\(\int \frac {(c+d x)^2 \sqrt {c^2-d^2 x^2}}{x^8} \, dx\) [103]

Optimal result

Integrand size = 27, antiderivative size = 196 \[ \int \frac {(c+d x)^2 \sqrt {c^2-d^2 x^2}}{x^8} \, dx=-\frac {c d \sqrt {c^2-d^2 x^2}}{3 x^6}+\frac {d^3 \sqrt {c^2-d^2 x^2}}{12 c x^4}+\frac {d^5 \sqrt {c^2-d^2 x^2}}{8 c^3 x^2}-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{7 x^7}-\frac {11 d^2 \left (c^2-d^2 x^2\right )^{3/2}}{35 c^2 x^5}-\frac {22 d^4 \left (c^2-d^2 x^2\right )^{3/2}}{105 c^4 x^3}+\frac {d^7 \text {arctanh}\left (\frac {\sqrt {c^2-d^2 x^2}}{c}\right )}{8 c^4} \] Output:

-1/3*c*d*(-d^2*x^2+c^2)^(1/2)/x^6+1/12*d^3*(-d^2*x^2+c^2)^(1/2)/c/x^4+1/8* 
d^5*(-d^2*x^2+c^2)^(1/2)/c^3/x^2-1/7*(-d^2*x^2+c^2)^(3/2)/x^7-11/35*d^2*(- 
d^2*x^2+c^2)^(3/2)/c^2/x^5-22/105*d^4*(-d^2*x^2+c^2)^(3/2)/c^4/x^3+1/8*d^7 
*arctanh((-d^2*x^2+c^2)^(1/2)/c)/c^4
 

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.74 \[ \int \frac {(c+d x)^2 \sqrt {c^2-d^2 x^2}}{x^8} \, dx=\frac {\frac {c \sqrt {c^2-d^2 x^2} \left (-120 c^6-280 c^5 d x-144 c^4 d^2 x^2+70 c^3 d^3 x^3+88 c^2 d^4 x^4+105 c d^5 x^5+176 d^6 x^6\right )}{x^7}+105 \sqrt {c^2} d^7 \log (x)-105 \sqrt {c^2} d^7 \log \left (\sqrt {c^2}-\sqrt {c^2-d^2 x^2}\right )}{840 c^5} \] Input:

Integrate[((c + d*x)^2*Sqrt[c^2 - d^2*x^2])/x^8,x]
 

Output:

((c*Sqrt[c^2 - d^2*x^2]*(-120*c^6 - 280*c^5*d*x - 144*c^4*d^2*x^2 + 70*c^3 
*d^3*x^3 + 88*c^2*d^4*x^4 + 105*c*d^5*x^5 + 176*d^6*x^6))/x^7 + 105*Sqrt[c 
^2]*d^7*Log[x] - 105*Sqrt[c^2]*d^7*Log[Sqrt[c^2] - Sqrt[c^2 - d^2*x^2]])/( 
840*c^5)
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.09, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.593, Rules used = {540, 25, 27, 539, 27, 539, 25, 27, 539, 25, 27, 534, 243, 51, 73, 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.

\(\displaystyle \int \frac {(c+d x)^2 \sqrt {c^2-d^2 x^2}}{x^8} \, dx\)

\(\Big \downarrow \) 540

\(\displaystyle -\frac {\int -\frac {c^2 d (14 c+11 d x) \sqrt {c^2-d^2 x^2}}{x^7}dx}{7 c^2}-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{7 x^7}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {c^2 d (14 c+11 d x) \sqrt {c^2-d^2 x^2}}{x^7}dx}{7 c^2}-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{7 x^7}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{7} d \int \frac {(14 c+11 d x) \sqrt {c^2-d^2 x^2}}{x^7}dx-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{7 x^7}\)

\(\Big \downarrow \) 539

\(\displaystyle \frac {1}{7} d \left (-\frac {\int -\frac {6 c d (11 c+7 d x) \sqrt {c^2-d^2 x^2}}{x^6}dx}{6 c^2}-\frac {7 \left (c^2-d^2 x^2\right )^{3/2}}{3 c x^6}\right )-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{7 x^7}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{7} d \left (\frac {d \int \frac {(11 c+7 d x) \sqrt {c^2-d^2 x^2}}{x^6}dx}{c}-\frac {7 \left (c^2-d^2 x^2\right )^{3/2}}{3 c x^6}\right )-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{7 x^7}\)

\(\Big \downarrow \) 539

\(\displaystyle \frac {1}{7} d \left (\frac {d \left (-\frac {\int -\frac {c d (35 c+22 d x) \sqrt {c^2-d^2 x^2}}{x^5}dx}{5 c^2}-\frac {11 \left (c^2-d^2 x^2\right )^{3/2}}{5 c x^5}\right )}{c}-\frac {7 \left (c^2-d^2 x^2\right )^{3/2}}{3 c x^6}\right )-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{7 x^7}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{7} d \left (\frac {d \left (\frac {\int \frac {c d (35 c+22 d x) \sqrt {c^2-d^2 x^2}}{x^5}dx}{5 c^2}-\frac {11 \left (c^2-d^2 x^2\right )^{3/2}}{5 c x^5}\right )}{c}-\frac {7 \left (c^2-d^2 x^2\right )^{3/2}}{3 c x^6}\right )-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{7 x^7}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{7} d \left (\frac {d \left (\frac {d \int \frac {(35 c+22 d x) \sqrt {c^2-d^2 x^2}}{x^5}dx}{5 c}-\frac {11 \left (c^2-d^2 x^2\right )^{3/2}}{5 c x^5}\right )}{c}-\frac {7 \left (c^2-d^2 x^2\right )^{3/2}}{3 c x^6}\right )-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{7 x^7}\)

\(\Big \downarrow \) 539

\(\displaystyle \frac {1}{7} d \left (\frac {d \left (\frac {d \left (-\frac {\int -\frac {c d (88 c+35 d x) \sqrt {c^2-d^2 x^2}}{x^4}dx}{4 c^2}-\frac {35 \left (c^2-d^2 x^2\right )^{3/2}}{4 c x^4}\right )}{5 c}-\frac {11 \left (c^2-d^2 x^2\right )^{3/2}}{5 c x^5}\right )}{c}-\frac {7 \left (c^2-d^2 x^2\right )^{3/2}}{3 c x^6}\right )-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{7 x^7}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{7} d \left (\frac {d \left (\frac {d \left (\frac {\int \frac {c d (88 c+35 d x) \sqrt {c^2-d^2 x^2}}{x^4}dx}{4 c^2}-\frac {35 \left (c^2-d^2 x^2\right )^{3/2}}{4 c x^4}\right )}{5 c}-\frac {11 \left (c^2-d^2 x^2\right )^{3/2}}{5 c x^5}\right )}{c}-\frac {7 \left (c^2-d^2 x^2\right )^{3/2}}{3 c x^6}\right )-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{7 x^7}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{7} d \left (\frac {d \left (\frac {d \left (\frac {d \int \frac {(88 c+35 d x) \sqrt {c^2-d^2 x^2}}{x^4}dx}{4 c}-\frac {35 \left (c^2-d^2 x^2\right )^{3/2}}{4 c x^4}\right )}{5 c}-\frac {11 \left (c^2-d^2 x^2\right )^{3/2}}{5 c x^5}\right )}{c}-\frac {7 \left (c^2-d^2 x^2\right )^{3/2}}{3 c x^6}\right )-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{7 x^7}\)

\(\Big \downarrow \) 534

\(\displaystyle \frac {1}{7} d \left (\frac {d \left (\frac {d \left (\frac {d \left (35 d \int \frac {\sqrt {c^2-d^2 x^2}}{x^3}dx-\frac {88 \left (c^2-d^2 x^2\right )^{3/2}}{3 c x^3}\right )}{4 c}-\frac {35 \left (c^2-d^2 x^2\right )^{3/2}}{4 c x^4}\right )}{5 c}-\frac {11 \left (c^2-d^2 x^2\right )^{3/2}}{5 c x^5}\right )}{c}-\frac {7 \left (c^2-d^2 x^2\right )^{3/2}}{3 c x^6}\right )-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{7 x^7}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {1}{7} d \left (\frac {d \left (\frac {d \left (\frac {d \left (\frac {35}{2} d \int \frac {\sqrt {c^2-d^2 x^2}}{x^4}dx^2-\frac {88 \left (c^2-d^2 x^2\right )^{3/2}}{3 c x^3}\right )}{4 c}-\frac {35 \left (c^2-d^2 x^2\right )^{3/2}}{4 c x^4}\right )}{5 c}-\frac {11 \left (c^2-d^2 x^2\right )^{3/2}}{5 c x^5}\right )}{c}-\frac {7 \left (c^2-d^2 x^2\right )^{3/2}}{3 c x^6}\right )-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{7 x^7}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {1}{7} d \left (\frac {d \left (\frac {d \left (\frac {d \left (\frac {35}{2} d \left (-\frac {1}{2} d^2 \int \frac {1}{x^2 \sqrt {c^2-d^2 x^2}}dx^2-\frac {\sqrt {c^2-d^2 x^2}}{x^2}\right )-\frac {88 \left (c^2-d^2 x^2\right )^{3/2}}{3 c x^3}\right )}{4 c}-\frac {35 \left (c^2-d^2 x^2\right )^{3/2}}{4 c x^4}\right )}{5 c}-\frac {11 \left (c^2-d^2 x^2\right )^{3/2}}{5 c x^5}\right )}{c}-\frac {7 \left (c^2-d^2 x^2\right )^{3/2}}{3 c x^6}\right )-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{7 x^7}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{7} d \left (\frac {d \left (\frac {d \left (\frac {d \left (\frac {35}{2} d \left (\int \frac {1}{\frac {c^2}{d^2}-\frac {x^4}{d^2}}d\sqrt {c^2-d^2 x^2}-\frac {\sqrt {c^2-d^2 x^2}}{x^2}\right )-\frac {88 \left (c^2-d^2 x^2\right )^{3/2}}{3 c x^3}\right )}{4 c}-\frac {35 \left (c^2-d^2 x^2\right )^{3/2}}{4 c x^4}\right )}{5 c}-\frac {11 \left (c^2-d^2 x^2\right )^{3/2}}{5 c x^5}\right )}{c}-\frac {7 \left (c^2-d^2 x^2\right )^{3/2}}{3 c x^6}\right )-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{7 x^7}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {1}{7} d \left (\frac {d \left (\frac {d \left (\frac {d \left (\frac {35}{2} d \left (\frac {d^2 \text {arctanh}\left (\frac {\sqrt {c^2-d^2 x^2}}{c}\right )}{c}-\frac {\sqrt {c^2-d^2 x^2}}{x^2}\right )-\frac {88 \left (c^2-d^2 x^2\right )^{3/2}}{3 c x^3}\right )}{4 c}-\frac {35 \left (c^2-d^2 x^2\right )^{3/2}}{4 c x^4}\right )}{5 c}-\frac {11 \left (c^2-d^2 x^2\right )^{3/2}}{5 c x^5}\right )}{c}-\frac {7 \left (c^2-d^2 x^2\right )^{3/2}}{3 c x^6}\right )-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{7 x^7}\)

Input:

Int[((c + d*x)^2*Sqrt[c^2 - d^2*x^2])/x^8,x]
 

Output:

-1/7*(c^2 - d^2*x^2)^(3/2)/x^7 + (d*((-7*(c^2 - d^2*x^2)^(3/2))/(3*c*x^6) 
+ (d*((-11*(c^2 - d^2*x^2)^(3/2))/(5*c*x^5) + (d*((-35*(c^2 - d^2*x^2)^(3/ 
2))/(4*c*x^4) + (d*((-88*(c^2 - d^2*x^2)^(3/2))/(3*c*x^3) + (35*d*(-(Sqrt[ 
c^2 - d^2*x^2]/x^2) + (d^2*ArcTanh[Sqrt[c^2 - d^2*x^2]/c])/c))/2))/(4*c))) 
/(5*c)))/c))/7
 

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 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) 
Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x 
] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

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]
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 534

Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
Simp[(-c)*x^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[d   Int[ 
x^(m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[m, 
0] && GtQ[p, -1] && EqQ[m + 2*p + 3, 0]
 

rule 539

Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
Simp[c*x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] + Simp[1/(a*(m + 1)) 
   Int[x^(m + 1)*(a + b*x^2)^p*(a*d*(m + 1) - b*c*(m + 2*p + 3)*x), x], x] 
/; FreeQ[{a, b, c, d, p}, x] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
 

rule 540

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol 
] :> With[{Qx = PolynomialQuotient[(c + d*x)^n, x, x], R = PolynomialRemain 
der[(c + d*x)^n, x, x]}, Simp[R*x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))) 
, x] + Simp[1/(a*(m + 1))   Int[x^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*(m + 
1)*Qx - b*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, p}, x] && IG 
tQ[n, 1] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
 

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.67

method	result	size
risch	\(-\frac {\sqrt {-d^{2} x^{2}+c^{2}}\, \left (-176 d^{6} x^{6}-105 c \,d^{5} x^{5}-88 c^{2} d^{4} x^{4}-70 d^{3} c^{3} x^{3}+144 c^{4} d^{2} x^{2}+280 c^{5} d x +120 c^{6}\right )}{840 x^{7} c^{4}}+\frac {d^{7} \ln \left (\frac {2 c^{2}+2 \sqrt {c^{2}}\, \sqrt {-d^{2} x^{2}+c^{2}}}{x}\right )}{8 c^{3} \sqrt {c^{2}}}\)	\(132\)
default	\(c^{2} \left (-\frac {\left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{7 c^{2} x^{7}}+\frac {4 d^{2} \left (-\frac {\left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{5 c^{2} x^{5}}-\frac {2 d^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{15 c^{4} x^{3}}\right )}{7 c^{2}}\right )+d^{2} \left (-\frac {\left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{5 c^{2} x^{5}}-\frac {2 d^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{15 c^{4} x^{3}}\right )+2 c d \left (-\frac {\left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{6 c^{2} x^{6}}+\frac {d^{2} \left (-\frac {\left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{4 c^{2} x^{4}}+\frac {d^{2} \left (-\frac {\left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{2 c^{2} x^{2}}-\frac {d^{2} \left (\sqrt {-d^{2} x^{2}+c^{2}}-\frac {c^{2} \ln \left (\frac {2 c^{2}+2 \sqrt {c^{2}}\, \sqrt {-d^{2} x^{2}+c^{2}}}{x}\right )}{\sqrt {c^{2}}}\right )}{2 c^{2}}\right )}{4 c^{2}}\right )}{2 c^{2}}\right )\)	\(291\)

Input:

int((d*x+c)^2*(-d^2*x^2+c^2)^(1/2)/x^8,x,method=_RETURNVERBOSE)
 

Output:

-1/840*(-d^2*x^2+c^2)^(1/2)*(-176*d^6*x^6-105*c*d^5*x^5-88*c^2*d^4*x^4-70* 
c^3*d^3*x^3+144*c^4*d^2*x^2+280*c^5*d*x+120*c^6)/x^7/c^4+1/8/c^3*d^7/(c^2) 
^(1/2)*ln((2*c^2+2*(c^2)^(1/2)*(-d^2*x^2+c^2)^(1/2))/x)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.61 \[ \int \frac {(c+d x)^2 \sqrt {c^2-d^2 x^2}}{x^8} \, dx=-\frac {105 \, d^{7} x^{7} \log \left (-\frac {c - \sqrt {-d^{2} x^{2} + c^{2}}}{x}\right ) - {\left (176 \, d^{6} x^{6} + 105 \, c d^{5} x^{5} + 88 \, c^{2} d^{4} x^{4} + 70 \, c^{3} d^{3} x^{3} - 144 \, c^{4} d^{2} x^{2} - 280 \, c^{5} d x - 120 \, c^{6}\right )} \sqrt {-d^{2} x^{2} + c^{2}}}{840 \, c^{4} x^{7}} \] Input:

integrate((d*x+c)^2*(-d^2*x^2+c^2)^(1/2)/x^8,x, algorithm="fricas")
 

Output:

-1/840*(105*d^7*x^7*log(-(c - sqrt(-d^2*x^2 + c^2))/x) - (176*d^6*x^6 + 10 
5*c*d^5*x^5 + 88*c^2*d^4*x^4 + 70*c^3*d^3*x^3 - 144*c^4*d^2*x^2 - 280*c^5* 
d*x - 120*c^6)*sqrt(-d^2*x^2 + c^2))/(c^4*x^7)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 6.82 (sec) , antiderivative size = 835, normalized size of antiderivative = 4.26 \[ \int \frac {(c+d x)^2 \sqrt {c^2-d^2 x^2}}{x^8} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)**2*(-d**2*x**2+c**2)**(1/2)/x**8,x)
 

Output:

c**2*Piecewise((-d*sqrt(c**2/(d**2*x**2) - 1)/(7*x**6) + d**3*sqrt(c**2/(d 
**2*x**2) - 1)/(35*c**2*x**4) + 4*d**5*sqrt(c**2/(d**2*x**2) - 1)/(105*c** 
4*x**2) + 8*d**7*sqrt(c**2/(d**2*x**2) - 1)/(105*c**6), Abs(c**2/(d**2*x** 
2)) > 1), (-I*d*sqrt(-c**2/(d**2*x**2) + 1)/(7*x**6) + I*d**3*sqrt(-c**2/( 
d**2*x**2) + 1)/(35*c**2*x**4) + 4*I*d**5*sqrt(-c**2/(d**2*x**2) + 1)/(105 
*c**4*x**2) + 8*I*d**7*sqrt(-c**2/(d**2*x**2) + 1)/(105*c**6), True)) + 2* 
c*d*Piecewise((-c**2/(6*d*x**7*sqrt(c**2/(d**2*x**2) - 1)) + 5*d/(24*x**5* 
sqrt(c**2/(d**2*x**2) - 1)) + d**3/(48*c**2*x**3*sqrt(c**2/(d**2*x**2) - 1 
)) - d**5/(16*c**4*x*sqrt(c**2/(d**2*x**2) - 1)) + d**6*acosh(c/(d*x))/(16 
*c**5), Abs(c**2/(d**2*x**2)) > 1), (I*c**2/(6*d*x**7*sqrt(-c**2/(d**2*x** 
2) + 1)) - 5*I*d/(24*x**5*sqrt(-c**2/(d**2*x**2) + 1)) - I*d**3/(48*c**2*x 
**3*sqrt(-c**2/(d**2*x**2) + 1)) + I*d**5/(16*c**4*x*sqrt(-c**2/(d**2*x**2 
) + 1)) - I*d**6*asin(c/(d*x))/(16*c**5), True)) + d**2*Piecewise((3*I*c** 
3*sqrt(-1 + d**2*x**2/c**2)/(-15*c**2*x**5 + 15*d**2*x**7) - 4*I*c*d**2*x* 
*2*sqrt(-1 + d**2*x**2/c**2)/(-15*c**2*x**5 + 15*d**2*x**7) + 2*I*d**6*x** 
6*sqrt(-1 + d**2*x**2/c**2)/(-15*c**5*x**5 + 15*c**3*d**2*x**7) - I*d**4*x 
**4*sqrt(-1 + d**2*x**2/c**2)/(-15*c**3*x**5 + 15*c*d**2*x**7), Abs(d**2*x 
**2/c**2) > 1), (3*c**3*sqrt(1 - d**2*x**2/c**2)/(-15*c**2*x**5 + 15*d**2* 
x**7) - 4*c*d**2*x**2*sqrt(1 - d**2*x**2/c**2)/(-15*c**2*x**5 + 15*d**2*x* 
*7) + 2*d**6*x**6*sqrt(1 - d**2*x**2/c**2)/(-15*c**5*x**5 + 15*c**3*d**...
                                                                                    
                                                                                    
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.05 \[ \int \frac {(c+d x)^2 \sqrt {c^2-d^2 x^2}}{x^8} \, dx=\frac {d^{7} \log \left (\frac {2 \, c^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-d^{2} x^{2} + c^{2}} c}{{\left | x \right |}}\right )}{8 \, c^{4}} - \frac {\sqrt {-d^{2} x^{2} + c^{2}} d^{7}}{8 \, c^{5}} - \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} d^{5}}{8 \, c^{5} x^{2}} - \frac {22 \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} d^{4}}{105 \, c^{4} x^{3}} - \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} d^{3}}{4 \, c^{3} x^{4}} - \frac {11 \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} d^{2}}{35 \, c^{2} x^{5}} - \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} d}{3 \, c x^{6}} - \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}}}{7 \, x^{7}} \] Input:

integrate((d*x+c)^2*(-d^2*x^2+c^2)^(1/2)/x^8,x, algorithm="maxima")
 

Output:

1/8*d^7*log(2*c^2/abs(x) + 2*sqrt(-d^2*x^2 + c^2)*c/abs(x))/c^4 - 1/8*sqrt 
(-d^2*x^2 + c^2)*d^7/c^5 - 1/8*(-d^2*x^2 + c^2)^(3/2)*d^5/(c^5*x^2) - 22/1 
05*(-d^2*x^2 + c^2)^(3/2)*d^4/(c^4*x^3) - 1/4*(-d^2*x^2 + c^2)^(3/2)*d^3/( 
c^3*x^4) - 11/35*(-d^2*x^2 + c^2)^(3/2)*d^2/(c^2*x^5) - 1/3*(-d^2*x^2 + c^ 
2)^(3/2)*d/(c*x^6) - 1/7*(-d^2*x^2 + c^2)^(3/2)/x^7
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 518 vs. \(2 (168) = 336\).

Time = 0.15 (sec) , antiderivative size = 518, normalized size of antiderivative = 2.64 \[ \int \frac {(c+d x)^2 \sqrt {c^2-d^2 x^2}}{x^8} \, dx=\frac {{\left (15 \, d^{8} + \frac {70 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )} d^{6}}{x} + \frac {147 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )}^{2} d^{4}}{x^{2}} + \frac {210 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )}^{3} d^{2}}{x^{3}} + \frac {175 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )}^{4}}{x^{4}} - \frac {210 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )}^{5}}{d^{2} x^{5}} - \frac {1365 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )}^{6}}{d^{4} x^{6}}\right )} d^{14} x^{7}}{13440 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )}^{7} c^{4} {\left | d \right |}} + \frac {d^{8} \log \left (\frac {{\left | -2 \, c d - 2 \, \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |} \right |}}{2 \, d^{2} {\left | x \right |}}\right )}{8 \, c^{4} {\left | d \right |}} + \frac {\frac {1365 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )} c^{24} d^{12}}{x} + \frac {210 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )}^{2} c^{24} d^{10}}{x^{2}} - \frac {175 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )}^{3} c^{24} d^{8}}{x^{3}} - \frac {210 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )}^{4} c^{24} d^{6}}{x^{4}} - \frac {147 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )}^{5} c^{24} d^{4}}{x^{5}} - \frac {70 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )}^{6} c^{24} d^{2}}{x^{6}} - \frac {15 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )}^{7} c^{24}}{x^{7}}}{13440 \, c^{28} d^{6} {\left | d \right |}} \] Input:

integrate((d*x+c)^2*(-d^2*x^2+c^2)^(1/2)/x^8,x, algorithm="giac")
 

Output:

1/13440*(15*d^8 + 70*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))*d^6/x + 147*(c*d 
+ sqrt(-d^2*x^2 + c^2)*abs(d))^2*d^4/x^2 + 210*(c*d + sqrt(-d^2*x^2 + c^2) 
*abs(d))^3*d^2/x^3 + 175*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^4/x^4 - 210*( 
c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^5/(d^2*x^5) - 1365*(c*d + sqrt(-d^2*x^2 
 + c^2)*abs(d))^6/(d^4*x^6))*d^14*x^7/((c*d + sqrt(-d^2*x^2 + c^2)*abs(d)) 
^7*c^4*abs(d)) + 1/8*d^8*log(1/2*abs(-2*c*d - 2*sqrt(-d^2*x^2 + c^2)*abs(d 
))/(d^2*abs(x)))/(c^4*abs(d)) + 1/13440*(1365*(c*d + sqrt(-d^2*x^2 + c^2)* 
abs(d))*c^24*d^12/x + 210*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^2*c^24*d^10/ 
x^2 - 175*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^3*c^24*d^8/x^3 - 210*(c*d + 
sqrt(-d^2*x^2 + c^2)*abs(d))^4*c^24*d^6/x^4 - 147*(c*d + sqrt(-d^2*x^2 + c 
^2)*abs(d))^5*c^24*d^4/x^5 - 70*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^6*c^24 
*d^2/x^6 - 15*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^7*c^24/x^7)/(c^28*d^6*ab 
s(d))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^2 \sqrt {c^2-d^2 x^2}}{x^8} \, dx=\int \frac {\sqrt {c^2-d^2\,x^2}\,{\left (c+d\,x\right )}^2}{x^8} \,d x \] Input:

int(((c^2 - d^2*x^2)^(1/2)*(c + d*x)^2)/x^8,x)
 

Output:

int(((c^2 - d^2*x^2)^(1/2)*(c + d*x)^2)/x^8, x)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.92 \[ \int \frac {(c+d x)^2 \sqrt {c^2-d^2 x^2}}{x^8} \, dx=\frac {-120 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{6}-280 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{5} d x -144 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{4} d^{2} x^{2}+70 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{3} d^{3} x^{3}+88 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{2} d^{4} x^{4}+105 \sqrt {-d^{2} x^{2}+c^{2}}\, c \,d^{5} x^{5}+176 \sqrt {-d^{2} x^{2}+c^{2}}\, d^{6} x^{6}-105 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {d x}{c}\right )}{2}\right )\right ) d^{7} x^{7}}{840 c^{4} x^{7}} \] Input:

int((d*x+c)^2*(-d^2*x^2+c^2)^(1/2)/x^8,x)
 

Output:

( - 120*sqrt(c**2 - d**2*x**2)*c**6 - 280*sqrt(c**2 - d**2*x**2)*c**5*d*x 
- 144*sqrt(c**2 - d**2*x**2)*c**4*d**2*x**2 + 70*sqrt(c**2 - d**2*x**2)*c* 
*3*d**3*x**3 + 88*sqrt(c**2 - d**2*x**2)*c**2*d**4*x**4 + 105*sqrt(c**2 - 
d**2*x**2)*c*d**5*x**5 + 176*sqrt(c**2 - d**2*x**2)*d**6*x**6 - 105*log(ta 
n(asin((d*x)/c)/2))*d**7*x**7)/(840*c**4*x**7)