\(\int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^7} \, dx\) [101]

Optimal result

Integrand size = 26, antiderivative size = 146 \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^7} \, dx=\frac {b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}{8 x^4}-\frac {b^4 c \sqrt {a+b x} \sqrt {a c-b c x}}{16 a^2 x^2}-\frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{6 x^6}-\frac {b^6 c^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{a \sqrt {c}}\right )}{16 a^3} \] Output:

1/8*b^2*c*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x^4-1/16*b^4*c*(b*x+a)^(1/2)*(- 
b*c*x+a*c)^(1/2)/a^2/x^2-1/6*(b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)/x^6-1/16*b^6 
*c^(3/2)*arctanh((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/a/c^(1/2))/a^3
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.74 \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^7} \, dx=-\frac {(c (a-b x))^{3/2} \left (a \sqrt {a-b x} \sqrt {a+b x} \left (8 a^4-14 a^2 b^2 x^2+3 b^4 x^4\right )+6 b^6 x^6 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a-b x}}\right )\right )}{48 a^3 x^6 (a-b x)^{3/2}} \] Input:

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

Output:

-1/48*((c*(a - b*x))^(3/2)*(a*Sqrt[a - b*x]*Sqrt[a + b*x]*(8*a^4 - 14*a^2* 
b^2*x^2 + 3*b^4*x^4) + 6*b^6*x^6*ArcTanh[Sqrt[a + b*x]/Sqrt[a - b*x]]))/(a 
^3*x^6*(a - b*x)^(3/2))
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.09, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {108, 27, 108, 25, 27, 114, 25, 27, 103, 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.

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

\(\Big \downarrow \) 108

\(\displaystyle \frac {1}{6} \int -\frac {3 b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}{x^5}dx-\frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{6 x^6}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{2} b^2 c \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^5}dx-\frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{6 x^6}\)

\(\Big \downarrow \) 108

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 103

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

\(\Big \downarrow \) 221

\(\displaystyle -\frac {1}{2} b^2 c \left (-\frac {1}{4} b^2 c \left (-\frac {b^2 \text {arctanh}\left (\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{a \sqrt {c}}\right )}{2 a^3 \sqrt {c}}-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{2 a^2 c x^2}\right )-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{4 x^4}\right )-\frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{6 x^6}\)

Input:

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

Output:

-1/6*((a + b*x)^(3/2)*(a*c - b*c*x)^(3/2))/x^6 - (b^2*c*(-1/4*(Sqrt[a + b* 
x]*Sqrt[a*c - b*c*x])/x^4 - (b^2*c*(-1/2*(Sqrt[a + b*x]*Sqrt[a*c - b*c*x]) 
/(a^2*c*x^2) - (b^2*ArcTanh[(Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/(a*Sqrt[c])] 
)/(2*a^3*Sqrt[c])))/4))/2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 103

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_ 
))), x_] :> Simp[b*f   Subst[Int[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sq 
rt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d 
*e - f*(b*c + a*d), 0]
 

rule 108

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/(b*(m + 1))) 
, x] - Simp[1/(b*(m + 1))   Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* 
x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c 
, d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 
*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
 

rule 114

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[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*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || 
 IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
 

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 [A] (verified)

Time = 0.23 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.06

method	result	size
risch	\(-\frac {\left (-b x +a \right ) \sqrt {b x +a}\, \left (3 b^{4} x^{4}-14 a^{2} b^{2} x^{2}+8 a^{4}\right ) c^{2}}{48 x^{6} a^{2} \sqrt {-c \left (b x -a \right )}}-\frac {b^{6} \ln \left (\frac {2 a^{2} c +2 \sqrt {a^{2} c}\, \sqrt {-b^{2} c \,x^{2}+a^{2} c}}{x}\right ) \sqrt {-\left (b x +a \right ) c \left (b x -a \right )}\, c^{2}}{16 a^{2} \sqrt {a^{2} c}\, \sqrt {b x +a}\, \sqrt {-c \left (b x -a \right )}}\)	\(155\)
default	\(-\frac {\sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}\, c \left (3 \ln \left (\frac {2 a^{2} c +2 \sqrt {a^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}}{x}\right ) b^{6} c \,x^{6}+3 \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {a^{2} c}\, b^{4} x^{4}-14 \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {a^{2} c}\, a^{2} b^{2} x^{2}+8 \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, a^{4} \sqrt {a^{2} c}\right )}{48 a^{2} \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, x^{6} \sqrt {a^{2} c}}\)	\(189\)

Input:

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

Output:

-1/48*(-b*x+a)*(b*x+a)^(1/2)*(3*b^4*x^4-14*a^2*b^2*x^2+8*a^4)/x^6/a^2/(-c* 
(b*x-a))^(1/2)*c^2-1/16*b^6/a^2/(a^2*c)^(1/2)*ln((2*a^2*c+2*(a^2*c)^(1/2)* 
(-b^2*c*x^2+a^2*c)^(1/2))/x)*(-(b*x+a)*c*(b*x-a))^(1/2)/(b*x+a)^(1/2)/(-c* 
(b*x-a))^(1/2)*c^2
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.57 \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^7} \, dx=\left [\frac {3 \, b^{6} c^{\frac {3}{2}} x^{6} \log \left (-\frac {b^{2} c x^{2} - 2 \, a^{2} c + 2 \, \sqrt {-b c x + a c} \sqrt {b x + a} a \sqrt {c}}{x^{2}}\right ) - 2 \, {\left (3 \, a b^{4} c x^{4} - 14 \, a^{3} b^{2} c x^{2} + 8 \, a^{5} c\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{96 \, a^{3} x^{6}}, -\frac {3 \, b^{6} \sqrt {-c} c x^{6} \arctan \left (\frac {\sqrt {-b c x + a c} \sqrt {b x + a} a \sqrt {-c}}{b^{2} c x^{2} - a^{2} c}\right ) + {\left (3 \, a b^{4} c x^{4} - 14 \, a^{3} b^{2} c x^{2} + 8 \, a^{5} c\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{48 \, a^{3} x^{6}}\right ] \] Input:

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

Output:

[1/96*(3*b^6*c^(3/2)*x^6*log(-(b^2*c*x^2 - 2*a^2*c + 2*sqrt(-b*c*x + a*c)* 
sqrt(b*x + a)*a*sqrt(c))/x^2) - 2*(3*a*b^4*c*x^4 - 14*a^3*b^2*c*x^2 + 8*a^ 
5*c)*sqrt(-b*c*x + a*c)*sqrt(b*x + a))/(a^3*x^6), -1/48*(3*b^6*sqrt(-c)*c* 
x^6*arctan(sqrt(-b*c*x + a*c)*sqrt(b*x + a)*a*sqrt(-c)/(b^2*c*x^2 - a^2*c) 
) + (3*a*b^4*c*x^4 - 14*a^3*b^2*c*x^2 + 8*a^5*c)*sqrt(-b*c*x + a*c)*sqrt(b 
*x + a))/(a^3*x^6)]
 

Sympy [F]

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

integrate((b*x+a)**(3/2)*(-b*c*x+a*c)**(3/2)/x**7,x)
 

Output:

Integral((-c*(-a + b*x))**(3/2)*(a + b*x)**(3/2)/x**7, x)
 

Maxima [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.32 \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^7} \, dx=-\frac {b^{6} c^{\frac {3}{2}} \log \left (\frac {2 \, a^{2} c}{{\left | x \right |}} + \frac {2 \, \sqrt {-b^{2} c x^{2} + a^{2} c} a \sqrt {c}}{{\left | x \right |}}\right )}{16 \, a^{3}} + \frac {\sqrt {-b^{2} c x^{2} + a^{2} c} b^{6} c}{16 \, a^{4}} + \frac {{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} b^{6}}{48 \, a^{6}} + \frac {{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {5}{2}} b^{4}}{48 \, a^{6} c x^{2}} - \frac {{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {5}{2}} b^{2}}{24 \, a^{4} c x^{4}} - \frac {{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {5}{2}}}{6 \, a^{2} c x^{6}} \] Input:

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

Output:

-1/16*b^6*c^(3/2)*log(2*a^2*c/abs(x) + 2*sqrt(-b^2*c*x^2 + a^2*c)*a*sqrt(c 
)/abs(x))/a^3 + 1/16*sqrt(-b^2*c*x^2 + a^2*c)*b^6*c/a^4 + 1/48*(-b^2*c*x^2 
 + a^2*c)^(3/2)*b^6/a^6 + 1/48*(-b^2*c*x^2 + a^2*c)^(5/2)*b^4/(a^6*c*x^2) 
- 1/24*(-b^2*c*x^2 + a^2*c)^(5/2)*b^2/(a^4*c*x^4) - 1/6*(-b^2*c*x^2 + a^2* 
c)^(5/2)/(a^2*c*x^6)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 403 vs. \(2 (118) = 236\).

Time = 0.24 (sec) , antiderivative size = 403, normalized size of antiderivative = 2.76 \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^7} \, dx=-\frac {\frac {3 \, b^{7} \sqrt {-c} c \arctan \left (\frac {{\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2}}{2 \, a c}\right )}{a^{3}} + \frac {2 \, {\left (3 \, b^{7} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{22} \sqrt {-c} c^{2} - 188 \, a^{2} b^{7} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{18} \sqrt {-c} c^{4} + 1248 \, a^{4} b^{7} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{14} \sqrt {-c} c^{6} - 4992 \, a^{6} b^{7} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{10} \sqrt {-c} c^{8} + 12032 \, a^{8} b^{7} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{6} \sqrt {-c} c^{10} - 3072 \, a^{10} b^{7} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2} \sqrt {-c} c^{12}\right )}}{{\left ({\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{4} + 4 \, a^{2} c^{2}\right )}^{6} a^{2}}}{24 \, b} \] Input:

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

Output:

-1/24*(3*b^7*sqrt(-c)*c*arctan(1/2*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + 
a)*c + 2*a*c))^2/(a*c))/a^3 + 2*(3*b^7*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b* 
x + a)*c + 2*a*c))^22*sqrt(-c)*c^2 - 188*a^2*b^7*(sqrt(b*x + a)*sqrt(-c) - 
 sqrt(-(b*x + a)*c + 2*a*c))^18*sqrt(-c)*c^4 + 1248*a^4*b^7*(sqrt(b*x + a) 
*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^14*sqrt(-c)*c^6 - 4992*a^6*b^7*(sq 
rt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^10*sqrt(-c)*c^8 + 12032 
*a^8*b^7*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^6*sqrt(-c)* 
c^10 - 3072*a^10*b^7*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c)) 
^2*sqrt(-c)*c^12)/(((sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^ 
4 + 4*a^2*c^2)^6*a^2))/b
 

Mupad [F(-1)]

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

int(((a*c - b*c*x)^(3/2)*(a + b*x)^(3/2))/x^7,x)
 

Output:

int(((a*c - b*c*x)^(3/2)*(a + b*x)^(3/2))/x^7, x)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.45 \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^7} \, dx=\frac {\sqrt {c}\, c \left (-8 \sqrt {b x +a}\, \sqrt {-b x +a}\, a^{5}+14 \sqrt {b x +a}\, \sqrt {-b x +a}\, a^{3} b^{2} x^{2}-3 \sqrt {b x +a}\, \sqrt {-b x +a}\, a \,b^{4} x^{4}-3 \,\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right )}{2}\right )-1\right ) b^{6} x^{6}+3 \,\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right )}{2}\right )+1\right ) b^{6} x^{6}-3 \,\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right )}{2}\right )-1\right ) b^{6} x^{6}+3 \,\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right )}{2}\right )+1\right ) b^{6} x^{6}\right )}{48 a^{3} x^{6}} \] Input:

int((b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)/x^7,x)
 

Output:

(sqrt(c)*c*( - 8*sqrt(a + b*x)*sqrt(a - b*x)*a**5 + 14*sqrt(a + b*x)*sqrt( 
a - b*x)*a**3*b**2*x**2 - 3*sqrt(a + b*x)*sqrt(a - b*x)*a*b**4*x**4 - 3*lo 
g( - sqrt(2) + tan(asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))/2) - 1)*b**6*x**6 
 + 3*log( - sqrt(2) + tan(asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))/2) + 1)*b* 
*6*x**6 - 3*log(sqrt(2) + tan(asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))/2) - 1 
)*b**6*x**6 + 3*log(sqrt(2) + tan(asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))/2) 
 + 1)*b**6*x**6))/(48*a**3*x**6)