3.228 \(\int \frac{1}{\left (d+e x^2\right )^{3/2} \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx\)

Optimal. Leaf size=149 \[ -\frac{c^2 \tanh ^{-1}\left (\frac{\sqrt{e} x \sqrt{2 c d-b e}}{\sqrt{d+e x^2} \sqrt{c d-b e}}\right )}{\sqrt{e} \sqrt{c d-b e} (2 c d-b e)^{5/2}}-\frac{x (7 c d-2 b e)}{3 d^2 \sqrt{d+e x^2} (2 c d-b e)^2}-\frac{x}{3 d \left (d+e x^2\right )^{3/2} (2 c d-b e)} \]

[Out]

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

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Rubi [A]  time = 0.604432, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.146 \[ -\frac{c^2 \tanh ^{-1}\left (\frac{\sqrt{e} x \sqrt{2 c d-b e}}{\sqrt{d+e x^2} \sqrt{c d-b e}}\right )}{\sqrt{e} \sqrt{c d-b e} (2 c d-b e)^{5/2}}-\frac{x (7 c d-2 b e)}{3 d^2 \sqrt{d+e x^2} (2 c d-b e)^2}-\frac{x}{3 d \left (d+e x^2\right )^{3/2} (2 c d-b e)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 118.803, size = 131, normalized size = 0.88 \[ \frac{c^{2} \operatorname{atanh}{\left (\frac{\sqrt{e} x \sqrt{b e - 2 c d}}{\sqrt{d + e x^{2}} \sqrt{b e - c d}} \right )}}{\sqrt{e} \left (b e - 2 c d\right )^{\frac{5}{2}} \sqrt{b e - c d}} + \frac{x}{3 d \left (d + e x^{2}\right )^{\frac{3}{2}} \left (b e - 2 c d\right )} + \frac{x \left (2 b e - 7 c d\right )}{3 d^{2} \sqrt{d + e x^{2}} \left (b e - 2 c d\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.552906, size = 134, normalized size = 0.9 \[ \frac{c^2 \tanh ^{-1}\left (\frac{\sqrt{e} x \sqrt{b e-2 c d}}{\sqrt{d+e x^2} \sqrt{b e-c d}}\right )}{\sqrt{e} (b e-2 c d)^{5/2} \sqrt{b e-c d}}-\frac{x \left (c d \left (9 d+7 e x^2\right )-b e \left (3 d+2 e x^2\right )\right )}{3 d^2 \left (d+e x^2\right )^{3/2} (b e-2 c d)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.032, size = 1647, normalized size = 11.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*c/d/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/((-d*e)^(1/2)*c-(-(b*e-c*d)*c*e)
^(1/2))/(x-1/e*(-d*e)^(1/2))/((x-1/e*(-d*e)^(1/2))^2*e+2*(-d*e)^(1/2)*(x-1/e*(-d
*e)^(1/2)))^(1/2)+1/3*e*c/d^2/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/((-d*e)^(1
/2)*c-(-(b*e-c*d)*c*e)^(1/2))/((x-1/e*(-d*e)^(1/2))^2*e+2*(-d*e)^(1/2)*(x-1/e*(-
d*e)^(1/2)))^(1/2)*x+1/6*c/d/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/((-d*e)^(1/
2)*c-(-(b*e-c*d)*c*e)^(1/2))/(x+1/e*(-d*e)^(1/2))/((x+1/e*(-d*e)^(1/2))^2*e-2*(-
d*e)^(1/2)*(x+1/e*(-d*e)^(1/2)))^(1/2)+1/3*e*c/d^2/((-d*e)^(1/2)*c+(-(b*e-c*d)*c
*e)^(1/2))/((-d*e)^(1/2)*c-(-(b*e-c*d)*c*e)^(1/2))/((x+1/e*(-d*e)^(1/2))^2*e-2*(
-d*e)^(1/2)*(x+1/e*(-d*e)^(1/2)))^(1/2)*x+1/2*e*c^3/((-d*e)^(1/2)*c+(-(b*e-c*d)*
c*e)^(1/2))/((-d*e)^(1/2)*c-(-(b*e-c*d)*c*e)^(1/2))/(-(b*e-c*d)*c*e)^(1/2)/(b*e-
2*c*d)/((x-(-(b*e-c*d)*c*e)^(1/2)/c/e)^2*e+2*(-(b*e-c*d)*c*e)^(1/2)/c*(x-(-(b*e-
c*d)*c*e)^(1/2)/c/e)-(b*e-2*c*d)/c)^(1/2)-1/2*e*c^2/((-d*e)^(1/2)*c+(-(b*e-c*d)*
c*e)^(1/2))/((-d*e)^(1/2)*c-(-(b*e-c*d)*c*e)^(1/2))/(b*e-2*c*d)/d/((x-(-(b*e-c*d
)*c*e)^(1/2)/c/e)^2*e+2*(-(b*e-c*d)*c*e)^(1/2)/c*(x-(-(b*e-c*d)*c*e)^(1/2)/c/e)-
(b*e-2*c*d)/c)^(1/2)*x-1/2*e*c^3/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/((-d*e)
^(1/2)*c-(-(b*e-c*d)*c*e)^(1/2))/(-(b*e-c*d)*c*e)^(1/2)/(b*e-2*c*d)/(-(b*e-2*c*d
)/c)^(1/2)*ln((-2*(b*e-2*c*d)/c+2*(-(b*e-c*d)*c*e)^(1/2)/c*(x-(-(b*e-c*d)*c*e)^(
1/2)/c/e)+2*(-(b*e-2*c*d)/c)^(1/2)*((x-(-(b*e-c*d)*c*e)^(1/2)/c/e)^2*e+2*(-(b*e-
c*d)*c*e)^(1/2)/c*(x-(-(b*e-c*d)*c*e)^(1/2)/c/e)-(b*e-2*c*d)/c)^(1/2))/(x-(-(b*e
-c*d)*c*e)^(1/2)/c/e))-1/2*e*c^3/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/((-d*e)
^(1/2)*c-(-(b*e-c*d)*c*e)^(1/2))/(-(b*e-c*d)*c*e)^(1/2)/(b*e-2*c*d)/((x+(-(b*e-c
*d)*c*e)^(1/2)/c/e)^2*e-2*(-(b*e-c*d)*c*e)^(1/2)/c*(x+(-(b*e-c*d)*c*e)^(1/2)/c/e
)-(b*e-2*c*d)/c)^(1/2)-1/2*e*c^2/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/((-d*e)
^(1/2)*c-(-(b*e-c*d)*c*e)^(1/2))/(b*e-2*c*d)/d/((x+(-(b*e-c*d)*c*e)^(1/2)/c/e)^2
*e-2*(-(b*e-c*d)*c*e)^(1/2)/c*(x+(-(b*e-c*d)*c*e)^(1/2)/c/e)-(b*e-2*c*d)/c)^(1/2
)*x+1/2*e*c^3/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/((-d*e)^(1/2)*c-(-(b*e-c*d
)*c*e)^(1/2))/(-(b*e-c*d)*c*e)^(1/2)/(b*e-2*c*d)/(-(b*e-2*c*d)/c)^(1/2)*ln((-2*(
b*e-2*c*d)/c-2*(-(b*e-c*d)*c*e)^(1/2)/c*(x+(-(b*e-c*d)*c*e)^(1/2)/c/e)+2*(-(b*e-
2*c*d)/c)^(1/2)*((x+(-(b*e-c*d)*c*e)^(1/2)/c/e)^2*e-2*(-(b*e-c*d)*c*e)^(1/2)/c*(
x+(-(b*e-c*d)*c*e)^(1/2)/c/e)-(b*e-2*c*d)/c)^(1/2))/(x+(-(b*e-c*d)*c*e)^(1/2)/c/
e))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c e^{2} x^{4} + b e^{2} x^{2} - c d^{2} + b d e\right )}{\left (e x^{2} + d\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.704455, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/12*(4*sqrt(2*c^2*d^2*e - 3*b*c*d*e^2 + b^2*e^3)*((7*c*d*e - 2*b*e^2)*x^3 + 3
*(3*c*d^2 - b*d*e)*x)*sqrt(e*x^2 + d) - 3*(c^2*d^2*e^2*x^4 + 2*c^2*d^3*e*x^2 + c
^2*d^4)*log(((c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + (17*c^2*d^2*e^2 - 24*b*c*d*e
^3 + 8*b^2*e^4)*x^4 + 2*(7*c^2*d^3*e - 11*b*c*d^2*e^2 + 4*b^2*d*e^3)*x^2)*sqrt(2
*c^2*d^2*e - 3*b*c*d*e^2 + b^2*e^3) - 4*((6*c^3*d^3*e^2 - 13*b*c^2*d^2*e^3 + 9*b
^2*c*d*e^4 - 2*b^3*e^5)*x^3 + (2*c^3*d^4*e - 5*b*c^2*d^3*e^2 + 4*b^2*c*d^2*e^3 -
b^3*d*e^4)*x)*sqrt(e*x^2 + d))/(c^2*e^2*x^4 + c^2*d^2 - 2*b*c*d*e + b^2*e^2 - 2
*(c^2*d*e - b*c*e^2)*x^2)))/((4*c^2*d^6 - 4*b*c*d^5*e + b^2*d^4*e^2 + (4*c^2*d^4
*e^2 - 4*b*c*d^3*e^3 + b^2*d^2*e^4)*x^4 + 2*(4*c^2*d^5*e - 4*b*c*d^4*e^2 + b^2*d
^3*e^3)*x^2)*sqrt(2*c^2*d^2*e - 3*b*c*d*e^2 + b^2*e^3)), -1/6*(2*sqrt(-2*c^2*d^2
*e + 3*b*c*d*e^2 - b^2*e^3)*((7*c*d*e - 2*b*e^2)*x^3 + 3*(3*c*d^2 - b*d*e)*x)*sq
rt(e*x^2 + d) - 3*(c^2*d^2*e^2*x^4 + 2*c^2*d^3*e*x^2 + c^2*d^4)*arctan(-1/2*sqrt
(-2*c^2*d^2*e + 3*b*c*d*e^2 - b^2*e^3)*(c*d^2 - b*d*e + (3*c*d*e - 2*b*e^2)*x^2)
/((2*c^2*d^2*e - 3*b*c*d*e^2 + b^2*e^3)*sqrt(e*x^2 + d)*x)))/((4*c^2*d^6 - 4*b*c
*d^5*e + b^2*d^4*e^2 + (4*c^2*d^4*e^2 - 4*b*c*d^3*e^3 + b^2*d^2*e^4)*x^4 + 2*(4*
c^2*d^5*e - 4*b*c*d^4*e^2 + b^2*d^3*e^3)*x^2)*sqrt(-2*c^2*d^2*e + 3*b*c*d*e^2 -
b^2*e^3))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (d + e x^{2}\right )^{\frac{5}{2}} \left (b e - c d + c e x^{2}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/((d + e*x**2)**(5/2)*(b*e - c*d + c*e*x**2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*e^2*x^4 + b*e^2*x^2 - c*d^2 + b*d*e)*(e*x^2 + d)^(3/2)),x, algorithm="giac")

[Out]

Timed out