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

Optimal. Leaf size=730 \[ \frac{1}{7} \left (\frac{c}{d}+x\right ) \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^{3/2}-\frac{16 c^3 \left (8 a d^2+c^3\right ) \left (\frac{c}{d}+x\right ) \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}{35 d^2 \sqrt{4 a d^2+c^3} \left (\frac{d^2 \left (\frac{c}{d}+x\right )^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right )}+\frac{2 c \left (\frac{c}{d}+x\right ) \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4} \left (20 a d^2+7 c^3-3 c d^2 \left (\frac{c}{d}+x\right )^2\right )}{35 d^2}+\frac{8 c^{7/4} \left (4 a d^2+c^3\right )^{3/4} \left (\sqrt{4 a d^2+c^3} \left (5 a d^2+c^3\right )-c^{3/2} \left (8 a d^2+c^3\right )\right ) \sqrt{\frac{d^2 \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}{\left (4 a d^2+c^3\right ) \left (\frac{d^2 \left (\frac{c}{d}+x\right )^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right )^2}} \left (\frac{d^2 \left (\frac{c}{d}+x\right )^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{c+d x}{\sqrt [4]{c} \sqrt [4]{c^3+4 a d^2}}\right )|\frac{1}{2} \left (\frac{c^{3/2}}{\sqrt{c^3+4 a d^2}}+1\right )\right )}{35 d^5 \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}+\frac{16 c^{13/4} \left (4 a d^2+c^3\right )^{3/4} \left (8 a d^2+c^3\right ) \sqrt{\frac{d^2 \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}{\left (4 a d^2+c^3\right ) \left (\frac{d^2 \left (\frac{c}{d}+x\right )^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right )^2}} \left (\frac{d^2 \left (\frac{c}{d}+x\right )^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right ) E\left (2 \tan ^{-1}\left (\frac{c+d x}{\sqrt [4]{c} \sqrt [4]{c^3+4 a d^2}}\right )|\frac{1}{2} \left (\frac{c^{3/2}}{\sqrt{c^3+4 a d^2}}+1\right )\right )}{35 d^5 \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}} \]

[Out]

((c/d + x)*(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4)^(3/2))/7 + (2*c*(c/d + x)*S
qrt[4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4]*(7*c^3 + 20*a*d^2 - 3*c*d^2*(c/d +
x)^2))/(35*d^2) - (16*c^3*(c^3 + 8*a*d^2)*(c/d + x)*Sqrt[4*a*c + 4*c^2*x^2 + 4*c
*d*x^3 + d^2*x^4])/(35*d^2*Sqrt[c^3 + 4*a*d^2]*(Sqrt[c] + (d^2*(c/d + x)^2)/Sqrt
[c^3 + 4*a*d^2])) + (16*c^(13/4)*(c^3 + 4*a*d^2)^(3/4)*(c^3 + 8*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])*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])/(35*d^5*Sqrt[4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4])
+ (8*c^(7/4)*(c^3 + 4*a*d^2)^(3/4)*(Sqrt[c^3 + 4*a*d^2]*(c^3 + 5*a*d^2) - c^(3/
2)*(c^3 + 8*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])/(35*d^5*Sqrt[4*a*c + 4*c^
2*x^2 + 4*c*d*x^3 + d^2*x^4])

_______________________________________________________________________________________

Rubi [A]  time = 1.9181, antiderivative size = 730, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194 \[ \frac{(c+d x) \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^{3/2}}{7 d}-\frac{16 c^3 \left (8 a d^2+c^3\right ) (c+d x) \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}{35 d^3 \sqrt{4 a d^2+c^3} \left (\frac{(c+d x)^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right )}+\frac{2 c (c+d x) \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4} \left (20 a d^2+7 c^3-3 c (c+d x)^2\right )}{35 d^3}+\frac{8 c^{7/4} \left (4 a d^2+c^3\right )^{3/4} \left (\sqrt{4 a d^2+c^3} \left (5 a d^2+c^3\right )-c^{3/2} \left (8 a d^2+c^3\right )\right ) \sqrt{\frac{d^2 \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}{\left (4 a d^2+c^3\right ) \left (\frac{(c+d x)^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right )^2}} \left (\frac{(c+d x)^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{c+d x}{\sqrt [4]{c} \sqrt [4]{c^3+4 a d^2}}\right )|\frac{1}{2} \left (\frac{c^{3/2}}{\sqrt{c^3+4 a d^2}}+1\right )\right )}{35 d^5 \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}+\frac{16 c^{13/4} \left (4 a d^2+c^3\right )^{3/4} \left (8 a d^2+c^3\right ) \sqrt{\frac{d^2 \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}{\left (4 a d^2+c^3\right ) \left (\frac{(c+d x)^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right )^2}} \left (\frac{(c+d x)^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right ) E\left (2 \tan ^{-1}\left (\frac{c+d x}{\sqrt [4]{c} \sqrt [4]{c^3+4 a d^2}}\right )|\frac{1}{2} \left (\frac{c^{3/2}}{\sqrt{c^3+4 a d^2}}+1\right )\right )}{35 d^5 \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}} \]

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)*(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4)^(3/2))/(7*d) + (2*c*(c + d*
x)*Sqrt[4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4]*(7*c^3 + 20*a*d^2 - 3*c*(c + d*
x)^2))/(35*d^3) - (16*c^3*(c^3 + 8*a*d^2)*(c + d*x)*Sqrt[4*a*c + 4*c^2*x^2 + 4*c
*d*x^3 + d^2*x^4])/(35*d^3*Sqrt[c^3 + 4*a*d^2]*(Sqrt[c] + (c + d*x)^2/Sqrt[c^3 +
4*a*d^2])) + (16*c^(13/4)*(c^3 + 4*a*d^2)^(3/4)*(c^3 + 8*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/Sq
rt[c^3 + 4*a*d^2])^2)]*(Sqrt[c] + (c + d*x)^2/Sqrt[c^3 + 4*a*d^2])*EllipticE[2*A
rcTan[(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])/(35*d^5*Sqrt[4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4]) + (8*c^(7/4)*(c^3
+ 4*a*d^2)^(3/4)*(Sqrt[c^3 + 4*a*d^2]*(c^3 + 5*a*d^2) - c^(3/2)*(c^3 + 8*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])/(35*d^5*Sqrt[4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 157.33, size = 734, normalized size = 1.01 \[ \text{result too large to display} \]

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)**(3/2),x)

[Out]

16*c**(13/4)*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)
*(8*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)/(35*d**5*sqrt(-2*c**2*(c/d + x)
**2 + c*(4*a + c**3/d**2) + d**2*(c/d + x)**4)) + 8*c**(7/4)*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)**(1/4)*(a*d**2*(20*a*d**2 + 9*c**3) -
c**(3/2)*sqrt(4*a*d**2 + c**3)*(8*a*d**2 + c**3) + c**6)*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)/(35*d**5*sqrt(-2*c**2*(c/d + x)**2 + c*(4*a + c**3/d**2) + d**2*(c/d + x)
**4)) - 16*c**3*(8*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)/(35*d**2*(sqrt(c) + d**2*(c/d + x)**2/sqrt(4*a*
d**2 + c**3))*sqrt(4*a*d**2 + c**3)) + c*(c/d + x)*(40*a*d**2 + 14*c**3 - 6*c*d*
*2*(c/d + x)**2)*sqrt(-2*c**2*(c/d + x)**2 + c*(4*a + c**3/d**2) + d**2*(c/d + x
)**4)/(35*d**2) + (c/d + x)*(-2*c**2*(c/d + x)**2 + c*(4*a + c**3/d**2) + d**2*(
c/d + x)**4)**(3/2)/7

_______________________________________________________________________________________

Mathematica [C]  time = 6.29824, size = 10468, normalized size = 14.34 \[ \text{Result too large to show} \]

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

_______________________________________________________________________________________

Maple [B]  time = 0.268, size = 5229, normalized size = 7.2 \[ \text{output too large to display} \]

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)^(3/2),x)

[Out]

result too large to display

_______________________________________________________________________________________

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

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)

_______________________________________________________________________________________

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

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)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((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)

_______________________________________________________________________________________

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

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)