3.2.79 \(\int \frac {x^5}{(d+e x)^3 \sqrt {d^2-e^2 x^2}} \, dx\) [179]

3.2.79.1 Optimal result

Integrand size = 27, antiderivative size = 177 \[ \int \frac {x^5}{(d+e x)^3 \sqrt {d^2-e^2 x^2}} \, dx=\frac {d^4 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {23 d^3 (d-e x)^2}{15 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {127 d^2 (d-e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}+\frac {3 d \sqrt {d^2-e^2 x^2}}{e^6}-\frac {x \sqrt {d^2-e^2 x^2}}{2 e^5}+\frac {13 d^2 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^6} \]

output

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

3.2.79.2 Mathematica [A] (verified)

Time = 0.47 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.63 \[ \int \frac {x^5}{(d+e x)^3 \sqrt {d^2-e^2 x^2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (304 d^4+717 d^3 e x+479 d^2 e^2 x^2+45 d e^3 x^3-15 e^4 x^4\right )}{30 e^6 (d+e x)^3}-\frac {13 d^2 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{e^6} \]

input

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

output

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

3.2.79.3 Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.16, 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.

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

\(\Big \downarrow \) 570

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

\(\Big \downarrow \) 529

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2166

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

\(\Big \downarrow \) 2166

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2346

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 455

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 216

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

input

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

output

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

3.2.79.3.1 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]
 

3.2.79.4 Maple [A] (verified)

Time = 0.47 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.12

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

input

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

output

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

3.2.79.5 Fricas [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.07 \[ \int \frac {x^5}{(d+e x)^3 \sqrt {d^2-e^2 x^2}} \, dx=\frac {304 \, d^{2} e^{3} x^{3} + 912 \, d^{3} e^{2} x^{2} + 912 \, d^{4} e x + 304 \, d^{5} - 390 \, {\left (d^{2} e^{3} x^{3} + 3 \, d^{3} e^{2} x^{2} + 3 \, d^{4} e x + d^{5}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - {\left (15 \, e^{4} x^{4} - 45 \, d e^{3} x^{3} - 479 \, d^{2} e^{2} x^{2} - 717 \, d^{3} e x - 304 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{30 \, {\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} \]

input

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

output

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

3.2.79.6 Sympy [F]

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

input

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

output

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

3.2.79.7 Maxima [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.05 \[ \int \frac {x^5}{(d+e x)^3 \sqrt {d^2-e^2 x^2}} \, dx=\frac {\sqrt {-e^{2} x^{2} + d^{2}} d^{4}}{5 \, {\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} - \frac {23 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3}}{15 \, {\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} + \frac {127 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2}}{15 \, {\left (e^{7} x + d e^{6}\right )}} + \frac {13 \, d^{2} \arcsin \left (\frac {e x}{d}\right )}{2 \, e^{6}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} x}{2 \, e^{5}} + \frac {3 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{e^{6}} \]

input

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

output

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

3.2.79.8 Giac [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.32 \[ \int \frac {x^5}{(d+e x)^3 \sqrt {d^2-e^2 x^2}} \, dx=-\frac {1}{2} \, \sqrt {-e^{2} x^{2} + d^{2}} {\left (\frac {x}{e^{5}} - \frac {6 \, d}{e^{6}}\right )} + \frac {13 \, d^{2} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{2 \, e^{5} {\left | e \right |}} - \frac {2 \, {\left (107 \, d^{2} + \frac {445 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{2}}{e^{2} x} + \frac {665 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{2}}{e^{4} x^{2}} + \frac {405 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{2}}{e^{6} x^{3}} + \frac {90 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d^{2}}{e^{8} x^{4}}\right )}}{15 \, e^{5} {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} + 1\right )}^{5} {\left | e \right |}} \]

input

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

output

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

3.2.79.9 Mupad [F(-1)]

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

input

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

output

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