3.2961 \(\int x^2 \sqrt{a+b \sqrt{c x^3}} \, dx\)

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 = 0.0352788, 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, Rules used = {369, 266, 43} \[ \frac{4 \left (a+b \sqrt{c x^3}\right )^{5/2}}{15 b^2 c}-\frac{4 a \left (a+b \sqrt{c x^3}\right )^{3/2}}{9 b^2 c} \]

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)

Rule 369

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbol] :> With[{k = Denominator[n]}, Su
bst[Int[(d*x)^m*(a + b*c^n*x^(n*q))^p, x], x^(1/k), (c*x^q)^(1/k)/(c^(1/k)*(x^(1/k))^(q - 1))]] /; FreeQ[{a, b
, c, d, m, p, q}, x] && FractionQ[n]

Rule 266

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x^2 \sqrt{a+b \sqrt{c x^3}} \, dx &=\operatorname{Subst}\left (\int x^2 \sqrt{a+b \sqrt{c} x^{3/2}} \, dx,\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=\operatorname{Subst}\left (\frac{2}{3} \operatorname{Subst}\left (\int x \sqrt{a+b \sqrt{c} x} \, dx,x,x^{3/2}\right ),\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=\operatorname{Subst}\left (\frac{2}{3} \operatorname{Subst}\left (\int \left (-\frac{a \sqrt{a+b \sqrt{c} x}}{b \sqrt{c}}+\frac{\left (a+b \sqrt{c} x\right )^{3/2}}{b \sqrt{c}}\right ) \, dx,x,x^{3/2}\right ),\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=-\frac{4 a \left (a+b \sqrt{c x^3}\right )^{3/2}}{9 b^2 c}+\frac{4 \left (a+b \sqrt{c x^3}\right )^{5/2}}{15 b^2 c}\\ \end{align*}

Mathematica [A]  time = 0.0326489, size = 43, normalized size = 0.77 \[ \frac{4 \left (a+b \sqrt{c x^3}\right )^{3/2} \left (3 b \sqrt{c x^3}-2 a\right )}{45 b^2 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.182, size = 65, normalized size = 1.2 \begin{align*}{\frac{4}{45\,{b}^{2}c}\sqrt{a+b\sqrt{c{x}^{3}}} \left ( 3\,{x}^{3}c\sqrt{c{x}^{3}}{b}^{2}+a{x}^{3}cb-2\,{a}^{2}\sqrt{c{x}^{3}} \right ){\frac{1}{\sqrt{c{x}^{3}}}}} \end{align*}

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

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Maxima [A]  time = 0.947457, size = 58, normalized size = 1.04 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*(c*x^3)^(1/2))^(1/2),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 = 1.91551, size = 105, normalized size = 1.88 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*(c*x^3)^(1/2))^(1/2),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. \begin{align*} \int x^{2} \sqrt{a + b \sqrt{c x^{3}}}\, dx \end{align*}

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 [A]  time = 1.18973, size = 89, normalized size = 1.59 \begin{align*} \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}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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