3.2.58 \(\int \frac {x^4 (d^2-e^2 x^2)^{5/2}}{(d+e x)^2} \, dx\) [158]

Optimal. Leaf size=200 \[ \frac {13 d^6 x \sqrt {d^2-e^2 x^2}}{128 e^4}+\frac {8 d^3 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^3}-\frac {13 d^2 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{48 e^2}+\frac {2 d x^4 \left (d^2-e^2 x^2\right )^{3/2}}{7 e}-\frac {1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}+\frac {d^4 (1024 d-1365 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6720 e^5}+\frac {13 d^8 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e^5} \]

[Out]

8/35*d^3*x^2*(-e^2*x^2+d^2)^(3/2)/e^3-13/48*d^2*x^3*(-e^2*x^2+d^2)^(3/2)/e^2+2/7*d*x^4*(-e^2*x^2+d^2)^(3/2)/e-
1/8*x^5*(-e^2*x^2+d^2)^(3/2)+1/6720*d^4*(-1365*e*x+1024*d)*(-e^2*x^2+d^2)^(3/2)/e^5+13/128*d^8*arctan(e*x/(-e^
2*x^2+d^2)^(1/2))/e^5+13/128*d^6*x*(-e^2*x^2+d^2)^(1/2)/e^4

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Rubi [A]
time = 0.17, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {866, 1823, 847, 794, 201, 223, 209} \begin {gather*} \frac {13 d^8 \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e^5}-\frac {1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}+\frac {2 d x^4 \left (d^2-e^2 x^2\right )^{3/2}}{7 e}-\frac {13 d^2 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{48 e^2}+\frac {13 d^6 x \sqrt {d^2-e^2 x^2}}{128 e^4}+\frac {d^4 (1024 d-1365 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6720 e^5}+\frac {8 d^3 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(13*d^6*x*Sqrt[d^2 - e^2*x^2])/(128*e^4) + (8*d^3*x^2*(d^2 - e^2*x^2)^(3/2))/(35*e^3) - (13*d^2*x^3*(d^2 - e^2
*x^2)^(3/2))/(48*e^2) + (2*d*x^4*(d^2 - e^2*x^2)^(3/2))/(7*e) - (x^5*(d^2 - e^2*x^2)^(3/2))/8 + (d^4*(1024*d -
1365*e*x)*(d^2 - e^2*x^2)^(3/2))/(6720*e^5) + (13*d^8*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(128*e^5)

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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 794

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*((a + c*x^2)^(p + 1)/(2*c*(p + 1)*(2*p + 3))), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

Rule 847

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[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*
Simp[c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p
}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p]) &&  !(IGtQ[m, 0] && EqQ[f, 0])

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 1823

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x,
Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Dist[1/(
b*(m + q + 2*p + 1)), Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*x^q
- a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x]
&& PolyQ[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])

Rubi steps

\begin {align*} \int \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx &=\int x^4 (d-e x)^2 \sqrt {d^2-e^2 x^2} \, dx\\ &=-\frac {1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}-\frac {\int x^4 \left (-13 d^2 e^2+16 d e^3 x\right ) \sqrt {d^2-e^2 x^2} \, dx}{8 e^2}\\ &=\frac {2 d x^4 \left (d^2-e^2 x^2\right )^{3/2}}{7 e}-\frac {1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}+\frac {\int x^3 \left (-64 d^3 e^3+91 d^2 e^4 x\right ) \sqrt {d^2-e^2 x^2} \, dx}{56 e^4}\\ &=-\frac {13 d^2 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{48 e^2}+\frac {2 d x^4 \left (d^2-e^2 x^2\right )^{3/2}}{7 e}-\frac {1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}-\frac {\int x^2 \left (-273 d^4 e^4+384 d^3 e^5 x\right ) \sqrt {d^2-e^2 x^2} \, dx}{336 e^6}\\ &=\frac {8 d^3 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^3}-\frac {13 d^2 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{48 e^2}+\frac {2 d x^4 \left (d^2-e^2 x^2\right )^{3/2}}{7 e}-\frac {1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}+\frac {\int x \left (-768 d^5 e^5+1365 d^4 e^6 x\right ) \sqrt {d^2-e^2 x^2} \, dx}{1680 e^8}\\ &=\frac {8 d^3 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^3}-\frac {13 d^2 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{48 e^2}+\frac {2 d x^4 \left (d^2-e^2 x^2\right )^{3/2}}{7 e}-\frac {1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}+\frac {d^4 (1024 d-1365 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6720 e^5}+\frac {\left (13 d^6\right ) \int \sqrt {d^2-e^2 x^2} \, dx}{64 e^4}\\ &=\frac {13 d^6 x \sqrt {d^2-e^2 x^2}}{128 e^4}+\frac {8 d^3 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^3}-\frac {13 d^2 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{48 e^2}+\frac {2 d x^4 \left (d^2-e^2 x^2\right )^{3/2}}{7 e}-\frac {1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}+\frac {d^4 (1024 d-1365 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6720 e^5}+\frac {\left (13 d^8\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{128 e^4}\\ &=\frac {13 d^6 x \sqrt {d^2-e^2 x^2}}{128 e^4}+\frac {8 d^3 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^3}-\frac {13 d^2 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{48 e^2}+\frac {2 d x^4 \left (d^2-e^2 x^2\right )^{3/2}}{7 e}-\frac {1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}+\frac {d^4 (1024 d-1365 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6720 e^5}+\frac {\left (13 d^8\right ) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e^4}\\ &=\frac {13 d^6 x \sqrt {d^2-e^2 x^2}}{128 e^4}+\frac {8 d^3 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^3}-\frac {13 d^2 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{48 e^2}+\frac {2 d x^4 \left (d^2-e^2 x^2\right )^{3/2}}{7 e}-\frac {1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}+\frac {d^4 (1024 d-1365 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6720 e^5}+\frac {13 d^8 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e^5}\\ \end {align*}

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Mathematica [A]
time = 0.25, size = 144, normalized size = 0.72 \begin {gather*} \frac {e \sqrt {d^2-e^2 x^2} \left (2048 d^7-1365 d^6 e x+1024 d^5 e^2 x^2-910 d^4 e^3 x^3+768 d^3 e^4 x^4+1960 d^2 e^5 x^5-3840 d e^6 x^6+1680 e^7 x^7\right )+1365 d^8 \sqrt {-e^2} \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{13440 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e*Sqrt[d^2 - e^2*x^2]*(2048*d^7 - 1365*d^6*e*x + 1024*d^5*e^2*x^2 - 910*d^4*e^3*x^3 + 768*d^3*e^4*x^4 + 1960*
d^2*e^5*x^5 - 3840*d*e^6*x^6 + 1680*e^7*x^7) + 1365*d^8*Sqrt[-e^2]*Log[-(Sqrt[-e^2]*x) + Sqrt[d^2 - e^2*x^2]])
/(13440*e^6)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(695\) vs. \(2(172)=344\).
time = 0.09, size = 696, normalized size = 3.48

method	result	size
risch	\(\frac {\left (1680 e^{7} x^{7}-3840 d \,e^{6} x^{6}+1960 d^{2} e^{5} x^{5}+768 d^{3} e^{4} x^{4}-910 d^{4} e^{3} x^{3}+1024 d^{5} e^{2} x^{2}-1365 d^{6} e x +2048 d^{7}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{13440 e^{5}}+\frac {13 d^{8} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{128 e^{4} \sqrt {e^{2}}}\)	\(130\)
default	\(\frac {-\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8 e^{2}}+\frac {d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{8 e^{2}}}{e^{2}}+\frac {2 d \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{7 e^{5}}+\frac {3 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{e^{4}}-\frac {4 d^{3} \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 )}{e^{5}}+\frac {d^{4} \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 )}{e^{6}}\)	\(696\)

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)^2,x,method=_RETURNVERBOSE)

[Out]

1/e^2*(-1/8*x*(-e^2*x^2+d^2)^(7/2)/e^2+1/8*d^2/e^2*(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))))))+2/
7*d/e^5*(-e^2*x^2+d^2)^(7/2)+3*d^2/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^3*(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^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 = 0.54, size = 256, normalized size = 1.28 \begin {gather*} \frac {7}{8} i \, d^{8} \arcsin \left (\frac {x e}{d} + 2\right ) e^{\left (-5\right )} + \frac {125}{128} \, d^{8} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-5\right )} - \frac {7}{8} \, \sqrt {x^{2} e^{2} + 4 \, d x e + 3 \, d^{2}} d^{6} x e^{\left (-4\right )} + \frac {125}{128} \, \sqrt {-x^{2} e^{2} + d^{2}} d^{6} x e^{\left (-4\right )} - \frac {7}{4} \, \sqrt {x^{2} e^{2} + 4 \, d x e + 3 \, d^{2}} d^{7} e^{\left (-5\right )} - \frac {67}{192} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4} x e^{\left (-4\right )} + \frac {5}{12} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5} e^{\left (-5\right )} + \frac {25}{48} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} x e^{\left (-4\right )} - \frac {4}{5} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{\left (-5\right )} + \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4}}{4 \, {\left (x e^{6} + d e^{5}\right )}} - \frac {1}{8} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} x e^{\left (-4\right )} + \frac {2}{7} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} d e^{\left (-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)^2,x, algorithm="maxima")

[Out]

7/8*I*d^8*arcsin(x*e/d + 2)*e^(-5) + 125/128*d^8*arcsin(x*e/d)*e^(-5) - 7/8*sqrt(x^2*e^2 + 4*d*x*e + 3*d^2)*d^
6*x*e^(-4) + 125/128*sqrt(-x^2*e^2 + d^2)*d^6*x*e^(-4) - 7/4*sqrt(x^2*e^2 + 4*d*x*e + 3*d^2)*d^7*e^(-5) - 67/1
92*(-x^2*e^2 + d^2)^(3/2)*d^4*x*e^(-4) + 5/12*(-x^2*e^2 + d^2)^(3/2)*d^5*e^(-5) + 25/48*(-x^2*e^2 + d^2)^(5/2)
*d^2*x*e^(-4) - 4/5*(-x^2*e^2 + d^2)^(5/2)*d^3*e^(-5) + 1/4*(-x^2*e^2 + d^2)^(5/2)*d^4/(x*e^6 + d*e^5) - 1/8*(
-x^2*e^2 + d^2)^(7/2)*x*e^(-4) + 2/7*(-x^2*e^2 + d^2)^(7/2)*d*e^(-5)

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Fricas [A]
time = 2.42, size = 119, normalized size = 0.60 \begin {gather*} -\frac {1}{13440} \, {\left (2730 \, d^{8} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) - {\left (1680 \, x^{7} e^{7} - 3840 \, d x^{6} e^{6} + 1960 \, d^{2} x^{5} e^{5} + 768 \, d^{3} x^{4} e^{4} - 910 \, d^{4} x^{3} e^{3} + 1024 \, d^{5} x^{2} e^{2} - 1365 \, d^{6} x e + 2048 \, d^{7}\right )} \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-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)^2,x, algorithm="fricas")

[Out]

-1/13440*(2730*d^8*arctan(-(d - sqrt(-x^2*e^2 + d^2))*e^(-1)/x) - (1680*x^7*e^7 - 3840*d*x^6*e^6 + 1960*d^2*x^
5*e^5 + 768*d^3*x^4*e^4 - 910*d^4*x^3*e^3 + 1024*d^5*x^2*e^2 - 1365*d^6*x*e + 2048*d^7)*sqrt(-x^2*e^2 + d^2))*
e^(-5)

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Sympy [C] Result contains complex when optimal does not.
time = 26.66, size = 690, normalized size = 3.45 \begin {gather*} d^{2} \left (\begin {cases} - \frac {i d^{6} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{16 e^{5}} + \frac {i d^{5} x}{16 e^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {i d^{3} x^{3}}{48 e^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {5 i d x^{5}}{24 \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {i e^{2} x^{7}}{6 d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {d^{6} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{16 e^{5}} - \frac {d^{5} x}{16 e^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {d^{3} x^{3}}{48 e^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {5 d x^{5}}{24 \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - \frac {e^{2} x^{7}}{6 d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) - 2 d e \left (\begin {cases} - \frac {8 d^{6} \sqrt {d^{2} - e^{2} x^{2}}}{105 e^{6}} - \frac {4 d^{4} x^{2} \sqrt {d^{2} - e^{2} x^{2}}}{105 e^{4}} - \frac {d^{2} x^{4} \sqrt {d^{2} - e^{2} x^{2}}}{35 e^{2}} + \frac {x^{6} \sqrt {d^{2} - e^{2} x^{2}}}{7} & \text {for}\: e \neq 0 \\\frac {x^{6} \sqrt {d^{2}}}{6} & \text {otherwise} \end {cases}\right ) + e^{2} \left (\begin {cases} - \frac {5 i d^{8} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{128 e^{7}} + \frac {5 i d^{7} x}{128 e^{6} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {5 i d^{5} x^{3}}{384 e^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {i d^{3} x^{5}}{192 e^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {7 i d x^{7}}{48 \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {i e^{2} x^{9}}{8 d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {5 d^{8} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{128 e^{7}} - \frac {5 d^{7} x}{128 e^{6} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {5 d^{5} x^{3}}{384 e^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {d^{3} x^{5}}{192 e^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {7 d x^{7}}{48 \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - \frac {e^{2} x^{9}}{8 d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\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)**2,x)

[Out]

d**2*Piecewise((-I*d**6*acosh(e*x/d)/(16*e**5) + I*d**5*x/(16*e**4*sqrt(-1 + e**2*x**2/d**2)) - I*d**3*x**3/(4
8*e**2*sqrt(-1 + e**2*x**2/d**2)) - 5*I*d*x**5/(24*sqrt(-1 + e**2*x**2/d**2)) + I*e**2*x**7/(6*d*sqrt(-1 + e**
2*x**2/d**2)), Abs(e**2*x**2/d**2) > 1), (d**6*asin(e*x/d)/(16*e**5) - d**5*x/(16*e**4*sqrt(1 - e**2*x**2/d**2
)) + d**3*x**3/(48*e**2*sqrt(1 - e**2*x**2/d**2)) + 5*d*x**5/(24*sqrt(1 - e**2*x**2/d**2)) - e**2*x**7/(6*d*sq
rt(1 - e**2*x**2/d**2)), True)) - 2*d*e*Piecewise((-8*d**6*sqrt(d**2 - e**2*x**2)/(105*e**6) - 4*d**4*x**2*sqr
t(d**2 - e**2*x**2)/(105*e**4) - d**2*x**4*sqrt(d**2 - e**2*x**2)/(35*e**2) + x**6*sqrt(d**2 - e**2*x**2)/7, N
e(e, 0)), (x**6*sqrt(d**2)/6, True)) + e**2*Piecewise((-5*I*d**8*acosh(e*x/d)/(128*e**7) + 5*I*d**7*x/(128*e**
6*sqrt(-1 + e**2*x**2/d**2)) - 5*I*d**5*x**3/(384*e**4*sqrt(-1 + e**2*x**2/d**2)) - I*d**3*x**5/(192*e**2*sqrt
(-1 + e**2*x**2/d**2)) - 7*I*d*x**7/(48*sqrt(-1 + e**2*x**2/d**2)) + I*e**2*x**9/(8*d*sqrt(-1 + e**2*x**2/d**2
)), Abs(e**2*x**2/d**2) > 1), (5*d**8*asin(e*x/d)/(128*e**7) - 5*d**7*x/(128*e**6*sqrt(1 - e**2*x**2/d**2)) +
5*d**5*x**3/(384*e**4*sqrt(1 - e**2*x**2/d**2)) + d**3*x**5/(192*e**2*sqrt(1 - e**2*x**2/d**2)) + 7*d*x**7/(48
*sqrt(1 - e**2*x**2/d**2)) - e**2*x**9/(8*d*sqrt(1 - e**2*x**2/d**2)), True))

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Giac [A]
time = 0.74, size = 301, normalized size = 1.50 \begin {gather*} -\frac {{\left (349440 \, d^{9} \arctan \left (\sqrt {\frac {2 \, d}{x e + d} - 1}\right ) e^{9} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + \frac {{\left (1365 \, d^{9} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {15}{2}} e^{9} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 61215 \, d^{9} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {13}{2}} e^{9} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 20517 \, d^{9} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {11}{2}} e^{9} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 141159 \, d^{9} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {9}{2}} e^{9} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 34969 \, d^{9} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {7}{2}} e^{9} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 34853 \, d^{9} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {5}{2}} e^{9} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 10465 \, d^{9} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} e^{9} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 1365 \, d^{9} \sqrt {\frac {2 \, d}{x e + d} - 1} e^{9} \mathrm {sgn}\left (\frac {1}{x e + d}\right )\right )} {\left (x e + d\right )}^{8}}{d^{8}}\right )} e^{\left (-14\right )}}{1720320 \, d} \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)^2,x, algorithm="giac")

[Out]

-1/1720320*(349440*d^9*arctan(sqrt(2*d/(x*e + d) - 1))*e^9*sgn(1/(x*e + d)) + (1365*d^9*(2*d/(x*e + d) - 1)^(1
5/2)*e^9*sgn(1/(x*e + d)) - 61215*d^9*(2*d/(x*e + d) - 1)^(13/2)*e^9*sgn(1/(x*e + d)) + 20517*d^9*(2*d/(x*e +
d) - 1)^(11/2)*e^9*sgn(1/(x*e + d)) - 141159*d^9*(2*d/(x*e + d) - 1)^(9/2)*e^9*sgn(1/(x*e + d)) - 34969*d^9*(2
*d/(x*e + d) - 1)^(7/2)*e^9*sgn(1/(x*e + d)) - 34853*d^9*(2*d/(x*e + d) - 1)^(5/2)*e^9*sgn(1/(x*e + d)) - 1046
5*d^9*(2*d/(x*e + d) - 1)^(3/2)*e^9*sgn(1/(x*e + d)) - 1365*d^9*sqrt(2*d/(x*e + d) - 1)*e^9*sgn(1/(x*e + d)))*
(x*e + d)^8/d^8)*e^(-14)/d

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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 )}^2} \,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)^2,x)

[Out]

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

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