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

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 = 0.555717, 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 \[ \frac{\left (d^2-e^2 x^2\right )^2 \left (a e^4+2 b d^2 e^2+3 c d^4\right )}{3 e^8 \sqrt{d-e x} \sqrt{d+e x}}-\frac{d^2 \left (d^2-e^2 x^2\right ) \left (a e^4+b d^2 e^2+c d^4\right )}{e^8 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (d^2-e^2 x^2\right )^3 \left (b e^2+3 c d^2\right )}{5 e^8 \sqrt{d-e x} \sqrt{d+e x}}+\frac{c \left (d^2-e^2 x^2\right )^4}{7 e^8 \sqrt{d-e x} \sqrt{d+e x}} \]

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]*Sq
rt[d + e*x]) + (c*(d^2 - e^2*x^2)^4)/(7*e^8*Sqrt[d - e*x]*Sqrt[d + e*x])

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Rubi in Sympy [A]  time = 29.0047, size = 177, normalized size = 1.11 \[ \frac{c \sqrt{d - e x} \sqrt{d + e x} \left (d^{2} - e^{2} x^{2}\right )^{3}}{7 e^{8}} - \frac{d^{2} \sqrt{d - e x} \sqrt{d + e x} \left (a e^{4} + b d^{2} e^{2} + c d^{4}\right )}{e^{8}} - \frac{\sqrt{d - e x} \sqrt{d + e x} \left (d^{2} - e^{2} x^{2}\right )^{2} \left (b e^{2} + 3 c d^{2}\right )}{5 e^{8}} + \frac{\sqrt{d - e x} \sqrt{d + e x} \left (d^{2} - e^{2} x^{2}\right ) \left (a e^{4} + 2 b d^{2} e^{2} + 3 c d^{4}\right )}{3 e^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(c*x**4+b*x**2+a)/(-e*x+d)**(1/2)/(e*x+d)**(1/2),x)

[Out]

c*sqrt(d - e*x)*sqrt(d + e*x)*(d**2 - e**2*x**2)**3/(7*e**8) - d**2*sqrt(d - e*x
)*sqrt(d + e*x)*(a*e**4 + b*d**2*e**2 + c*d**4)/e**8 - sqrt(d - e*x)*sqrt(d + e*
x)*(d**2 - e**2*x**2)**2*(b*e**2 + 3*c*d**2)/(5*e**8) + sqrt(d - e*x)*sqrt(d + e
*x)*(d**2 - e**2*x**2)*(a*e**4 + 2*b*d**2*e**2 + 3*c*d**4)/(3*e**8)

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Mathematica [A]  time = 0.128465, size = 116, normalized size = 0.73 \[ -\frac{\sqrt{d-e x} \sqrt{d+e x} \left (35 a e^4 \left (2 d^2+e^2 x^2\right )+7 b \left (8 d^4 e^2+4 d^2 e^4 x^2+3 e^6 x^4\right )+3 c \left (16 d^6+8 d^4 e^2 x^2+6 d^2 e^4 x^4+5 e^6 x^6\right )\right )}{105 e^8} \]

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)))/(105*e^8)

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Maple [A]  time = 0.01, size = 109, normalized size = 0.7 \[ -{\frac{15\,c{x}^{6}{e}^{6}+21\,b{e}^{6}{x}^{4}+18\,c{d}^{2}{e}^{4}{x}^{4}+35\,a{e}^{6}{x}^{2}+28\,b{d}^{2}{e}^{4}{x}^{2}+24\,c{d}^{4}{e}^{2}{x}^{2}+70\,a{d}^{2}{e}^{4}+56\,b{d}^{4}{e}^{2}+48\,c{d}^{6}}{105\,{e}^{8}}\sqrt{-ex+d}\sqrt{ex+d}} \]

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 = 0.789057, size = 293, normalized size = 1.84 \[ -\frac{\sqrt{-e^{2} x^{2} + d^{2}} c x^{6}}{7 \, e^{2}} - \frac{6 \, \sqrt{-e^{2} x^{2} + d^{2}} c d^{2} x^{4}}{35 \, e^{4}} - \frac{\sqrt{-e^{2} x^{2} + d^{2}} b x^{4}}{5 \, e^{2}} - \frac{8 \, \sqrt{-e^{2} x^{2} + d^{2}} c d^{4} x^{2}}{35 \, e^{6}} - \frac{4 \, \sqrt{-e^{2} x^{2} + d^{2}} b d^{2} x^{2}}{15 \, e^{4}} - \frac{\sqrt{-e^{2} x^{2} + d^{2}} a x^{2}}{3 \, e^{2}} - \frac{16 \, \sqrt{-e^{2} x^{2} + d^{2}} c d^{6}}{35 \, e^{8}} - \frac{8 \, \sqrt{-e^{2} x^{2} + d^{2}} b d^{4}}{15 \, e^{6}} - \frac{2 \, \sqrt{-e^{2} x^{2} + d^{2}} a d^{2}}{3 \, e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)*x^3/(sqrt(e*x + d)*sqrt(-e*x + d)),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*s
qrt(-e^2*x^2 + d^2)*a*d^2/e^4

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Fricas [A]  time = 0.274299, size = 435, normalized size = 2.74 \[ -\frac{15 \, c e^{6} x^{14} - 21 \,{\left (17 \, c d^{2} e^{4} - b e^{6}\right )} x^{12} - 1680 \, a d^{6} x^{4} + 7 \,{\left (162 \, c d^{4} e^{2} - 71 \, b d^{2} e^{4} + 5 \, a e^{6}\right )} x^{10} - 35 \,{\left (24 \, c d^{6} - 44 \, b d^{4} e^{2} + 23 \, a d^{2} e^{4}\right )} x^{8} - 140 \,{\left (8 \, b d^{6} - 17 \, a d^{4} e^{2}\right )} x^{6} + 7 \,{\left (15 \, c d e^{4} x^{12} - 3 \,{\left (34 \, c d^{3} e^{2} - 7 \, b d e^{4}\right )} x^{10} + 240 \, a d^{5} x^{4} + 5 \,{\left (24 \, c d^{5} - 28 \, b d^{3} e^{2} + 7 \, a d e^{4}\right )} x^{8} + 20 \,{\left (8 \, b d^{5} - 11 \, a d^{3} e^{2}\right )} x^{6}\right )} \sqrt{e x + d} \sqrt{-e x + d}}{105 \,{\left (7 \, d e^{6} x^{6} - 56 \, d^{3} e^{4} x^{4} + 112 \, d^{5} e^{2} x^{2} - 64 \, d^{7} -{\left (e^{6} x^{6} - 24 \, d^{2} e^{4} x^{4} + 80 \, d^{4} e^{2} x^{2} - 64 \, d^{6}\right )} \sqrt{e x + d} \sqrt{-e x + d}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/105*(15*c*e^6*x^14 - 21*(17*c*d^2*e^4 - b*e^6)*x^12 - 1680*a*d^6*x^4 + 7*(162
*c*d^4*e^2 - 71*b*d^2*e^4 + 5*a*e^6)*x^10 - 35*(24*c*d^6 - 44*b*d^4*e^2 + 23*a*d
^2*e^4)*x^8 - 140*(8*b*d^6 - 17*a*d^4*e^2)*x^6 + 7*(15*c*d*e^4*x^12 - 3*(34*c*d^
3*e^2 - 7*b*d*e^4)*x^10 + 240*a*d^5*x^4 + 5*(24*c*d^5 - 28*b*d^3*e^2 + 7*a*d*e^4
)*x^8 + 20*(8*b*d^5 - 11*a*d^3*e^2)*x^6)*sqrt(e*x + d)*sqrt(-e*x + d))/(7*d*e^6*
x^6 - 56*d^3*e^4*x^4 + 112*d^5*e^2*x^2 - 64*d^7 - (e^6*x^6 - 24*d^2*e^4*x^4 + 80
*d^4*e^2*x^2 - 64*d^6)*sqrt(e*x + d)*sqrt(-e*x + d))

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

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/XCAS [A]  time = 0.295257, size = 239, normalized size = 1.5 \[ -\frac{1}{44728320} \,{\left (105 \, c d^{6} e^{49} + 105 \, b d^{4} e^{51} + 105 \, a d^{2} e^{53} -{\left (210 \, c d^{5} e^{49} + 140 \, b d^{3} e^{51} + 70 \, a d e^{53} -{\left (357 \, c d^{4} e^{49} + 154 \, b d^{2} e^{51} - 3 \,{\left (124 \, c d^{3} e^{49} + 28 \, b d e^{51} -{\left (81 \, c d^{2} e^{49} + 5 \,{\left ({\left (x e + d\right )} c e^{49} - 6 \, c d e^{49}\right )}{\left (x e + d\right )} + 7 \, b e^{51}\right )}{\left (x e + d\right )}\right )}{\left (x e + d\right )} + 35 \, a e^{53}\right )}{\left (x e + d\right )}\right )}{\left (x e + d\right )}\right )} \sqrt{x e + d} \sqrt{-x e + d} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)*x^3/(sqrt(e*x + d)*sqrt(-e*x + d)),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)