3.620 \(\int \frac{1}{\sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}} \, dx\)

Optimal. Leaf size=227 \[ \frac{\sqrt [4]{4 a d^2+c^3} \sqrt{\frac{d^2 \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}{\left (4 a d^2+c^3\right ) \left (\frac{d^2 \left (\frac{c}{d}+x\right )^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right )^2}} \left (\frac{d^2 \left (\frac{c}{d}+x\right )^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{c+d x}{\sqrt [4]{c} \sqrt [4]{c^3+4 a d^2}}\right )|\frac{1}{2} \left (\frac{c^{3/2}}{\sqrt{c^3+4 a d^2}}+1\right )\right )}{2 \sqrt [4]{c} d \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}} \]

[Out]

((c^3 + 4*a*d^2)^(1/4)*Sqrt[(d^2*(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4))/((c^
3 + 4*a*d^2)*(Sqrt[c] + (d^2*(c/d + x)^2)/Sqrt[c^3 + 4*a*d^2])^2)]*(Sqrt[c] + (d
^2*(c/d + x)^2)/Sqrt[c^3 + 4*a*d^2])*EllipticF[2*ArcTan[(c + d*x)/(c^(1/4)*(c^3
+ 4*a*d^2)^(1/4))], (1 + c^(3/2)/Sqrt[c^3 + 4*a*d^2])/2])/(2*c^(1/4)*d*Sqrt[4*a*
c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4])

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Rubi [A]  time = 0.468725, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ \frac{\sqrt [4]{4 a d^2+c^3} \sqrt{\frac{d^2 \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}{\left (4 a d^2+c^3\right ) \left (\frac{(c+d x)^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right )^2}} \left (\frac{(c+d x)^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{c+d x}{\sqrt [4]{c} \sqrt [4]{c^3+4 a d^2}}\right )|\frac{1}{2} \left (\frac{c^{3/2}}{\sqrt{c^3+4 a d^2}}+1\right )\right )}{2 \sqrt [4]{c} d \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4],x]

[Out]

((c^3 + 4*a*d^2)^(1/4)*Sqrt[(d^2*(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4))/((c^
3 + 4*a*d^2)*(Sqrt[c] + (c + d*x)^2/Sqrt[c^3 + 4*a*d^2])^2)]*(Sqrt[c] + (c + d*x
)^2/Sqrt[c^3 + 4*a*d^2])*EllipticF[2*ArcTan[(c + d*x)/(c^(1/4)*(c^3 + 4*a*d^2)^(
1/4))], (1 + c^(3/2)/Sqrt[c^3 + 4*a*d^2])/2])/(2*c^(1/4)*d*Sqrt[4*a*c + 4*c^2*x^
2 + 4*c*d*x^3 + d^2*x^4])

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Rubi in Sympy [A]  time = 56.1069, size = 221, normalized size = 0.97 \[ \frac{\sqrt{\frac{d^{2} \left (- 2 c^{2} \left (\frac{c}{d} + x\right )^{2} + c \left (4 a + \frac{c^{3}}{d^{2}}\right ) + d^{2} \left (\frac{c}{d} + x\right )^{4}\right )}{\left (\sqrt{c} + \frac{d^{2} \left (\frac{c}{d} + x\right )^{2}}{\sqrt{4 a d^{2} + c^{3}}}\right )^{2} \left (4 a d^{2} + c^{3}\right )}} \left (\sqrt{c} + \frac{d^{2} \left (\frac{c}{d} + x\right )^{2}}{\sqrt{4 a d^{2} + c^{3}}}\right ) \sqrt [4]{4 a d^{2} + c^{3}} F\left (2 \operatorname{atan}{\left (\frac{d \left (\frac{c}{d} + x\right )}{\sqrt [4]{c} \sqrt [4]{4 a d^{2} + c^{3}}} \right )}\middle | \frac{c^{\frac{3}{2}}}{2 \sqrt{4 a d^{2} + c^{3}}} + \frac{1}{2}\right )}{2 \sqrt [4]{c} d \sqrt{- 2 c^{2} \left (\frac{c}{d} + x\right )^{2} + c \left (4 a + \frac{c^{3}}{d^{2}}\right ) + d^{2} \left (\frac{c}{d} + x\right )^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(d**2*(-2*c**2*(c/d + x)**2 + c*(4*a + c**3/d**2) + d**2*(c/d + x)**4)/((sqr
t(c) + d**2*(c/d + x)**2/sqrt(4*a*d**2 + c**3))**2*(4*a*d**2 + c**3)))*(sqrt(c)
+ d**2*(c/d + x)**2/sqrt(4*a*d**2 + c**3))*(4*a*d**2 + c**3)**(1/4)*elliptic_f(2
*atan(d*(c/d + x)/(c**(1/4)*(4*a*d**2 + c**3)**(1/4))), c**(3/2)/(2*sqrt(4*a*d**
2 + c**3)) + 1/2)/(2*c**(1/4)*d*sqrt(-2*c**2*(c/d + x)**2 + c*(4*a + c**3/d**2)
+ d**2*(c/d + x)**4))

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Mathematica [C]  time = 3.75966, size = 822, normalized size = 3.62 \[ \frac{2 \left (-c-d x+\sqrt{c^2-2 i \sqrt{a} \sqrt{c} d}\right ) \left (c+d x+\sqrt{c^2-2 i \sqrt{a} \sqrt{c} d}\right ) \sqrt{-\frac{\sqrt{c^2-2 i \sqrt{a} \sqrt{c} d} \left (c+d x-\sqrt{c^2+2 i \sqrt{a} d \sqrt{c}}\right )}{\left (\sqrt{c^2-2 i \sqrt{a} \sqrt{c} d}+\sqrt{c^2+2 i \sqrt{a} d \sqrt{c}}\right ) \left (-c-d x+\sqrt{c^2-2 i \sqrt{a} \sqrt{c} d}\right )}} \sqrt{-\frac{\sqrt{c^2-2 i \sqrt{a} \sqrt{c} d} \left (c+d x+\sqrt{c^2+2 i \sqrt{a} d \sqrt{c}}\right )}{\left (\sqrt{c^2-2 i \sqrt{a} \sqrt{c} d}-\sqrt{c^2+2 i \sqrt{a} d \sqrt{c}}\right ) \left (-c-d x+\sqrt{c^2-2 i \sqrt{a} \sqrt{c} d}\right )}} F\left (\sin ^{-1}\left (\sqrt{\frac{\left (\sqrt{c^2-2 i \sqrt{a} \sqrt{c} d}-\sqrt{c^2+2 i \sqrt{a} d \sqrt{c}}\right ) \left (c+d x+\sqrt{c^2-2 i \sqrt{a} \sqrt{c} d}\right )}{\left (\sqrt{c^2-2 i \sqrt{a} \sqrt{c} d}+\sqrt{c^2+2 i \sqrt{a} d \sqrt{c}}\right ) \left (-c-d x+\sqrt{c^2-2 i \sqrt{a} \sqrt{c} d}\right )}}\right )|\frac{\left (\sqrt{c^2-2 i \sqrt{a} \sqrt{c} d}+\sqrt{c^2+2 i \sqrt{a} d \sqrt{c}}\right )^2}{\left (\sqrt{c^2-2 i \sqrt{a} \sqrt{c} d}-\sqrt{c^2+2 i \sqrt{a} d \sqrt{c}}\right )^2}\right )}{d \sqrt{c^2-2 i \sqrt{a} \sqrt{c} d} \sqrt{\frac{\left (\sqrt{c^2-2 i \sqrt{a} \sqrt{c} d}-\sqrt{c^2+2 i \sqrt{a} d \sqrt{c}}\right ) \left (c+d x+\sqrt{c^2-2 i \sqrt{a} \sqrt{c} d}\right )}{\left (\sqrt{c^2-2 i \sqrt{a} \sqrt{c} d}+\sqrt{c^2+2 i \sqrt{a} d \sqrt{c}}\right ) \left (-c-d x+\sqrt{c^2-2 i \sqrt{a} \sqrt{c} d}\right )}} \sqrt{x^2 (2 c+d x)^2+4 a c}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4],x]

[Out]

(2*(-c + Sqrt[c^2 - (2*I)*Sqrt[a]*Sqrt[c]*d] - d*x)*(c + Sqrt[c^2 - (2*I)*Sqrt[a
]*Sqrt[c]*d] + d*x)*Sqrt[-((Sqrt[c^2 - (2*I)*Sqrt[a]*Sqrt[c]*d]*(c - Sqrt[c^2 +
(2*I)*Sqrt[a]*Sqrt[c]*d] + d*x))/((Sqrt[c^2 - (2*I)*Sqrt[a]*Sqrt[c]*d] + Sqrt[c^
2 + (2*I)*Sqrt[a]*Sqrt[c]*d])*(-c + Sqrt[c^2 - (2*I)*Sqrt[a]*Sqrt[c]*d] - d*x)))
]*Sqrt[-((Sqrt[c^2 - (2*I)*Sqrt[a]*Sqrt[c]*d]*(c + Sqrt[c^2 + (2*I)*Sqrt[a]*Sqrt
[c]*d] + d*x))/((Sqrt[c^2 - (2*I)*Sqrt[a]*Sqrt[c]*d] - Sqrt[c^2 + (2*I)*Sqrt[a]*
Sqrt[c]*d])*(-c + Sqrt[c^2 - (2*I)*Sqrt[a]*Sqrt[c]*d] - d*x)))]*EllipticF[ArcSin
[Sqrt[((Sqrt[c^2 - (2*I)*Sqrt[a]*Sqrt[c]*d] - Sqrt[c^2 + (2*I)*Sqrt[a]*Sqrt[c]*d
])*(c + Sqrt[c^2 - (2*I)*Sqrt[a]*Sqrt[c]*d] + d*x))/((Sqrt[c^2 - (2*I)*Sqrt[a]*S
qrt[c]*d] + Sqrt[c^2 + (2*I)*Sqrt[a]*Sqrt[c]*d])*(-c + Sqrt[c^2 - (2*I)*Sqrt[a]*
Sqrt[c]*d] - d*x))]], (Sqrt[c^2 - (2*I)*Sqrt[a]*Sqrt[c]*d] + Sqrt[c^2 + (2*I)*Sq
rt[a]*Sqrt[c]*d])^2/(Sqrt[c^2 - (2*I)*Sqrt[a]*Sqrt[c]*d] - Sqrt[c^2 + (2*I)*Sqrt
[a]*Sqrt[c]*d])^2])/(d*Sqrt[c^2 - (2*I)*Sqrt[a]*Sqrt[c]*d]*Sqrt[((Sqrt[c^2 - (2*
I)*Sqrt[a]*Sqrt[c]*d] - Sqrt[c^2 + (2*I)*Sqrt[a]*Sqrt[c]*d])*(c + Sqrt[c^2 - (2*
I)*Sqrt[a]*Sqrt[c]*d] + d*x))/((Sqrt[c^2 - (2*I)*Sqrt[a]*Sqrt[c]*d] + Sqrt[c^2 +
(2*I)*Sqrt[a]*Sqrt[c]*d])*(-c + Sqrt[c^2 - (2*I)*Sqrt[a]*Sqrt[c]*d] - d*x))]*Sq
rt[4*a*c + x^2*(2*c + d*x)^2])

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Maple [B]  time = 0.046, size = 1056, normalized size = 4.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*((-c+(2*d*(-a*c)^(1/2)+c^2)^(1/2))/d+(c+(-2*d*(-a*c)^(1/2)+c^2)^(1/2))/d)*((-(
c+(-2*d*(-a*c)^(1/2)+c^2)^(1/2))/d+(c+(2*d*(-a*c)^(1/2)+c^2)^(1/2))/d)*(x-(-c+(2
*d*(-a*c)^(1/2)+c^2)^(1/2))/d)/(-(c+(-2*d*(-a*c)^(1/2)+c^2)^(1/2))/d-(-c+(2*d*(-
a*c)^(1/2)+c^2)^(1/2))/d)/(x+(c+(2*d*(-a*c)^(1/2)+c^2)^(1/2))/d))^(1/2)*(x+(c+(2
*d*(-a*c)^(1/2)+c^2)^(1/2))/d)^2*((-(c+(2*d*(-a*c)^(1/2)+c^2)^(1/2))/d-(-c+(2*d*
(-a*c)^(1/2)+c^2)^(1/2))/d)*(x-(-c+(-2*d*(-a*c)^(1/2)+c^2)^(1/2))/d)/((-c+(-2*d*
(-a*c)^(1/2)+c^2)^(1/2))/d-(-c+(2*d*(-a*c)^(1/2)+c^2)^(1/2))/d)/(x+(c+(2*d*(-a*c
)^(1/2)+c^2)^(1/2))/d))^(1/2)*((-(c+(2*d*(-a*c)^(1/2)+c^2)^(1/2))/d-(-c+(2*d*(-a
*c)^(1/2)+c^2)^(1/2))/d)*(x+(c+(-2*d*(-a*c)^(1/2)+c^2)^(1/2))/d)/(-(c+(-2*d*(-a*
c)^(1/2)+c^2)^(1/2))/d-(-c+(2*d*(-a*c)^(1/2)+c^2)^(1/2))/d)/(x+(c+(2*d*(-a*c)^(1
/2)+c^2)^(1/2))/d))^(1/2)/(-(c+(-2*d*(-a*c)^(1/2)+c^2)^(1/2))/d+(c+(2*d*(-a*c)^(
1/2)+c^2)^(1/2))/d)/(-(c+(2*d*(-a*c)^(1/2)+c^2)^(1/2))/d-(-c+(2*d*(-a*c)^(1/2)+c
^2)^(1/2))/d)/(d^2*(x-(-c+(2*d*(-a*c)^(1/2)+c^2)^(1/2))/d)*(x+(c+(2*d*(-a*c)^(1/
2)+c^2)^(1/2))/d)*(x-(-c+(-2*d*(-a*c)^(1/2)+c^2)^(1/2))/d)*(x+(c+(-2*d*(-a*c)^(1
/2)+c^2)^(1/2))/d))^(1/2)*EllipticF(((-(c+(-2*d*(-a*c)^(1/2)+c^2)^(1/2))/d+(c+(2
*d*(-a*c)^(1/2)+c^2)^(1/2))/d)*(x-(-c+(2*d*(-a*c)^(1/2)+c^2)^(1/2))/d)/(-(c+(-2*
d*(-a*c)^(1/2)+c^2)^(1/2))/d-(-c+(2*d*(-a*c)^(1/2)+c^2)^(1/2))/d)/(x+(c+(2*d*(-a
*c)^(1/2)+c^2)^(1/2))/d))^(1/2),((-(c+(2*d*(-a*c)^(1/2)+c^2)^(1/2))/d-(-c+(-2*d*
(-a*c)^(1/2)+c^2)^(1/2))/d)*((-c+(2*d*(-a*c)^(1/2)+c^2)^(1/2))/d+(c+(-2*d*(-a*c)
^(1/2)+c^2)^(1/2))/d)/((-c+(2*d*(-a*c)^(1/2)+c^2)^(1/2))/d-(-c+(-2*d*(-a*c)^(1/2
)+c^2)^(1/2))/d)/(-(c+(2*d*(-a*c)^(1/2)+c^2)^(1/2))/d+(c+(-2*d*(-a*c)^(1/2)+c^2)
^(1/2))/d))^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{d^{2} x^{4} + 4 \, c d x^{3} + 4 \, c^{2} x^{2} + 4 \, a c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/sqrt(d^2*x^4 + 4*c*d*x^3 + 4*c^2*x^2 + 4*a*c), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{d^{2} x^{4} + 4 \, c d x^{3} + 4 \, c^{2} x^{2} + 4 \, a c}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/sqrt(d^2*x^4 + 4*c*d*x^3 + 4*c^2*x^2 + 4*a*c), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{4 a c + 4 c^{2} x^{2} + 4 c d x^{3} + d^{2} x^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/sqrt(4*a*c + 4*c**2*x**2 + 4*c*d*x**3 + d**2*x**4), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{d^{2} x^{4} + 4 \, c d x^{3} + 4 \, c^{2} x^{2} + 4 \, a c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/sqrt(d^2*x^4 + 4*c*d*x^3 + 4*c^2*x^2 + 4*a*c), x)