\(\int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^7} \, dx\) [86]

Optimal result

Integrand size = 26, antiderivative size = 147 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^7} \, dx=-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{6 x^6}+\frac {b^2 \sqrt {a+b x} \sqrt {a c-b c x}}{24 a^2 x^4}+\frac {b^4 \sqrt {a+b x} \sqrt {a c-b c x}}{16 a^4 x^2}+\frac {b^6 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{a \sqrt {c}}\right )}{16 a^5} \] Output:

-1/6*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x^6+1/24*b^2*(b*x+a)^(1/2)*(-b*c*x+a 
*c)^(1/2)/a^2/x^4+1/16*b^4*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/a^4/x^2+1/16*b 
^6*c^(1/2)*arctanh((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/a/c^(1/2))/a^5
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.73 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^7} \, dx=\frac {\sqrt {c (a-b x)} \left (a \sqrt {a-b x} \sqrt {a+b x} \left (-8 a^4+2 a^2 b^2 x^2+3 b^4 x^4\right )+6 b^6 x^6 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a-b x}}\right )\right )}{48 a^5 x^6 \sqrt {a-b x}} \] Input:

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

Output:

(Sqrt[c*(a - b*x)]*(a*Sqrt[a - b*x]*Sqrt[a + b*x]*(-8*a^4 + 2*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]]))/(48*a^5*x^ 
6*Sqrt[a - b*x])
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.14, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {108, 25, 27, 114, 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.

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

\(\Big \downarrow \) 108

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 103

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

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 = 0.22 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.03

method	result	size
risch	\(-\frac {\left (-b x +a \right ) \sqrt {b x +a}\, \left (-3 b^{4} x^{4}-2 a^{2} b^{2} x^{2}+8 a^{4}\right ) c}{48 x^{6} a^{4} \sqrt {-c \left (b x -a \right )}}+\frac {b^{6} \ln \left (\frac {2 a^{2} c +2 \sqrt {a^{2} c}\, \sqrt {-b^{2} c \,x^{2}+a^{2} c}}{x}\right ) \sqrt {-\left (b x +a \right ) c \left (b x -a \right )}\, c}{16 a^{4} \sqrt {a^{2} c}\, \sqrt {b x +a}\, \sqrt {-c \left (b x -a \right )}}\)	\(151\)
default	\(\frac {\sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}\, \left (3 \ln \left (\frac {2 a^{2} c +2 \sqrt {a^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}}{x}\right ) b^{6} c \,x^{6}+3 \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {a^{2} c}\, b^{4} x^{4}+2 \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {a^{2} c}\, a^{2} b^{2} x^{2}-8 \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, a^{4} \sqrt {a^{2} c}\right )}{48 a^{4} \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, x^{6} \sqrt {a^{2} c}}\)	\(188\)

Input:

int((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x^7,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.51 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^7} \, dx=\left [\frac {3 \, b^{6} \sqrt {c} x^{6} \log \left (-\frac {b^{2} c x^{2} - 2 \, a^{2} c - 2 \, \sqrt {-b c x + a c} \sqrt {b x + a} a \sqrt {c}}{x^{2}}\right ) + 2 \, {\left (3 \, a b^{4} x^{4} + 2 \, a^{3} b^{2} x^{2} - 8 \, a^{5}\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{96 \, a^{5} x^{6}}, \frac {3 \, b^{6} \sqrt {-c} x^{6} \arctan \left (\frac {\sqrt {-b c x + a c} \sqrt {b x + a} a \sqrt {-c}}{b^{2} c x^{2} - a^{2} c}\right ) + {\left (3 \, a b^{4} x^{4} + 2 \, a^{3} b^{2} x^{2} - 8 \, a^{5}\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{48 \, a^{5} x^{6}}\right ] \] Input:

integrate((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x^7,x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

integrate((b*x+a)**(1/2)*(-b*c*x+a*c)**(1/2)/x**7,x)
 

Output:

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

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.13 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^7} \, dx=\frac {b^{6} \sqrt {c} \log \left (\frac {2 \, a^{2} c}{{\left | x \right |}} + \frac {2 \, \sqrt {-b^{2} c x^{2} + a^{2} c} a \sqrt {c}}{{\left | x \right |}}\right )}{16 \, a^{5}} - \frac {\sqrt {-b^{2} c x^{2} + a^{2} c} b^{6}}{16 \, a^{6}} - \frac {{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} b^{4}}{16 \, a^{6} c x^{2}} - \frac {{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} b^{2}}{8 \, a^{4} c x^{4}} - \frac {{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}}}{6 \, a^{2} c x^{6}} \] Input:

integrate((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x^7,x, algorithm="maxima")
 

Output:

1/16*b^6*sqrt(c)*log(2*a^2*c/abs(x) + 2*sqrt(-b^2*c*x^2 + a^2*c)*a*sqrt(c) 
/abs(x))/a^5 - 1/16*sqrt(-b^2*c*x^2 + a^2*c)*b^6/a^6 - 1/16*(-b^2*c*x^2 + 
a^2*c)^(3/2)*b^4/(a^6*c*x^2) - 1/8*(-b^2*c*x^2 + a^2*c)^(3/2)*b^2/(a^4*c*x 
^4) - 1/6*(-b^2*c*x^2 + a^2*c)^(3/2)/(a^2*c*x^6)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 400 vs. \(2 (119) = 238\).

Time = 0.19 (sec) , antiderivative size = 400, normalized size of antiderivative = 2.72 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^7} \, dx=\frac {\frac {3 \, b^{7} \sqrt {-c} \arctan \left (\frac {{\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2}}{2 \, a c}\right )}{a^{5}} + \frac {2 \, {\left (3 \, b^{7} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{22} \sqrt {-c} c + 68 \, a^{2} b^{7} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{18} \sqrt {-c} c^{3} - 1824 \, a^{4} b^{7} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{14} \sqrt {-c} c^{5} + 7296 \, a^{6} b^{7} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{10} \sqrt {-c} c^{7} - 4352 \, a^{8} b^{7} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{6} \sqrt {-c} c^{9} - 3072 \, a^{10} b^{7} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2} \sqrt {-c} c^{11}\right )}}{{\left ({\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{4} + 4 \, a^{2} c^{2}\right )}^{6} a^{4}}}{24 \, b} \] Input:

integrate((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x^7,x, algorithm="giac")
 

Output:

1/24*(3*b^7*sqrt(-c)*arctan(1/2*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)* 
c + 2*a*c))^2/(a*c))/a^5 + 2*(3*b^7*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + 
 a)*c + 2*a*c))^22*sqrt(-c)*c + 68*a^2*b^7*(sqrt(b*x + a)*sqrt(-c) - sqrt( 
-(b*x + a)*c + 2*a*c))^18*sqrt(-c)*c^3 - 1824*a^4*b^7*(sqrt(b*x + a)*sqrt( 
-c) - sqrt(-(b*x + a)*c + 2*a*c))^14*sqrt(-c)*c^5 + 7296*a^6*b^7*(sqrt(b*x 
 + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^10*sqrt(-c)*c^7 - 4352*a^8*b^ 
7*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^6*sqrt(-c)*c^9 - 3 
072*a^10*b^7*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^2*sqrt( 
-c)*c^11)/(((sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^4 + 4*a^ 
2*c^2)^6*a^4))/b
 

Mupad [B] (verification not implemented)

Time = 9.76 (sec) , antiderivative size = 922, normalized size of antiderivative = 6.27 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^7} \, dx =\text {Too large to display} \] Input:

int(((a*c - b*c*x)^(1/2)*(a + b*x)^(1/2))/x^7,x)
 

Output:

((b^6*c^9*(a*c)^(1/2))/(24576*a^(11/2)) + (229*b^6*c*(a*c)^(1/2)*((a*c - b 
*c*x)^(1/2) - (a*c)^(1/2))^16)/(8192*a^(11/2)*((a + b*x)^(1/2) - a^(1/2))^ 
16) - (17*b^6*c^7*(a*c)^(1/2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^4)/(4096 
*a^(11/2)*((a + b*x)^(1/2) - a^(1/2))^4) + (139*b^6*c^6*(a*c)^(1/2)*((a*c 
- b*c*x)^(1/2) - (a*c)^(1/2))^6)/(6144*a^(11/2)*((a + b*x)^(1/2) - a^(1/2) 
)^6) - (23*b^6*c^5*(a*c)^(1/2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^8)/(102 
4*a^(11/2)*((a + b*x)^(1/2) - a^(1/2))^8) - (185*b^6*c^4*(a*c)^(1/2)*((a*c 
 - b*c*x)^(1/2) - (a*c)^(1/2))^10)/(1024*a^(11/2)*((a + b*x)^(1/2) - a^(1/ 
2))^10) + (901*b^6*c^3*(a*c)^(1/2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^12) 
/(4096*a^(11/2)*((a + b*x)^(1/2) - a^(1/2))^12) - (471*b^6*c^2*(a*c)^(1/2) 
*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^14)/(2048*a^(11/2)*((a + b*x)^(1/2) - 
 a^(1/2))^14))/(((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^18/((a + b*x)^(1/2) - 
a^(1/2))^18 - (6*c*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^16)/((a + b*x)^(1/2 
) - a^(1/2))^16 + (c^6*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^6)/((a + b*x)^( 
1/2) - a^(1/2))^6 - (6*c^5*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^8)/((a + b* 
x)^(1/2) - a^(1/2))^8 + (15*c^4*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^10)/(( 
a + b*x)^(1/2) - a^(1/2))^10 - (20*c^3*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)) 
^12)/((a + b*x)^(1/2) - a^(1/2))^12 + (15*c^2*((a*c - b*c*x)^(1/2) - (a*c) 
^(1/2))^14)/((a + b*x)^(1/2) - a^(1/2))^14) - (b^6*log(((c*(a - b*x))^(1/2 
) - (a*c)^(1/2))^2/((a + b*x)^(1/2) - a^(1/2))^2 - c)*(a*c)^(1/2))/(16*...
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.44 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x^7} \, dx=\frac {\sqrt {c}\, \left (-8 \sqrt {b x +a}\, \sqrt {-b x +a}\, a^{5}+2 \sqrt {b x +a}\, \sqrt {-b x +a}\, a^{3} b^{2} x^{2}+3 \sqrt {b x +a}\, \sqrt {-b x +a}\, a \,b^{4} x^{4}+3 \,\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right )}{2}\right )-1\right ) b^{6} x^{6}-3 \,\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right )}{2}\right )+1\right ) b^{6} x^{6}+3 \,\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right )}{2}\right )-1\right ) b^{6} x^{6}-3 \,\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right )}{2}\right )+1\right ) b^{6} x^{6}\right )}{48 a^{5} x^{6}} \] Input:

int((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/x^7,x)
 

Output:

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