3.2.100 \(\int \frac {x^3 (d^2-e^2 x^2)^{5/2}}{(d+e x)^4} \, dx\) [200]

Optimal. Leaf size=192 \[ \frac {d^2 (d-e x)^4}{e^4 \sqrt {d^2-e^2 x^2}}+\frac {101 d^4 \sqrt {d^2-e^2 x^2}}{5 e^4}-\frac {19 d^3 x \sqrt {d^2-e^2 x^2}}{2 e^3}+\frac {18 d^2 x^2 \sqrt {d^2-e^2 x^2}}{5 e^2}-\frac {d x^3 \sqrt {d^2-e^2 x^2}}{e}+\frac {1}{5} x^4 \sqrt {d^2-e^2 x^2}+\frac {27 d^5 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^4} \]

[Out]

27/2*d^5*arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e^4+d^2*(-e*x+d)^4/e^4/(-e^2*x^2+d^2)^(1/2)+101/5*d^4*(-e^2*x^2+d^2)
^(1/2)/e^4-19/2*d^3*x*(-e^2*x^2+d^2)^(1/2)/e^3+18/5*d^2*x^2*(-e^2*x^2+d^2)^(1/2)/e^2-d*x^3*(-e^2*x^2+d^2)^(1/2
)/e+1/5*x^4*(-e^2*x^2+d^2)^(1/2)

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Rubi [A]
time = 0.26, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 9, 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 {27 d^5 \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^4}+\frac {18 d^2 x^2 \sqrt {d^2-e^2 x^2}}{5 e^2}+\frac {1}{5} x^4 \sqrt {d^2-e^2 x^2}-\frac {d x^3 \sqrt {d^2-e^2 x^2}}{e}+\frac {d^2 (d-e x)^4}{e^4 \sqrt {d^2-e^2 x^2}}+\frac {101 d^4 \sqrt {d^2-e^2 x^2}}{5 e^4}-\frac {19 d^3 x \sqrt {d^2-e^2 x^2}}{2 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^3*(d^2 - e^2*x^2)^(5/2))/(d + e*x)^4,x]

[Out]

(d^2*(d - e*x)^4)/(e^4*Sqrt[d^2 - e^2*x^2]) + (101*d^4*Sqrt[d^2 - e^2*x^2])/(5*e^4) - (19*d^3*x*Sqrt[d^2 - e^2
*x^2])/(2*e^3) + (18*d^2*x^2*Sqrt[d^2 - e^2*x^2])/(5*e^2) - (d*x^3*Sqrt[d^2 - e^2*x^2])/e + (x^4*Sqrt[d^2 - e^
2*x^2])/5 + (27*d^5*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(2*e^4)

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^3 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx &=\int \frac {x^3 (d-e x)^4}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx\\ &=\frac {d^2 (d-e x)^4}{e^4 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {(d-e x)^3 \left (-\frac {4 d^3}{e^3}+\frac {d^2 x}{e^2}-\frac {d x^2}{e}\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{d}\\ &=\frac {d^2 (d-e x)^4}{e^4 \sqrt {d^2-e^2 x^2}}+\frac {1}{5} x^4 \sqrt {d^2-e^2 x^2}+\frac {\int \frac {\frac {20 d^6}{e}-65 d^5 x+80 d^4 e x^2-54 d^3 e^2 x^3+20 d^2 e^3 x^4}{\sqrt {d^2-e^2 x^2}} \, dx}{5 d e^2}\\ &=\frac {d^2 (d-e x)^4}{e^4 \sqrt {d^2-e^2 x^2}}-\frac {d x^3 \sqrt {d^2-e^2 x^2}}{e}+\frac {1}{5} x^4 \sqrt {d^2-e^2 x^2}-\frac {\int \frac {-80 d^6 e+260 d^5 e^2 x-380 d^4 e^3 x^2+216 d^3 e^4 x^3}{\sqrt {d^2-e^2 x^2}} \, dx}{20 d e^4}\\ &=\frac {d^2 (d-e x)^4}{e^4 \sqrt {d^2-e^2 x^2}}+\frac {18 d^2 x^2 \sqrt {d^2-e^2 x^2}}{5 e^2}-\frac {d x^3 \sqrt {d^2-e^2 x^2}}{e}+\frac {1}{5} x^4 \sqrt {d^2-e^2 x^2}+\frac {\int \frac {240 d^6 e^3-1212 d^5 e^4 x+1140 d^4 e^5 x^2}{\sqrt {d^2-e^2 x^2}} \, dx}{60 d e^6}\\ &=\frac {d^2 (d-e x)^4}{e^4 \sqrt {d^2-e^2 x^2}}-\frac {19 d^3 x \sqrt {d^2-e^2 x^2}}{2 e^3}+\frac {18 d^2 x^2 \sqrt {d^2-e^2 x^2}}{5 e^2}-\frac {d x^3 \sqrt {d^2-e^2 x^2}}{e}+\frac {1}{5} x^4 \sqrt {d^2-e^2 x^2}-\frac {\int \frac {-1620 d^6 e^5+2424 d^5 e^6 x}{\sqrt {d^2-e^2 x^2}} \, dx}{120 d e^8}\\ &=\frac {d^2 (d-e x)^4}{e^4 \sqrt {d^2-e^2 x^2}}+\frac {101 d^4 \sqrt {d^2-e^2 x^2}}{5 e^4}-\frac {19 d^3 x \sqrt {d^2-e^2 x^2}}{2 e^3}+\frac {18 d^2 x^2 \sqrt {d^2-e^2 x^2}}{5 e^2}-\frac {d x^3 \sqrt {d^2-e^2 x^2}}{e}+\frac {1}{5} x^4 \sqrt {d^2-e^2 x^2}+\frac {\left (27 d^5\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{2 e^3}\\ &=\frac {d^2 (d-e x)^4}{e^4 \sqrt {d^2-e^2 x^2}}+\frac {101 d^4 \sqrt {d^2-e^2 x^2}}{5 e^4}-\frac {19 d^3 x \sqrt {d^2-e^2 x^2}}{2 e^3}+\frac {18 d^2 x^2 \sqrt {d^2-e^2 x^2}}{5 e^2}-\frac {d x^3 \sqrt {d^2-e^2 x^2}}{e}+\frac {1}{5} x^4 \sqrt {d^2-e^2 x^2}+\frac {\left (27 d^5\right ) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^3}\\ &=\frac {d^2 (d-e x)^4}{e^4 \sqrt {d^2-e^2 x^2}}+\frac {101 d^4 \sqrt {d^2-e^2 x^2}}{5 e^4}-\frac {19 d^3 x \sqrt {d^2-e^2 x^2}}{2 e^3}+\frac {18 d^2 x^2 \sqrt {d^2-e^2 x^2}}{5 e^2}-\frac {d x^3 \sqrt {d^2-e^2 x^2}}{e}+\frac {1}{5} x^4 \sqrt {d^2-e^2 x^2}+\frac {27 d^5 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^4}\\ \end {align*}

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Mathematica [A]
time = 0.32, size = 129, normalized size = 0.67 \begin {gather*} \frac {\frac {e \sqrt {d^2-e^2 x^2} \left (212 d^5+77 d^4 e x-29 d^3 e^2 x^2+16 d^2 e^3 x^3-8 d e^4 x^4+2 e^5 x^5\right )}{d+e x}+135 d^5 \sqrt {-e^2} \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{10 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^3*(d^2 - e^2*x^2)^(5/2))/(d + e*x)^4,x]

[Out]

((e*Sqrt[d^2 - e^2*x^2]*(212*d^5 + 77*d^4*e*x - 29*d^3*e^2*x^2 + 16*d^2*e^3*x^3 - 8*d*e^4*x^4 + 2*e^5*x^5))/(d
+ e*x) + 135*d^5*Sqrt[-e^2]*Log[-(Sqrt[-e^2]*x) + Sqrt[d^2 - e^2*x^2]])/(10*e^5)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1085\) vs. \(2(168)=336\).
time = 0.08, size = 1086, normalized size = 5.66

method	result	size
risch	\(\frac {\left (2 e^{4} x^{4}-10 d \,e^{3} x^{3}+26 d^{2} x^{2} e^{2}-55 d^{3} e x +132 d^{4}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{10 e^{4}}+\frac {27 d^{5} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 e^{3} \sqrt {e^{2}}}+\frac {8 d^{5} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{e^{5} \left (x +\frac {d}{e}\right )}\)	\(142\)
default	\(\text {Expression too large to display}\)	\(1086\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(-e^2*x^2+d^2)^(5/2)/(e*x+d)^4,x,method=_RETURNVERBOSE)

[Out]

1/e^4*(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)))))+3/e^6*d^2*(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)))))))-d^3/e^7*(-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))))))))-3*d/e^5*(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 = 0.53, size = 375, normalized size = 1.95 \begin {gather*} \frac {3}{2} i \, d^{5} \arcsin \left (\frac {x e}{d} + 2\right ) e^{\left (-4\right )} + 15 \, d^{5} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-4\right )} - \frac {3}{2} \, \sqrt {x^{2} e^{2} + 4 \, d x e + 3 \, d^{2}} d^{3} x e^{\left (-3\right )} - 3 \, \sqrt {x^{2} e^{2} + 4 \, d x e + 3 \, d^{2}} d^{4} e^{\left (-4\right )} + \frac {15}{2} \, \sqrt {-x^{2} e^{2} + d^{2}} d^{4} e^{\left (-4\right )} - \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3}}{2 \, {\left (x^{3} e^{7} + 3 \, d x^{2} e^{6} + 3 \, d^{2} x e^{5} + d^{3} e^{4}\right )}} - \frac {5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4}}{2 \, {\left (x^{2} e^{6} + 2 \, d x e^{5} + d^{2} e^{4}\right )}} + \frac {15 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{5}}{x e^{5} + d e^{4}} + \frac {1}{4} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d x e^{\left (-3\right )} - \frac {5}{4} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2} e^{\left (-4\right )} + \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2}}{x^{2} e^{6} + 2 \, d x e^{5} + d^{2} e^{4}} + \frac {5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3}}{2 \, {\left (x e^{5} + d e^{4}\right )}} + \frac {1}{5} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} e^{\left (-4\right )} - \frac {3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d}{4 \, {\left (x e^{5} + d e^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(-e^2*x^2+d^2)^(5/2)/(e*x+d)^4,x, algorithm="maxima")

[Out]

3/2*I*d^5*arcsin(x*e/d + 2)*e^(-4) + 15*d^5*arcsin(x*e/d)*e^(-4) - 3/2*sqrt(x^2*e^2 + 4*d*x*e + 3*d^2)*d^3*x*e
^(-3) - 3*sqrt(x^2*e^2 + 4*d*x*e + 3*d^2)*d^4*e^(-4) + 15/2*sqrt(-x^2*e^2 + d^2)*d^4*e^(-4) - 1/2*(-x^2*e^2 +
d^2)^(5/2)*d^3/(x^3*e^7 + 3*d*x^2*e^6 + 3*d^2*x*e^5 + d^3*e^4) - 5/2*(-x^2*e^2 + d^2)^(3/2)*d^4/(x^2*e^6 + 2*d
*x*e^5 + d^2*e^4) + 15*sqrt(-x^2*e^2 + d^2)*d^5/(x*e^5 + d*e^4) + 1/4*(-x^2*e^2 + d^2)^(3/2)*d*x*e^(-3) - 5/4*
(-x^2*e^2 + d^2)^(3/2)*d^2*e^(-4) + (-x^2*e^2 + d^2)^(5/2)*d^2/(x^2*e^6 + 2*d*x*e^5 + d^2*e^4) + 5/2*(-x^2*e^2
+ d^2)^(3/2)*d^3/(x*e^5 + d*e^4) + 1/5*(-x^2*e^2 + d^2)^(5/2)*e^(-4) - 3/4*(-x^2*e^2 + d^2)^(5/2)*d/(x*e^5 +
d*e^4)

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Fricas [A]
time = 2.09, size = 128, normalized size = 0.67 \begin {gather*} \frac {212 \, d^{5} x e + 212 \, d^{6} - 270 \, {\left (d^{5} x e + d^{6}\right )} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) + {\left (2 \, x^{5} e^{5} - 8 \, d x^{4} e^{4} + 16 \, d^{2} x^{3} e^{3} - 29 \, d^{3} x^{2} e^{2} + 77 \, d^{4} x e + 212 \, d^{5}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{10 \, {\left (x e^{5} + d e^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(-e^2*x^2+d^2)^(5/2)/(e*x+d)^4,x, algorithm="fricas")

[Out]

1/10*(212*d^5*x*e + 212*d^6 - 270*(d^5*x*e + d^6)*arctan(-(d - sqrt(-x^2*e^2 + d^2))*e^(-1)/x) + (2*x^5*e^5 -
8*d*x^4*e^4 + 16*d^2*x^3*e^3 - 29*d^3*x^2*e^2 + 77*d^4*x*e + 212*d^5)*sqrt(-x^2*e^2 + d^2))/(x*e^5 + d*e^4)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3} \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**3*(-e**2*x**2+d**2)**(5/2)/(e*x+d)**4,x)

[Out]

Integral(x**3*(-(-d + e*x)*(d + e*x))**(5/2)/(d + e*x)**4, x)

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Giac [A]
time = 1.69, size = 113, normalized size = 0.59 \begin {gather*} \frac {27}{2} \, d^{5} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-4\right )} \mathrm {sgn}\left (d\right ) - \frac {16 \, d^{5} e^{\left (-4\right )}}{\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1} + \frac {1}{10} \, {\left (132 \, d^{4} e^{\left (-4\right )} - {\left (55 \, d^{3} e^{\left (-3\right )} - 2 \, {\left (13 \, d^{2} e^{\left (-2\right )} - {\left (5 \, d e^{\left (-1\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^3*(-e^2*x^2+d^2)^(5/2)/(e*x+d)^4,x, algorithm="giac")

[Out]

27/2*d^5*arcsin(x*e/d)*e^(-4)*sgn(d) - 16*d^5*e^(-4)/((d*e + sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/x + 1) + 1/10*(132
*d^4*e^(-4) - (55*d^3*e^(-3) - 2*(13*d^2*e^(-2) - (5*d*e^(-1) - x)*x)*x)*x)*sqrt(-x^2*e^2 + d^2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^3\,{\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^3*(d^2 - e^2*x^2)^(5/2))/(d + e*x)^4,x)

[Out]

int((x^3*(d^2 - e^2*x^2)^(5/2))/(d + e*x)^4, x)

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