3.7.56 \(\int \frac {a+b x}{x^{4/3}} \, dx\) [656]

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

[Out]

-3*a/x^(1/3)+3/2*b*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 = {45} \begin {gather*} \frac {3}{2} b x^{2/3}-\frac {3 a}{\sqrt [3]{x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/x^(4/3),x]

[Out]

(-3*a)/x^(1/3) + (3*b*x^(2/3))/2

Rule 45

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

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

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/x^(4/3),x]

[Out]

(-3*(2*a - b*x))/(2*x^(1/3))

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Maple [A]
time = 0.03, size = 14, normalized size = 0.74

method	result	size
gosper	\(-\frac {3 \left (-b x +2 a \right )}{2 x^{\frac {1}{3}}}\)	\(14\)
derivativedivides	\(-\frac {3 a}{x^{\frac {1}{3}}}+\frac {3 b \,x^{\frac {2}{3}}}{2}\)	\(14\)
default	\(-\frac {3 a}{x^{\frac {1}{3}}}+\frac {3 b \,x^{\frac {2}{3}}}{2}\)	\(14\)
trager	\(-\frac {3 \left (-b x +2 a \right )}{2 x^{\frac {1}{3}}}\)	\(14\)
risch	\(-\frac {3 \left (-b x +2 a \right )}{2 x^{\frac {1}{3}}}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 13, normalized size = 0.68 \begin {gather*} \frac {3}{2} \, b x^{\frac {2}{3}} - \frac {3 \, a}{x^{\frac {1}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*b*x^(2/3) - 3*a/x^(1/3)

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Fricas [A]
time = 0.83, size = 12, normalized size = 0.63 \begin {gather*} \frac {3 \, {\left (b x - 2 \, a\right )}}{2 \, x^{\frac {1}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*(b*x - 2*a)/x^(1/3)

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Sympy [A]
time = 0.15, size = 17, normalized size = 0.89 \begin {gather*} - \frac {3 a}{\sqrt [3]{x}} + \frac {3 b x^{\frac {2}{3}}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*a/x**(1/3) + 3*b*x**(2/3)/2

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Giac [A]
time = 1.89, size = 13, normalized size = 0.68 \begin {gather*} \frac {3}{2} \, b x^{\frac {2}{3}} - \frac {3 \, a}{x^{\frac {1}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*b*x^(2/3) - 3*a/x^(1/3)

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Mupad [B]
time = 0.03, size = 13, normalized size = 0.68 \begin {gather*} -\frac {6\,a-3\,b\,x}{2\,x^{1/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)/x^(4/3),x)

[Out]

-(6*a - 3*b*x)/(2*x^(1/3))

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