3.422 \(\int \frac {x^8}{(a+b x^3)^{3/2}} \, dx\)

Optimal. Leaf size=59 \[ -\frac {2 a^2}{3 b^3 \sqrt {a+b x^3}}-\frac {4 a \sqrt {a+b x^3}}{3 b^3}+\frac {2 \left (a+b x^3\right )^{3/2}}{9 b^3} \]

[Out]

2/9*(b*x^3+a)^(3/2)/b^3-2/3*a^2/b^3/(b*x^3+a)^(1/2)-4/3*a*(b*x^3+a)^(1/2)/b^3

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Rubi [A]  time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 43} \[ -\frac {2 a^2}{3 b^3 \sqrt {a+b x^3}}-\frac {4 a \sqrt {a+b x^3}}{3 b^3}+\frac {2 \left (a+b x^3\right )^{3/2}}{9 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^8/(a + b*x^3)^(3/2),x]

[Out]

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

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])

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]]

Rubi steps

\begin {align*} \int \frac {x^8}{\left (a+b x^3\right )^{3/2}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^2}{(a+b x)^{3/2}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {a^2}{b^2 (a+b x)^{3/2}}-\frac {2 a}{b^2 \sqrt {a+b x}}+\frac {\sqrt {a+b x}}{b^2}\right ) \, dx,x,x^3\right )\\ &=-\frac {2 a^2}{3 b^3 \sqrt {a+b x^3}}-\frac {4 a \sqrt {a+b x^3}}{3 b^3}+\frac {2 \left (a+b x^3\right )^{3/2}}{9 b^3}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 38, normalized size = 0.64 \[ \frac {2 \left (-8 a^2-4 a b x^3+b^2 x^6\right )}{9 b^3 \sqrt {a+b x^3}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^8/(a + b*x^3)^(3/2),x]

[Out]

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

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fricas [A]  time = 0.85, size = 46, normalized size = 0.78 \[ \frac {2 \, {\left (b^{2} x^{6} - 4 \, a b x^{3} - 8 \, a^{2}\right )} \sqrt {b x^{3} + a}}{9 \, {\left (b^{4} x^{3} + a b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(b*x^3+a)^(3/2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.16, size = 52, normalized size = 0.88 \[ -\frac {2 \, a^{2}}{3 \, \sqrt {b x^{3} + a} b^{3}} + \frac {2 \, {\left ({\left (b x^{3} + a\right )}^{\frac {3}{2}} b^{6} - 6 \, \sqrt {b x^{3} + a} a b^{6}\right )}}{9 \, b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(b*x^3+a)^(3/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.01, size = 36, normalized size = 0.61 \[ -\frac {2 \left (-b^{2} x^{6}+4 a b \,x^{3}+8 a^{2}\right )}{9 \sqrt {b \,x^{3}+a}\, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8/(b*x^3+a)^(3/2),x)

[Out]

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

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maxima [A]  time = 1.32, size = 47, normalized size = 0.80 \[ \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}}}{9 \, b^{3}} - \frac {4 \, \sqrt {b x^{3} + a} a}{3 \, b^{3}} - \frac {2 \, a^{2}}{3 \, \sqrt {b x^{3} + a} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(b*x^3+a)^(3/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.18, size = 41, normalized size = 0.69 \[ -\frac {12\,a\,\left (b\,x^3+a\right )-2\,{\left (b\,x^3+a\right )}^2+6\,a^2}{9\,b^3\,\sqrt {b\,x^3+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8/(a + b*x^3)^(3/2),x)

[Out]

-(12*a*(a + b*x^3) - 2*(a + b*x^3)^2 + 6*a^2)/(9*b^3*(a + b*x^3)^(1/2))

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sympy [A]  time = 4.92, size = 70, normalized size = 1.19 \[ \begin {cases} - \frac {16 a^{2}}{9 b^{3} \sqrt {a + b x^{3}}} - \frac {8 a x^{3}}{9 b^{2} \sqrt {a + b x^{3}}} + \frac {2 x^{6}}{9 b \sqrt {a + b x^{3}}} & \text {for}\: b \neq 0 \\\frac {x^{9}}{9 a^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8/(b*x**3+a)**(3/2),x)

[Out]

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

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