3.205 \(\int x^2 \left (a+b x^3+c x^6\right )^{3/2} \, dx\)

Optimal. Leaf size=124 \[ \frac{\left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{128 c^{5/2}}-\frac{\left (b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{64 c^2}+\frac{\left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{24 c} \]

[Out]

-((b^2 - 4*a*c)*(b + 2*c*x^3)*Sqrt[a + b*x^3 + c*x^6])/(64*c^2) + ((b + 2*c*x^3)
*(a + b*x^3 + c*x^6)^(3/2))/(24*c) + ((b^2 - 4*a*c)^2*ArcTanh[(b + 2*c*x^3)/(2*S
qrt[c]*Sqrt[a + b*x^3 + c*x^6])])/(128*c^(5/2))

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Rubi [A]  time = 0.174603, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{\left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{128 c^{5/2}}-\frac{\left (b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{64 c^2}+\frac{\left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{24 c} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(a + b*x^3 + c*x^6)^(3/2),x]

[Out]

-((b^2 - 4*a*c)*(b + 2*c*x^3)*Sqrt[a + b*x^3 + c*x^6])/(64*c^2) + ((b + 2*c*x^3)
*(a + b*x^3 + c*x^6)^(3/2))/(24*c) + ((b^2 - 4*a*c)^2*ArcTanh[(b + 2*c*x^3)/(2*S
qrt[c]*Sqrt[a + b*x^3 + c*x^6])])/(128*c^(5/2))

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Rubi in Sympy [A]  time = 15.1663, size = 112, normalized size = 0.9 \[ \frac{\left (b + 2 c x^{3}\right ) \left (a + b x^{3} + c x^{6}\right )^{\frac{3}{2}}}{24 c} - \frac{\left (b + 2 c x^{3}\right ) \left (- 4 a c + b^{2}\right ) \sqrt{a + b x^{3} + c x^{6}}}{64 c^{2}} + \frac{\left (- 4 a c + b^{2}\right )^{2} \operatorname{atanh}{\left (\frac{b + 2 c x^{3}}{2 \sqrt{c} \sqrt{a + b x^{3} + c x^{6}}} \right )}}{128 c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(c*x**6+b*x**3+a)**(3/2),x)

[Out]

(b + 2*c*x**3)*(a + b*x**3 + c*x**6)**(3/2)/(24*c) - (b + 2*c*x**3)*(-4*a*c + b*
*2)*sqrt(a + b*x**3 + c*x**6)/(64*c**2) + (-4*a*c + b**2)**2*atanh((b + 2*c*x**3
)/(2*sqrt(c)*sqrt(a + b*x**3 + c*x**6)))/(128*c**(5/2))

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Mathematica [A]  time = 0.106789, size = 111, normalized size = 0.9 \[ \frac{2 \sqrt{c} \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6} \left (4 c \left (5 a+2 c x^6\right )-3 b^2+8 b c x^3\right )+3 \left (b^2-4 a c\right )^2 \log \left (2 \sqrt{c} \sqrt{a+b x^3+c x^6}+b+2 c x^3\right )}{384 c^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*(a + b*x^3 + c*x^6)^(3/2),x]

[Out]

(2*Sqrt[c]*(b + 2*c*x^3)*Sqrt[a + b*x^3 + c*x^6]*(-3*b^2 + 8*b*c*x^3 + 4*c*(5*a
+ 2*c*x^6)) + 3*(b^2 - 4*a*c)^2*Log[b + 2*c*x^3 + 2*Sqrt[c]*Sqrt[a + b*x^3 + c*x
^6]])/(384*c^(5/2))

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Maple [F]  time = 0.028, size = 0, normalized size = 0. \[ \int{x}^{2} \left ( c{x}^{6}+b{x}^{3}+a \right ) ^{{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(c*x^6+b*x^3+a)^(3/2),x)

[Out]

int(x^2*(c*x^6+b*x^3+a)^(3/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)^(3/2)*x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.294575, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (16 \, c^{3} x^{9} + 24 \, b c^{2} x^{6} + 2 \,{\left (b^{2} c + 20 \, a c^{2}\right )} x^{3} - 3 \, b^{3} + 20 \, a b c\right )} \sqrt{c x^{6} + b x^{3} + a} \sqrt{c} + 3 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \log \left (-4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (2 \, c^{2} x^{3} + b c\right )} -{\left (8 \, c^{2} x^{6} + 8 \, b c x^{3} + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{768 \, c^{\frac{5}{2}}}, \frac{2 \,{\left (16 \, c^{3} x^{9} + 24 \, b c^{2} x^{6} + 2 \,{\left (b^{2} c + 20 \, a c^{2}\right )} x^{3} - 3 \, b^{3} + 20 \, a b c\right )} \sqrt{c x^{6} + b x^{3} + a} \sqrt{-c} + 3 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \arctan \left (\frac{{\left (2 \, c x^{3} + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{6} + b x^{3} + a} c}\right )}{384 \, \sqrt{-c} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)^(3/2)*x^2,x, algorithm="fricas")

[Out]

[1/768*(4*(16*c^3*x^9 + 24*b*c^2*x^6 + 2*(b^2*c + 20*a*c^2)*x^3 - 3*b^3 + 20*a*b
*c)*sqrt(c*x^6 + b*x^3 + a)*sqrt(c) + 3*(b^4 - 8*a*b^2*c + 16*a^2*c^2)*log(-4*sq
rt(c*x^6 + b*x^3 + a)*(2*c^2*x^3 + b*c) - (8*c^2*x^6 + 8*b*c*x^3 + b^2 + 4*a*c)*
sqrt(c)))/c^(5/2), 1/384*(2*(16*c^3*x^9 + 24*b*c^2*x^6 + 2*(b^2*c + 20*a*c^2)*x^
3 - 3*b^3 + 20*a*b*c)*sqrt(c*x^6 + b*x^3 + a)*sqrt(-c) + 3*(b^4 - 8*a*b^2*c + 16
*a^2*c^2)*arctan(1/2*(2*c*x^3 + b)*sqrt(-c)/(sqrt(c*x^6 + b*x^3 + a)*c)))/(sqrt(
-c)*c^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{2} \left (a + b x^{3} + c x^{6}\right )^{\frac{3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(c*x**6+b*x**3+a)**(3/2),x)

[Out]

Integral(x**2*(a + b*x**3 + c*x**6)**(3/2), x)

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GIAC/XCAS [A]  time = 0.290269, size = 182, normalized size = 1.47 \[ \frac{1}{192} \, \sqrt{c x^{6} + b x^{3} + a}{\left (2 \,{\left (4 \,{\left (2 \, c x^{3} + 3 \, b\right )} x^{3} + \frac{b^{2} c^{2} + 20 \, a c^{3}}{c^{3}}\right )} x^{3} - \frac{3 \, b^{3} c - 20 \, a b c^{2}}{c^{3}}\right )} - \frac{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x^{3} - \sqrt{c x^{6} + b x^{3} + a}\right )} \sqrt{c} - b \right |}\right )}{128 \, c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)^(3/2)*x^2,x, algorithm="giac")

[Out]

1/192*sqrt(c*x^6 + b*x^3 + a)*(2*(4*(2*c*x^3 + 3*b)*x^3 + (b^2*c^2 + 20*a*c^3)/c
^3)*x^3 - (3*b^3*c - 20*a*b*c^2)/c^3) - 1/128*(b^4 - 8*a*b^2*c + 16*a^2*c^2)*ln(
abs(-2*(sqrt(c)*x^3 - sqrt(c*x^6 + b*x^3 + a))*sqrt(c) - b))/c^(5/2)