3.5.16 \(\int \frac {1}{(a+b x)^{4/3}} \, dx\)

Optimal. Leaf size=14 \[ -\frac {3}{b \sqrt [3]{a+b x}} \]

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Rubi [A]  time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {32} \begin {gather*} -\frac {3}{b \sqrt [3]{a+b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(-4/3),x]

[Out]

-3/(b*(a + b*x)^(1/3))

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} -\frac {3}{b \sqrt [3]{a+b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(-4/3),x]

[Out]

-3/(b*(a + b*x)^(1/3))

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IntegrateAlgebraic [A]  time = 0.01, size = 14, normalized size = 1.00 \begin {gather*} -\frac {3}{b \sqrt [3]{a+b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x)^(-4/3),x]

[Out]

-3/(b*(a + b*x)^(1/3))

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fricas [A]  time = 1.05, size = 20, normalized size = 1.43 \begin {gather*} -\frac {3 \, {\left (b x + a\right )}^{\frac {2}{3}}}{b^{2} x + a b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*(b*x + a)^(2/3)/(b^2*x + a*b)

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giac [A]  time = 0.76, size = 12, normalized size = 0.86 \begin {gather*} -\frac {3}{{\left (b x + a\right )}^{\frac {1}{3}} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/((b*x + a)^(1/3)*b)

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maple [A]  time = 0.00, size = 13, normalized size = 0.93 \begin {gather*} -\frac {3}{\left (b x +a \right )^{\frac {1}{3}} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.35, size = 12, normalized size = 0.86 \begin {gather*} -\frac {3}{{\left (b x + a\right )}^{\frac {1}{3}} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/((b*x + a)^(1/3)*b)

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mupad [B]  time = 0.02, size = 12, normalized size = 0.86 \begin {gather*} -\frac {3}{b\,{\left (a+b\,x\right )}^{1/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*x)^(4/3),x)

[Out]

-3/(b*(a + b*x)^(1/3))

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sympy [A]  time = 0.07, size = 12, normalized size = 0.86 \begin {gather*} - \frac {3}{b \sqrt [3]{a + b x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/(b*(a + b*x)**(1/3))

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