3.4.42 \(\int \frac {x^5}{(a+b x^3)^3} \, dx\) [342]

Optimal. Leaf size=19 \[ \frac {x^6}{6 a \left (a+b x^3\right )^2} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {270} \begin {gather*} \frac {x^6}{6 a \left (a+b x^3\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5/(a + b*x^3)^3,x]

[Out]

x^6/(6*a*(a + b*x^3)^2)

Rule 270

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

Rubi steps

\begin {align*} \int \frac {x^5}{\left (a+b x^3\right )^3} \, dx &=\frac {x^6}{6 a \left (a+b x^3\right )^2}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 24, normalized size = 1.26 \begin {gather*} -\frac {a+2 b x^3}{6 b^2 \left (a+b x^3\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5/(a + b*x^3)^3,x]

[Out]

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

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Maple [A]
time = 0.12, size = 31, normalized size = 1.63

method	result	size
gosper	\(-\frac {2 b \,x^{3}+a}{6 b^{2} \left (b \,x^{3}+a \right )^{2}}\)	\(23\)
norman	\(\frac {-\frac {x^{3}}{3 b}-\frac {a}{6 b^{2}}}{\left (b \,x^{3}+a \right )^{2}}\)	\(26\)
risch	\(\frac {-\frac {x^{3}}{3 b}-\frac {a}{6 b^{2}}}{\left (b \,x^{3}+a \right )^{2}}\)	\(26\)
default	\(-\frac {1}{3 b^{2} \left (b \,x^{3}+a \right )}+\frac {a}{6 b^{2} \left (b \,x^{3}+a \right )^{2}}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(b*x^3+a)^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (17) = 34\).
time = 0.29, size = 36, normalized size = 1.89 \begin {gather*} -\frac {2 \, b x^{3} + a}{6 \, {\left (b^{4} x^{6} + 2 \, a b^{3} x^{3} + a^{2} b^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (17) = 34\).
time = 0.35, size = 36, normalized size = 1.89 \begin {gather*} -\frac {2 \, b x^{3} + a}{6 \, {\left (b^{4} x^{6} + 2 \, a b^{3} x^{3} + a^{2} b^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (14) = 28\).
time = 0.15, size = 36, normalized size = 1.89 \begin {gather*} \frac {- a - 2 b x^{3}}{6 a^{2} b^{2} + 12 a b^{3} x^{3} + 6 b^{4} x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-a - 2*b*x**3)/(6*a**2*b**2 + 12*a*b**3*x**3 + 6*b**4*x**6)

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Giac [A]
time = 1.67, size = 22, normalized size = 1.16 \begin {gather*} -\frac {2 \, b x^{3} + a}{6 \, {\left (b x^{3} + a\right )}^{2} b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.03, size = 37, normalized size = 1.95 \begin {gather*} -\frac {\frac {a}{6\,b^2}+\frac {x^3}{3\,b}}{a^2+2\,a\,b\,x^3+b^2\,x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(a + b*x^3)^3,x)

[Out]

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

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