\(\int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^5} \, dx\) [100]

Optimal result

Integrand size = 26, antiderivative size = 110 \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^5} \, dx=\frac {3 b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}{8 x^2}-\frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{4 x^4}-\frac {3 b^4 c^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{a \sqrt {c}}\right )}{8 a} \] Output:

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

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.19 \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^5} \, dx=-\frac {(c (a-b x))^{3/2} \left (a \sqrt {a-b x} \sqrt {a+b x} \left (2 a^2-5 b^2 x^2\right )-3 b^4 x^4 \log \left (-1+\frac {\sqrt {a+b x}}{\sqrt {a-b x}}\right )+3 b^4 x^4 \log \left (a+\frac {a \sqrt {a+b x}}{\sqrt {a-b x}}\right )\right )}{8 a x^4 (a-b x)^{3/2}} \] Input:

Integrate[((a + b*x)^(3/2)*(a*c - b*c*x)^(3/2))/x^5,x]
 

Output:

-1/8*((c*(a - b*x))^(3/2)*(a*Sqrt[a - b*x]*Sqrt[a + b*x]*(2*a^2 - 5*b^2*x^ 
2) - 3*b^4*x^4*Log[-1 + Sqrt[a + b*x]/Sqrt[a - b*x]] + 3*b^4*x^4*Log[a + ( 
a*Sqrt[a + b*x])/Sqrt[a - b*x]]))/(a*x^4*(a - b*x)^(3/2))
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.76, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {105, 105, 105, 105, 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.

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

\(\Big \downarrow \) 105

\(\displaystyle \frac {3}{4} b \int \frac {\sqrt {a+b x} (a c-b c x)^{3/2}}{x^4}dx-\frac {(a+b x)^{3/2} (a c-b c x)^{5/2}}{4 a c x^4}\)

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 103

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

\(\Big \downarrow \) 221

\(\displaystyle \frac {3}{4} b \left (\frac {1}{3} b \left (-\frac {3}{2} b c \left (\frac {b \sqrt {c} \text {arctanh}\left (\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{a \sqrt {c}}\right )}{a}-\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{a x}\right )-\frac {\sqrt {a+b x} (a c-b c x)^{3/2}}{2 a x^2}\right )-\frac {\sqrt {a+b x} (a c-b c x)^{5/2}}{3 a c x^3}\right )-\frac {(a+b x)^{3/2} (a c-b c x)^{5/2}}{4 a c x^4}\)

Input:

Int[((a + b*x)^(3/2)*(a*c - b*c*x)^(3/2))/x^5,x]
 

Output:

-1/4*((a + b*x)^(3/2)*(a*c - b*c*x)^(5/2))/(a*c*x^4) + (3*b*(-1/3*(Sqrt[a 
+ b*x]*(a*c - b*c*x)^(5/2))/(a*c*x^3) + (b*(-1/2*(Sqrt[a + b*x]*(a*c - b*c 
*x)^(3/2))/(a*x^2) - (3*b*c*(-((Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/(a*x)) + 
(b*Sqrt[c]*ArcTanh[(Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/(a*Sqrt[c])])/a))/2)) 
/3))/4
 

Defintions of rubi rules used

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 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 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 = 0.24 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.25

method	result	size
risch	\(-\frac {\left (-b x +a \right ) \sqrt {b x +a}\, \left (-5 b^{2} x^{2}+2 a^{2}\right ) c^{2}}{8 x^{4} \sqrt {-c \left (b x -a \right )}}-\frac {3 b^{4} \ln \left (\frac {2 a^{2} c +2 \sqrt {a^{2} c}\, \sqrt {-b^{2} c \,x^{2}+a^{2} c}}{x}\right ) \sqrt {-\left (b x +a \right ) c \left (b x -a \right )}\, c^{2}}{8 \sqrt {a^{2} c}\, \sqrt {b x +a}\, \sqrt {-c \left (b x -a \right )}}\)	\(138\)
default	\(-\frac {\sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}\, c \left (3 \ln \left (\frac {2 a^{2} c +2 \sqrt {a^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}}{x}\right ) b^{4} c \,x^{4}-5 b^{2} x^{2} \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {a^{2} c}+2 \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {a^{2} c}\, a^{2}\right )}{8 \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, x^{4} \sqrt {a^{2} c}}\)	\(152\)

Input:

int((b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)/x^5,x,method=_RETURNVERBOSE)
 

Output:

-1/8*(-b*x+a)*(b*x+a)^(1/2)*(-5*b^2*x^2+2*a^2)/x^4/(-c*(b*x-a))^(1/2)*c^2- 
3/8*b^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^2
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.87 \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^5} \, dx=\left [\frac {3 \, b^{4} c^{\frac {3}{2}} x^{4} \log \left (-\frac {b^{2} c x^{2} - 2 \, a^{2} c + 2 \, \sqrt {-b c x + a c} \sqrt {b x + a} a \sqrt {c}}{x^{2}}\right ) + 2 \, {\left (5 \, a b^{2} c x^{2} - 2 \, a^{3} c\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{16 \, a x^{4}}, -\frac {3 \, b^{4} \sqrt {-c} c x^{4} \arctan \left (\frac {\sqrt {-b c x + a c} \sqrt {b x + a} a \sqrt {-c}}{b^{2} c x^{2} - a^{2} c}\right ) - {\left (5 \, a b^{2} c x^{2} - 2 \, a^{3} c\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{8 \, a x^{4}}\right ] \] Input:

integrate((b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)/x^5,x, algorithm="fricas")
 

Output:

[1/16*(3*b^4*c^(3/2)*x^4*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*(5*a*b^2*c*x^2 - 2*a^3*c)*sqrt(-b*c*x + 
a*c)*sqrt(b*x + a))/(a*x^4), -1/8*(3*b^4*sqrt(-c)*c*x^4*arctan(sqrt(-b*c*x 
 + a*c)*sqrt(b*x + a)*a*sqrt(-c)/(b^2*c*x^2 - a^2*c)) - (5*a*b^2*c*x^2 - 2 
*a^3*c)*sqrt(-b*c*x + a*c)*sqrt(b*x + a))/(a*x^4)]
 

Sympy [F]

\[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^5} \, dx=\int \frac {\left (- c \left (- a + b x\right )\right )^{\frac {3}{2}} \left (a + b x\right )^{\frac {3}{2}}}{x^{5}}\, dx \] Input:

integrate((b*x+a)**(3/2)*(-b*c*x+a*c)**(3/2)/x**5,x)
 

Output:

Integral((-c*(-a + b*x))**(3/2)*(a + b*x)**(3/2)/x**5, x)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.46 \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^5} \, dx=-\frac {3 \, b^{4} c^{\frac {3}{2}} \log \left (\frac {2 \, a^{2} c}{{\left | x \right |}} + \frac {2 \, \sqrt {-b^{2} c x^{2} + a^{2} c} a \sqrt {c}}{{\left | x \right |}}\right )}{8 \, a} + \frac {3 \, \sqrt {-b^{2} c x^{2} + a^{2} c} b^{4} c}{8 \, a^{2}} + \frac {{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} b^{4}}{8 \, a^{4}} + \frac {{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {5}{2}} b^{2}}{8 \, a^{4} c x^{2}} - \frac {{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {5}{2}}}{4 \, a^{2} c x^{4}} \] Input:

integrate((b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)/x^5,x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 302 vs. \(2 (88) = 176\).

Time = 0.22 (sec) , antiderivative size = 302, normalized size of antiderivative = 2.75 \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^5} \, dx=-\frac {\frac {3 \, b^{5} \sqrt {-c} c \arctan \left (\frac {{\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2}}{2 \, a c}\right )}{a} - \frac {2 \, {\left (5 \, b^{5} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{14} \sqrt {-c} c^{2} - 12 \, a^{2} b^{5} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{10} \sqrt {-c} c^{4} + 48 \, a^{4} b^{5} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{6} \sqrt {-c} c^{6} - 320 \, a^{6} b^{5} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2} \sqrt {-c} c^{8}\right )}}{{\left ({\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{4} + 4 \, a^{2} c^{2}\right )}^{4}}}{4 \, b} \] Input:

integrate((b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)/x^5,x, algorithm="giac")
 

Output:

-1/4*(3*b^5*sqrt(-c)*c*arctan(1/2*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a 
)*c + 2*a*c))^2/(a*c))/a - 2*(5*b^5*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + 
 a)*c + 2*a*c))^14*sqrt(-c)*c^2 - 12*a^2*b^5*(sqrt(b*x + a)*sqrt(-c) - sqr 
t(-(b*x + a)*c + 2*a*c))^10*sqrt(-c)*c^4 + 48*a^4*b^5*(sqrt(b*x + a)*sqrt( 
-c) - sqrt(-(b*x + a)*c + 2*a*c))^6*sqrt(-c)*c^6 - 320*a^6*b^5*(sqrt(b*x + 
 a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^2*sqrt(-c)*c^8)/((sqrt(b*x + a) 
*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^4 + 4*a^2*c^2)^4)/b
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^5} \, dx=\int \frac {{\left (a\,c-b\,c\,x\right )}^{3/2}\,{\left (a+b\,x\right )}^{3/2}}{x^5} \,d x \] Input:

int(((a*c - b*c*x)^(3/2)*(a + b*x)^(3/2))/x^5,x)
 

Output:

int(((a*c - b*c*x)^(3/2)*(a + b*x)^(3/2))/x^5, x)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.71 \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x^5} \, dx=\frac {\sqrt {c}\, c \left (-2 \sqrt {b x +a}\, \sqrt {-b x +a}\, a^{3}+5 \sqrt {b x +a}\, \sqrt {-b x +a}\, a \,b^{2} x^{2}-3 \,\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right )}{2}\right )-1\right ) b^{4} x^{4}+3 \,\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right )}{2}\right )+1\right ) b^{4} x^{4}-3 \,\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right )}{2}\right )-1\right ) b^{4} x^{4}+3 \,\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right )}{2}\right )+1\right ) b^{4} x^{4}\right )}{8 a \,x^{4}} \] Input:

int((b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)/x^5,x)
 

Output:

(sqrt(c)*c*( - 2*sqrt(a + b*x)*sqrt(a - b*x)*a**3 + 5*sqrt(a + b*x)*sqrt(a 
 - b*x)*a*b**2*x**2 - 3*log( - sqrt(2) + tan(asin(sqrt(a - b*x)/(sqrt(a)*s 
qrt(2)))/2) - 1)*b**4*x**4 + 3*log( - sqrt(2) + tan(asin(sqrt(a - b*x)/(sq 
rt(a)*sqrt(2)))/2) + 1)*b**4*x**4 - 3*log(sqrt(2) + tan(asin(sqrt(a - b*x) 
/(sqrt(a)*sqrt(2)))/2) - 1)*b**4*x**4 + 3*log(sqrt(2) + tan(asin(sqrt(a - 
b*x)/(sqrt(a)*sqrt(2)))/2) + 1)*b**4*x**4))/(8*a*x**4)