3.621 \(\int \frac{1}{\left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=674 \[ -\frac{d^2 \left (\frac{c}{d}+x\right ) \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}{8 a \left (4 a d^2+c^3\right )^{3/2} \left (\frac{d^2 \left (\frac{c}{d}+x\right )^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right )}-\frac{\left (\frac{c}{d}+x\right ) \left (-4 a d^2+c^3-c d^2 \left (\frac{c}{d}+x\right )^2\right )}{8 a c \left (4 a d^2+c^3\right ) \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}+\frac{\left (-c^{3/2} \sqrt{4 a d^2+c^3}+4 a d^2+c^3\right ) \sqrt{\frac{d^2 \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}{\left (4 a d^2+c^3\right ) \left (\frac{d^2 \left (\frac{c}{d}+x\right )^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right )^2}} \left (\frac{d^2 \left (\frac{c}{d}+x\right )^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{c+d x}{\sqrt [4]{c} \sqrt [4]{c^3+4 a d^2}}\right )|\frac{1}{2} \left (\frac{c^{3/2}}{\sqrt{c^3+4 a d^2}}+1\right )\right )}{16 a c^{5/4} d \left (4 a d^2+c^3\right )^{3/4} \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}+\frac{\sqrt [4]{c} \sqrt{\frac{d^2 \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}{\left (4 a d^2+c^3\right ) \left (\frac{d^2 \left (\frac{c}{d}+x\right )^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right )^2}} \left (\frac{d^2 \left (\frac{c}{d}+x\right )^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right ) E\left (2 \tan ^{-1}\left (\frac{c+d x}{\sqrt [4]{c} \sqrt [4]{c^3+4 a d^2}}\right )|\frac{1}{2} \left (\frac{c^{3/2}}{\sqrt{c^3+4 a d^2}}+1\right )\right )}{8 a d \sqrt [4]{4 a d^2+c^3} \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}} \]

[Out]

-((c/d + x)*(c^3 - 4*a*d^2 - c*d^2*(c/d + x)^2))/(8*a*c*(c^3 + 4*a*d^2)*Sqrt[4*a
*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4]) - (d^2*(c/d + x)*Sqrt[4*a*c + 4*c^2*x^2 +
4*c*d*x^3 + d^2*x^4])/(8*a*(c^3 + 4*a*d^2)^(3/2)*(Sqrt[c] + (d^2*(c/d + x)^2)/S
qrt[c^3 + 4*a*d^2])) + (c^(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)]*(Sqr
t[c] + (d^2*(c/d + x)^2)/Sqrt[c^3 + 4*a*d^2])*EllipticE[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])/(8*a*d*(c^3 +
4*a*d^2)^(1/4)*Sqrt[4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4]) + ((c^3 + 4*a*d^2
- c^(3/2)*Sqrt[c^3 + 4*a*d^2])*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)]*(Sqr
t[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])/(16*a*c^(5/4)
*d*(c^3 + 4*a*d^2)^(3/4)*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 = 1.70507, antiderivative size = 674, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161 \[ -\frac{(c+d x) \left (-4 a d^2+c^3-c (c+d x)^2\right )}{8 a c d \left (4 a d^2+c^3\right ) \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}-\frac{d (c+d x) \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}{8 a \left (4 a d^2+c^3\right )^{3/2} \left (\frac{(c+d x)^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right )}+\frac{\left (-c^{3/2} \sqrt{4 a d^2+c^3}+4 a d^2+c^3\right ) \sqrt{\frac{d^2 \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}{\left (4 a d^2+c^3\right ) \left (\frac{(c+d x)^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right )^2}} \left (\frac{(c+d x)^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{c+d x}{\sqrt [4]{c} \sqrt [4]{c^3+4 a d^2}}\right )|\frac{1}{2} \left (\frac{c^{3/2}}{\sqrt{c^3+4 a d^2}}+1\right )\right )}{16 a c^{5/4} d \left (4 a d^2+c^3\right )^{3/4} \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}+\frac{\sqrt [4]{c} \sqrt{\frac{d^2 \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}{\left (4 a d^2+c^3\right ) \left (\frac{(c+d x)^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right )^2}} \left (\frac{(c+d x)^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right ) E\left (2 \tan ^{-1}\left (\frac{c+d x}{\sqrt [4]{c} \sqrt [4]{c^3+4 a d^2}}\right )|\frac{1}{2} \left (\frac{c^{3/2}}{\sqrt{c^3+4 a d^2}}+1\right )\right )}{8 a d \sqrt [4]{4 a d^2+c^3} \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}} \]

Warning: Unable to verify antiderivative.

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

[Out]

-((c + d*x)*(c^3 - 4*a*d^2 - c*(c + d*x)^2))/(8*a*c*d*(c^3 + 4*a*d^2)*Sqrt[4*a*c
+ 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4]) - (d*(c + d*x)*Sqrt[4*a*c + 4*c^2*x^2 + 4*c
*d*x^3 + d^2*x^4])/(8*a*(c^3 + 4*a*d^2)^(3/2)*(Sqrt[c] + (c + d*x)^2/Sqrt[c^3 +
4*a*d^2])) + (c^(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])*EllipticE[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])/(8*a*d*(c^3 + 4*a*d^2)^(1/4)*Sqrt[4
*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4]) + ((c^3 + 4*a*d^2 - c^(3/2)*Sqrt[c^3 +
4*a*d^2])*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])/(16*a*c^(5/4)*d*(c^3 + 4*a*d^2)^(3/4)*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 = 131.157, size = 651, normalized size = 0.97 \[ \text{result too large to display} \]

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)**(3/2),x)

[Out]

c**(1/4)*sqrt(d**2*(-2*c**2*(c/d + x)**2 + c*(4*a + c**3/d**2) + d**2*(c/d + x)*
*4)/((sqrt(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))*elliptic_e(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
)/(8*a*d*(4*a*d**2 + c**3)**(1/4)*sqrt(-2*c**2*(c/d + x)**2 + c*(4*a + c**3/d**2
) + d**2*(c/d + x)**4)) - d**2*(c/d + x)*sqrt(-2*c**2*(c/d + x)**2 + c*(4*a + c*
*3/d**2) + d**2*(c/d + x)**4)/(8*a*(sqrt(c) + d**2*(c/d + x)**2/sqrt(4*a*d**2 +
c**3))*(4*a*d**2 + c**3)**(3/2)) - (c/d + x)*(-8*a*d**2 + 2*c**3 - 2*c*d**2*(c/d
+ x)**2)/(16*a*c*(4*a*d**2 + c**3)*sqrt(-2*c**2*(c/d + x)**2 + c*(4*a + c**3/d*
*2) + d**2*(c/d + x)**4)) + sqrt(d**2*(-2*c**2*(c/d + x)**2 + c*(4*a + c**3/d**2
) + d**2*(c/d + x)**4)/((sqrt(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**(3/2)*sqrt(4*a*d**2 + c**3) + c**3)*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)/(16*a*c*
*(5/4)*d*(4*a*d**2 + c**3)**(3/4)*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 = 6.20092, size = 5276, normalized size = 7.83 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

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

[Out]

Result too large to show

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Maple [B]  time = 0.067, size = 5024, normalized size = 7.5 \[ \text{output too large to display} \]

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)^(3/2),x)

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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)**(3/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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