### 3.132 $$\int \frac{x^5 (a+b x^2+c x^4)}{\sqrt{d-e x} \sqrt{d+e x}} \, dx$$

Optimal. Leaf size=210 $-\frac{(d-e x)^{5/2} (d+e x)^{5/2} \left (a e^4+3 b d^2 e^2+6 c d^4\right )}{5 e^{10}}+\frac{d^2 (d-e x)^{3/2} (d+e x)^{3/2} \left (2 a e^4+3 b d^2 e^2+4 c d^4\right )}{3 e^{10}}-\frac{d^4 \sqrt{d-e x} \sqrt{d+e x} \left (a e^4+b d^2 e^2+c d^4\right )}{e^{10}}+\frac{(d-e x)^{7/2} (d+e x)^{7/2} \left (b e^2+4 c d^2\right )}{7 e^{10}}-\frac{c (d-e x)^{9/2} (d+e x)^{9/2}}{9 e^{10}}$

[Out]

-((d^4*(c*d^4 + b*d^2*e^2 + a*e^4)*Sqrt[d - e*x]*Sqrt[d + e*x])/e^10) + (d^2*(4*c*d^4 + 3*b*d^2*e^2 + 2*a*e^4)
*(d - e*x)^(3/2)*(d + e*x)^(3/2))/(3*e^10) - ((6*c*d^4 + 3*b*d^2*e^2 + a*e^4)*(d - e*x)^(5/2)*(d + e*x)^(5/2))
/(5*e^10) + ((4*c*d^2 + b*e^2)*(d - e*x)^(7/2)*(d + e*x)^(7/2))/(7*e^10) - (c*(d - e*x)^(9/2)*(d + e*x)^(9/2))
/(9*e^10)

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Rubi [A]  time = 0.314532, antiderivative size = 278, normalized size of antiderivative = 1.32, number of steps used = 5, number of rules used = 4, integrand size = 35, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.114, Rules used = {520, 1251, 897, 1153} $-\frac{\left (d^2-e^2 x^2\right )^3 \left (a e^4+3 b d^2 e^2+6 c d^4\right )}{5 e^{10} \sqrt{d-e x} \sqrt{d+e x}}+\frac{d^2 \left (d^2-e^2 x^2\right )^2 \left (2 a e^4+3 b d^2 e^2+4 c d^4\right )}{3 e^{10} \sqrt{d-e x} \sqrt{d+e x}}-\frac{d^4 \left (d^2-e^2 x^2\right ) \left (a e^4+b d^2 e^2+c d^4\right )}{e^{10} \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (d^2-e^2 x^2\right )^4 \left (b e^2+4 c d^2\right )}{7 e^{10} \sqrt{d-e x} \sqrt{d+e x}}-\frac{c \left (d^2-e^2 x^2\right )^5}{9 e^{10} \sqrt{d-e x} \sqrt{d+e x}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-((d^4*(c*d^4 + b*d^2*e^2 + a*e^4)*(d^2 - e^2*x^2))/(e^10*Sqrt[d - e*x]*Sqrt[d + e*x])) + (d^2*(4*c*d^4 + 3*b*
d^2*e^2 + 2*a*e^4)*(d^2 - e^2*x^2)^2)/(3*e^10*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((6*c*d^4 + 3*b*d^2*e^2 + a*e^4)*
(d^2 - e^2*x^2)^3)/(5*e^10*Sqrt[d - e*x]*Sqrt[d + e*x]) + ((4*c*d^2 + b*e^2)*(d^2 - e^2*x^2)^4)/(7*e^10*Sqrt[d
- e*x]*Sqrt[d + e*x]) - (c*(d^2 - e^2*x^2)^5)/(9*e^10*Sqrt[d - e*x]*Sqrt[d + e*x])

Rule 520

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.) + (e_.)*(x_)^(n2_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b
2_.)*(x_)^(non2_.))^(p_.), x_Symbol] :> Dist[((a1 + b1*x^(n/2))^FracPart[p]*(a2 + b2*x^(n/2))^FracPart[p])/(a1
*a2 + b1*b2*x^n)^FracPart[p], Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n + e*x^(2*n))^q, x], x] /; FreeQ[{a1, b1,
a2, b2, c, d, e, n, p, q}, x] && EqQ[non2, n/2] && EqQ[n2, 2*n] && EqQ[a2*b1 + a1*b2, 0]

Rule 1251

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rule 897

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{q = Denominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + (g*x^q)/e)^n*((c*d^2 - b*d
*e + a*e^2)/e^2 - ((2*c*d - b*e)*x^q)/e^2 + (c*x^(2*q))/e^2)^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegersQ[n,
p] && FractionQ[m]

Rule 1153

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rubi steps

\begin{align*} \int \frac{x^5 \left (a+b x^2+c x^4\right )}{\sqrt{d-e x} \sqrt{d+e x}} \, dx &=\frac{\sqrt{d^2-e^2 x^2} \int \frac{x^5 \left (a+b x^2+c x^4\right )}{\sqrt{d^2-e^2 x^2}} \, dx}{\sqrt{d-e x} \sqrt{d+e x}}\\ &=\frac{\sqrt{d^2-e^2 x^2} \operatorname{Subst}\left (\int \frac{x^2 \left (a+b x+c x^2\right )}{\sqrt{d^2-e^2 x}} \, dx,x,x^2\right )}{2 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{\sqrt{d^2-e^2 x^2} \operatorname{Subst}\left (\int \left (\frac{d^2}{e^2}-\frac{x^2}{e^2}\right )^2 \left (\frac{c d^4+b d^2 e^2+a e^4}{e^4}-\frac{\left (2 c d^2+b e^2\right ) x^2}{e^4}+\frac{c x^4}{e^4}\right ) \, dx,x,\sqrt{d^2-e^2 x^2}\right )}{e^2 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{\sqrt{d^2-e^2 x^2} \operatorname{Subst}\left (\int \left (\frac{c d^8+b d^6 e^2+a d^4 e^4}{e^8}-\frac{d^2 \left (4 c d^4+3 b d^2 e^2+2 a e^4\right ) x^2}{e^8}+\frac{\left (6 c d^4+3 b d^2 e^2+a e^4\right ) x^4}{e^8}-\frac{\left (4 c d^2+b e^2\right ) x^6}{e^8}+\frac{c x^8}{e^8}\right ) \, dx,x,\sqrt{d^2-e^2 x^2}\right )}{e^2 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{d^4 \left (c d^4+b d^2 e^2+a e^4\right ) \left (d^2-e^2 x^2\right )}{e^{10} \sqrt{d-e x} \sqrt{d+e x}}+\frac{d^2 \left (4 c d^4+3 b d^2 e^2+2 a e^4\right ) \left (d^2-e^2 x^2\right )^2}{3 e^{10} \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (6 c d^4+3 b d^2 e^2+a e^4\right ) \left (d^2-e^2 x^2\right )^3}{5 e^{10} \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (4 c d^2+b e^2\right ) \left (d^2-e^2 x^2\right )^4}{7 e^{10} \sqrt{d-e x} \sqrt{d+e x}}-\frac{c \left (d^2-e^2 x^2\right )^5}{9 e^{10} \sqrt{d-e x} \sqrt{d+e x}}\\ \end{align*}

Mathematica [C]  time = 1.43718, size = 265, normalized size = 1.26 $-\frac{\sqrt{d-e x} \sqrt{d+e x} \left (21 a e^4 \left (4 d^2 e^2 x^2+8 d^4+3 e^4 x^4\right )+9 b \left (6 d^2 e^6 x^4+8 d^4 e^4 x^2+16 d^6 e^2+5 e^8 x^6\right )+c \left (64 d^6 e^2 x^2+48 d^4 e^4 x^4+40 d^2 e^6 x^6+128 d^8+35 e^8 x^8\right )\right )+\frac{630 d^{9/2} \sqrt{d+e x} \sin ^{-1}\left (\frac{\sqrt{d-e x}}{\sqrt{2} \sqrt{d}}\right ) \left (a e^4+b d^2 e^2+c d^4\right )}{\sqrt{\frac{e x}{d}+1}}-630 d^5 \tan ^{-1}\left (\frac{\sqrt{d-e x}}{\sqrt{d+e x}}\right ) \left (a e^4+b d^2 e^2+c d^4\right )}{315 e^{10}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(Sqrt[d - e*x]*Sqrt[d + e*x]*(21*a*e^4*(8*d^4 + 4*d^2*e^2*x^2 + 3*e^4*x^4) + 9*b*(16*d^6*e^2 + 8*d^4*e^4*x^2
+ 6*d^2*e^6*x^4 + 5*e^8*x^6) + c*(128*d^8 + 64*d^6*e^2*x^2 + 48*d^4*e^4*x^4 + 40*d^2*e^6*x^6 + 35*e^8*x^8)) +
(630*d^(9/2)*(c*d^4 + b*d^2*e^2 + a*e^4)*Sqrt[d + e*x]*ArcSin[Sqrt[d - e*x]/(Sqrt[2]*Sqrt[d])])/Sqrt[1 + (e*x)
/d] - 630*d^5*(c*d^4 + b*d^2*e^2 + a*e^4)*ArcTan[Sqrt[d - e*x]/Sqrt[d + e*x]])/(315*e^10)

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Maple [A]  time = 0.006, size = 145, normalized size = 0.7 \begin{align*} -{\frac{35\,c{x}^{8}{e}^{8}+45\,b{e}^{8}{x}^{6}+40\,c{d}^{2}{e}^{6}{x}^{6}+63\,a{e}^{8}{x}^{4}+54\,b{d}^{2}{e}^{6}{x}^{4}+48\,c{d}^{4}{e}^{4}{x}^{4}+84\,a{d}^{2}{e}^{6}{x}^{2}+72\,b{d}^{4}{e}^{4}{x}^{2}+64\,c{d}^{6}{e}^{2}{x}^{2}+168\,a{d}^{4}{e}^{4}+144\,b{d}^{6}{e}^{2}+128\,c{d}^{8}}{315\,{e}^{10}}\sqrt{ex+d}\sqrt{-ex+d}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(c*x^4+b*x^2+a)/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x)

[Out]

-1/315*(e*x+d)^(1/2)*(-e*x+d)^(1/2)*(35*c*e^8*x^8+45*b*e^8*x^6+40*c*d^2*e^6*x^6+63*a*e^8*x^4+54*b*d^2*e^6*x^4+
48*c*d^4*e^4*x^4+84*a*d^2*e^6*x^2+72*b*d^4*e^4*x^2+64*c*d^6*e^2*x^2+168*a*d^4*e^4+144*b*d^6*e^2+128*c*d^8)/e^1
0

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Maxima [A]  time = 1.64729, size = 398, normalized size = 1.9 \begin{align*} -\frac{\sqrt{-e^{2} x^{2} + d^{2}} c x^{8}}{9 \, e^{2}} - \frac{8 \, \sqrt{-e^{2} x^{2} + d^{2}} c d^{2} x^{6}}{63 \, e^{4}} - \frac{\sqrt{-e^{2} x^{2} + d^{2}} b x^{6}}{7 \, e^{2}} - \frac{16 \, \sqrt{-e^{2} x^{2} + d^{2}} c d^{4} x^{4}}{105 \, e^{6}} - \frac{6 \, \sqrt{-e^{2} x^{2} + d^{2}} b d^{2} x^{4}}{35 \, e^{4}} - \frac{\sqrt{-e^{2} x^{2} + d^{2}} a x^{4}}{5 \, e^{2}} - \frac{64 \, \sqrt{-e^{2} x^{2} + d^{2}} c d^{6} x^{2}}{315 \, e^{8}} - \frac{8 \, \sqrt{-e^{2} x^{2} + d^{2}} b d^{4} x^{2}}{35 \, e^{6}} - \frac{4 \, \sqrt{-e^{2} x^{2} + d^{2}} a d^{2} x^{2}}{15 \, e^{4}} - \frac{128 \, \sqrt{-e^{2} x^{2} + d^{2}} c d^{8}}{315 \, e^{10}} - \frac{16 \, \sqrt{-e^{2} x^{2} + d^{2}} b d^{6}}{35 \, e^{8}} - \frac{8 \, \sqrt{-e^{2} x^{2} + d^{2}} a d^{4}}{15 \, e^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(c*x^4+b*x^2+a)/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

-1/9*sqrt(-e^2*x^2 + d^2)*c*x^8/e^2 - 8/63*sqrt(-e^2*x^2 + d^2)*c*d^2*x^6/e^4 - 1/7*sqrt(-e^2*x^2 + d^2)*b*x^6
/e^2 - 16/105*sqrt(-e^2*x^2 + d^2)*c*d^4*x^4/e^6 - 6/35*sqrt(-e^2*x^2 + d^2)*b*d^2*x^4/e^4 - 1/5*sqrt(-e^2*x^2
+ d^2)*a*x^4/e^2 - 64/315*sqrt(-e^2*x^2 + d^2)*c*d^6*x^2/e^8 - 8/35*sqrt(-e^2*x^2 + d^2)*b*d^4*x^2/e^6 - 4/15
*sqrt(-e^2*x^2 + d^2)*a*d^2*x^2/e^4 - 128/315*sqrt(-e^2*x^2 + d^2)*c*d^8/e^10 - 16/35*sqrt(-e^2*x^2 + d^2)*b*d
^6/e^8 - 8/15*sqrt(-e^2*x^2 + d^2)*a*d^4/e^6

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Fricas [A]  time = 1.77388, size = 317, normalized size = 1.51 \begin{align*} -\frac{{\left (35 \, c e^{8} x^{8} + 128 \, c d^{8} + 144 \, b d^{6} e^{2} + 168 \, a d^{4} e^{4} + 5 \,{\left (8 \, c d^{2} e^{6} + 9 \, b e^{8}\right )} x^{6} + 3 \,{\left (16 \, c d^{4} e^{4} + 18 \, b d^{2} e^{6} + 21 \, a e^{8}\right )} x^{4} + 4 \,{\left (16 \, c d^{6} e^{2} + 18 \, b d^{4} e^{4} + 21 \, a d^{2} e^{6}\right )} x^{2}\right )} \sqrt{e x + d} \sqrt{-e x + d}}{315 \, e^{10}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(c*x^4+b*x^2+a)/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

-1/315*(35*c*e^8*x^8 + 128*c*d^8 + 144*b*d^6*e^2 + 168*a*d^4*e^4 + 5*(8*c*d^2*e^6 + 9*b*e^8)*x^6 + 3*(16*c*d^4
*e^4 + 18*b*d^2*e^6 + 21*a*e^8)*x^4 + 4*(16*c*d^6*e^2 + 18*b*d^4*e^4 + 21*a*d^2*e^6)*x^2)*sqrt(e*x + d)*sqrt(-
e*x + d)/e^10

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(c*x**4+b*x**2+a)/(-e*x+d)**(1/2)/(e*x+d)**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 1.25458, size = 328, normalized size = 1.56 \begin{align*} -\frac{1}{2807562240} \,{\left (315 \, c d^{8} e^{81} + 315 \, b d^{6} e^{83} + 315 \, a d^{4} e^{85} -{\left (840 \, c d^{7} e^{81} + 630 \, b d^{5} e^{83} + 420 \, a d^{3} e^{85} -{\left (1932 \, c d^{6} e^{81} + 1071 \, b d^{4} e^{83} + 462 \, a d^{2} e^{85} -{\left (2952 \, c d^{5} e^{81} + 1116 \, b d^{3} e^{83} + 252 \, a d e^{85} -{\left (3098 \, c d^{4} e^{81} + 729 \, b d^{2} e^{83} - 5 \,{\left (440 \, c d^{3} e^{81} + 54 \, b d e^{83} -{\left (204 \, c d^{2} e^{81} + 7 \,{\left ({\left (x e + d\right )} c e^{81} - 8 \, c d e^{81}\right )}{\left (x e + d\right )} + 9 \, b e^{83}\right )}{\left (x e + d\right )}\right )}{\left (x e + d\right )} + 63 \, a e^{85}\right )}{\left (x e + d\right )}\right )}{\left (x e + d\right )}\right )}{\left (x e + d\right )}\right )}{\left (x e + d\right )}\right )} \sqrt{x e + d} \sqrt{-x e + d} e^{\left (-1\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(c*x^4+b*x^2+a)/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

-1/2807562240*(315*c*d^8*e^81 + 315*b*d^6*e^83 + 315*a*d^4*e^85 - (840*c*d^7*e^81 + 630*b*d^5*e^83 + 420*a*d^3
*e^85 - (1932*c*d^6*e^81 + 1071*b*d^4*e^83 + 462*a*d^2*e^85 - (2952*c*d^5*e^81 + 1116*b*d^3*e^83 + 252*a*d*e^8
5 - (3098*c*d^4*e^81 + 729*b*d^2*e^83 - 5*(440*c*d^3*e^81 + 54*b*d*e^83 - (204*c*d^2*e^81 + 7*((x*e + d)*c*e^8
1 - 8*c*d*e^81)*(x*e + d) + 9*b*e^83)*(x*e + d))*(x*e + d) + 63*a*e^85)*(x*e + d))*(x*e + d))*(x*e + d))*(x*e
+ d))*sqrt(x*e + d)*sqrt(-x*e + d)*e^(-1)