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

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

[Out]

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

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Rubi [A]  time = 0.0365055, 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} \[ \frac{2 B \sqrt{a+b x^3}}{3 b^2}-\frac{2 (A b-a B)}{3 b^2 \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0221462, size = 33, normalized size = 0.72 \[ \frac{2 \left (2 a B-A b+b B x^3\right )}{3 b^2 \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.006, size = 30, normalized size = 0.7 \begin{align*} -{\frac{-2\,bB{x}^{3}+2\,Ab-4\,Ba}{3\,{b}^{2}}{\frac{1}{\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)^(3/2),x)

[Out]

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

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Maxima [A]  time = 0.942499, size = 63, normalized size = 1.37 \begin{align*} \frac{2}{3} \, B{\left (\frac{\sqrt{b x^{3} + a}}{b^{2}} + \frac{a}{\sqrt{b x^{3} + a} b^{2}}\right )} - \frac{2 \, A}{3 \, \sqrt{b x^{3} + a} 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)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.75286, size = 85, normalized size = 1.85 \begin{align*} \frac{2 \,{\left (B b x^{3} + 2 \, B a - A b\right )} \sqrt{b x^{3} + a}}{3 \,{\left (b^{3} x^{3} + a b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.884662, size = 75, normalized size = 1.63 \begin{align*} \begin{cases} - \frac{2 A}{3 b \sqrt{a + b x^{3}}} + \frac{4 B a}{3 b^{2} \sqrt{a + b x^{3}}} + \frac{2 B x^{3}}{3 b \sqrt{a + b x^{3}}} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{3}}{3} + \frac{B x^{6}}{6}}{a^{\frac{3}{2}}} & \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)**(3/2),x)

[Out]

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

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Giac [A]  time = 1.1392, size = 47, normalized size = 1.02 \begin{align*} \frac{2 \,{\left (\sqrt{b x^{3} + a} B + \frac{B a - A b}{\sqrt{b x^{3} + a}}\right )}}{3 \, 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)^(3/2),x, algorithm="giac")

[Out]

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