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

Optimal. Leaf size=32 \[ \frac{3 a}{b^2 \sqrt [3]{a+b x}}+\frac{3 (a+b x)^{2/3}}{2 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0085798, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {43} \[ \frac{3 a}{b^2 \sqrt [3]{a+b x}}+\frac{3 (a+b x)^{2/3}}{2 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*x)^(4/3),x]

[Out]

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x}{(a+b x)^{4/3}} \, dx &=\int \left (-\frac{a}{b (a+b x)^{4/3}}+\frac{1}{b \sqrt [3]{a+b x}}\right ) \, dx\\ &=\frac{3 a}{b^2 \sqrt [3]{a+b x}}+\frac{3 (a+b x)^{2/3}}{2 b^2}\\ \end{align*}

Mathematica [A]  time = 0.04836, size = 23, normalized size = 0.72 \[ \frac{3 (3 a+b x)}{2 b^2 \sqrt [3]{a+b x}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*x)^(4/3),x]

[Out]

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

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Maple [A]  time = 0.003, size = 20, normalized size = 0.6 \begin{align*}{\frac{3\,bx+9\,a}{2\,{b}^{2}}{\frac{1}{\sqrt [3]{bx+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02825, size = 35, normalized size = 1.09 \begin{align*} \frac{3 \,{\left (b x + a\right )}^{\frac{2}{3}}}{2 \, b^{2}} + \frac{3 \, a}{{\left (b x + a\right )}^{\frac{1}{3}} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.60112, size = 66, normalized size = 2.06 \begin{align*} \frac{3 \,{\left (b x + 3 \, a\right )}{\left (b x + a\right )}^{\frac{2}{3}}}{2 \,{\left (b^{3} x + a b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.04138, size = 41, normalized size = 1.28 \begin{align*} \begin{cases} \frac{9 a}{2 b^{2} \sqrt [3]{a + b x}} + \frac{3 x}{2 b \sqrt [3]{a + b x}} & \text{for}\: b \neq 0 \\\frac{x^{2}}{2 a^{\frac{4}{3}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((9*a/(2*b**2*(a + b*x)**(1/3)) + 3*x/(2*b*(a + b*x)**(1/3)), Ne(b, 0)), (x**2/(2*a**(4/3)), True))

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Giac [A]  time = 1.21717, size = 41, normalized size = 1.28 \begin{align*} \frac{3 \,{\left (\frac{{\left (b x + a\right )}^{\frac{2}{3}}}{b} + \frac{2 \, a}{{\left (b x + a\right )}^{\frac{1}{3}} b}\right )}}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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