3.246 \(\int \frac{x^2 (A+B x^3)}{(a+b x^3)^{5/2}} \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0359458, 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} \[ -\frac{2 (A b-a B)}{9 b^2 \left (a+b x^3\right )^{3/2}}-\frac{2 B}{3 b^2 \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

[In]

Int[(x^2*(A + B*x^3))/(a + b*x^3)^(5/2),x]

[Out]

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

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 )^{5/2}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{A+B x}{(a+b x)^{5/2}} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{A b-a B}{b (a+b x)^{5/2}}+\frac{B}{b (a+b x)^{3/2}}\right ) \, dx,x,x^3\right )\\ &=-\frac{2 (A b-a B)}{9 b^2 \left (a+b x^3\right )^{3/2}}-\frac{2 B}{3 b^2 \sqrt{a+b x^3}}\\ \end{align*}

Mathematica [A]  time = 0.0228209, size = 33, normalized size = 0.72 \[ -\frac{2 \left (2 a B+A b+3 b B x^3\right )}{9 b^2 \left (a+b x^3\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^2*(A + B*x^3))/(a + b*x^3)^(5/2),x]

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(B*x^3+A)/(b*x^3+a)^(5/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.93893, size = 66, normalized size = 1.43 \begin{align*} -\frac{2}{9} \, B{\left (\frac{3}{\sqrt{b x^{3} + a} b^{2}} - \frac{a}{{\left (b x^{3} + a\right )}^{\frac{3}{2}} b^{2}}\right )} - \frac{2 \, A}{9 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} 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)^(5/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.68313, size = 111, normalized size = 2.41 \begin{align*} -\frac{2 \,{\left (3 \, B b x^{3} + 2 \, B a + A b\right )} \sqrt{b x^{3} + a}}{9 \,{\left (b^{4} x^{6} + 2 \, a b^{3} x^{3} + a^{2} 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)^(5/2),x, algorithm="fricas")

[Out]

-2/9*(3*B*b*x^3 + 2*B*a + A*b)*sqrt(b*x^3 + a)/(b^4*x^6 + 2*a*b^3*x^3 + a^2*b^2)

________________________________________________________________________________________

Sympy [A]  time = 1.16649, size = 144, normalized size = 3.13 \begin{align*} \begin{cases} - \frac{2 A b}{9 a b^{2} \sqrt{a + b x^{3}} + 9 b^{3} x^{3} \sqrt{a + b x^{3}}} - \frac{4 B a}{9 a b^{2} \sqrt{a + b x^{3}} + 9 b^{3} x^{3} \sqrt{a + b x^{3}}} - \frac{6 B b x^{3}}{9 a b^{2} \sqrt{a + b x^{3}} + 9 b^{3} x^{3} \sqrt{a + b x^{3}}} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{3}}{3} + \frac{B x^{6}}{6}}{a^{\frac{5}{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)**(5/2),x)

[Out]

Piecewise((-2*A*b/(9*a*b**2*sqrt(a + b*x**3) + 9*b**3*x**3*sqrt(a + b*x**3)) - 4*B*a/(9*a*b**2*sqrt(a + b*x**3
) + 9*b**3*x**3*sqrt(a + b*x**3)) - 6*B*b*x**3/(9*a*b**2*sqrt(a + b*x**3) + 9*b**3*x**3*sqrt(a + b*x**3)), Ne(
b, 0)), ((A*x**3/3 + B*x**6/6)/a**(5/2), True))

________________________________________________________________________________________

Giac [A]  time = 1.14252, size = 43, normalized size = 0.93 \begin{align*} -\frac{2 \,{\left (3 \,{\left (b x^{3} + a\right )} B - B a + A b\right )}}{9 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} 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)^(5/2),x, algorithm="giac")

[Out]

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