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

Optimal. Leaf size=109 \[ -\frac{\sqrt{d-e x} \sqrt{d+e x} \left (a e^4+b d^2 e^2+c d^4\right )}{e^6}+\frac{(d-e x)^{3/2} (d+e x)^{3/2} \left (b e^2+2 c d^2\right )}{3 e^6}-\frac{c (d-e x)^{5/2} (d+e x)^{5/2}}{5 e^6} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.35672, antiderivative size = 149, normalized size of antiderivative = 1.37, number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{\left (d^2-e^2 x^2\right ) \left (a e^4+b d^2 e^2+c d^4\right )}{e^6 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (d^2-e^2 x^2\right )^2 \left (b e^2+2 c d^2\right )}{3 e^6 \sqrt{d-e x} \sqrt{d+e x}}-\frac{c \left (d^2-e^2 x^2\right )^3}{5 e^6 \sqrt{d-e x} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 19.777, size = 117, normalized size = 1.07 \[ - \frac{c \sqrt{d - e x} \sqrt{d + e x} \left (d^{2} - e^{2} x^{2}\right )^{2}}{5 e^{6}} + \frac{\sqrt{d - e x} \sqrt{d + e x} \left (d^{2} - e^{2} x^{2}\right ) \left (b e^{2} + 2 c d^{2}\right )}{3 e^{6}} - \frac{\sqrt{d - e x} \sqrt{d + e x} \left (a e^{4} + b d^{2} e^{2} + c d^{4}\right )}{e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(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)**2/(5*e**6) + sqrt(d - e*x)*sq
rt(d + e*x)*(d**2 - e**2*x**2)*(b*e**2 + 2*c*d**2)/(3*e**6) - sqrt(d - e*x)*sqrt
(d + e*x)*(a*e**4 + b*d**2*e**2 + c*d**4)/e**6

_______________________________________________________________________________________

Mathematica [A]  time = 0.0930085, size = 80, normalized size = 0.73 \[ -\frac{\sqrt{d-e x} \sqrt{d+e x} \left (5 \left (3 a e^4+2 b d^2 e^2+b e^4 x^2\right )+c \left (8 d^4+4 d^2 e^2 x^2+3 e^4 x^4\right )\right )}{15 e^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(x*(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]*(5*(2*b*d^2*e^2 + 3*a*e^4 + b*e^4*x^2) + c*(8*d^4
+ 4*d^2*e^2*x^2 + 3*e^4*x^4)))/(15*e^6)

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 73, normalized size = 0.7 \[ -{\frac{3\,c{x}^{4}{e}^{4}+5\,b{e}^{4}{x}^{2}+4\,c{d}^{2}{e}^{2}{x}^{2}+15\,a{e}^{4}+10\,b{d}^{2}{e}^{2}+8\,c{d}^{4}}{15\,{e}^{6}}\sqrt{-ex+d}\sqrt{ex+d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*(-e*x+d)^(1/2)*(e*x+d)^(1/2)*(3*c*e^4*x^4+5*b*e^4*x^2+4*c*d^2*e^2*x^2+15*a
*e^4+10*b*d^2*e^2+8*c*d^4)/e^6

_______________________________________________________________________________________

Maxima [A]  time = 0.797571, size = 188, normalized size = 1.72 \[ -\frac{\sqrt{-e^{2} x^{2} + d^{2}} c x^{4}}{5 \, e^{2}} - \frac{4 \, \sqrt{-e^{2} x^{2} + d^{2}} c d^{2} x^{2}}{15 \, e^{4}} - \frac{\sqrt{-e^{2} x^{2} + d^{2}} b x^{2}}{3 \, e^{2}} - \frac{8 \, \sqrt{-e^{2} x^{2} + d^{2}} c d^{4}}{15 \, e^{6}} - \frac{2 \, \sqrt{-e^{2} x^{2} + d^{2}} b d^{2}}{3 \, e^{4}} - \frac{\sqrt{-e^{2} x^{2} + d^{2}} a}{e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.271901, size = 313, normalized size = 2.87 \[ -\frac{3 \, c e^{4} x^{10} - 5 \,{\left (7 \, c d^{2} e^{2} - b e^{4}\right )} x^{8} + 120 \, a d^{4} x^{2} + 5 \,{\left (8 \, c d^{4} - 11 \, b d^{2} e^{2} + 3 \, a e^{4}\right )} x^{6} + 60 \,{\left (b d^{4} - 2 \, a d^{2} e^{2}\right )} x^{4} + 5 \,{\left (3 \, c d e^{2} x^{8} -{\left (8 \, c d^{3} - 5 \, b d e^{2}\right )} x^{6} - 24 \, a d^{3} x^{2} - 12 \,{\left (b d^{3} - a d e^{2}\right )} x^{4}\right )} \sqrt{e x + d} \sqrt{-e x + d}}{15 \,{\left (5 \, d e^{4} x^{4} - 20 \, d^{3} e^{2} x^{2} + 16 \, d^{5} -{\left (e^{4} x^{4} - 12 \, d^{2} e^{2} x^{2} + 16 \, d^{4}\right )} \sqrt{e x + d} \sqrt{-e x + d}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*(3*c*e^4*x^10 - 5*(7*c*d^2*e^2 - b*e^4)*x^8 + 120*a*d^4*x^2 + 5*(8*c*d^4 -
11*b*d^2*e^2 + 3*a*e^4)*x^6 + 60*(b*d^4 - 2*a*d^2*e^2)*x^4 + 5*(3*c*d*e^2*x^8 -
(8*c*d^3 - 5*b*d*e^2)*x^6 - 24*a*d^3*x^2 - 12*(b*d^3 - a*d*e^2)*x^4)*sqrt(e*x +
d)*sqrt(-e*x + d))/(5*d*e^4*x^4 - 20*d^3*e^2*x^2 + 16*d^5 - (e^4*x^4 - 12*d^2*e
^2*x^2 + 16*d^4)*sqrt(e*x + d)*sqrt(-e*x + d))

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.286287, size = 153, normalized size = 1.4 \[ -\frac{1}{276480} \,{\left (15 \, c d^{4} e^{25} + 15 \, b d^{2} e^{27} -{\left (20 \, c d^{3} e^{25} + 10 \, b d e^{27} -{\left (22 \, c d^{2} e^{25} + 3 \,{\left ({\left (x e + d\right )} c e^{25} - 4 \, c d e^{25}\right )}{\left (x e + d\right )} + 5 \, b e^{27}\right )}{\left (x e + d\right )}\right )}{\left (x e + d\right )} + 15 \, a e^{29}\right )} \sqrt{x e + d} \sqrt{-x e + d} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/276480*(15*c*d^4*e^25 + 15*b*d^2*e^27 - (20*c*d^3*e^25 + 10*b*d*e^27 - (22*c*
d^2*e^25 + 3*((x*e + d)*c*e^25 - 4*c*d*e^25)*(x*e + d) + 5*b*e^27)*(x*e + d))*(x
*e + d) + 15*a*e^29)*sqrt(x*e + d)*sqrt(-x*e + d)*e^(-1)