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

Optimal. Leaf size=224 \[ -\frac {d^3 (d-e x)^4}{e^5 \sqrt {d^2-e^2 x^2}}-\frac {337 d^5 \sqrt {d^2-e^2 x^2}}{15 e^5}+\frac {175 d^4 x \sqrt {d^2-e^2 x^2}}{16 e^4}-\frac {71 d^3 x^2 \sqrt {d^2-e^2 x^2}}{15 e^3}+\frac {47 d^2 x^3 \sqrt {d^2-e^2 x^2}}{24 e^2}-\frac {4 d x^4 \sqrt {d^2-e^2 x^2}}{5 e}+\frac {1}{6} x^5 \sqrt {d^2-e^2 x^2}-\frac {239 d^6 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e^5} \]

[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)

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Rubi [A]
time = 0.33, antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {866, 1649, 1829, 655, 223, 209} \begin {gather*} -\frac {239 d^6 \text {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} \end {gather*}

Antiderivative was successfully verified.

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

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Mathematica [A]
time = 0.40, size = 143, normalized size = 0.64 \begin {gather*} \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 )}{240 e^5 (d+e x)}+\frac {239 d^6 \left (-e^2\right )^{3/2} \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{16 e^8} \end {gather*}

Antiderivative was successfully verified.

[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))/(240*e^5*(d + e*x)) + (239*d^6*(-e^2)^(3/2)*Log[-(Sqrt[-e^2]*x) + Sqrt[d^2 - e^2*x^2]])/
(16*e^8)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1188\) vs. \(2(194)=388\).
time = 0.07, size = 1189, normalized size = 5.31

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/e^4*(1/6*x*(-e^2*x^2+d^2)^(5/2)+5/6*d^2*(1/4*x*(-e^2*x^2+d^2)^(3/2)+3/4*d^2*(1/2*x*(-e^2*x^2+d^2)^(1/2)+1/2*
d^2/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2)))))-4/e^5*d*(1/5*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(5/2
)+d*e*(-1/8*(-2*e^2*(x+d/e)+2*d*e)/e^2*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)+3/4*d^2*(-1/4*(-2*e^2*(x+d/e)+2*d*
e)/e^2*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)+1/2*d^2/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-(x+d/e)^2*e^2+2*d*e*(x+
d/e))^(1/2)))))-4/e^7*d^3*(1/d/e/(x+d/e)^3*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(7/2)+4*e/d*(1/3/d/e/(x+d/e)^2*(-(x+
d/e)^2*e^2+2*d*e*(x+d/e))^(7/2)+5/3*e/d*(1/5*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(5/2)+d*e*(-1/8*(-2*e^2*(x+d/e)+2*
d*e)/e^2*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)+3/4*d^2*(-1/4*(-2*e^2*(x+d/e)+2*d*e)/e^2*(-(x+d/e)^2*e^2+2*d*e*(
x+d/e))^(1/2)+1/2*d^2/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)))))))+1/e^8*d^4*(-
1/d/e/(x+d/e)^4*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(7/2)-3*e/d*(1/d/e/(x+d/e)^3*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(7/
2)+4*e/d*(1/3/d/e/(x+d/e)^2*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(7/2)+5/3*e/d*(1/5*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(
5/2)+d*e*(-1/8*(-2*e^2*(x+d/e)+2*d*e)/e^2*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)+3/4*d^2*(-1/4*(-2*e^2*(x+d/e)+2
*d*e)/e^2*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)+1/2*d^2/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-(x+d/e)^2*e^2+2*d*e*
(x+d/e))^(1/2))))))))+6/e^6*d^2*(1/3/d/e/(x+d/e)^2*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(7/2)+5/3*e/d*(1/5*(-(x+d/e)
^2*e^2+2*d*e*(x+d/e))^(5/2)+d*e*(-1/8*(-2*e^2*(x+d/e)+2*d*e)/e^2*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)+3/4*d^2*
(-1/4*(-2*e^2*(x+d/e)+2*d*e)/e^2*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)+1/2*d^2/(e^2)^(1/2)*arctan((e^2)^(1/2)*x
/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2))))))

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Maxima [C] Result contains complex when optimal does not.
time = 0.52, size = 420, normalized size = 1.88 \begin {gather*} -\frac {9}{4} i \, d^{6} \arcsin \left (\frac {x e}{d} + 2\right ) e^{\left (-5\right )} - \frac {275}{16} \, d^{6} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-5\right )} + \frac {9}{4} \, \sqrt {x^{2} e^{2} + 4 \, d x e + 3 \, d^{2}} d^{4} x e^{\left (-4\right )} + \frac {5}{16} \, \sqrt {-x^{2} e^{2} + d^{2}} d^{4} x e^{\left (-4\right )} + \frac {9}{2} \, \sqrt {x^{2} e^{2} + 4 \, d x e + 3 \, d^{2}} d^{5} e^{\left (-5\right )} - 10 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{5} e^{\left (-5\right )} + \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4}}{2 \, {\left (x^{3} e^{8} + 3 \, d x^{2} e^{7} + 3 \, d^{2} x e^{6} + d^{3} e^{5}\right )}} + \frac {5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5}}{2 \, {\left (x^{2} e^{7} + 2 \, d x e^{6} + d^{2} e^{5}\right )}} - \frac {15 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{6}}{x e^{6} + d e^{5}} - \frac {19}{24} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2} x e^{\left (-4\right )} + \frac {5}{2} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} e^{\left (-5\right )} - \frac {4 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3}}{3 \, {\left (x^{2} e^{7} + 2 \, d x e^{6} + d^{2} e^{5}\right )}} - \frac {10 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4}}{3 \, {\left (x e^{6} + d e^{5}\right )}} + \frac {1}{6} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} x e^{\left (-4\right )} - \frac {4}{5} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d e^{\left (-5\right )} + \frac {3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2}}{2 \, {\left (x e^{6} + d e^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 1.51, size = 139, normalized size = 0.62 \begin {gather*} -\frac {5632 \, d^{6} x e + 5632 \, d^{7} - 7170 \, {\left (d^{6} x e + d^{7}\right )} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) - {\left (40 \, x^{6} e^{6} - 152 \, d x^{5} e^{5} + 278 \, d^{2} x^{4} e^{4} - 426 \, d^{3} x^{3} e^{3} + 769 \, d^{4} x^{2} e^{2} - 2047 \, d^{5} x e - 5632 \, d^{6}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{240 \, {\left (x e^{6} + d e^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]
time = 4.65, size = 124, normalized size = 0.55 \begin {gather*} -\frac {239}{16} \, d^{6} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-5\right )} \mathrm {sgn}\left (d\right ) + \frac {16 \, d^{6} e^{\left (-5\right )}}{\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1} - \frac {1}{240} \, {\left (3712 \, d^{5} e^{\left (-5\right )} - {\left (1665 \, d^{4} e^{\left (-4\right )} - 2 \, {\left (448 \, d^{3} e^{\left (-3\right )} - {\left (235 \, d^{2} e^{\left (-2\right )} - 4 \, {\left (24 \, d e^{\left (-1\right )} - 5 \, x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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(x*e/d)*e^(-5)*sgn(d) + 16*d^6*e^(-5)/((d*e + sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/x + 1) - 1/240*
(3712*d^5*e^(-5) - (1665*d^4*e^(-4) - 2*(448*d^3*e^(-3) - (235*d^2*e^(-2) - 4*(24*d*e^(-1) - 5*x)*x)*x)*x)*x)*
sqrt(-x^2*e^2 + d^2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^4\,{\left (d^2-e^2\,x^2\right )}^{5/2}}{{\left (d+e\,x\right )}^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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