3.2954 \(\int x^2 \sqrt{a+b \sqrt{c x^3}} \, dx\)

Optimal. Leaf size=56 \[ \frac{4 \left (a+b \sqrt{c x^3}\right )^{5/2}}{15 b^2 c}-\frac{4 a \left (a+b \sqrt{c x^3}\right )^{3/2}}{9 b^2 c} \]

[Out]

(-4*a*(a + b*Sqrt[c*x^3])^(3/2))/(9*b^2*c) + (4*(a + b*Sqrt[c*x^3])^(5/2))/(15*b
^2*c)

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Rubi [A]  time = 0.0859186, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{4 \left (a+b \sqrt{c x^3}\right )^{5/2}}{15 b^2 c}-\frac{4 a \left (a+b \sqrt{c x^3}\right )^{3/2}}{9 b^2 c} \]

Antiderivative was successfully verified.

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

[Out]

(-4*a*(a + b*Sqrt[c*x^3])^(3/2))/(9*b^2*c) + (4*(a + b*Sqrt[c*x^3])^(5/2))/(15*b
^2*c)

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Rubi in Sympy [A]  time = 8.4167, size = 48, normalized size = 0.86 \[ - \frac{4 a \left (a + b \sqrt{c x^{3}}\right )^{\frac{3}{2}}}{9 b^{2} c} + \frac{4 \left (a + b \sqrt{c x^{3}}\right )^{\frac{5}{2}}}{15 b^{2} c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*a*(a + b*sqrt(c*x**3))**(3/2)/(9*b**2*c) + 4*(a + b*sqrt(c*x**3))**(5/2)/(15*
b**2*c)

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Mathematica [A]  time = 0.0294877, size = 54, normalized size = 0.96 \[ \frac{4 \sqrt{a+b \sqrt{c x^3}} \left (-2 a^2+a b \sqrt{c x^3}+3 b^2 c x^3\right )}{45 b^2 c} \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[a + b*Sqrt[c*x^3]]*(-2*a^2 + 3*b^2*c*x^3 + a*b*Sqrt[c*x^3]))/(45*b^2*c)

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Maple [A]  time = 0.171, size = 65, normalized size = 1.2 \[{\frac{4}{45\,{b}^{2}c}\sqrt{a+b\sqrt{c{x}^{3}}} \left ( 3\,c{x}^{3}{b}^{2}\sqrt{c{x}^{3}}+a{x}^{3}cb-2\,{a}^{2}\sqrt{c{x}^{3}} \right ){\frac{1}{\sqrt{c{x}^{3}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/45/c*(a+b*(c*x^3)^(1/2))^(1/2)*(3*c*x^3*b^2*(c*x^3)^(1/2)+a*x^3*c*b-2*a^2*(c*x
^3)^(1/2))/b^2/(c*x^3)^(1/2)

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Maxima [A]  time = 1.34639, size = 58, normalized size = 1.04 \[ \frac{4 \,{\left (\frac{3 \,{\left (\sqrt{c x^{3}} b + a\right )}^{\frac{5}{2}}}{b^{2}} - \frac{5 \,{\left (\sqrt{c x^{3}} b + a\right )}^{\frac{3}{2}} a}{b^{2}}\right )}}{45 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/45*(3*(sqrt(c*x^3)*b + a)^(5/2)/b^2 - 5*(sqrt(c*x^3)*b + a)^(3/2)*a/b^2)/c

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Fricas [A]  time = 0.300414, size = 62, normalized size = 1.11 \[ \frac{4 \,{\left (3 \, b^{2} c x^{3} + \sqrt{c x^{3}} a b - 2 \, a^{2}\right )} \sqrt{\sqrt{c x^{3}} b + a}}{45 \, b^{2} c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/45*(3*b^2*c*x^3 + sqrt(c*x^3)*a*b - 2*a^2)*sqrt(sqrt(c*x^3)*b + a)/(b^2*c)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.221095, size = 89, normalized size = 1.59 \[ \frac{4 \,{\left (\frac{2 \, \sqrt{a c} a^{2}}{b^{2}} - \frac{5 \,{\left (\sqrt{c x} b c x + a c\right )}^{\frac{3}{2}} a c - 3 \,{\left (\sqrt{c x} b c x + a c\right )}^{\frac{5}{2}}}{b^{2} c^{2}}\right )}{\left | c \right |}}{45 \, c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/45*(2*sqrt(a*c)*a^2/b^2 - (5*(sqrt(c*x)*b*c*x + a*c)^(3/2)*a*c - 3*(sqrt(c*x)*
b*c*x + a*c)^(5/2))/(b^2*c^2))*abs(c)/c^(5/2)