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

Optimal. Leaf size=622 \[ \frac{1}{3} \left (\frac{c}{d}+x\right ) \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}-\frac{2 c^2 \left (\frac{c}{d}+x\right ) \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}{3 \sqrt{4 a d^2+c^3} \left (\frac{d^2 \left (\frac{c}{d}+x\right )^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right )}+\frac{c^{3/4} \sqrt [4]{4 a d^2+c^3} \left (-c^{3/2} \sqrt{4 a d^2+c^3}+4 a d^2+c^3\right ) \sqrt{\frac{d^2 \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}{\left (4 a d^2+c^3\right ) \left (\frac{d^2 \left (\frac{c}{d}+x\right )^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right )^2}} \left (\frac{d^2 \left (\frac{c}{d}+x\right )^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{c+d x}{\sqrt [4]{c} \sqrt [4]{c^3+4 a d^2}}\right )|\frac{1}{2} \left (\frac{c^{3/2}}{\sqrt{c^3+4 a d^2}}+1\right )\right )}{3 d^3 \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}+\frac{2 c^{9/4} \left (4 a d^2+c^3\right )^{3/4} \sqrt{\frac{d^2 \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}{\left (4 a d^2+c^3\right ) \left (\frac{d^2 \left (\frac{c}{d}+x\right )^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right )^2}} \left (\frac{d^2 \left (\frac{c}{d}+x\right )^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right ) E\left (2 \tan ^{-1}\left (\frac{c+d x}{\sqrt [4]{c} \sqrt [4]{c^3+4 a d^2}}\right )|\frac{1}{2} \left (\frac{c^{3/2}}{\sqrt{c^3+4 a d^2}}+1\right )\right )}{3 d^3 \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}} \]

[Out]

((c/d + x)*Sqrt[4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4])/3 - (2*c^2*(c/d + x)*S
qrt[4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4])/(3*Sqrt[c^3 + 4*a*d^2]*(Sqrt[c] +
(d^2*(c/d + x)^2)/Sqrt[c^3 + 4*a*d^2])) + (2*c^(9/4)*(c^3 + 4*a*d^2)^(3/4)*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])*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])/(3*d^3*Sqrt[4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^
4]) + (c^(3/4)*(c^3 + 4*a*d^2)^(1/4)*(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)]*(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])/(3*d^3*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 = 1.55441, antiderivative size = 622, 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 \[ \frac{(c+d x) \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}{3 d}-\frac{2 c^2 (c+d x) \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}{3 d \sqrt{4 a d^2+c^3} \left (\frac{(c+d x)^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right )}+\frac{c^{3/4} \sqrt [4]{4 a d^2+c^3} \left (-c^{3/2} \sqrt{4 a d^2+c^3}+4 a d^2+c^3\right ) \sqrt{\frac{d^2 \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}{\left (4 a d^2+c^3\right ) \left (\frac{(c+d x)^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right )^2}} \left (\frac{(c+d x)^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{c+d x}{\sqrt [4]{c} \sqrt [4]{c^3+4 a d^2}}\right )|\frac{1}{2} \left (\frac{c^{3/2}}{\sqrt{c^3+4 a d^2}}+1\right )\right )}{3 d^3 \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}+\frac{2 c^{9/4} \left (4 a d^2+c^3\right )^{3/4} \sqrt{\frac{d^2 \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}{\left (4 a d^2+c^3\right ) \left (\frac{(c+d x)^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right )^2}} \left (\frac{(c+d x)^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right ) E\left (2 \tan ^{-1}\left (\frac{c+d x}{\sqrt [4]{c} \sqrt [4]{c^3+4 a d^2}}\right )|\frac{1}{2} \left (\frac{c^{3/2}}{\sqrt{c^3+4 a d^2}}+1\right )\right )}{3 d^3 \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((c + d*x)*Sqrt[4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4])/(3*d) - (2*c^2*(c + d*
x)*Sqrt[4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4])/(3*d*Sqrt[c^3 + 4*a*d^2]*(Sqrt
[c] + (c + d*x)^2/Sqrt[c^3 + 4*a*d^2])) + (2*c^(9/4)*(c^3 + 4*a*d^2)^(3/4)*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])*Ell
ipticE[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])/(3*d^3*Sqrt[4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4]) + (c^(3/
4)*(c^3 + 4*a*d^2)^(1/4)*(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])*Ellipti
cF[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])/(3*d^3*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 = 127.364, size = 604, normalized size = 0.97 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*c**(9/4)*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)**(3/4)*e
lliptic_e(2*atan(d*(c/d + x)/(c**(1/4)*(4*a*d**2 + c**3)**(1/4))), c**(3/2)/(2*s
qrt(4*a*d**2 + c**3)) + 1/2)/(3*d**3*sqrt(-2*c**2*(c/d + x)**2 + c*(4*a + c**3/d
**2) + d**2*(c/d + x)**4)) - c**(3/4)*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))
*(c**(3/2) - sqrt(4*a*d**2 + c**3))*(4*a*d**2 + c**3)**(3/4)*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)/(3*d**3*sqrt(-2*c**2*(c/d + x)**2 + c*(4*a + c**3/d**2) + d**2*(c/d +
x)**4)) - 2*c**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)/(3*(sqrt(c) + d**2*(c/d + x)**2/sqrt(4*a*d**2 + c**3))*sqrt(4*a
*d**2 + c**3)) + (c/d + x)*sqrt(-2*c**2*(c/d + x)**2 + c*(4*a + c**3/d**2) + d**
2*(c/d + x)**4)/3

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Mathematica [C]  time = 6.14725, size = 5218, normalized size = 8.39 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

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

[Out]

Result too large to show

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Maple [B]  time = 0.05, size = 4890, normalized size = 7.9 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*x*(d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c)^(1/2)+1/3*c/d*(d^2*x^4+4*c*d*x^3+4*c^2
*x^2+4*a*c)^(1/2)+16/3*a*c*((-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))-8/3*c^3/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)*(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)*(-
(c+(2*d*(-a*c)^(1/2)+c^2)^(1/2))/d*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))+((-c+(2*d*(-a*c)^(1/2)+c^2)^(1/2))/d+(c+(2*d*(-a*c)^
(1/2)+c^2)^(1/2))/d)*EllipticPi(((-(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)/((-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)))-2/3*c^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)+((-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^2*(-c+(2*d*(-a*c)^(1/2)+c^2)^(1/2))+(c+(-2*
d*(-a*c)^(1/2)+c^2)^(1/2))/d^2*(-c+(2*d*(-a*c)^(1/2)+c^2)^(1/2))+(c+(-2*d*(-a*c)
^(1/2)+c^2)^(1/2))/d^2*(c+(2*d*(-a*c)^(1/2)+c^2)^(1/2))+(c+(2*d*(-a*c)^(1/2)+c^2
)^(1/2))^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)/(-(c+(2*d*(-a*c)^(1/2)+c^2)^(1/2))/d-(-c+(2*d*(-a*c)^(1/2)+c^2)^(1/2))
/d)*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))+((-
c+(2*d*(-a*c)^(1/2)+c^2)^(1/2))/d-(-c+(-2*d*(-a*c)^(1/2)+c^2)^(1/2))/d)*Elliptic
E(((-(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))/(-(c+(2*d*(-a*c
)^(1/2)+c^2)^(1/2))/d-(-c+(2*d*(-a*c)^(1/2)+c^2)^(1/2))/d)+4*c/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)*EllipticPi(((-(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)/((-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))))/(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)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(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. \[{\rm integral}\left (\sqrt{d^{2} x^{4} + 4 \, c d x^{3} + 4 \, c^{2} x^{2} + 4 \, a c}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(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. \[ \int \sqrt{4 a c + 4 c^{2} x^{2} + 4 c d x^{3} + d^{2} x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(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. \[ \int \sqrt{d^{2} x^{4} + 4 \, c d x^{3} + 4 \, c^{2} x^{2} + 4 \, a c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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