\(\int \frac {\sqrt {c-d x} (c+d x)^{3/2}}{x^6} \, dx\) [257]

Optimal result

Integrand size = 23, antiderivative size = 155 \[ \int \frac {\sqrt {c-d x} (c+d x)^{3/2}}{x^6} \, dx=-\frac {d \sqrt {c-d x} \sqrt {c+d x}}{4 x^4}+\frac {d^3 \sqrt {c-d x} \sqrt {c+d x}}{8 c^2 x^2}-\frac {(c-d x)^{3/2} (c+d x)^{3/2}}{5 c x^5}-\frac {2 d^2 (c-d x)^{3/2} (c+d x)^{3/2}}{15 c^3 x^3}+\frac {d^5 \text {arctanh}\left (\frac {\sqrt {c-d x} \sqrt {c+d x}}{c}\right )}{8 c^3} \] Output:

-1/4*d*(-d*x+c)^(1/2)*(d*x+c)^(1/2)/x^4+1/8*d^3*(-d*x+c)^(1/2)*(d*x+c)^(1/ 
2)/c^2/x^2-1/5*(-d*x+c)^(3/2)*(d*x+c)^(3/2)/c/x^5-2/15*d^2*(-d*x+c)^(3/2)* 
(d*x+c)^(3/2)/c^3/x^3+1/8*d^5*arctanh((-d*x+c)^(1/2)*(d*x+c)^(1/2)/c)/c^3
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {c-d x} (c+d x)^{3/2}}{x^6} \, dx=\frac {\frac {\sqrt {c-d x} \left (-24 c^5-54 c^4 d x-22 c^3 d^2 x^2+23 c^2 d^3 x^3+31 c d^4 x^4+16 d^5 x^5\right )}{x^5 \sqrt {c+d x}}+30 d^5 \text {arctanh}\left (\frac {\sqrt {c-d x}}{\sqrt {c+d x}}\right )}{120 c^3} \] Input:

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

Output:

((Sqrt[c - d*x]*(-24*c^5 - 54*c^4*d*x - 22*c^3*d^2*x^2 + 23*c^2*d^3*x^3 + 
31*c*d^4*x^4 + 16*d^5*x^5))/(x^5*Sqrt[c + d*x]) + 30*d^5*ArcTanh[Sqrt[c - 
d*x]/Sqrt[c + d*x]])/(120*c^3)
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.28, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {108, 27, 166, 25, 27, 168, 25, 27, 168, 25, 27, 168, 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.

\(\displaystyle \int \frac {\sqrt {c-d x} (c+d x)^{3/2}}{x^6} \, dx\)

\(\Big \downarrow \) 108

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 166

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{5} d \left (-\frac {1}{4} d \left (-\frac {\int -\frac {c d (15 c+8 d x)}{x^3 \sqrt {c-d x} \sqrt {c+d x}}dx}{3 c^2}-\frac {4 \sqrt {c-d x} \sqrt {c+d x}}{3 c x^3}\right )-\frac {\sqrt {c-d x} \sqrt {c+d x}}{4 x^4}\right )-\frac {\sqrt {c-d x} (c+d x)^{3/2}}{5 x^5}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{5} d \left (-\frac {1}{4} d \left (\frac {\int \frac {c d (15 c+8 d x)}{x^3 \sqrt {c-d x} \sqrt {c+d x}}dx}{3 c^2}-\frac {4 \sqrt {c-d x} \sqrt {c+d x}}{3 c x^3}\right )-\frac {\sqrt {c-d x} \sqrt {c+d x}}{4 x^4}\right )-\frac {\sqrt {c-d x} (c+d x)^{3/2}}{5 x^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{5} d \left (-\frac {1}{4} d \left (\frac {d \int \frac {15 c+8 d x}{x^3 \sqrt {c-d x} \sqrt {c+d x}}dx}{3 c}-\frac {4 \sqrt {c-d x} \sqrt {c+d x}}{3 c x^3}\right )-\frac {\sqrt {c-d x} \sqrt {c+d x}}{4 x^4}\right )-\frac {\sqrt {c-d x} (c+d x)^{3/2}}{5 x^5}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{5} d \left (-\frac {1}{4} d \left (\frac {d \left (-\frac {\int -\frac {c d (16 c+15 d x)}{x^2 \sqrt {c-d x} \sqrt {c+d x}}dx}{2 c^2}-\frac {15 \sqrt {c-d x} \sqrt {c+d x}}{2 c x^2}\right )}{3 c}-\frac {4 \sqrt {c-d x} \sqrt {c+d x}}{3 c x^3}\right )-\frac {\sqrt {c-d x} \sqrt {c+d x}}{4 x^4}\right )-\frac {\sqrt {c-d x} (c+d x)^{3/2}}{5 x^5}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{5} d \left (-\frac {1}{4} d \left (\frac {d \left (\frac {\int \frac {c d (16 c+15 d x)}{x^2 \sqrt {c-d x} \sqrt {c+d x}}dx}{2 c^2}-\frac {15 \sqrt {c-d x} \sqrt {c+d x}}{2 c x^2}\right )}{3 c}-\frac {4 \sqrt {c-d x} \sqrt {c+d x}}{3 c x^3}\right )-\frac {\sqrt {c-d x} \sqrt {c+d x}}{4 x^4}\right )-\frac {\sqrt {c-d x} (c+d x)^{3/2}}{5 x^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{5} d \left (-\frac {1}{4} d \left (\frac {d \left (\frac {d \int \frac {16 c+15 d x}{x^2 \sqrt {c-d x} \sqrt {c+d x}}dx}{2 c}-\frac {15 \sqrt {c-d x} \sqrt {c+d x}}{2 c x^2}\right )}{3 c}-\frac {4 \sqrt {c-d x} \sqrt {c+d x}}{3 c x^3}\right )-\frac {\sqrt {c-d x} \sqrt {c+d x}}{4 x^4}\right )-\frac {\sqrt {c-d x} (c+d x)^{3/2}}{5 x^5}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{5} d \left (-\frac {1}{4} d \left (\frac {d \left (\frac {d \left (-\frac {\int -\frac {15 c^2 d}{x \sqrt {c-d x} \sqrt {c+d x}}dx}{c^2}-\frac {16 \sqrt {c-d x} \sqrt {c+d x}}{c x}\right )}{2 c}-\frac {15 \sqrt {c-d x} \sqrt {c+d x}}{2 c x^2}\right )}{3 c}-\frac {4 \sqrt {c-d x} \sqrt {c+d x}}{3 c x^3}\right )-\frac {\sqrt {c-d x} \sqrt {c+d x}}{4 x^4}\right )-\frac {\sqrt {c-d x} (c+d x)^{3/2}}{5 x^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{5} d \left (-\frac {1}{4} d \left (\frac {d \left (\frac {d \left (15 d \int \frac {1}{x \sqrt {c-d x} \sqrt {c+d x}}dx-\frac {16 \sqrt {c-d x} \sqrt {c+d x}}{c x}\right )}{2 c}-\frac {15 \sqrt {c-d x} \sqrt {c+d x}}{2 c x^2}\right )}{3 c}-\frac {4 \sqrt {c-d x} \sqrt {c+d x}}{3 c x^3}\right )-\frac {\sqrt {c-d x} \sqrt {c+d x}}{4 x^4}\right )-\frac {\sqrt {c-d x} (c+d x)^{3/2}}{5 x^5}\)

\(\Big \downarrow \) 103

\(\displaystyle \frac {1}{5} d \left (-\frac {1}{4} d \left (\frac {d \left (\frac {d \left (-15 d^2 \int \frac {1}{c^2 d-d (c-d x) (c+d x)}d\left (\sqrt {c-d x} \sqrt {c+d x}\right )-\frac {16 \sqrt {c-d x} \sqrt {c+d x}}{c x}\right )}{2 c}-\frac {15 \sqrt {c-d x} \sqrt {c+d x}}{2 c x^2}\right )}{3 c}-\frac {4 \sqrt {c-d x} \sqrt {c+d x}}{3 c x^3}\right )-\frac {\sqrt {c-d x} \sqrt {c+d x}}{4 x^4}\right )-\frac {\sqrt {c-d x} (c+d x)^{3/2}}{5 x^5}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {1}{5} d \left (-\frac {1}{4} d \left (\frac {d \left (\frac {d \left (-\frac {15 d \text {arctanh}\left (\frac {\sqrt {c-d x} \sqrt {c+d x}}{c}\right )}{c}-\frac {16 \sqrt {c-d x} \sqrt {c+d x}}{c x}\right )}{2 c}-\frac {15 \sqrt {c-d x} \sqrt {c+d x}}{2 c x^2}\right )}{3 c}-\frac {4 \sqrt {c-d x} \sqrt {c+d x}}{3 c x^3}\right )-\frac {\sqrt {c-d x} \sqrt {c+d x}}{4 x^4}\right )-\frac {\sqrt {c-d x} (c+d x)^{3/2}}{5 x^5}\)

Input:

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

Output:

-1/5*(Sqrt[c - d*x]*(c + d*x)^(3/2))/x^5 + (d*(-1/4*(Sqrt[c - d*x]*Sqrt[c 
+ d*x])/x^4 - (d*((-4*Sqrt[c - d*x]*Sqrt[c + d*x])/(3*c*x^3) + (d*((-15*Sq 
rt[c - d*x]*Sqrt[c + d*x])/(2*c*x^2) + (d*((-16*Sqrt[c - d*x]*Sqrt[c + d*x 
])/(c*x) - (15*d*ArcTanh[(Sqrt[c - d*x]*Sqrt[c + d*x])/c])/c))/(2*c)))/(3* 
c)))/4))/5
 

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 166

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - 
a*f)*(m + 1))   Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* 
c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h 
)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, 
e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]
 

rule 168

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(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] + S 
imp[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*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[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 = 140, normalized size of antiderivative = 0.90

method	result	size
risch	\(-\frac {\sqrt {-x d +c}\, \sqrt {x d +c}\, \left (-16 x^{4} d^{4}-15 c \,x^{3} d^{3}-8 c^{2} x^{2} d^{2}+30 c^{3} x d +24 c^{4}\right )}{120 x^{5} c^{3}}+\frac {d^{5} \ln \left (\frac {2 c^{2}+2 \sqrt {c^{2}}\, \sqrt {-d^{2} x^{2}+c^{2}}}{x}\right ) \sqrt {\left (-x d +c \right ) \left (x d +c \right )}}{8 c^{2} \sqrt {c^{2}}\, \sqrt {-x d +c}\, \sqrt {x d +c}}\)	\(140\)
default	\(-\frac {\sqrt {-x d +c}\, \sqrt {x d +c}\, \left (-15 \ln \left (\frac {2 c \left (\sqrt {-d^{2} x^{2}+c^{2}}\, \operatorname {csgn}\left (c \right )+c \right )}{x}\right ) d^{5} x^{5}-16 \sqrt {-d^{2} x^{2}+c^{2}}\, \operatorname {csgn}\left (c \right ) d^{4} x^{4}-15 \,\operatorname {csgn}\left (c \right ) c \sqrt {-d^{2} x^{2}+c^{2}}\, d^{3} x^{3}-8 \,\operatorname {csgn}\left (c \right ) c^{2} d^{2} x^{2} \sqrt {-d^{2} x^{2}+c^{2}}+30 \,\operatorname {csgn}\left (c \right ) c^{3} d x \sqrt {-d^{2} x^{2}+c^{2}}+24 \,\operatorname {csgn}\left (c \right ) c^{4} \sqrt {-d^{2} x^{2}+c^{2}}\right ) \operatorname {csgn}\left (c \right )}{120 c^{3} \sqrt {-d^{2} x^{2}+c^{2}}\, x^{5}}\)	\(195\)

Input:

int((-d*x+c)^(1/2)*(d*x+c)^(3/2)/x^6,x,method=_RETURNVERBOSE)
 

Output:

-1/120*(-d*x+c)^(1/2)*(d*x+c)^(1/2)*(-16*d^4*x^4-15*c*d^3*x^3-8*c^2*d^2*x^ 
2+30*c^3*d*x+24*c^4)/x^5/c^3+1/8*d^5/c^2/(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 = 100, normalized size of antiderivative = 0.65 \[ \int \frac {\sqrt {c-d x} (c+d x)^{3/2}}{x^6} \, dx=-\frac {15 \, d^{5} x^{5} \log \left (\frac {\sqrt {d x + c} \sqrt {-d x + c} - c}{x}\right ) - {\left (16 \, d^{4} x^{4} + 15 \, c d^{3} x^{3} + 8 \, c^{2} d^{2} x^{2} - 30 \, c^{3} d x - 24 \, c^{4}\right )} \sqrt {d x + c} \sqrt {-d x + c}}{120 \, c^{3} x^{5}} \] Input:

integrate((-d*x+c)^(1/2)*(d*x+c)^(3/2)/x^6,x, algorithm="fricas")
 

Output:

-1/120*(15*d^5*x^5*log((sqrt(d*x + c)*sqrt(-d*x + c) - c)/x) - (16*d^4*x^4 
 + 15*c*d^3*x^3 + 8*c^2*d^2*x^2 - 30*c^3*d*x - 24*c^4)*sqrt(d*x + c)*sqrt( 
-d*x + c))/(c^3*x^5)
 

Sympy [F]

\[ \int \frac {\sqrt {c-d x} (c+d x)^{3/2}}{x^6} \, dx=\int \frac {\sqrt {c - d x} \left (c + d x\right )^{\frac {3}{2}}}{x^{6}}\, dx \] Input:

integrate((-d*x+c)**(1/2)*(d*x+c)**(3/2)/x**6,x)
 

Output:

Integral(sqrt(c - d*x)*(c + d*x)**(3/2)/x**6, x)
 

Maxima [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt {c-d x} (c+d x)^{3/2}}{x^6} \, dx=\frac {d^{5} \log \left (\frac {2 \, c^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-d^{2} x^{2} + c^{2}} c}{{\left | x \right |}}\right )}{8 \, c^{3}} - \frac {\sqrt {-d^{2} x^{2} + c^{2}} d^{5}}{8 \, c^{4}} - \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} d^{3}}{8 \, c^{4} x^{2}} - \frac {2 \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} d^{2}}{15 \, c^{3} x^{3}} - \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} d}{4 \, c^{2} x^{4}} - \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}}}{5 \, c x^{5}} \] Input:

integrate((-d*x+c)^(1/2)*(d*x+c)^(3/2)/x^6,x, algorithm="maxima")
 

Output:

1/8*d^5*log(2*c^2/abs(x) + 2*sqrt(-d^2*x^2 + c^2)*c/abs(x))/c^3 - 1/8*sqrt 
(-d^2*x^2 + c^2)*d^5/c^4 - 1/8*(-d^2*x^2 + c^2)^(3/2)*d^3/(c^4*x^2) - 2/15 
*(-d^2*x^2 + c^2)^(3/2)*d^2/(c^3*x^3) - 1/4*(-d^2*x^2 + c^2)^(3/2)*d/(c^2* 
x^4) - 1/5*(-d^2*x^2 + c^2)^(3/2)/(c*x^5)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 521 vs. \(2 (125) = 250\).

Time = 0.43 (sec) , antiderivative size = 521, normalized size of antiderivative = 3.36 \[ \int \frac {\sqrt {c-d x} (c+d x)^{3/2}}{x^6} \, dx=\frac {\frac {15 \, d^{6} \log \left ({\left | -\frac {\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}{\sqrt {d x + c}} + \frac {\sqrt {d x + c}}{\sqrt {2} \sqrt {c} - \sqrt {-d x + c}} + 2 \right |}\right )}{c^{3}} - \frac {15 \, d^{6} \log \left ({\left | -\frac {\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}{\sqrt {d x + c}} + \frac {\sqrt {d x + c}}{\sqrt {2} \sqrt {c} - \sqrt {-d x + c}} - 2 \right |}\right )}{c^{3}} + \frac {4 \, {\left (15 \, d^{6} {\left (\frac {\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}{\sqrt {d x + c}} - \frac {\sqrt {d x + c}}{\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}\right )}^{9} - 280 \, d^{6} {\left (\frac {\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}{\sqrt {d x + c}} - \frac {\sqrt {d x + c}}{\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}\right )}^{7} - 1024 \, d^{6} {\left (\frac {\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}{\sqrt {d x + c}} - \frac {\sqrt {d x + c}}{\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}\right )}^{5} - 16000 \, d^{6} {\left (\frac {\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}{\sqrt {d x + c}} - \frac {\sqrt {d x + c}}{\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}\right )}^{3} - 3840 \, d^{6} {\left (\frac {\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}{\sqrt {d x + c}} - \frac {\sqrt {d x + c}}{\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}\right )}\right )}}{{\left ({\left (\frac {\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}{\sqrt {d x + c}} - \frac {\sqrt {d x + c}}{\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}\right )}^{2} - 4\right )}^{5} c^{3}}}{120 \, d} \] Input:

integrate((-d*x+c)^(1/2)*(d*x+c)^(3/2)/x^6,x, algorithm="giac")
 

Output:

1/120*(15*d^6*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^3 - 15*d^6*log(ab 
s(-(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^3 + 4*(15*d^6*((sqrt(2)*sqrt(c) - sqr 
t(-d*x + c))/sqrt(d*x + c) - sqrt(d*x + c)/(sqrt(2)*sqrt(c) - sqrt(-d*x + 
c)))^9 - 280*d^6*((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 - 1024*d^6*((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 - 16000*d^6*((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 - 3840*d^6*((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)^5*c^3))/d
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c-d x} (c+d x)^{3/2}}{x^6} \, dx=\int \frac {{\left (c+d\,x\right )}^{3/2}\,\sqrt {c-d\,x}}{x^6} \,d x \] Input:

int(((c + d*x)^(3/2)*(c - d*x)^(1/2))/x^6,x)
                                                                                    
                                                                                    
 

Output:

int(((c + d*x)^(3/2)*(c - d*x)^(1/2))/x^6, x)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.61 \[ \int \frac {\sqrt {c-d x} (c+d x)^{3/2}}{x^6} \, dx=\frac {-24 \sqrt {d x +c}\, \sqrt {-d x +c}\, c^{4}-30 \sqrt {d x +c}\, \sqrt {-d x +c}\, c^{3} d x +8 \sqrt {d x +c}\, \sqrt {-d x +c}\, c^{2} d^{2} x^{2}+15 \sqrt {d x +c}\, \sqrt {-d x +c}\, c \,d^{3} x^{3}+16 \sqrt {d x +c}\, \sqrt {-d x +c}\, d^{4} x^{4}+15 \,\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-d x +c}}{\sqrt {c}\, \sqrt {2}}\right )}{2}\right )-1\right ) d^{5} x^{5}-15 \,\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-d x +c}}{\sqrt {c}\, \sqrt {2}}\right )}{2}\right )+1\right ) d^{5} x^{5}+15 \,\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-d x +c}}{\sqrt {c}\, \sqrt {2}}\right )}{2}\right )-1\right ) d^{5} x^{5}-15 \,\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-d x +c}}{\sqrt {c}\, \sqrt {2}}\right )}{2}\right )+1\right ) d^{5} x^{5}}{120 c^{3} x^{5}} \] Input:

int((-d*x+c)^(1/2)*(d*x+c)^(3/2)/x^6,x)
 

Output:

( - 24*sqrt(c + d*x)*sqrt(c - d*x)*c**4 - 30*sqrt(c + d*x)*sqrt(c - d*x)*c 
**3*d*x + 8*sqrt(c + d*x)*sqrt(c - d*x)*c**2*d**2*x**2 + 15*sqrt(c + d*x)* 
sqrt(c - d*x)*c*d**3*x**3 + 16*sqrt(c + d*x)*sqrt(c - d*x)*d**4*x**4 + 15* 
log( - sqrt(2) + tan(asin(sqrt(c - d*x)/(sqrt(c)*sqrt(2)))/2) - 1)*d**5*x* 
*5 - 15*log( - sqrt(2) + tan(asin(sqrt(c - d*x)/(sqrt(c)*sqrt(2)))/2) + 1) 
*d**5*x**5 + 15*log(sqrt(2) + tan(asin(sqrt(c - d*x)/(sqrt(c)*sqrt(2)))/2) 
 - 1)*d**5*x**5 - 15*log(sqrt(2) + tan(asin(sqrt(c - d*x)/(sqrt(c)*sqrt(2) 
))/2) + 1)*d**5*x**5)/(120*c**3*x**5)