3.6.19 \(\int \frac {\sqrt [3]{a+b x^3}}{x^5} \, dx\) [519]

Optimal. Leaf size=21 \[ -\frac {\left (a+b x^3\right )^{4/3}}{4 a x^4} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {270} \begin {gather*} -\frac {\left (a+b x^3\right )^{4/3}}{4 a x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^(1/3)/x^5,x]

[Out]

-1/4*(a + b*x^3)^(4/3)/(a*x^4)

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 {\sqrt [3]{a+b x^3}}{x^5} \, dx &=-\frac {\left (a+b x^3\right )^{4/3}}{4 a x^4}\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 21, normalized size = 1.00 \begin {gather*} -\frac {\left (a+b x^3\right )^{4/3}}{4 a x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^(1/3)/x^5,x]

[Out]

-1/4*(a + b*x^3)^(4/3)/(a*x^4)

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Maple [A]
time = 0.13, size = 18, normalized size = 0.86

method	result	size
gosper	\(-\frac {\left (b \,x^{3}+a \right )^{\frac {4}{3}}}{4 a \,x^{4}}\)	\(18\)
trager	\(-\frac {\left (b \,x^{3}+a \right )^{\frac {4}{3}}}{4 a \,x^{4}}\)	\(18\)
risch	\(-\frac {\left (b \,x^{3}+a \right )^{\frac {4}{3}}}{4 a \,x^{4}}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^(1/3)/x^5,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 17, normalized size = 0.81 \begin {gather*} -\frac {{\left (b x^{3} + a\right )}^{\frac {4}{3}}}{4 \, a x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(1/3)/x^5,x, algorithm="maxima")

[Out]

-1/4*(b*x^3 + a)^(4/3)/(a*x^4)

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Fricas [A]
time = 0.36, size = 17, normalized size = 0.81 \begin {gather*} -\frac {{\left (b x^{3} + a\right )}^{\frac {4}{3}}}{4 \, a x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(1/3)/x^5,x, algorithm="fricas")

[Out]

-1/4*(b*x^3 + a)^(4/3)/(a*x^4)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (17) = 34\).
time = 0.38, size = 68, normalized size = 3.24 \begin {gather*} \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b x^{3}} + 1} \Gamma \left (- \frac {4}{3}\right )}{3 x^{3} \Gamma \left (- \frac {1}{3}\right )} + \frac {b^{\frac {4}{3}} \sqrt [3]{\frac {a}{b x^{3}} + 1} \Gamma \left (- \frac {4}{3}\right )}{3 a \Gamma \left (- \frac {1}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**(1/3)/x**5,x)

[Out]

b**(1/3)*(a/(b*x**3) + 1)**(1/3)*gamma(-4/3)/(3*x**3*gamma(-1/3)) + b**(4/3)*(a/(b*x**3) + 1)**(1/3)*gamma(-4/
3)/(3*a*gamma(-1/3))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(1/3)/x^5,x, algorithm="giac")

[Out]

integrate((b*x^3 + a)^(1/3)/x^5, x)

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Mupad [B]
time = 1.11, size = 17, normalized size = 0.81 \begin {gather*} -\frac {{\left (b\,x^3+a\right )}^{4/3}}{4\,a\,x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^(1/3)/x^5,x)

[Out]

-(a + b*x^3)^(4/3)/(4*a*x^4)

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