\(\int \frac {x^4 (d^2-e^2 x^2)^{5/2}}{(d+e x)^4} \, dx\) [199]

Optimal result

Integrand size = 27, antiderivative size = 224 \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx=-\frac {d^3 (d-e x)^4}{e^5 \sqrt {d^2-e^2 x^2}}-\frac {337 d^5 \sqrt {d^2-e^2 x^2}}{15 e^5}+\frac {175 d^4 x \sqrt {d^2-e^2 x^2}}{16 e^4}-\frac {71 d^3 x^2 \sqrt {d^2-e^2 x^2}}{15 e^3}+\frac {47 d^2 x^3 \sqrt {d^2-e^2 x^2}}{24 e^2}-\frac {4 d x^4 \sqrt {d^2-e^2 x^2}}{5 e}+\frac {1}{6} x^5 \sqrt {d^2-e^2 x^2}-\frac {239 d^6 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e^5} \]

[Out]

-239/16*d^6*arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e^5-d^3*(-e*x+d)^4/e^5/(-e^2*x^2+d^2)^(1/2)-337/15*d^5*(-e^2*x^2+
d^2)^(1/2)/e^5+175/16*d^4*x*(-e^2*x^2+d^2)^(1/2)/e^4-71/15*d^3*x^2*(-e^2*x^2+d^2)^(1/2)/e^3+47/24*d^2*x^3*(-e^
2*x^2+d^2)^(1/2)/e^2-4/5*d*x^4*(-e^2*x^2+d^2)^(1/2)/e+1/6*x^5*(-e^2*x^2+d^2)^(1/2)

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {866, 1649, 1829, 655, 223, 209} \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx=-\frac {239 d^6 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e^5}+\frac {1}{6} x^5 \sqrt {d^2-e^2 x^2}-\frac {4 d x^4 \sqrt {d^2-e^2 x^2}}{5 e}+\frac {47 d^2 x^3 \sqrt {d^2-e^2 x^2}}{24 e^2}-\frac {337 d^5 \sqrt {d^2-e^2 x^2}}{15 e^5}+\frac {175 d^4 x \sqrt {d^2-e^2 x^2}}{16 e^4}-\frac {d^3 (d-e x)^4}{e^5 \sqrt {d^2-e^2 x^2}}-\frac {71 d^3 x^2 \sqrt {d^2-e^2 x^2}}{15 e^3} \]

[In]

Int[(x^4*(d^2 - e^2*x^2)^(5/2))/(d + e*x)^4,x]

[Out]

-((d^3*(d - e*x)^4)/(e^5*Sqrt[d^2 - e^2*x^2])) - (337*d^5*Sqrt[d^2 - e^2*x^2])/(15*e^5) + (175*d^4*x*Sqrt[d^2
- e^2*x^2])/(16*e^4) - (71*d^3*x^2*Sqrt[d^2 - e^2*x^2])/(15*e^3) + (47*d^2*x^3*Sqrt[d^2 - e^2*x^2])/(24*e^2) -
(4*d*x^4*Sqrt[d^2 - e^2*x^2])/(5*e) + (x^5*Sqrt[d^2 - e^2*x^2])/6 - (239*d^6*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]
])/(16*e^5)

Rule 209

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

Rule 223

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 655

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

Rule 866

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[d^(2*m)/a
^m, Int[(f + g*x)^n*((a + c*x^2)^(m + p)/(d - e*x)^m), x], x] /; FreeQ[{a, c, d, e, f, g, n, p}, x] && NeQ[e*f
- d*g, 0] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[f, 0] && ILtQ[m, -1] &&  !(IGtQ[n, 0] && ILtQ[m +
n, 0] &&  !GtQ[p, 1])

Rule 1649

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq,
a*e + c*d*x, x], f = PolynomialRemainder[Pq, a*e + c*d*x, x]}, Simp[(-d)*f*(d + e*x)^m*((a + c*x^2)^(p + 1)/(2
*a*e*(p + 1))), x] + Dist[d/(2*a*(p + 1)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1)*ExpandToSum[2*a*e*(p + 1)
*Q + f*(m + 2*p + 2), x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && EqQ[c*d^2 + a*e^2, 0] && ILtQ[p
+ 1/2, 0] && GtQ[m, 0]

Rule 1829

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Si
mp[e*x^(q - 1)*((a + b*x^2)^(p + 1)/(b*(q + 2*p + 1))), x] + Dist[1/(b*(q + 2*p + 1)), Int[(a + b*x^2)^p*Expan
dToSum[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]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^4 (d-e x)^4}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx \\ & = -\frac {d^3 (d-e x)^4}{e^5 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {(d-e x)^3 \left (\frac {4 d^4}{e^4}-\frac {d^3 x}{e^3}+\frac {d^2 x^2}{e^2}-\frac {d x^3}{e}\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{d} \\ & = -\frac {d^3 (d-e x)^4}{e^5 \sqrt {d^2-e^2 x^2}}+\frac {1}{6} x^5 \sqrt {d^2-e^2 x^2}+\frac {\int \frac {-\frac {24 d^7}{e^2}+\frac {78 d^6 x}{e}-96 d^5 x^2+66 d^4 e x^3-47 d^3 e^2 x^4+24 d^2 e^3 x^5}{\sqrt {d^2-e^2 x^2}} \, dx}{6 d e^2} \\ & = -\frac {d^3 (d-e x)^4}{e^5 \sqrt {d^2-e^2 x^2}}-\frac {4 d x^4 \sqrt {d^2-e^2 x^2}}{5 e}+\frac {1}{6} x^5 \sqrt {d^2-e^2 x^2}-\frac {\int \frac {120 d^7-390 d^6 e x+480 d^5 e^2 x^2-426 d^4 e^3 x^3+235 d^3 e^4 x^4}{\sqrt {d^2-e^2 x^2}} \, dx}{30 d e^4} \\ & = -\frac {d^3 (d-e x)^4}{e^5 \sqrt {d^2-e^2 x^2}}+\frac {47 d^2 x^3 \sqrt {d^2-e^2 x^2}}{24 e^2}-\frac {4 d x^4 \sqrt {d^2-e^2 x^2}}{5 e}+\frac {1}{6} x^5 \sqrt {d^2-e^2 x^2}+\frac {\int \frac {-480 d^7 e^2+1560 d^6 e^3 x-2625 d^5 e^4 x^2+1704 d^4 e^5 x^3}{\sqrt {d^2-e^2 x^2}} \, dx}{120 d e^6} \\ & = -\frac {d^3 (d-e x)^4}{e^5 \sqrt {d^2-e^2 x^2}}-\frac {71 d^3 x^2 \sqrt {d^2-e^2 x^2}}{15 e^3}+\frac {47 d^2 x^3 \sqrt {d^2-e^2 x^2}}{24 e^2}-\frac {4 d x^4 \sqrt {d^2-e^2 x^2}}{5 e}+\frac {1}{6} x^5 \sqrt {d^2-e^2 x^2}-\frac {\int \frac {1440 d^7 e^4-8088 d^6 e^5 x+7875 d^5 e^6 x^2}{\sqrt {d^2-e^2 x^2}} \, dx}{360 d e^8} \\ & = -\frac {d^3 (d-e x)^4}{e^5 \sqrt {d^2-e^2 x^2}}+\frac {175 d^4 x \sqrt {d^2-e^2 x^2}}{16 e^4}-\frac {71 d^3 x^2 \sqrt {d^2-e^2 x^2}}{15 e^3}+\frac {47 d^2 x^3 \sqrt {d^2-e^2 x^2}}{24 e^2}-\frac {4 d x^4 \sqrt {d^2-e^2 x^2}}{5 e}+\frac {1}{6} x^5 \sqrt {d^2-e^2 x^2}+\frac {\int \frac {-10755 d^7 e^6+16176 d^6 e^7 x}{\sqrt {d^2-e^2 x^2}} \, dx}{720 d e^{10}} \\ & = -\frac {d^3 (d-e x)^4}{e^5 \sqrt {d^2-e^2 x^2}}-\frac {337 d^5 \sqrt {d^2-e^2 x^2}}{15 e^5}+\frac {175 d^4 x \sqrt {d^2-e^2 x^2}}{16 e^4}-\frac {71 d^3 x^2 \sqrt {d^2-e^2 x^2}}{15 e^3}+\frac {47 d^2 x^3 \sqrt {d^2-e^2 x^2}}{24 e^2}-\frac {4 d x^4 \sqrt {d^2-e^2 x^2}}{5 e}+\frac {1}{6} x^5 \sqrt {d^2-e^2 x^2}-\frac {\left (239 d^6\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{16 e^4} \\ & = -\frac {d^3 (d-e x)^4}{e^5 \sqrt {d^2-e^2 x^2}}-\frac {337 d^5 \sqrt {d^2-e^2 x^2}}{15 e^5}+\frac {175 d^4 x \sqrt {d^2-e^2 x^2}}{16 e^4}-\frac {71 d^3 x^2 \sqrt {d^2-e^2 x^2}}{15 e^3}+\frac {47 d^2 x^3 \sqrt {d^2-e^2 x^2}}{24 e^2}-\frac {4 d x^4 \sqrt {d^2-e^2 x^2}}{5 e}+\frac {1}{6} x^5 \sqrt {d^2-e^2 x^2}-\frac {\left (239 d^6\right ) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e^4} \\ & = -\frac {d^3 (d-e x)^4}{e^5 \sqrt {d^2-e^2 x^2}}-\frac {337 d^5 \sqrt {d^2-e^2 x^2}}{15 e^5}+\frac {175 d^4 x \sqrt {d^2-e^2 x^2}}{16 e^4}-\frac {71 d^3 x^2 \sqrt {d^2-e^2 x^2}}{15 e^3}+\frac {47 d^2 x^3 \sqrt {d^2-e^2 x^2}}{24 e^2}-\frac {4 d x^4 \sqrt {d^2-e^2 x^2}}{5 e}+\frac {1}{6} x^5 \sqrt {d^2-e^2 x^2}-\frac {239 d^6 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.59 \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx=\frac {\frac {\sqrt {d^2-e^2 x^2} \left (-5632 d^6-2047 d^5 e x+769 d^4 e^2 x^2-426 d^3 e^3 x^3+278 d^2 e^4 x^4-152 d e^5 x^5+40 e^6 x^6\right )}{d+e x}+7170 d^6 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{240 e^5} \]

[In]

Integrate[(x^4*(d^2 - e^2*x^2)^(5/2))/(d + e*x)^4,x]

[Out]

((Sqrt[d^2 - e^2*x^2]*(-5632*d^6 - 2047*d^5*e*x + 769*d^4*e^2*x^2 - 426*d^3*e^3*x^3 + 278*d^2*e^4*x^4 - 152*d*
e^5*x^5 + 40*e^6*x^6))/(d + e*x) + 7170*d^6*ArcTan[(e*x)/(Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])])/(240*e^5)

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.68

method	result	size
risch	\(-\frac {\left (-40 e^{5} x^{5}+192 d \,e^{4} x^{4}-470 d^{2} e^{3} x^{3}+896 d^{3} e^{2} x^{2}-1665 d^{4} e x +3712 d^{5}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{240 e^{5}}-\frac {239 d^{6} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{16 e^{4} \sqrt {e^{2}}}-\frac {8 d^{6} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{e^{6} \left (x +\frac {d}{e}\right )}\)	\(153\)
default	\(\text {Expression too large to display}\)	\(1189\)

[In]

int(x^4*(-e^2*x^2+d^2)^(5/2)/(e*x+d)^4,x,method=_RETURNVERBOSE)

[Out]

-1/240*(-40*e^5*x^5+192*d*e^4*x^4-470*d^2*e^3*x^3+896*d^3*e^2*x^2-1665*d^4*e*x+3712*d^5)/e^5*(-e^2*x^2+d^2)^(1
/2)-239/16*d^6/e^4/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-8*d^6/e^6/(x+d/e)*(-(x+d/e)^2*e^2+2*
d*e*(x+d/e))^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.65 \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx=-\frac {5632 \, d^{6} e x + 5632 \, d^{7} - 7170 \, {\left (d^{6} e x + d^{7}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - {\left (40 \, e^{6} x^{6} - 152 \, d e^{5} x^{5} + 278 \, d^{2} e^{4} x^{4} - 426 \, d^{3} e^{3} x^{3} + 769 \, d^{4} e^{2} x^{2} - 2047 \, d^{5} e x - 5632 \, d^{6}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{240 \, {\left (e^{6} x + d e^{5}\right )}} \]

[In]

integrate(x^4*(-e^2*x^2+d^2)^(5/2)/(e*x+d)^4,x, algorithm="fricas")

[Out]

-1/240*(5632*d^6*e*x + 5632*d^7 - 7170*(d^6*e*x + d^7)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - (40*e^6*x^6
- 152*d*e^5*x^5 + 278*d^2*e^4*x^4 - 426*d^3*e^3*x^3 + 769*d^4*e^2*x^2 - 2047*d^5*e*x - 5632*d^6)*sqrt(-e^2*x^
2 + d^2))/(e^6*x + d*e^5)

Sympy [F]

\[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx=\int \frac {x^{4} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}}}{\left (d + e x\right )^{4}}\, dx \]

[In]

integrate(x**4*(-e**2*x**2+d**2)**(5/2)/(e*x+d)**4,x)

[Out]

Integral(x**4*(-(-d + e*x)*(d + e*x))**(5/2)/(d + e*x)**4, x)

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.29 (sec) , antiderivative size = 456, normalized size of antiderivative = 2.04 \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx=\frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4}}{2 \, {\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} + \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5}}{2 \, {\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )}} - \frac {15 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{6}}{e^{6} x + d e^{5}} - \frac {4 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3}}{3 \, {\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )}} - \frac {10 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4}}{3 \, {\left (e^{6} x + d e^{5}\right )}} + \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2}}{2 \, {\left (e^{6} x + d e^{5}\right )}} - \frac {9 i \, d^{6} \arcsin \left (\frac {e x}{d} + 2\right )}{4 \, e^{5}} - \frac {275 \, d^{6} \arcsin \left (\frac {e x}{d}\right )}{16 \, e^{5}} + \frac {9 \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{4} x}{4 \, e^{4}} + \frac {5 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{4} x}{16 \, e^{4}} + \frac {9 \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{5}}{2 \, e^{5}} - \frac {10 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{5}}{e^{5}} - \frac {19 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2} x}{24 \, e^{4}} + \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3}}{2 \, e^{5}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} x}{6 \, e^{4}} - \frac {4 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d}{5 \, e^{5}} \]

[In]

integrate(x^4*(-e^2*x^2+d^2)^(5/2)/(e*x+d)^4,x, algorithm="maxima")

[Out]

1/2*(-e^2*x^2 + d^2)^(5/2)*d^4/(e^8*x^3 + 3*d*e^7*x^2 + 3*d^2*e^6*x + d^3*e^5) + 5/2*(-e^2*x^2 + d^2)^(3/2)*d^
5/(e^7*x^2 + 2*d*e^6*x + d^2*e^5) - 15*sqrt(-e^2*x^2 + d^2)*d^6/(e^6*x + d*e^5) - 4/3*(-e^2*x^2 + d^2)^(5/2)*d
^3/(e^7*x^2 + 2*d*e^6*x + d^2*e^5) - 10/3*(-e^2*x^2 + d^2)^(3/2)*d^4/(e^6*x + d*e^5) + 3/2*(-e^2*x^2 + d^2)^(5
/2)*d^2/(e^6*x + d*e^5) - 9/4*I*d^6*arcsin(e*x/d + 2)/e^5 - 275/16*d^6*arcsin(e*x/d)/e^5 + 9/4*sqrt(e^2*x^2 +
4*d*e*x + 3*d^2)*d^4*x/e^4 + 5/16*sqrt(-e^2*x^2 + d^2)*d^4*x/e^4 + 9/2*sqrt(e^2*x^2 + 4*d*e*x + 3*d^2)*d^5/e^5
- 10*sqrt(-e^2*x^2 + d^2)*d^5/e^5 - 19/24*(-e^2*x^2 + d^2)^(3/2)*d^2*x/e^4 + 5/2*(-e^2*x^2 + d^2)^(3/2)*d^3/e
^5 + 1/6*(-e^2*x^2 + d^2)^(5/2)*x/e^4 - 4/5*(-e^2*x^2 + d^2)^(5/2)*d/e^5

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.62 \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx=-\frac {239 \, d^{6} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{16 \, e^{4} {\left | e \right |}} + \frac {1}{240} \, \sqrt {-e^{2} x^{2} + d^{2}} {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, x - \frac {24 \, d}{e}\right )} x + \frac {235 \, d^{2}}{e^{2}}\right )} x - \frac {448 \, d^{3}}{e^{3}}\right )} x + \frac {1665 \, d^{4}}{e^{4}}\right )} x - \frac {3712 \, d^{5}}{e^{5}}\right )} + \frac {16 \, d^{6}}{e^{4} {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} + 1\right )} {\left | e \right |}} \]

[In]

integrate(x^4*(-e^2*x^2+d^2)^(5/2)/(e*x+d)^4,x, algorithm="giac")

[Out]

-239/16*d^6*arcsin(e*x/d)*sgn(d)*sgn(e)/(e^4*abs(e)) + 1/240*sqrt(-e^2*x^2 + d^2)*((2*((4*(5*x - 24*d/e)*x + 2
35*d^2/e^2)*x - 448*d^3/e^3)*x + 1665*d^4/e^4)*x - 3712*d^5/e^5) + 16*d^6/(e^4*((d*e + sqrt(-e^2*x^2 + d^2)*ab
s(e))/(e^2*x) + 1)*abs(e))

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx=\int \frac {x^4\,{\left (d^2-e^2\,x^2\right )}^{5/2}}{{\left (d+e\,x\right )}^4} \,d x \]

[In]

int((x^4*(d^2 - e^2*x^2)^(5/2))/(d + e*x)^4,x)

[Out]

int((x^4*(d^2 - e^2*x^2)^(5/2))/(d + e*x)^4, x)