3.133 $$\int \frac{x^3 (a+b x^2+c x^4)}{\sqrt{d-e x} \sqrt{d+e x}} \, dx$$

Optimal. Leaf size=159 $\frac{(d-e x)^{3/2} (d+e x)^{3/2} \left (a e^4+2 b d^2 e^2+3 c d^4\right )}{3 e^8}-\frac{d^2 \sqrt{d-e x} \sqrt{d+e x} \left (a e^4+b d^2 e^2+c d^4\right )}{e^8}-\frac{(d-e x)^{5/2} (d+e x)^{5/2} \left (b e^2+3 c d^2\right )}{5 e^8}+\frac{c (d-e x)^{7/2} (d+e x)^{7/2}}{7 e^8}$

[Out]

-((d^2*(c*d^4 + b*d^2*e^2 + a*e^4)*Sqrt[d - e*x]*Sqrt[d + e*x])/e^8) + ((3*c*d^4 + 2*b*d^2*e^2 + a*e^4)*(d - e
*x)^(3/2)*(d + e*x)^(3/2))/(3*e^8) - ((3*c*d^2 + b*e^2)*(d - e*x)^(5/2)*(d + e*x)^(5/2))/(5*e^8) + (c*(d - e*x
)^(7/2)*(d + e*x)^(7/2))/(7*e^8)

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Rubi [A]  time = 0.189442, antiderivative size = 213, normalized size of antiderivative = 1.34, number of steps used = 4, number of rules used = 3, integrand size = 35, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.086, Rules used = {520, 1251, 771} $\frac{\left (d^2-e^2 x^2\right )^2 \left (a e^4+2 b d^2 e^2+3 c d^4\right )}{3 e^8 \sqrt{d-e x} \sqrt{d+e x}}-\frac{d^2 \left (d^2-e^2 x^2\right ) \left (a e^4+b d^2 e^2+c d^4\right )}{e^8 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (d^2-e^2 x^2\right )^3 \left (b e^2+3 c d^2\right )}{5 e^8 \sqrt{d-e x} \sqrt{d+e x}}+\frac{c \left (d^2-e^2 x^2\right )^4}{7 e^8 \sqrt{d-e x} \sqrt{d+e x}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(x^3*(a + b*x^2 + c*x^4))/(Sqrt[d - e*x]*Sqrt[d + e*x]),x]

[Out]

-((d^2*(c*d^4 + b*d^2*e^2 + a*e^4)*(d^2 - e^2*x^2))/(e^8*Sqrt[d - e*x]*Sqrt[d + e*x])) + ((3*c*d^4 + 2*b*d^2*e
^2 + a*e^4)*(d^2 - e^2*x^2)^2)/(3*e^8*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((3*c*d^2 + b*e^2)*(d^2 - e^2*x^2)^3)/(5*
e^8*Sqrt[d - e*x]*Sqrt[d + e*x]) + (c*(d^2 - e^2*x^2)^4)/(7*e^8*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 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

Mathematica [C]  time = 1.0886, size = 232, normalized size = 1.46 $-\frac{\sqrt{d-e x} \sqrt{d+e x} \left (35 a e^4 \left (2 d^2+e^2 x^2\right )+7 b \left (4 d^2 e^4 x^2+8 d^4 e^2+3 e^6 x^4\right )+3 c \left (8 d^4 e^2 x^2+6 d^2 e^4 x^4+16 d^6+5 e^6 x^6\right )\right )+\frac{210 d^{5/2} \sqrt{d+e x} \sin ^{-1}\left (\frac{\sqrt{d-e x}}{\sqrt{2} \sqrt{d}}\right ) \left (a e^4+b d^2 e^2+c d^4\right )}{\sqrt{\frac{e x}{d}+1}}-210 d^3 \tan ^{-1}\left (\frac{\sqrt{d-e x}}{\sqrt{d+e x}}\right ) \left (a e^4+b d^2 e^2+c d^4\right )}{105 e^8}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(x^3*(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]*(35*a*e^4*(2*d^2 + e^2*x^2) + 7*b*(8*d^4*e^2 + 4*d^2*e^4*x^2 + 3*e^6*x^4) + 3*c*
(16*d^6 + 8*d^4*e^2*x^2 + 6*d^2*e^4*x^4 + 5*e^6*x^6)) + (210*d^(5/2)*(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] - 210*d^3*(c*d^4 + b*d^2*e^2 + a*e^4)*ArcTan[Sqrt[
d - e*x]/Sqrt[d + e*x]])/(105*e^8)

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Maple [A]  time = 0.007, size = 109, normalized size = 0.7 \begin{align*} -{\frac{15\,c{x}^{6}{e}^{6}+21\,b{e}^{6}{x}^{4}+18\,c{d}^{2}{e}^{4}{x}^{4}+35\,a{e}^{6}{x}^{2}+28\,b{d}^{2}{e}^{4}{x}^{2}+24\,c{d}^{4}{e}^{2}{x}^{2}+70\,a{d}^{2}{e}^{4}+56\,b{d}^{4}{e}^{2}+48\,c{d}^{6}}{105\,{e}^{8}}\sqrt{ex+d}\sqrt{-ex+d}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(c*x^4+b*x^2+a)/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x)

[Out]

-1/105*(e*x+d)^(1/2)*(-e*x+d)^(1/2)*(15*c*e^6*x^6+21*b*e^6*x^4+18*c*d^2*e^4*x^4+35*a*e^6*x^2+28*b*d^2*e^4*x^2+
24*c*d^4*e^2*x^2+70*a*d^2*e^4+56*b*d^4*e^2+48*c*d^6)/e^8

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Maxima [A]  time = 1.49211, size = 293, normalized size = 1.84 \begin{align*} -\frac{\sqrt{-e^{2} x^{2} + d^{2}} c x^{6}}{7 \, e^{2}} - \frac{6 \, \sqrt{-e^{2} x^{2} + d^{2}} c d^{2} x^{4}}{35 \, e^{4}} - \frac{\sqrt{-e^{2} x^{2} + d^{2}} b x^{4}}{5 \, e^{2}} - \frac{8 \, \sqrt{-e^{2} x^{2} + d^{2}} c d^{4} x^{2}}{35 \, e^{6}} - \frac{4 \, \sqrt{-e^{2} x^{2} + d^{2}} b d^{2} x^{2}}{15 \, e^{4}} - \frac{\sqrt{-e^{2} x^{2} + d^{2}} a x^{2}}{3 \, e^{2}} - \frac{16 \, \sqrt{-e^{2} x^{2} + d^{2}} c d^{6}}{35 \, e^{8}} - \frac{8 \, \sqrt{-e^{2} x^{2} + d^{2}} b d^{4}}{15 \, e^{6}} - \frac{2 \, \sqrt{-e^{2} x^{2} + d^{2}} a d^{2}}{3 \, e^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(c*x^4+b*x^2+a)/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

-1/7*sqrt(-e^2*x^2 + d^2)*c*x^6/e^2 - 6/35*sqrt(-e^2*x^2 + d^2)*c*d^2*x^4/e^4 - 1/5*sqrt(-e^2*x^2 + d^2)*b*x^4
/e^2 - 8/35*sqrt(-e^2*x^2 + d^2)*c*d^4*x^2/e^6 - 4/15*sqrt(-e^2*x^2 + d^2)*b*d^2*x^2/e^4 - 1/3*sqrt(-e^2*x^2 +
d^2)*a*x^2/e^2 - 16/35*sqrt(-e^2*x^2 + d^2)*c*d^6/e^8 - 8/15*sqrt(-e^2*x^2 + d^2)*b*d^4/e^6 - 2/3*sqrt(-e^2*x
^2 + d^2)*a*d^2/e^4

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Fricas [A]  time = 1.91068, size = 238, normalized size = 1.5 \begin{align*} -\frac{{\left (15 \, c e^{6} x^{6} + 48 \, c d^{6} + 56 \, b d^{4} e^{2} + 70 \, a d^{2} e^{4} + 3 \,{\left (6 \, c d^{2} e^{4} + 7 \, b e^{6}\right )} x^{4} +{\left (24 \, c d^{4} e^{2} + 28 \, b d^{2} e^{4} + 35 \, a e^{6}\right )} x^{2}\right )} \sqrt{e x + d} \sqrt{-e x + d}}{105 \, e^{8}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(c*x^4+b*x^2+a)/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

-1/105*(15*c*e^6*x^6 + 48*c*d^6 + 56*b*d^4*e^2 + 70*a*d^2*e^4 + 3*(6*c*d^2*e^4 + 7*b*e^6)*x^4 + (24*c*d^4*e^2
+ 28*b*d^2*e^4 + 35*a*e^6)*x^2)*sqrt(e*x + d)*sqrt(-e*x + d)/e^8

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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**3*(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 = 1.14837, size = 239, normalized size = 1.5 \begin{align*} -\frac{1}{44728320} \,{\left (105 \, c d^{6} e^{49} + 105 \, b d^{4} e^{51} + 105 \, a d^{2} e^{53} -{\left (210 \, c d^{5} e^{49} + 140 \, b d^{3} e^{51} + 70 \, a d e^{53} -{\left (357 \, c d^{4} e^{49} + 154 \, b d^{2} e^{51} - 3 \,{\left (124 \, c d^{3} e^{49} + 28 \, b d e^{51} -{\left (81 \, c d^{2} e^{49} + 5 \,{\left ({\left (x e + d\right )} c e^{49} - 6 \, c d e^{49}\right )}{\left (x e + d\right )} + 7 \, b e^{51}\right )}{\left (x e + d\right )}\right )}{\left (x e + d\right )} + 35 \, a e^{53}\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^3*(c*x^4+b*x^2+a)/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

-1/44728320*(105*c*d^6*e^49 + 105*b*d^4*e^51 + 105*a*d^2*e^53 - (210*c*d^5*e^49 + 140*b*d^3*e^51 + 70*a*d*e^53
- (357*c*d^4*e^49 + 154*b*d^2*e^51 - 3*(124*c*d^3*e^49 + 28*b*d*e^51 - (81*c*d^2*e^49 + 5*((x*e + d)*c*e^49 -
6*c*d*e^49)*(x*e + d) + 7*b*e^51)*(x*e + d))*(x*e + d) + 35*a*e^53)*(x*e + d))*(x*e + d))*sqrt(x*e + d)*sqrt(
-x*e + d)*e^(-1)