3.2.17 \(\int \frac {1}{\sqrt [3]{b x^4}} \, dx\) [117]

Optimal. Leaf size=12 \[ -\frac {3 x}{\sqrt [3]{b x^4}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {15, 30} \begin {gather*} -\frac {3 x}{\sqrt [3]{b x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x^4)^(-1/3),x]

[Out]

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

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[a^IntPart[m]*((a*x^n)^FracPart[m]/x^(n*FracPart[m])), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt [3]{b x^4}} \, dx &=\frac {x^{4/3} \int \frac {1}{x^{4/3}} \, dx}{\sqrt [3]{b x^4}}\\ &=-\frac {3 x}{\sqrt [3]{b x^4}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} -\frac {3 x}{\sqrt [3]{b x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*x^4)^(-1/3),x]

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.04, size = 11, normalized size = 0.92

method	result	size
gosper	\(-\frac {3 x}{\left (b \,x^{4}\right )^{\frac {1}{3}}}\)	\(11\)
risch	\(-\frac {3 x}{\left (b \,x^{4}\right )^{\frac {1}{3}}}\)	\(11\)
trager	\(-\frac {3 \left (b \,x^{4}\right )^{\frac {2}{3}}}{b \,x^{3}}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^4)^(1/3),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.29, size = 10, normalized size = 0.83 \begin {gather*} -\frac {3 \, x}{\left (b x^{4}\right )^{\frac {1}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^4)^(1/3),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.35, size = 15, normalized size = 1.25 \begin {gather*} -\frac {3 \, \left (b x^{4}\right )^{\frac {2}{3}}}{b x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^4)^(1/3),x, algorithm="fricas")

[Out]

-3*(b*x^4)^(2/3)/(b*x^3)

________________________________________________________________________________________

Sympy [A]
time = 0.16, size = 12, normalized size = 1.00 \begin {gather*} - \frac {3 x}{\sqrt [3]{b x^{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x**4)**(1/3),x)

[Out]

-3*x/(b*x**4)**(1/3)

________________________________________________________________________________________

Giac [A]
time = 1.52, size = 10, normalized size = 0.83 \begin {gather*} -\frac {3 \, x}{\left (b x^{4}\right )^{\frac {1}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^4)^(1/3),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 0.93, size = 13, normalized size = 1.08 \begin {gather*} -\frac {3\,{\left (x^8\right )}^{1/3}}{b^{1/3}\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^4)^(1/3),x)

[Out]

-(3*(x^8)^(1/3))/(b^(1/3)*x^3)

________________________________________________________________________________________