3.3.1 \(\int \frac {x^2 (d^2-e^2 x^2)^{5/2}}{(d+e x)^4} \, dx\) [201]

Optimal. Leaf size=182 \[ -\frac {d (d-e x)^4}{e^3 \sqrt {d^2-e^2 x^2}}-\frac {95 d^3 \sqrt {d^2-e^2 x^2}}{8 e^3}-\frac {95 d^2 (d-e x) \sqrt {d^2-e^2 x^2}}{24 e^3}-\frac {19 d (d-e x)^2 \sqrt {d^2-e^2 x^2}}{12 e^3}-\frac {(d-e x)^3 \sqrt {d^2-e^2 x^2}}{4 e^3}-\frac {95 d^4 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e^3} \]

[Out]

-95/8*d^4*arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e^3-d*(-e*x+d)^4/e^3/(-e^2*x^2+d^2)^(1/2)-95/8*d^3*(-e^2*x^2+d^2)^(
1/2)/e^3-95/24*d^2*(-e*x+d)*(-e^2*x^2+d^2)^(1/2)/e^3-19/12*d*(-e*x+d)^2*(-e^2*x^2+d^2)^(1/2)/e^3-1/4*(-e*x+d)^
3*(-e^2*x^2+d^2)^(1/2)/e^3

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Rubi [A]
time = 0.13, antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {866, 1649, 809, 685, 655, 223, 209} \begin {gather*} -\frac {95 d^4 \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e^3}-\frac {95 d^2 (d-e x) \sqrt {d^2-e^2 x^2}}{24 e^3}-\frac {19 d (d-e x)^2 \sqrt {d^2-e^2 x^2}}{12 e^3}-\frac {d (d-e x)^4}{e^3 \sqrt {d^2-e^2 x^2}}-\frac {(d-e x)^3 \sqrt {d^2-e^2 x^2}}{4 e^3}-\frac {95 d^3 \sqrt {d^2-e^2 x^2}}{8 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((d*(d - e*x)^4)/(e^3*Sqrt[d^2 - e^2*x^2])) - (95*d^3*Sqrt[d^2 - e^2*x^2])/(8*e^3) - (95*d^2*(d - e*x)*Sqrt[d
^2 - e^2*x^2])/(24*e^3) - (19*d*(d - e*x)^2*Sqrt[d^2 - e^2*x^2])/(12*e^3) - ((d - e*x)^3*Sqrt[d^2 - e^2*x^2])/
(4*e^3) - (95*d^4*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(8*e^3)

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 685

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

Rule 809

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

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]

Rubi steps

\begin {align*} \int \frac {x^2 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx &=\int \frac {x^2 (d-e x)^4}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx\\ &=-\frac {d (d-e x)^4}{e^3 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {\left (\frac {4 d^2}{e^2}-\frac {d x}{e}\right ) (d-e x)^3}{\sqrt {d^2-e^2 x^2}} \, dx}{d}\\ &=-\frac {d (d-e x)^4}{e^3 \sqrt {d^2-e^2 x^2}}-\frac {(d-e x)^3 \sqrt {d^2-e^2 x^2}}{4 e^3}-\frac {(19 d) \int \frac {(d-e x)^3}{\sqrt {d^2-e^2 x^2}} \, dx}{4 e^2}\\ &=-\frac {d (d-e x)^4}{e^3 \sqrt {d^2-e^2 x^2}}-\frac {19 d (d-e x)^2 \sqrt {d^2-e^2 x^2}}{12 e^3}-\frac {(d-e x)^3 \sqrt {d^2-e^2 x^2}}{4 e^3}-\frac {\left (95 d^2\right ) \int \frac {(d-e x)^2}{\sqrt {d^2-e^2 x^2}} \, dx}{12 e^2}\\ &=-\frac {d (d-e x)^4}{e^3 \sqrt {d^2-e^2 x^2}}-\frac {95 d^2 (d-e x) \sqrt {d^2-e^2 x^2}}{24 e^3}-\frac {19 d (d-e x)^2 \sqrt {d^2-e^2 x^2}}{12 e^3}-\frac {(d-e x)^3 \sqrt {d^2-e^2 x^2}}{4 e^3}-\frac {\left (95 d^3\right ) \int \frac {d-e x}{\sqrt {d^2-e^2 x^2}} \, dx}{8 e^2}\\ &=-\frac {d (d-e x)^4}{e^3 \sqrt {d^2-e^2 x^2}}-\frac {95 d^3 \sqrt {d^2-e^2 x^2}}{8 e^3}-\frac {95 d^2 (d-e x) \sqrt {d^2-e^2 x^2}}{24 e^3}-\frac {19 d (d-e x)^2 \sqrt {d^2-e^2 x^2}}{12 e^3}-\frac {(d-e x)^3 \sqrt {d^2-e^2 x^2}}{4 e^3}-\frac {\left (95 d^4\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{8 e^2}\\ &=-\frac {d (d-e x)^4}{e^3 \sqrt {d^2-e^2 x^2}}-\frac {95 d^3 \sqrt {d^2-e^2 x^2}}{8 e^3}-\frac {95 d^2 (d-e x) \sqrt {d^2-e^2 x^2}}{24 e^3}-\frac {19 d (d-e x)^2 \sqrt {d^2-e^2 x^2}}{12 e^3}-\frac {(d-e x)^3 \sqrt {d^2-e^2 x^2}}{4 e^3}-\frac {\left (95 d^4\right ) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e^2}\\ &=-\frac {d (d-e x)^4}{e^3 \sqrt {d^2-e^2 x^2}}-\frac {95 d^3 \sqrt {d^2-e^2 x^2}}{8 e^3}-\frac {95 d^2 (d-e x) \sqrt {d^2-e^2 x^2}}{24 e^3}-\frac {19 d (d-e x)^2 \sqrt {d^2-e^2 x^2}}{12 e^3}-\frac {(d-e x)^3 \sqrt {d^2-e^2 x^2}}{4 e^3}-\frac {95 d^4 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e^3}\\ \end {align*}

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Mathematica [A]
time = 0.36, size = 121, normalized size = 0.66 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-448 d^4-163 d^3 e x+61 d^2 e^2 x^2-26 d e^3 x^3+6 e^4 x^4\right )}{24 e^3 (d+e x)}+\frac {95 d^4 \left (-e^2\right )^{3/2} \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{8 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(-448*d^4 - 163*d^3*e*x + 61*d^2*e^2*x^2 - 26*d*e^3*x^3 + 6*e^4*x^4))/(24*e^3*(d + e*x))
+ (95*d^4*(-e^2)^(3/2)*Log[-(Sqrt[-e^2]*x) + Sqrt[d^2 - e^2*x^2]])/(8*e^6)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(889\) vs. \(2(160)=320\).
time = 0.07, size = 890, normalized size = 4.89

method	result	size
risch	\(-\frac {\left (-6 e^{3} x^{3}+32 d \,e^{2} x^{2}-93 d^{2} e x +256 d^{3}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{24 e^{3}}-\frac {95 d^{4} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{8 e^{2} \sqrt {e^{2}}}-\frac {8 d^{4} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{e^{4} \left (x +\frac {d}{e}\right )}\)	\(131\)
default	\(-\frac {2 d \left (\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{d e \left (x +\frac {d}{e}\right )^{3}}+\frac {4 e \left (\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{3 d e \left (x +\frac {d}{e}\right )^{2}}+\frac {5 e \left (\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{5}+d e \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 e^{2}}+\frac {3 d^{2} \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{4 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )\right )}{3 d}\right )}{d}\right )}{e^{5}}+\frac {d^{2} \left (-\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{d e \left (x +\frac {d}{e}\right )^{4}}-\frac {3 e \left (\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{d e \left (x +\frac {d}{e}\right )^{3}}+\frac {4 e \left (\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{3 d e \left (x +\frac {d}{e}\right )^{2}}+\frac {5 e \left (\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{5}+d e \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 e^{2}}+\frac {3 d^{2} \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{4 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )\right )}{3 d}\right )}{d}\right )}{d}\right )}{e^{6}}+\frac {\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{3 d e \left (x +\frac {d}{e}\right )^{2}}+\frac {5 e \left (\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{5}+d e \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 e^{2}}+\frac {3 d^{2} \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{4 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )\right )}{3 d}}{e^{4}}\)	\(890\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*d/e^5*(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^6*d^2*(-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))))
))))+1/e^4*(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 = 0.57, size = 335, normalized size = 1.84 \begin {gather*} -\frac {5}{8} i \, d^{4} \arcsin \left (\frac {x e}{d} + 2\right ) e^{\left (-3\right )} - \frac {25}{2} \, d^{4} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-3\right )} + \frac {5}{8} \, \sqrt {x^{2} e^{2} + 4 \, d x e + 3 \, d^{2}} d^{2} x e^{\left (-2\right )} + \frac {5}{4} \, \sqrt {x^{2} e^{2} + 4 \, d x e + 3 \, d^{2}} d^{3} e^{\left (-3\right )} - 5 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{3} e^{\left (-3\right )} + \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2}}{2 \, {\left (x^{3} e^{6} + 3 \, d x^{2} e^{5} + 3 \, d^{2} x e^{4} + d^{3} e^{3}\right )}} + \frac {5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3}}{2 \, {\left (x^{2} e^{5} + 2 \, d x e^{4} + d^{2} e^{3}\right )}} - \frac {15 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{4}}{x e^{4} + d e^{3}} + \frac {5}{12} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d e^{\left (-3\right )} - \frac {2 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d}{3 \, {\left (x^{2} e^{5} + 2 \, d x e^{4} + d^{2} e^{3}\right )}} - \frac {5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}}{3 \, {\left (x e^{4} + d e^{3}\right )}} + \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}}{4 \, {\left (x e^{4} + d e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/8*I*d^4*arcsin(x*e/d + 2)*e^(-3) - 25/2*d^4*arcsin(x*e/d)*e^(-3) + 5/8*sqrt(x^2*e^2 + 4*d*x*e + 3*d^2)*d^2*
x*e^(-2) + 5/4*sqrt(x^2*e^2 + 4*d*x*e + 3*d^2)*d^3*e^(-3) - 5*sqrt(-x^2*e^2 + d^2)*d^3*e^(-3) + 1/2*(-x^2*e^2
+ d^2)^(5/2)*d^2/(x^3*e^6 + 3*d*x^2*e^5 + 3*d^2*x*e^4 + d^3*e^3) + 5/2*(-x^2*e^2 + d^2)^(3/2)*d^3/(x^2*e^5 + 2
*d*x*e^4 + d^2*e^3) - 15*sqrt(-x^2*e^2 + d^2)*d^4/(x*e^4 + d*e^3) + 5/12*(-x^2*e^2 + d^2)^(3/2)*d*e^(-3) - 2/3
*(-x^2*e^2 + d^2)^(5/2)*d/(x^2*e^5 + 2*d*x*e^4 + d^2*e^3) - 5/3*(-x^2*e^2 + d^2)^(3/2)*d^2/(x*e^4 + d*e^3) + 1
/4*(-x^2*e^2 + d^2)^(5/2)/(x*e^4 + d*e^3)

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Fricas [A]
time = 1.86, size = 119, normalized size = 0.65 \begin {gather*} -\frac {448 \, d^{4} x e + 448 \, d^{5} - 570 \, {\left (d^{4} x e + d^{5}\right )} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) - {\left (6 \, x^{4} e^{4} - 26 \, d x^{3} e^{3} + 61 \, d^{2} x^{2} e^{2} - 163 \, d^{3} x e - 448 \, d^{4}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{24 \, {\left (x e^{4} + d e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(448*d^4*x*e + 448*d^5 - 570*(d^4*x*e + d^5)*arctan(-(d - sqrt(-x^2*e^2 + d^2))*e^(-1)/x) - (6*x^4*e^4 -
26*d*x^3*e^3 + 61*d^2*x^2*e^2 - 163*d^3*x*e - 448*d^4)*sqrt(-x^2*e^2 + d^2))/(x*e^4 + d*e^3)

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

[Out]

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

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Giac [A]
time = 1.99, size = 102, normalized size = 0.56 \begin {gather*} -\frac {95}{8} \, d^{4} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-3\right )} \mathrm {sgn}\left (d\right ) + \frac {16 \, d^{4} e^{\left (-3\right )}}{\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1} - \frac {1}{24} \, {\left (256 \, d^{3} e^{\left (-3\right )} - {\left (93 \, d^{2} e^{\left (-2\right )} - 2 \, {\left (16 \, d e^{\left (-1\right )} - 3 \, 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^2*(-e^2*x^2+d^2)^(5/2)/(e*x+d)^4,x, algorithm="giac")

[Out]

-95/8*d^4*arcsin(x*e/d)*e^(-3)*sgn(d) + 16*d^4*e^(-3)/((d*e + sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/x + 1) - 1/24*(25
6*d^3*e^(-3) - (93*d^2*e^(-2) - 2*(16*d*e^(-1) - 3*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^2\,{\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^2*(d^2 - e^2*x^2)^(5/2))/(d + e*x)^4,x)

[Out]

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

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