3.30.44 \(\int x^2 \sqrt {a+b (c x^2)^{3/2}} \, dx\) [2944]

Optimal. Leaf size=36 \[ \frac {2 x^3 \left (a+b \left (c x^2\right )^{3/2}\right )^{3/2}}{9 b \left (c x^2\right )^{3/2}} \]

[Out]

2/9*x^3*(a+b*(c*x^2)^(3/2))^(3/2)/b/(c*x^2)^(3/2)

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Rubi [A]
time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {375, 267} \begin {gather*} \frac {2 x^3 \left (a+b \left (c x^2\right )^{3/2}\right )^{3/2}}{9 b \left (c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*Sqrt[a + b*(c*x^2)^(3/2)],x]

[Out]

(2*x^3*(a + b*(c*x^2)^(3/2))^(3/2))/(9*b*(c*x^2)^(3/2))

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 375

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbol] :> Dist[(d*x)^(m + 1)/(d*((c*x^q
)^(1/q))^(m + 1)), Subst[Int[x^m*(a + b*x^(n*q))^p, x], x, (c*x^q)^(1/q)], x] /; FreeQ[{a, b, c, d, m, n, p, q
}, x] && IntegerQ[n*q] && NeQ[x, (c*x^q)^(1/q)]

Rubi steps

\begin {align*} \int x^2 \sqrt {a+b \left (c x^2\right )^{3/2}} \, dx &=\frac {x^3 \text {Subst}\left (\int x^2 \sqrt {a+b x^3} \, dx,x,\sqrt {c x^2}\right )}{\left (c x^2\right )^{3/2}}\\ &=\frac {2 x^3 \left (a+b \left (c x^2\right )^{3/2}\right )^{3/2}}{9 b \left (c x^2\right )^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 1.30, size = 36, normalized size = 1.00 \begin {gather*} \frac {2 x^3 \left (a+b \left (c x^2\right )^{3/2}\right )^{3/2}}{9 b \left (c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^3*(a + b*(c*x^2)^(3/2))^(3/2))/(9*b*(c*x^2)^(3/2))

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Maple [A]
time = 0.24, size = 29, normalized size = 0.81

method	result	size
default	\(\frac {2 x^{3} \left (a +b \left (c \,x^{2}\right )^{\frac {3}{2}}\right )^{\frac {3}{2}}}{9 b \left (c \,x^{2}\right )^{\frac {3}{2}}}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a+b*(c*x^2)^(3/2))^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/9*x^3*(a+b*(c*x^2)^(3/2))^(3/2)/b/(c*x^2)^(3/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*(c*x^2)^(3/2))^(1/2),x, algorithm="maxima")

[Out]

(c - sqrt(c))*integrate(sqrt(b*c^(3/2)*x^3 + a)*x^2, x)/(c + 1) + 1/3*(b*c^(3/2)*x^3 + a)^(3/2)/((c^2 + c)*b*s
qrt(c))

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Fricas [A]
time = 0.49, size = 46, normalized size = 1.28 \begin {gather*} \frac {2 \, {\left (b c^{2} x^{4} + \sqrt {c x^{2}} a\right )} \sqrt {\sqrt {c x^{2}} b c x^{2} + a}}{9 \, b c^{2} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*(c*x^2)^(3/2))^(1/2),x, algorithm="fricas")

[Out]

2/9*(b*c^2*x^4 + sqrt(c*x^2)*a)*sqrt(sqrt(c*x^2)*b*c*x^2 + a)/(b*c^2*x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \sqrt {a + b \left (c x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.21, size = 20, normalized size = 0.56 \begin {gather*} \frac {2 \, {\left (b c^{\frac {3}{2}} x^{3} + a\right )}^{\frac {3}{2}}}{9 \, b c^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*(c*x^2)^(3/2))^(1/2),x, algorithm="giac")

[Out]

2/9*(b*c^(3/2)*x^3 + a)^(3/2)/(b*c^(3/2))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int x^2\,\sqrt {a+b\,{\left (c\,x^2\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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