3.373 \(\int x \sqrt [3]{a+b x} \, dx\)

Optimal. Leaf size=34 \[ \frac{3 (a+b x)^{7/3}}{7 b^2}-\frac{3 a (a+b x)^{4/3}}{4 b^2} \]

[Out]

(-3*a*(a + b*x)^(4/3))/(4*b^2) + (3*(a + b*x)^(7/3))/(7*b^2)

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Rubi [A]  time = 0.0082427, antiderivative size = 34, 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} \[ \frac{3 (a+b x)^{7/3}}{7 b^2}-\frac{3 a (a+b x)^{4/3}}{4 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x)^(1/3),x]

[Out]

(-3*a*(a + b*x)^(4/3))/(4*b^2) + (3*(a + b*x)^(7/3))/(7*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 x \sqrt [3]{a+b x} \, dx &=\int \left (-\frac{a \sqrt [3]{a+b x}}{b}+\frac{(a+b x)^{4/3}}{b}\right ) \, dx\\ &=-\frac{3 a (a+b x)^{4/3}}{4 b^2}+\frac{3 (a+b x)^{7/3}}{7 b^2}\\ \end{align*}

Mathematica [A]  time = 0.0245567, size = 24, normalized size = 0.71 \[ \frac{3 (a+b x)^{4/3} (4 b x-3 a)}{28 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x)^(1/3),x]

[Out]

(3*(a + b*x)^(4/3)*(-3*a + 4*b*x))/(28*b^2)

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Maple [A]  time = 0.003, size = 21, normalized size = 0.6 \begin{align*} -{\frac{-12\,bx+9\,a}{28\,{b}^{2}} \left ( bx+a \right ) ^{{\frac{4}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/28*(b*x+a)^(4/3)*(-4*b*x+3*a)/b^2

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Maxima [A]  time = 1.08618, size = 35, normalized size = 1.03 \begin{align*} \frac{3 \,{\left (b x + a\right )}^{\frac{7}{3}}}{7 \, b^{2}} - \frac{3 \,{\left (b x + a\right )}^{\frac{4}{3}} a}{4 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/7*(b*x + a)^(7/3)/b^2 - 3/4*(b*x + a)^(4/3)*a/b^2

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Fricas [A]  time = 1.70608, size = 73, normalized size = 2.15 \begin{align*} \frac{3 \,{\left (4 \, b^{2} x^{2} + a b x - 3 \, a^{2}\right )}{\left (b x + a\right )}^{\frac{1}{3}}}{28 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/28*(4*b^2*x^2 + a*b*x - 3*a^2)*(b*x + a)^(1/3)/b^2

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Sympy [B]  time = 1.63821, size = 202, normalized size = 5.94 \begin{align*} - \frac{9 a^{\frac{13}{3}} \sqrt [3]{1 + \frac{b x}{a}}}{28 a^{2} b^{2} + 28 a b^{3} x} + \frac{9 a^{\frac{13}{3}}}{28 a^{2} b^{2} + 28 a b^{3} x} - \frac{6 a^{\frac{10}{3}} b x \sqrt [3]{1 + \frac{b x}{a}}}{28 a^{2} b^{2} + 28 a b^{3} x} + \frac{9 a^{\frac{10}{3}} b x}{28 a^{2} b^{2} + 28 a b^{3} x} + \frac{15 a^{\frac{7}{3}} b^{2} x^{2} \sqrt [3]{1 + \frac{b x}{a}}}{28 a^{2} b^{2} + 28 a b^{3} x} + \frac{12 a^{\frac{4}{3}} b^{3} x^{3} \sqrt [3]{1 + \frac{b x}{a}}}{28 a^{2} b^{2} + 28 a b^{3} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9*a**(13/3)*(1 + b*x/a)**(1/3)/(28*a**2*b**2 + 28*a*b**3*x) + 9*a**(13/3)/(28*a**2*b**2 + 28*a*b**3*x) - 6*a*
*(10/3)*b*x*(1 + b*x/a)**(1/3)/(28*a**2*b**2 + 28*a*b**3*x) + 9*a**(10/3)*b*x/(28*a**2*b**2 + 28*a*b**3*x) + 1
5*a**(7/3)*b**2*x**2*(1 + b*x/a)**(1/3)/(28*a**2*b**2 + 28*a*b**3*x) + 12*a**(4/3)*b**3*x**3*(1 + b*x/a)**(1/3
)/(28*a**2*b**2 + 28*a*b**3*x)

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Giac [A]  time = 1.22147, size = 34, normalized size = 1. \begin{align*} \frac{3 \,{\left (4 \,{\left (b x + a\right )}^{\frac{7}{3}} - 7 \,{\left (b x + a\right )}^{\frac{4}{3}} a\right )}}{28 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/28*(4*(b*x + a)^(7/3) - 7*(b*x + a)^(4/3)*a)/b^2