
qrt(2) - 3*a*f + b*e)*sqrt(a*f + b*e)*sqrt(a*f - b*e)*sqrt(2)*atan((tan(as 
in(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))/2)*a*f + tan(asin(sqrt(a - b*x)/(sqrt( 
a)*sqrt(2)))/2)*b*e)/(sqrt(2*sqrt(f)*sqrt(a)*sqrt(a*f - b*e)*sqrt(2) - 3*a 
*f + b*e)*sqrt(a*f + b*e)))*a**6*b**6*e**6 + 180*sqrt(f)*sqrt(a)*sqrt(2*sq 
rt(f)*sqrt(a)*sqrt(a*f - b*e)*sqrt(2) - 3*a*f + b*e)*sqrt(a*f + b*e)*sqrt( 
a*f - b*e)*sqrt(2)*atan((tan(asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))/2)*a*f 
+ tan(asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))/2)*b*e)/(sqrt(2*sqrt(f)*sqrt(a 
)*sqrt(a*f - b*e)*sqrt(2) - 3*a*f + b*e)*sqrt(a*f + b*e)))*a**6*b**6*e**5* 
f*x + 450*sqrt(f)*sqrt(a)*sqrt(2*sqrt(f)*sqrt(a)*sqrt(a*f - b*e)*sqrt(2) - 
 3*a*f + b*e)*sqrt(a*f + b*e)*sqrt(a*f - b*e)*sqrt(2)*atan((tan(asin(sqrt( 
a - b*x)/(sqrt(a)*sqrt(2)))/2)*a*f + tan(asin(sqrt(a - b*x)/(sqrt(a)*sqrt( 
2)))/2)*b*e)/(sqrt(2*sqrt(f)*sqrt(a)*sqrt(a*f - b*e)*sqrt(2) - 3*a*f + b*e 
)*sqrt(a*f + b*e)))*a**6*b**6*e**4*f**2*x**2 + 600*sqrt(f)*sqrt(a)*sqrt(2* 
sqrt(f)*sqrt(a)*sqrt(a*f - b*e)*sqrt(2) - 3*a*f + b*e)*sqrt(a*f + b*e)*sqr 
t(a*f - b*e)*sqrt(2)*atan((tan(asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))/2)*a* 
f + tan(asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))/2)*b*e)/(sqrt(2*sqrt(f)*sqrt 
(a)*sqrt(a*f - b*e)*sqrt(2) - 3*a*f + b*e)*sqrt(a*f + b*e)))*a**6*b**6*e** 
3*f**3*x**3 + 450*sqrt(f)*sqrt(a)*sqrt(2*sqrt(f)*sqrt(a)*sqrt(a*f - b*e)*s 
qrt(2) - 3*a*f + b*e)*sqrt(a*f + b*e)*sqrt(a*f - b*e)*sqrt(2)*atan((tan...