\(\int \frac {\sqrt {c-d x} (c+d x)^{3/2}}{x^5} \, dx\) [256]

Optimal result

Integrand size = 23, antiderivative size = 124 \[ \int \frac {\sqrt {c-d x} (c+d x)^{3/2}}{x^5} \, dx=-\frac {c \sqrt {c-d x} \sqrt {c+d x}}{4 x^4}+\frac {d^2 \sqrt {c-d x} \sqrt {c+d x}}{8 c x^2}-\frac {d (c-d x)^{3/2} (c+d x)^{3/2}}{3 c^2 x^3}+\frac {d^4 \text {arctanh}\left (\frac {\sqrt {c-d x} \sqrt {c+d x}}{c}\right )}{8 c^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt {c-d x} (c+d x)^{3/2}}{x^5} \, dx=\frac {\frac {\sqrt {c-d x} \left (-6 c^4-14 c^3 d x-5 c^2 d^2 x^2+11 c d^3 x^3+8 d^4 x^4\right )}{x^4 \sqrt {c+d x}}+6 d^4 \text {arctanh}\left (\frac {\sqrt {c-d x}}{\sqrt {c+d x}}\right )}{24 c^2} \] Input:

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

Output:

((Sqrt[c - d*x]*(-6*c^4 - 14*c^3*d*x - 5*c^2*d^2*x^2 + 11*c*d^3*x^3 + 8*d^ 
4*x^4))/(x^4*Sqrt[c + d*x]) + 6*d^4*ArcTanh[Sqrt[c - d*x]/Sqrt[c + d*x]])/ 
(24*c^2)
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.34, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {107, 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.

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

\(\Big \downarrow \) 107

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 103

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

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 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 [A] (verified)

Time = 0.24 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.04

method	result	size
risch	\(-\frac {\sqrt {-x d +c}\, \sqrt {x d +c}\, \left (-8 x^{3} d^{3}-3 c \,d^{2} x^{2}+8 c^{2} d x +6 c^{3}\right )}{24 c^{2} x^{4}}+\frac {d^{4} \ln \left (\frac {2 c^{2}+2 \sqrt {c^{2}}\, \sqrt {-d^{2} x^{2}+c^{2}}}{x}\right ) \sqrt {\left (-x d +c \right ) \left (x d +c \right )}}{8 c \sqrt {c^{2}}\, \sqrt {-x d +c}\, \sqrt {x d +c}}\)	\(129\)
default	\(-\frac {\sqrt {-x d +c}\, \sqrt {x d +c}\, \left (-3 \ln \left (\frac {2 c \left (\sqrt {-d^{2} x^{2}+c^{2}}\, \operatorname {csgn}\left (c \right )+c \right )}{x}\right ) d^{4} x^{4}-8 \sqrt {-d^{2} x^{2}+c^{2}}\, \operatorname {csgn}\left (c \right ) d^{3} x^{3}-3 \,\operatorname {csgn}\left (c \right ) c \,d^{2} x^{2} \sqrt {-d^{2} x^{2}+c^{2}}+8 \,\operatorname {csgn}\left (c \right ) c^{2} d x \sqrt {-d^{2} x^{2}+c^{2}}+6 \,\operatorname {csgn}\left (c \right ) c^{3} \sqrt {-d^{2} x^{2}+c^{2}}\right ) \operatorname {csgn}\left (c \right )}{24 c^{2} \sqrt {-d^{2} x^{2}+c^{2}}\, x^{4}}\)	\(168\)

Input:

int((-d*x+c)^(1/2)*(d*x+c)^(3/2)/x^5,x,method=_RETURNVERBOSE)
 

Output:

-1/24*(-d*x+c)^(1/2)*(d*x+c)^(1/2)*(-8*d^3*x^3-3*c*d^2*x^2+8*c^2*d*x+6*c^3 
)/c^2/x^4+1/8*d^4/c/(c^2)^(1/2)*ln((2*c^2+2*(c^2)^(1/2)*(-d^2*x^2+c^2)^(1/ 
2))/x)*((-d*x+c)*(d*x+c))^(1/2)/(-d*x+c)^(1/2)/(d*x+c)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.72 \[ \int \frac {\sqrt {c-d x} (c+d x)^{3/2}}{x^5} \, dx=-\frac {3 \, d^{4} x^{4} \log \left (\frac {\sqrt {d x + c} \sqrt {-d x + c} - c}{x}\right ) - {\left (8 \, d^{3} x^{3} + 3 \, c d^{2} x^{2} - 8 \, c^{2} d x - 6 \, c^{3}\right )} \sqrt {d x + c} \sqrt {-d x + c}}{24 \, c^{2} x^{4}} \] Input:

integrate((-d*x+c)^(1/2)*(d*x+c)^(3/2)/x^5,x, algorithm="fricas")
 

Output:

-1/24*(3*d^4*x^4*log((sqrt(d*x + c)*sqrt(-d*x + c) - c)/x) - (8*d^3*x^3 + 
3*c*d^2*x^2 - 8*c^2*d*x - 6*c^3)*sqrt(d*x + c)*sqrt(-d*x + c))/(c^2*x^4)
 

Sympy [F]

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

integrate((-d*x+c)**(1/2)*(d*x+c)**(3/2)/x**5,x)
 

Output:

Integral(sqrt(c - d*x)*(c + d*x)**(3/2)/x**5, x)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {c-d x} (c+d x)^{3/2}}{x^5} \, dx=\frac {d^{4} \log \left (\frac {2 \, c^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-d^{2} x^{2} + c^{2}} c}{{\left | x \right |}}\right )}{8 \, c^{2}} - \frac {\sqrt {-d^{2} x^{2} + c^{2}} d^{4}}{8 \, c^{3}} - \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} d^{2}}{8 \, c^{3} x^{2}} - \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} d}{3 \, c^{2} x^{3}} - \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}}}{4 \, c x^{4}} \] Input:

integrate((-d*x+c)^(1/2)*(d*x+c)^(3/2)/x^5,x, algorithm="maxima")
 

Output:

1/8*d^4*log(2*c^2/abs(x) + 2*sqrt(-d^2*x^2 + c^2)*c/abs(x))/c^2 - 1/8*sqrt 
(-d^2*x^2 + c^2)*d^4/c^3 - 1/8*(-d^2*x^2 + c^2)^(3/2)*d^2/(c^3*x^2) - 1/3* 
(-d^2*x^2 + c^2)^(3/2)*d/(c^2*x^3) - 1/4*(-d^2*x^2 + c^2)^(3/2)/(c*x^4)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 458 vs. \(2 (100) = 200\).

Time = 0.43 (sec) , antiderivative size = 458, normalized size of antiderivative = 3.69 \[ \int \frac {\sqrt {c-d x} (c+d x)^{3/2}}{x^5} \, dx=\frac {\frac {3 \, d^{5} \log \left ({\left | -\frac {\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}{\sqrt {d x + c}} + \frac {\sqrt {d x + c}}{\sqrt {2} \sqrt {c} - \sqrt {-d x + c}} + 2 \right |}\right )}{c^{2}} - \frac {3 \, d^{5} \log \left ({\left | -\frac {\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}{\sqrt {d x + c}} + \frac {\sqrt {d x + c}}{\sqrt {2} \sqrt {c} - \sqrt {-d x + c}} - 2 \right |}\right )}{c^{2}} + \frac {4 \, {\left (3 \, d^{5} {\left (\frac {\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}{\sqrt {d x + c}} - \frac {\sqrt {d x + c}}{\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}\right )}^{7} - 44 \, d^{5} {\left (\frac {\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}{\sqrt {d x + c}} - \frac {\sqrt {d x + c}}{\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}\right )}^{5} + 848 \, d^{5} {\left (\frac {\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}{\sqrt {d x + c}} - \frac {\sqrt {d x + c}}{\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}\right )}^{3} + 192 \, d^{5} {\left (\frac {\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}{\sqrt {d x + c}} - \frac {\sqrt {d x + c}}{\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}\right )}\right )}}{{\left ({\left (\frac {\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}{\sqrt {d x + c}} - \frac {\sqrt {d x + c}}{\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}\right )}^{2} - 4\right )}^{4} c^{2}}}{24 \, d} \] Input:

integrate((-d*x+c)^(1/2)*(d*x+c)^(3/2)/x^5,x, algorithm="giac")
 

Output:

1/24*(3*d^5*log(abs(-(sqrt(2)*sqrt(c) - sqrt(-d*x + c))/sqrt(d*x + c) + sq 
rt(d*x + c)/(sqrt(2)*sqrt(c) - sqrt(-d*x + c)) + 2))/c^2 - 3*d^5*log(abs(- 
(sqrt(2)*sqrt(c) - sqrt(-d*x + c))/sqrt(d*x + c) + sqrt(d*x + c)/(sqrt(2)* 
sqrt(c) - sqrt(-d*x + c)) - 2))/c^2 + 4*(3*d^5*((sqrt(2)*sqrt(c) - sqrt(-d 
*x + c))/sqrt(d*x + c) - sqrt(d*x + c)/(sqrt(2)*sqrt(c) - sqrt(-d*x + c))) 
^7 - 44*d^5*((sqrt(2)*sqrt(c) - sqrt(-d*x + c))/sqrt(d*x + c) - sqrt(d*x + 
 c)/(sqrt(2)*sqrt(c) - sqrt(-d*x + c)))^5 + 848*d^5*((sqrt(2)*sqrt(c) - sq 
rt(-d*x + c))/sqrt(d*x + c) - sqrt(d*x + c)/(sqrt(2)*sqrt(c) - sqrt(-d*x + 
 c)))^3 + 192*d^5*((sqrt(2)*sqrt(c) - sqrt(-d*x + c))/sqrt(d*x + c) - sqrt 
(d*x + c)/(sqrt(2)*sqrt(c) - sqrt(-d*x + c))))/((((sqrt(2)*sqrt(c) - sqrt( 
-d*x + c))/sqrt(d*x + c) - sqrt(d*x + c)/(sqrt(2)*sqrt(c) - sqrt(-d*x + c) 
))^2 - 4)^4*c^2))/d
 

Mupad [F(-1)]

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

int(((c + d*x)^(3/2)*(c - d*x)^(1/2))/x^5,x)
 

Output:

int(((c + d*x)^(3/2)*(c - d*x)^(1/2))/x^5, x)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.82 \[ \int \frac {\sqrt {c-d x} (c+d x)^{3/2}}{x^5} \, dx=\frac {-6 \sqrt {d x +c}\, \sqrt {-d x +c}\, c^{3}-8 \sqrt {d x +c}\, \sqrt {-d x +c}\, c^{2} d x +3 \sqrt {d x +c}\, \sqrt {-d x +c}\, c \,d^{2} x^{2}+8 \sqrt {d x +c}\, \sqrt {-d x +c}\, d^{3} x^{3}+3 \,\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-d x +c}}{\sqrt {c}\, \sqrt {2}}\right )}{2}\right )-1\right ) d^{4} x^{4}-3 \,\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-d x +c}}{\sqrt {c}\, \sqrt {2}}\right )}{2}\right )+1\right ) d^{4} x^{4}+3 \,\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-d x +c}}{\sqrt {c}\, \sqrt {2}}\right )}{2}\right )-1\right ) d^{4} x^{4}-3 \,\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-d x +c}}{\sqrt {c}\, \sqrt {2}}\right )}{2}\right )+1\right ) d^{4} x^{4}}{24 c^{2} x^{4}} \] Input:

int((-d*x+c)^(1/2)*(d*x+c)^(3/2)/x^5,x)
 

Output:

( - 6*sqrt(c + d*x)*sqrt(c - d*x)*c**3 - 8*sqrt(c + d*x)*sqrt(c - d*x)*c** 
2*d*x + 3*sqrt(c + d*x)*sqrt(c - d*x)*c*d**2*x**2 + 8*sqrt(c + d*x)*sqrt(c 
 - d*x)*d**3*x**3 + 3*log( - sqrt(2) + tan(asin(sqrt(c - d*x)/(sqrt(c)*sqr 
t(2)))/2) - 1)*d**4*x**4 - 3*log( - sqrt(2) + tan(asin(sqrt(c - d*x)/(sqrt 
(c)*sqrt(2)))/2) + 1)*d**4*x**4 + 3*log(sqrt(2) + tan(asin(sqrt(c - d*x)/( 
sqrt(c)*sqrt(2)))/2) - 1)*d**4*x**4 - 3*log(sqrt(2) + tan(asin(sqrt(c - d* 
x)/(sqrt(c)*sqrt(2)))/2) + 1)*d**4*x**4)/(24*c**2*x**4)