3.7.57 \(\int \frac {a+b x}{x^{5/3}} \, dx\)

Optimal. Leaf size=19 \[ 3 b \sqrt [3]{x}-\frac {3 a}{2 x^{2/3}} \]

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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} \begin {gather*} 3 b \sqrt [3]{x}-\frac {3 a}{2 x^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/x^(5/3),x]

[Out]

(-3*a)/(2*x^(2/3)) + 3*b*x^(1/3)

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^{5/3}} \, dx &=\int \left (\frac {a}{x^{5/3}}+\frac {b}{x^{2/3}}\right ) \, dx\\ &=-\frac {3 a}{2 x^{2/3}}+3 b \sqrt [3]{x}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 19, normalized size = 1.00 \begin {gather*} 3 b \sqrt [3]{x}-\frac {3 a}{2 x^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/x^(5/3),x]

[Out]

(-3*a)/(2*x^(2/3)) + 3*b*x^(1/3)

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IntegrateAlgebraic [A]  time = 0.01, size = 17, normalized size = 0.89 \begin {gather*} \frac {3 (2 b x-a)}{2 x^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x)/x^(5/3),x]

[Out]

(3*(-a + 2*b*x))/(2*x^(2/3))

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fricas [A]  time = 1.17, size = 13, normalized size = 0.68 \begin {gather*} \frac {3 \, {\left (2 \, b x - a\right )}}{2 \, x^{\frac {2}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/x^(5/3),x, algorithm="fricas")

[Out]

3/2*(2*b*x - a)/x^(2/3)

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giac [A]  time = 1.10, size = 13, normalized size = 0.68 \begin {gather*} 3 \, b x^{\frac {1}{3}} - \frac {3 \, a}{2 \, x^{\frac {2}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/x^(5/3),x, algorithm="giac")

[Out]

3*b*x^(1/3) - 3/2*a/x^(2/3)

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maple [A]  time = 0.00, size = 12, normalized size = 0.63 \begin {gather*} -\frac {3 \left (-2 b x +a \right )}{2 x^{\frac {2}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)/x^(5/3),x)

[Out]

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

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maxima [A]  time = 1.30, size = 13, normalized size = 0.68 \begin {gather*} 3 \, b x^{\frac {1}{3}} - \frac {3 \, a}{2 \, x^{\frac {2}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/x^(5/3),x, algorithm="maxima")

[Out]

3*b*x^(1/3) - 3/2*a/x^(2/3)

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mupad [B]  time = 0.03, size = 13, normalized size = 0.68 \begin {gather*} -\frac {3\,a-6\,b\,x}{2\,x^{2/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)/x^(5/3),x)

[Out]

-(3*a - 6*b*x)/(2*x^(2/3))

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sympy [A]  time = 0.45, size = 17, normalized size = 0.89 \begin {gather*} - \frac {3 a}{2 x^{\frac {2}{3}}} + 3 b \sqrt [3]{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/x**(5/3),x)

[Out]

-3*a/(2*x**(2/3)) + 3*b*x**(1/3)

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