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

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

[Out]

-3/(b*(a + b*x)^(1/3))

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Rubi [A]  time = 0.0013637, antiderivative size = 14, normalized size of antiderivative = 1., 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} \[ -\frac{3}{b \sqrt [3]{a+b x}} \]

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*}

Mathematica [A]  time = 0.0122369, size = 14, normalized size = 1. \[ -\frac{3}{b \sqrt [3]{a+b x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(-4/3),x]

[Out]

-3/(b*(a + b*x)^(1/3))

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Maple [A]  time = 0.002, size = 13, normalized size = 0.9 \begin{align*} -3\,{\frac{1}{b\sqrt [3]{bx+a}}} \end{align*}

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 = 1.01713, size = 16, normalized size = 1.14 \begin{align*} -\frac{3}{{\left (b x + a\right )}^{\frac{1}{3}} b} \end{align*}

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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Fricas [A]  time = 1.44181, size = 46, normalized size = 3.29 \begin{align*} -\frac{3 \,{\left (b x + a\right )}^{\frac{2}{3}}}{b^{2} x + a b} \end{align*}

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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Sympy [A]  time = 0.107064, size = 12, normalized size = 0.86 \begin{align*} - \frac{3}{b \sqrt [3]{a + b x}} \end{align*}

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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Giac [A]  time = 1.20436, size = 16, normalized size = 1.14 \begin{align*} -\frac{3}{{\left (b x + a\right )}^{\frac{1}{3}} b} \end{align*}

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)