3.655 \(\int \frac {a+b x}{x^{2/3}} \, dx\)

Optimal. Leaf size=19 \[ 3 a \sqrt [3]{x}+\frac {3}{4} b x^{4/3} \]

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, 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} \[ 3 a \sqrt [3]{x}+\frac {3}{4} b x^{4/3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/x^(2/3),x]

[Out]

3*a*x^(1/3) + (3*b*x^(4/3))/4

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

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Mathematica [A]  time = 0.00, size = 16, normalized size = 0.84 \[ \frac {3}{4} \sqrt [3]{x} (4 a+b x) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/x^(2/3),x]

[Out]

(3*x^(1/3)*(4*a + b*x))/4

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fricas [A]  time = 0.42, size = 12, normalized size = 0.63 \[ \frac {3}{4} \, {\left (b x + 4 \, a\right )} x^{\frac {1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/x^(2/3),x, algorithm="fricas")

[Out]

3/4*(b*x + 4*a)*x^(1/3)

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giac [A]  time = 1.12, size = 13, normalized size = 0.68 \[ \frac {3}{4} \, b x^{\frac {4}{3}} + 3 \, a x^{\frac {1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/x^(2/3),x, algorithm="giac")

[Out]

3/4*b*x^(4/3) + 3*a*x^(1/3)

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maple [A]  time = 0.00, size = 13, normalized size = 0.68 \[ \frac {3 \left (b x +4 a \right ) x^{\frac {1}{3}}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.28, size = 13, normalized size = 0.68 \[ \frac {3}{4} \, b x^{\frac {4}{3}} + 3 \, a x^{\frac {1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/x^(2/3),x, algorithm="maxima")

[Out]

3/4*b*x^(4/3) + 3*a*x^(1/3)

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mupad [B]  time = 0.02, size = 12, normalized size = 0.63 \[ \frac {3\,x^{1/3}\,\left (4\,a+b\,x\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)/x^(2/3),x)

[Out]

(3*x^(1/3)*(4*a + b*x))/4

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sympy [A]  time = 1.49, size = 17, normalized size = 0.89 \[ 3 a \sqrt [3]{x} + \frac {3 b x^{\frac {4}{3}}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/x**(2/3),x)

[Out]

3*a*x**(1/3) + 3*b*x**(4/3)/4

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