\(\int \frac {\sqrt {c-d x} (c+d x)^{3/2}}{x^4} \, dx\) [255]

Optimal result

Integrand size = 23, antiderivative size = 91 \[ \int \frac {\sqrt {c-d x} (c+d x)^{3/2}}{x^4} \, dx=-\frac {d \sqrt {c-d x} \sqrt {c+d x}}{2 x^2}-\frac {(c-d x)^{3/2} (c+d x)^{3/2}}{3 c x^3}+\frac {d^3 \text {arctanh}\left (\frac {\sqrt {c-d x} \sqrt {c+d x}}{c}\right )}{2 c} \] Output:

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {c-d x} (c+d x)^{3/2}}{x^4} \, dx=\frac {-\frac {\sqrt {c-d x} \left (2 c^3+5 c^2 d x+c d^2 x^2-2 d^3 x^3\right )}{x^3 \sqrt {c+d x}}+6 d^3 \text {arctanh}\left (\frac {\sqrt {c-d x}}{\sqrt {c+d x}}\right )}{6 c} \] Input:

Integrate[(Sqrt[c - d*x]*(c + d*x)^(3/2))/x^4,x]
 

Output:

(-((Sqrt[c - d*x]*(2*c^3 + 5*c^2*d*x + c*d^2*x^2 - 2*d^3*x^3))/(x^3*Sqrt[c 
 + d*x])) + 6*d^3*ArcTanh[Sqrt[c - d*x]/Sqrt[c + d*x]])/(6*c)
 

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.41, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {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.

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 103

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

\(\Big \downarrow \) 221

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

Input:

Int[(Sqrt[c - d*x]*(c + d*x)^(3/2))/x^4,x]
 

Output:

-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
 

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 = 0.23 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.26

method	result	size
risch	\(-\frac {\sqrt {-x d +c}\, \sqrt {x d +c}\, \left (-2 d^{2} x^{2}+3 c d x +2 c^{2}\right )}{6 x^{3} c}+\frac {d^{3} \ln \left (\frac {2 c^{2}+2 \sqrt {c^{2}}\, \sqrt {-d^{2} x^{2}+c^{2}}}{x}\right ) \sqrt {\left (-x d +c \right ) \left (x d +c \right )}}{2 \sqrt {c^{2}}\, \sqrt {-x d +c}\, \sqrt {x d +c}}\)	\(115\)
default	\(-\frac {\sqrt {-x d +c}\, \sqrt {x d +c}\, \left (-3 \ln \left (\frac {2 c \left (\sqrt {-d^{2} x^{2}+c^{2}}\, \operatorname {csgn}\left (c \right )+c \right )}{x}\right ) d^{3} x^{3}-2 \,\operatorname {csgn}\left (c \right ) d^{2} x^{2} \sqrt {-d^{2} x^{2}+c^{2}}+3 \,\operatorname {csgn}\left (c \right ) c d x \sqrt {-d^{2} x^{2}+c^{2}}+2 \,\operatorname {csgn}\left (c \right ) c^{2} \sqrt {-d^{2} x^{2}+c^{2}}\right ) \operatorname {csgn}\left (c \right )}{6 c \sqrt {-d^{2} x^{2}+c^{2}}\, x^{3}}\)	\(141\)

Input:

int((-d*x+c)^(1/2)*(d*x+c)^(3/2)/x^4,x,method=_RETURNVERBOSE)
 

Output:

-1/6*(-d*x+c)^(1/2)*(d*x+c)^(1/2)*(-2*d^2*x^2+3*c*d*x+2*c^2)/x^3/c+1/2*d^3 
/(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 = 78, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {c-d x} (c+d x)^{3/2}}{x^4} \, dx=-\frac {3 \, d^{3} x^{3} \log \left (\frac {\sqrt {d x + c} \sqrt {-d x + c} - c}{x}\right ) - {\left (2 \, d^{2} x^{2} - 3 \, c d x - 2 \, c^{2}\right )} \sqrt {d x + c} \sqrt {-d x + c}}{6 \, c x^{3}} \] Input:

integrate((-d*x+c)^(1/2)*(d*x+c)^(3/2)/x^4,x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {\sqrt {c-d x} (c+d x)^{3/2}}{x^4} \, dx=\int \frac {\sqrt {c - d x} \left (c + d x\right )^{\frac {3}{2}}}{x^{4}}\, dx \] Input:

integrate((-d*x+c)**(1/2)*(d*x+c)**(3/2)/x**4,x)
 

Output:

Integral(sqrt(c - d*x)*(c + d*x)**(3/2)/x**4, x)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.19 \[ \int \frac {\sqrt {c-d x} (c+d x)^{3/2}}{x^4} \, dx=\frac {d^{3} \log \left (\frac {2 \, c^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-d^{2} x^{2} + c^{2}} c}{{\left | x \right |}}\right )}{2 \, c} - \frac {\sqrt {-d^{2} x^{2} + c^{2}} d^{3}}{2 \, c^{2}} - \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} d}{2 \, c^{2} x^{2}} - \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}}}{3 \, c x^{3}} \] Input:

integrate((-d*x+c)^(1/2)*(d*x+c)^(3/2)/x^4,x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 395 vs. \(2 (73) = 146\).

Time = 0.39 (sec) , antiderivative size = 395, normalized size of antiderivative = 4.34 \[ \int \frac {\sqrt {c-d x} (c+d x)^{3/2}}{x^4} \, dx=\frac {\frac {3 \, d^{4} \log \left ({\left | -\frac {\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}{\sqrt {d x + c}} + \frac {\sqrt {d x + c}}{\sqrt {2} \sqrt {c} - \sqrt {-d x + c}} + 2 \right |}\right )}{c} - \frac {3 \, d^{4} \log \left ({\left | -\frac {\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}{\sqrt {d x + c}} + \frac {\sqrt {d x + c}}{\sqrt {2} \sqrt {c} - \sqrt {-d x + c}} - 2 \right |}\right )}{c} + \frac {4 \, {\left (3 \, d^{4} {\left (\frac {\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}{\sqrt {d x + c}} - \frac {\sqrt {d x + c}}{\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}\right )}^{5} - 32 \, d^{4} {\left (\frac {\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}{\sqrt {d x + c}} - \frac {\sqrt {d x + c}}{\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}\right )}^{3} - 48 \, d^{4} {\left (\frac {\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}{\sqrt {d x + c}} - \frac {\sqrt {d x + c}}{\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}\right )}\right )}}{{\left ({\left (\frac {\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}{\sqrt {d x + c}} - \frac {\sqrt {d x + c}}{\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}\right )}^{2} - 4\right )}^{3} c}}{6 \, d} \] Input:

integrate((-d*x+c)^(1/2)*(d*x+c)^(3/2)/x^4,x, algorithm="giac")
 

Output:

1/6*(3*d^4*log(abs(-(sqrt(2)*sqrt(c) - sqrt(-d*x + c))/sqrt(d*x + c) + sqr 
t(d*x + c)/(sqrt(2)*sqrt(c) - sqrt(-d*x + c)) + 2))/c - 3*d^4*log(abs(-(sq 
rt(2)*sqrt(c) - sqrt(-d*x + c))/sqrt(d*x + c) + sqrt(d*x + c)/(sqrt(2)*sqr 
t(c) - sqrt(-d*x + c)) - 2))/c + 4*(3*d^4*((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 - 
32*d^4*((sqrt(2)*sqrt(c) - sqrt(-d*x + c))/sqrt(d*x + c) - sqrt(d*x + c)/( 
sqrt(2)*sqrt(c) - sqrt(-d*x + c)))^3 - 48*d^4*((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)^3*c))/d
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c-d x} (c+d x)^{3/2}}{x^4} \, dx=\int \frac {{\left (c+d\,x\right )}^{3/2}\,\sqrt {c-d\,x}}{x^4} \,d x \] Input:

int(((c + d*x)^(3/2)*(c - d*x)^(1/2))/x^4,x)
 

Output:

int(((c + d*x)^(3/2)*(c - d*x)^(1/2))/x^4, x)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 202, normalized size of antiderivative = 2.22 \[ \int \frac {\sqrt {c-d x} (c+d x)^{3/2}}{x^4} \, dx=\frac {-2 \sqrt {d x +c}\, \sqrt {-d x +c}\, c^{2}-3 \sqrt {d x +c}\, \sqrt {-d x +c}\, c d x +2 \sqrt {d x +c}\, \sqrt {-d x +c}\, d^{2} x^{2}+3 \,\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-d x +c}}{\sqrt {c}\, \sqrt {2}}\right )}{2}\right )-1\right ) d^{3} x^{3}-3 \,\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-d x +c}}{\sqrt {c}\, \sqrt {2}}\right )}{2}\right )+1\right ) d^{3} x^{3}+3 \,\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-d x +c}}{\sqrt {c}\, \sqrt {2}}\right )}{2}\right )-1\right ) d^{3} x^{3}-3 \,\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-d x +c}}{\sqrt {c}\, \sqrt {2}}\right )}{2}\right )+1\right ) d^{3} x^{3}}{6 c \,x^{3}} \] Input:

int((-d*x+c)^(1/2)*(d*x+c)^(3/2)/x^4,x)
 

Output:

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