3.215 \(\int \frac{x^2 (A+B x^3)}{\sqrt{a+b x^3}} \, dx\)

Optimal. Leaf size=46 \[ \frac{2 \sqrt{a+b x^3} (A b-a B)}{3 b^2}+\frac{2 B \left (a+b x^3\right )^{3/2}}{9 b^2} \]

[Out]

(2*(A*b - a*B)*Sqrt[a + b*x^3])/(3*b^2) + (2*B*(a + b*x^3)^(3/2))/(9*b^2)

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Rubi [A]  time = 0.0378856, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {444, 43} \[ \frac{2 \sqrt{a+b x^3} (A b-a B)}{3 b^2}+\frac{2 B \left (a+b x^3\right )^{3/2}}{9 b^2} \]

Antiderivative was successfully verified.

[In]

Int[(x^2*(A + B*x^3))/Sqrt[a + b*x^3],x]

[Out]

(2*(A*b - a*B)*Sqrt[a + b*x^3])/(3*b^2) + (2*B*(a + b*x^3)^(3/2))/(9*b^2)

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

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 \frac{x^2 \left (A+B x^3\right )}{\sqrt{a+b x^3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{A+B x}{\sqrt{a+b x}} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{A b-a B}{b \sqrt{a+b x}}+\frac{B \sqrt{a+b x}}{b}\right ) \, dx,x,x^3\right )\\ &=\frac{2 (A b-a B) \sqrt{a+b x^3}}{3 b^2}+\frac{2 B \left (a+b x^3\right )^{3/2}}{9 b^2}\\ \end{align*}

Mathematica [A]  time = 0.0221966, size = 33, normalized size = 0.72 \[ \frac{2 \sqrt{a+b x^3} \left (-2 a B+3 A b+b B x^3\right )}{9 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[a + b*x^3]*(3*A*b - 2*a*B + b*B*x^3))/(9*b^2)

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Maple [A]  time = 0.006, size = 30, normalized size = 0.7 \begin{align*}{\frac{2\,bB{x}^{3}+6\,Ab-4\,Ba}{9\,{b}^{2}}\sqrt{b{x}^{3}+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*(b*x^3+a)^(1/2)*(B*b*x^3+3*A*b-2*B*a)/b^2

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Maxima [A]  time = 0.938191, size = 65, normalized size = 1.41 \begin{align*} \frac{2}{9} \, B{\left (\frac{{\left (b x^{3} + a\right )}^{\frac{3}{2}}}{b^{2}} - \frac{3 \, \sqrt{b x^{3} + a} a}{b^{2}}\right )} + \frac{2 \, \sqrt{b x^{3} + a} A}{3 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*B*((b*x^3 + a)^(3/2)/b^2 - 3*sqrt(b*x^3 + a)*a/b^2) + 2/3*sqrt(b*x^3 + a)*A/b

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Fricas [A]  time = 1.74681, size = 69, normalized size = 1.5 \begin{align*} \frac{2 \,{\left (B b x^{3} - 2 \, B a + 3 \, A b\right )} \sqrt{b x^{3} + a}}{9 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*(B*b*x^3 - 2*B*a + 3*A*b)*sqrt(b*x^3 + a)/b^2

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Sympy [A]  time = 0.768402, size = 75, normalized size = 1.63 \begin{align*} \begin{cases} \frac{2 A \sqrt{a + b x^{3}}}{3 b} - \frac{4 B a \sqrt{a + b x^{3}}}{9 b^{2}} + \frac{2 B x^{3} \sqrt{a + b x^{3}}}{9 b} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{3}}{3} + \frac{B x^{6}}{6}}{\sqrt{a}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2*A*sqrt(a + b*x**3)/(3*b) - 4*B*a*sqrt(a + b*x**3)/(9*b**2) + 2*B*x**3*sqrt(a + b*x**3)/(9*b), Ne(
b, 0)), ((A*x**3/3 + B*x**6/6)/sqrt(a), True))

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Giac [A]  time = 1.11092, size = 58, normalized size = 1.26 \begin{align*} \frac{2 \,{\left ({\left (b x^{3} + a\right )}^{\frac{3}{2}} B - 3 \, \sqrt{b x^{3} + a} B a + 3 \, \sqrt{b x^{3} + a} A b\right )}}{9 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*((b*x^3 + a)^(3/2)*B - 3*sqrt(b*x^3 + a)*B*a + 3*sqrt(b*x^3 + a)*A*b)/b^2