\(\int \frac {x^5}{(c+d x)^3 \sqrt {c^2-d^2 x^2}} \, dx\) [366]

Optimal result

Integrand size = 27, antiderivative size = 176 \[ \int \frac {x^5}{(c+d x)^3 \sqrt {c^2-d^2 x^2}} \, dx=\frac {3 c \sqrt {c^2-d^2 x^2}}{d^6}-\frac {x \sqrt {c^2-d^2 x^2}}{2 d^5}+\frac {c^4 \sqrt {c^2-d^2 x^2}}{5 d^6 (c+d x)^3}-\frac {23 c^3 \sqrt {c^2-d^2 x^2}}{15 d^6 (c+d x)^2}+\frac {127 c^2 \sqrt {c^2-d^2 x^2}}{15 d^6 (c+d x)}+\frac {13 c^2 \arctan \left (\frac {d x}{\sqrt {c^2-d^2 x^2}}\right )}{2 d^6} \] Output:

3*c*(-d^2*x^2+c^2)^(1/2)/d^6-1/2*x*(-d^2*x^2+c^2)^(1/2)/d^5+1/5*c^4*(-d^2* 
x^2+c^2)^(1/2)/d^6/(d*x+c)^3-23/15*c^3*(-d^2*x^2+c^2)^(1/2)/d^6/(d*x+c)^2+ 
127/15*c^2*(-d^2*x^2+c^2)^(1/2)/d^6/(d*x+c)+13/2*c^2*arctan(d*x/(-d^2*x^2+ 
c^2)^(1/2))/d^6
 

Mathematica [A] (verified)

Time = 0.57 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.64 \[ \int \frac {x^5}{(c+d x)^3 \sqrt {c^2-d^2 x^2}} \, dx=\frac {\sqrt {c^2-d^2 x^2} \left (304 c^4+717 c^3 d x+479 c^2 d^2 x^2+45 c d^3 x^3-15 d^4 x^4\right )}{30 d^6 (c+d x)^3}-\frac {13 c^2 \arctan \left (\frac {d x}{\sqrt {c^2}-\sqrt {c^2-d^2 x^2}}\right )}{d^6} \] Input:

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

Output:

(Sqrt[c^2 - d^2*x^2]*(304*c^4 + 717*c^3*d*x + 479*c^2*d^2*x^2 + 45*c*d^3*x 
^3 - 15*d^4*x^4))/(30*d^6*(c + d*x)^3) - (13*c^2*ArcTan[(d*x)/(Sqrt[c^2] - 
 Sqrt[c^2 - d^2*x^2])])/d^6
 

Rubi [A] (verified)

Time = 1.28 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.17, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {570, 529, 25, 2166, 2166, 27, 2346, 25, 27, 455, 224, 216}

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.

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

\(\Big \downarrow \) 570

\(\displaystyle \int \frac {x^5 (c-d x)^3}{\left (c^2-d^2 x^2\right )^{7/2}}dx\)

\(\Big \downarrow \) 529

\(\displaystyle \frac {c^4 (c-d x)^3}{5 d^6 \left (c^2-d^2 x^2\right )^{5/2}}-\frac {\int -\frac {(c-d x)^2 \left (\frac {3 c^5}{d^5}-\frac {5 x c^4}{d^4}+\frac {5 x^2 c^3}{d^3}-\frac {5 x^3 c^2}{d^2}+\frac {5 x^4 c}{d}\right )}{\left (c^2-d^2 x^2\right )^{5/2}}dx}{5 c}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {(c-d x)^2 \left (\frac {3 c^5}{d^5}-\frac {5 x c^4}{d^4}+\frac {5 x^2 c^3}{d^3}-\frac {5 x^3 c^2}{d^2}+\frac {5 x^4 c}{d}\right )}{\left (c^2-d^2 x^2\right )^{5/2}}dx}{5 c}+\frac {c^4 (c-d x)^3}{5 d^6 \left (c^2-d^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 2166

\(\displaystyle \frac {-\frac {\int \frac {(c-d x) \left (\frac {37 c^5}{d^5}-\frac {45 x c^4}{d^4}+\frac {30 x^2 c^3}{d^3}-\frac {15 x^3 c^2}{d^2}\right )}{\left (c^2-d^2 x^2\right )^{3/2}}dx}{3 c}-\frac {23 c^4 (c-d x)^2}{3 d^6 \left (c^2-d^2 x^2\right )^{3/2}}}{5 c}+\frac {c^4 (c-d x)^3}{5 d^6 \left (c^2-d^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 2166

\(\displaystyle \frac {-\frac {-\frac {\int \frac {15 \left (\frac {6 c^5}{d^5}-\frac {3 x c^4}{d^4}+\frac {x^2 c^3}{d^3}\right )}{\sqrt {c^2-d^2 x^2}}dx}{c}-\frac {127 c^4 (c-d x)}{d^6 \sqrt {c^2-d^2 x^2}}}{3 c}-\frac {23 c^4 (c-d x)^2}{3 d^6 \left (c^2-d^2 x^2\right )^{3/2}}}{5 c}+\frac {c^4 (c-d x)^3}{5 d^6 \left (c^2-d^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {-\frac {15 \int \frac {\frac {6 c^5}{d^5}-\frac {3 x c^4}{d^4}+\frac {x^2 c^3}{d^3}}{\sqrt {c^2-d^2 x^2}}dx}{c}-\frac {127 c^4 (c-d x)}{d^6 \sqrt {c^2-d^2 x^2}}}{3 c}-\frac {23 c^4 (c-d x)^2}{3 d^6 \left (c^2-d^2 x^2\right )^{3/2}}}{5 c}+\frac {c^4 (c-d x)^3}{5 d^6 \left (c^2-d^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 2346

\(\displaystyle \frac {-\frac {-\frac {15 \left (-\frac {\int -\frac {c^4 (13 c-6 d x)}{d^3 \sqrt {c^2-d^2 x^2}}dx}{2 d^2}-\frac {c^3 x \sqrt {c^2-d^2 x^2}}{2 d^5}\right )}{c}-\frac {127 c^4 (c-d x)}{d^6 \sqrt {c^2-d^2 x^2}}}{3 c}-\frac {23 c^4 (c-d x)^2}{3 d^6 \left (c^2-d^2 x^2\right )^{3/2}}}{5 c}+\frac {c^4 (c-d x)^3}{5 d^6 \left (c^2-d^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {-\frac {-\frac {15 \left (\frac {\int \frac {c^4 (13 c-6 d x)}{d^3 \sqrt {c^2-d^2 x^2}}dx}{2 d^2}-\frac {c^3 x \sqrt {c^2-d^2 x^2}}{2 d^5}\right )}{c}-\frac {127 c^4 (c-d x)}{d^6 \sqrt {c^2-d^2 x^2}}}{3 c}-\frac {23 c^4 (c-d x)^2}{3 d^6 \left (c^2-d^2 x^2\right )^{3/2}}}{5 c}+\frac {c^4 (c-d x)^3}{5 d^6 \left (c^2-d^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {-\frac {15 \left (\frac {c^4 \int \frac {13 c-6 d x}{\sqrt {c^2-d^2 x^2}}dx}{2 d^5}-\frac {c^3 x \sqrt {c^2-d^2 x^2}}{2 d^5}\right )}{c}-\frac {127 c^4 (c-d x)}{d^6 \sqrt {c^2-d^2 x^2}}}{3 c}-\frac {23 c^4 (c-d x)^2}{3 d^6 \left (c^2-d^2 x^2\right )^{3/2}}}{5 c}+\frac {c^4 (c-d x)^3}{5 d^6 \left (c^2-d^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 455

\(\displaystyle \frac {-\frac {-\frac {15 \left (\frac {c^4 \left (13 c \int \frac {1}{\sqrt {c^2-d^2 x^2}}dx+\frac {6 \sqrt {c^2-d^2 x^2}}{d}\right )}{2 d^5}-\frac {c^3 x \sqrt {c^2-d^2 x^2}}{2 d^5}\right )}{c}-\frac {127 c^4 (c-d x)}{d^6 \sqrt {c^2-d^2 x^2}}}{3 c}-\frac {23 c^4 (c-d x)^2}{3 d^6 \left (c^2-d^2 x^2\right )^{3/2}}}{5 c}+\frac {c^4 (c-d x)^3}{5 d^6 \left (c^2-d^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {-\frac {-\frac {15 \left (\frac {c^4 \left (13 c \int \frac {1}{\frac {d^2 x^2}{c^2-d^2 x^2}+1}d\frac {x}{\sqrt {c^2-d^2 x^2}}+\frac {6 \sqrt {c^2-d^2 x^2}}{d}\right )}{2 d^5}-\frac {c^3 x \sqrt {c^2-d^2 x^2}}{2 d^5}\right )}{c}-\frac {127 c^4 (c-d x)}{d^6 \sqrt {c^2-d^2 x^2}}}{3 c}-\frac {23 c^4 (c-d x)^2}{3 d^6 \left (c^2-d^2 x^2\right )^{3/2}}}{5 c}+\frac {c^4 (c-d x)^3}{5 d^6 \left (c^2-d^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {-\frac {-\frac {15 \left (\frac {c^4 \left (\frac {13 c \arctan \left (\frac {d x}{\sqrt {c^2-d^2 x^2}}\right )}{d}+\frac {6 \sqrt {c^2-d^2 x^2}}{d}\right )}{2 d^5}-\frac {c^3 x \sqrt {c^2-d^2 x^2}}{2 d^5}\right )}{c}-\frac {127 c^4 (c-d x)}{d^6 \sqrt {c^2-d^2 x^2}}}{3 c}-\frac {23 c^4 (c-d x)^2}{3 d^6 \left (c^2-d^2 x^2\right )^{3/2}}}{5 c}+\frac {c^4 (c-d x)^3}{5 d^6 \left (c^2-d^2 x^2\right )^{5/2}}\)

Input:

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

Output:

(c^4*(c - d*x)^3)/(5*d^6*(c^2 - d^2*x^2)^(5/2)) + ((-23*c^4*(c - d*x)^2)/( 
3*d^6*(c^2 - d^2*x^2)^(3/2)) - ((-127*c^4*(c - d*x))/(d^6*Sqrt[c^2 - d^2*x 
^2]) - (15*(-1/2*(c^3*x*Sqrt[c^2 - d^2*x^2])/d^5 + (c^4*((6*Sqrt[c^2 - d^2 
*x^2])/d + (13*c*ArcTan[(d*x)/Sqrt[c^2 - d^2*x^2]])/d))/(2*d^5)))/c)/(3*c) 
)/(5*c)
 

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 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 455

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

rule 529

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{Qx = PolynomialQuotient[x^m, a*d + b*c*x, x], R = PolynomialRem 
ainder[x^m, a*d + b*c*x, x]}, Simp[(-c)*R*(c + d*x)^n*((a + b*x^2)^(p + 1)/ 
(2*a*d*(p + 1))), x] + Simp[c/(2*a*(p + 1))   Int[(c + d*x)^(n - 1)*(a + b* 
x^2)^(p + 1)*ExpandToSum[2*a*d*(p + 1)*Qx + R*(n + 2*p + 2), x], x], x]] /; 
 FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && IGtQ[m, 1] && LtQ[p, -1] && EqQ[b* 
c^2 + a*d^2, 0]
 

rule 570

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

rule 2166

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{Qx = PolynomialQuotient[Pq, a*e + b*d*x, x], R = PolynomialRemainde 
r[Pq, a*e + b*d*x, x]}, Simp[(-d)*R*(d + e*x)^m*((a + b*x^2)^(p + 1)/(2*a*e 
*(p + 1))), x] + Simp[d/(2*a*(p + 1))   Int[(d + e*x)^(m - 1)*(a + b*x^2)^( 
p + 1)*ExpandToSum[2*a*e*(p + 1)*Qx + R*(m + 2*p + 2), x], x], x]] /; FreeQ 
[{a, b, d, e}, x] && PolyQ[Pq, x] && EqQ[b*d^2 + a*e^2, 0] && ILtQ[p + 1/2, 
 0] && GtQ[m, 0]
 

rule 2346

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], 
e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x^2)^(p + 1)/(b*( 
q + 2*p + 1))), x] + Simp[1/(b*(q + 2*p + 1))   Int[(a + b*x^2)^p*ExpandToS 
um[b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], 
x], x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] &&  !LeQ[p, -1]
 

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.13

method	result	size
risch	\(\frac {\left (-d x +6 c \right ) \sqrt {-d^{2} x^{2}+c^{2}}}{2 d^{6}}+\frac {13 c^{2} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+c^{2}}}\right )}{2 d^{5} \sqrt {d^{2}}}+\frac {127 c^{2} \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}{15 d^{7} \left (x +\frac {c}{d}\right )}-\frac {23 c^{3} \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}{15 d^{8} \left (x +\frac {c}{d}\right )^{2}}+\frac {c^{4} \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}{5 d^{9} \left (x +\frac {c}{d}\right )^{3}}\)	\(199\)
default	\(\frac {-\frac {x \sqrt {-d^{2} x^{2}+c^{2}}}{2 d^{2}}+\frac {c^{2} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+c^{2}}}\right )}{2 d^{2} \sqrt {d^{2}}}}{d^{3}}+\frac {6 c^{2} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+c^{2}}}\right )}{d^{5} \sqrt {d^{2}}}+\frac {3 c \sqrt {-d^{2} x^{2}+c^{2}}}{d^{6}}+\frac {10 c^{2} \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}{d^{7} \left (x +\frac {c}{d}\right )}+\frac {5 c^{4} \left (-\frac {\sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}{3 c d \left (x +\frac {c}{d}\right )^{2}}-\frac {\sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}{3 c^{2} \left (x +\frac {c}{d}\right )}\right )}{d^{7}}-\frac {c^{5} \left (-\frac {\sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}{5 c d \left (x +\frac {c}{d}\right )^{3}}+\frac {2 d \left (-\frac {\sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}{3 c d \left (x +\frac {c}{d}\right )^{2}}-\frac {\sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}{3 c^{2} \left (x +\frac {c}{d}\right )}\right )}{5 c}\right )}{d^{8}}\)	\(406\)

Input:

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

Output:

1/2*(-d*x+6*c)/d^6*(-d^2*x^2+c^2)^(1/2)+13/2*c^2/d^5/(d^2)^(1/2)*arctan((d 
^2)^(1/2)*x/(-d^2*x^2+c^2)^(1/2))+127/15/d^7*c^2/(x+c/d)*(-d^2*(x+c/d)^2+2 
*c*d*(x+c/d))^(1/2)-23/15*c^3/d^8/(x+c/d)^2*(-d^2*(x+c/d)^2+2*c*d*(x+c/d)) 
^(1/2)+1/5*c^4/d^9/(x+c/d)^3*(-d^2*(x+c/d)^2+2*c*d*(x+c/d))^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.08 \[ \int \frac {x^5}{(c+d x)^3 \sqrt {c^2-d^2 x^2}} \, dx=\frac {304 \, c^{2} d^{3} x^{3} + 912 \, c^{3} d^{2} x^{2} + 912 \, c^{4} d x + 304 \, c^{5} - 390 \, {\left (c^{2} d^{3} x^{3} + 3 \, c^{3} d^{2} x^{2} + 3 \, c^{4} d x + c^{5}\right )} \arctan \left (-\frac {c - \sqrt {-d^{2} x^{2} + c^{2}}}{d x}\right ) - {\left (15 \, d^{4} x^{4} - 45 \, c d^{3} x^{3} - 479 \, c^{2} d^{2} x^{2} - 717 \, c^{3} d x - 304 \, c^{4}\right )} \sqrt {-d^{2} x^{2} + c^{2}}}{30 \, {\left (d^{9} x^{3} + 3 \, c d^{8} x^{2} + 3 \, c^{2} d^{7} x + c^{3} d^{6}\right )}} \] Input:

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

Output:

1/30*(304*c^2*d^3*x^3 + 912*c^3*d^2*x^2 + 912*c^4*d*x + 304*c^5 - 390*(c^2 
*d^3*x^3 + 3*c^3*d^2*x^2 + 3*c^4*d*x + c^5)*arctan(-(c - sqrt(-d^2*x^2 + c 
^2))/(d*x)) - (15*d^4*x^4 - 45*c*d^3*x^3 - 479*c^2*d^2*x^2 - 717*c^3*d*x - 
 304*c^4)*sqrt(-d^2*x^2 + c^2))/(d^9*x^3 + 3*c*d^8*x^2 + 3*c^2*d^7*x + c^3 
*d^6)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.05 \[ \int \frac {x^5}{(c+d x)^3 \sqrt {c^2-d^2 x^2}} \, dx=\frac {\sqrt {-d^{2} x^{2} + c^{2}} c^{4}}{5 \, {\left (d^{9} x^{3} + 3 \, c d^{8} x^{2} + 3 \, c^{2} d^{7} x + c^{3} d^{6}\right )}} - \frac {23 \, \sqrt {-d^{2} x^{2} + c^{2}} c^{3}}{15 \, {\left (d^{8} x^{2} + 2 \, c d^{7} x + c^{2} d^{6}\right )}} + \frac {127 \, \sqrt {-d^{2} x^{2} + c^{2}} c^{2}}{15 \, {\left (d^{7} x + c d^{6}\right )}} + \frac {13 \, c^{2} \arcsin \left (\frac {d x}{c}\right )}{2 \, d^{6}} - \frac {\sqrt {-d^{2} x^{2} + c^{2}} x}{2 \, d^{5}} + \frac {3 \, \sqrt {-d^{2} x^{2} + c^{2}} c}{d^{6}} \] Input:

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

Output:

1/5*sqrt(-d^2*x^2 + c^2)*c^4/(d^9*x^3 + 3*c*d^8*x^2 + 3*c^2*d^7*x + c^3*d^ 
6) - 23/15*sqrt(-d^2*x^2 + c^2)*c^3/(d^8*x^2 + 2*c*d^7*x + c^2*d^6) + 127/ 
15*sqrt(-d^2*x^2 + c^2)*c^2/(d^7*x + c*d^6) + 13/2*c^2*arcsin(d*x/c)/d^6 - 
 1/2*sqrt(-d^2*x^2 + c^2)*x/d^5 + 3*sqrt(-d^2*x^2 + c^2)*c/d^6
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.32 \[ \int \frac {x^5}{(c+d x)^3 \sqrt {c^2-d^2 x^2}} \, dx=-\frac {1}{2} \, \sqrt {-d^{2} x^{2} + c^{2}} {\left (\frac {x}{d^{5}} - \frac {6 \, c}{d^{6}}\right )} + \frac {13 \, c^{2} \arcsin \left (\frac {d x}{c}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right )}{2 \, d^{5} {\left | d \right |}} - \frac {2 \, {\left (107 \, c^{2} + \frac {445 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )} c^{2}}{d^{2} x} + \frac {665 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )}^{2} c^{2}}{d^{4} x^{2}} + \frac {405 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )}^{3} c^{2}}{d^{6} x^{3}} + \frac {90 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )}^{4} c^{2}}{d^{8} x^{4}}\right )}}{15 \, d^{5} {\left (\frac {c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}}{d^{2} x} + 1\right )}^{5} {\left | d \right |}} \] Input:

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

Output:

-1/2*sqrt(-d^2*x^2 + c^2)*(x/d^5 - 6*c/d^6) + 13/2*c^2*arcsin(d*x/c)*sgn(c 
)*sgn(d)/(d^5*abs(d)) - 2/15*(107*c^2 + 445*(c*d + sqrt(-d^2*x^2 + c^2)*ab 
s(d))*c^2/(d^2*x) + 665*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^2*c^2/(d^4*x^2 
) + 405*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^3*c^2/(d^6*x^3) + 90*(c*d + sq 
rt(-d^2*x^2 + c^2)*abs(d))^4*c^2/(d^8*x^4))/(d^5*((c*d + sqrt(-d^2*x^2 + c 
^2)*abs(d))/(d^2*x) + 1)^5*abs(d))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 394, normalized size of antiderivative = 2.24 \[ \int \frac {x^5}{(c+d x)^3 \sqrt {c^2-d^2 x^2}} \, dx=\frac {195 \sqrt {-d^{2} x^{2}+c^{2}}\, \mathit {asin} \left (\frac {d x}{c}\right ) c^{4}+390 \sqrt {-d^{2} x^{2}+c^{2}}\, \mathit {asin} \left (\frac {d x}{c}\right ) c^{3} d x +195 \sqrt {-d^{2} x^{2}+c^{2}}\, \mathit {asin} \left (\frac {d x}{c}\right ) c^{2} d^{2} x^{2}-195 \mathit {asin} \left (\frac {d x}{c}\right ) c^{5}-585 \mathit {asin} \left (\frac {d x}{c}\right ) c^{4} d x -585 \mathit {asin} \left (\frac {d x}{c}\right ) c^{3} d^{2} x^{2}-195 \mathit {asin} \left (\frac {d x}{c}\right ) c^{2} d^{3} x^{3}-72 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{4}-253 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{3} d x -247 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{2} d^{2} x^{2}-45 \sqrt {-d^{2} x^{2}+c^{2}}\, c \,d^{3} x^{3}+15 \sqrt {-d^{2} x^{2}+c^{2}}\, d^{4} x^{4}+72 c^{5}-283 c^{4} d x -934 c^{3} d^{2} x^{2}-666 c^{2} d^{3} x^{3}-60 c \,d^{4} x^{4}+15 d^{5} x^{5}}{30 d^{6} \left (\sqrt {-d^{2} x^{2}+c^{2}}\, c^{2}+2 \sqrt {-d^{2} x^{2}+c^{2}}\, c d x +\sqrt {-d^{2} x^{2}+c^{2}}\, d^{2} x^{2}-c^{3}-3 c^{2} d x -3 c \,d^{2} x^{2}-d^{3} x^{3}\right )} \] Input:

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

Output:

(195*sqrt(c**2 - d**2*x**2)*asin((d*x)/c)*c**4 + 390*sqrt(c**2 - d**2*x**2 
)*asin((d*x)/c)*c**3*d*x + 195*sqrt(c**2 - d**2*x**2)*asin((d*x)/c)*c**2*d 
**2*x**2 - 195*asin((d*x)/c)*c**5 - 585*asin((d*x)/c)*c**4*d*x - 585*asin( 
(d*x)/c)*c**3*d**2*x**2 - 195*asin((d*x)/c)*c**2*d**3*x**3 - 72*sqrt(c**2 
- d**2*x**2)*c**4 - 253*sqrt(c**2 - d**2*x**2)*c**3*d*x - 247*sqrt(c**2 - 
d**2*x**2)*c**2*d**2*x**2 - 45*sqrt(c**2 - d**2*x**2)*c*d**3*x**3 + 15*sqr 
t(c**2 - d**2*x**2)*d**4*x**4 + 72*c**5 - 283*c**4*d*x - 934*c**3*d**2*x** 
2 - 666*c**2*d**3*x**3 - 60*c*d**4*x**4 + 15*d**5*x**5)/(30*d**6*(sqrt(c** 
2 - d**2*x**2)*c**2 + 2*sqrt(c**2 - d**2*x**2)*c*d*x + sqrt(c**2 - d**2*x* 
*2)*d**2*x**2 - c**3 - 3*c**2*d*x - 3*c*d**2*x**2 - d**3*x**3))