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

Optimal. Leaf size=73 \[ \frac{2 \sqrt{a+b x^3} (A b-2 a B)}{3 b^3}+\frac{2 a (A b-a B)}{3 b^3 \sqrt{a+b x^3}}+\frac{2 B \left (a+b x^3\right )^{3/2}}{9 b^3} \]

[Out]

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

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Rubi [A]  time = 0.0548598, antiderivative size = 73, 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 = {446, 77} \[ \frac{2 \sqrt{a+b x^3} (A b-2 a B)}{3 b^3}+\frac{2 a (A b-a B)}{3 b^3 \sqrt{a+b x^3}}+\frac{2 B \left (a+b x^3\right )^{3/2}}{9 b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(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] && IntegerQ[Simplify[(m + 1)/n]]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

Mathematica [A]  time = 0.033604, size = 55, normalized size = 0.75 \[ \frac{2 \left (-8 a^2 B+a \left (6 A b-4 b B x^3\right )+b^2 x^3 \left (3 A+B x^3\right )\right )}{9 b^3 \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-8*a^2*B + b^2*x^3*(3*A + B*x^3) + a*(6*A*b - 4*b*B*x^3)))/(9*b^3*Sqrt[a + b*x^3])

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Maple [A]  time = 0.007, size = 52, normalized size = 0.7 \begin{align*}{\frac{2\,{b}^{2}B{x}^{6}+6\,A{x}^{3}{b}^{2}-8\,B{x}^{3}ab+12\,abA-16\,{a}^{2}B}{9\,{b}^{3}}{\frac{1}{\sqrt{b{x}^{3}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9/(b*x^3+a)^(1/2)*(B*b^2*x^6+3*A*b^2*x^3-4*B*a*b*x^3+6*A*a*b-8*B*a^2)/b^3

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Maxima [A]  time = 0.936323, size = 109, normalized size = 1.49 \begin{align*} \frac{2}{9} \, B{\left (\frac{{\left (b x^{3} + a\right )}^{\frac{3}{2}}}{b^{3}} - \frac{6 \, \sqrt{b x^{3} + a} a}{b^{3}} - \frac{3 \, a^{2}}{\sqrt{b x^{3} + a} b^{3}}\right )} + \frac{2}{3} \, A{\left (\frac{\sqrt{b x^{3} + a}}{b^{2}} + \frac{a}{\sqrt{b x^{3} + a} b^{2}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(B*x^3+A)/(b*x^3+a)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.73956, size = 131, normalized size = 1.79 \begin{align*} \frac{2 \,{\left (B b^{2} x^{6} -{\left (4 \, B a b - 3 \, A b^{2}\right )} x^{3} - 8 \, B a^{2} + 6 \, A a b\right )} \sqrt{b x^{3} + a}}{9 \,{\left (b^{4} x^{3} + a b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(B*x^3+A)/(b*x^3+a)^(3/2),x, algorithm="fricas")

[Out]

2/9*(B*b^2*x^6 - (4*B*a*b - 3*A*b^2)*x^3 - 8*B*a^2 + 6*A*a*b)*sqrt(b*x^3 + a)/(b^4*x^3 + a*b^3)

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Sympy [A]  time = 1.8081, size = 124, normalized size = 1.7 \begin{align*} \begin{cases} \frac{4 A a}{3 b^{2} \sqrt{a + b x^{3}}} + \frac{2 A x^{3}}{3 b \sqrt{a + b x^{3}}} - \frac{16 B a^{2}}{9 b^{3} \sqrt{a + b x^{3}}} - \frac{8 B a x^{3}}{9 b^{2} \sqrt{a + b x^{3}}} + \frac{2 B x^{6}}{9 b \sqrt{a + b x^{3}}} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{6}}{6} + \frac{B x^{9}}{9}}{a^{\frac{3}{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((4*A*a/(3*b**2*sqrt(a + b*x**3)) + 2*A*x**3/(3*b*sqrt(a + b*x**3)) - 16*B*a**2/(9*b**3*sqrt(a + b*x*
*3)) - 8*B*a*x**3/(9*b**2*sqrt(a + b*x**3)) + 2*B*x**6/(9*b*sqrt(a + b*x**3)), Ne(b, 0)), ((A*x**6/6 + B*x**9/
9)/a**(3/2), True))

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Giac [A]  time = 1.12547, size = 88, normalized size = 1.21 \begin{align*} \frac{2 \,{\left ({\left (b x^{3} + a\right )}^{\frac{3}{2}} B - 6 \, \sqrt{b x^{3} + a} B a + 3 \, \sqrt{b x^{3} + a} A b - \frac{3 \,{\left (B a^{2} - A a b\right )}}{\sqrt{b x^{3} + a}}\right )}}{9 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(B*x^3+A)/(b*x^3+a)^(3/2),x, algorithm="giac")

[Out]

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