3.7.54 \(\int \frac {a+b x}{\sqrt [3]{x}} \, dx\) [654]

Optimal. Leaf size=21 \[ \frac {3}{2} a x^{2/3}+\frac {3}{5} b x^{5/3} \]

[Out]

3/2*a*x^(2/3)+3/5*b*x^(5/3)

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Rubi [A]
time = 0.00, antiderivative size = 21, 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} a x^{2/3}+\frac {3}{5} b x^{5/3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/x^(1/3),x]

[Out]

(3*a*x^(2/3))/2 + (3*b*x^(5/3))/5

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

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Mathematica [A]
time = 0.01, size = 17, normalized size = 0.81 \begin {gather*} \frac {3}{10} x^{2/3} (5 a+2 b x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/x^(1/3),x]

[Out]

(3*x^(2/3)*(5*a + 2*b*x))/10

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Mathics [A]
time = 2.21, size = 13, normalized size = 0.62 \begin {gather*} \frac {3 x^{\frac {2}{3}} \left (5 a+2 b x\right )}{10} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[(a + b*x)/x^(1/3),x]')

[Out]

3 x ^ (2 / 3) (5 a + 2 b x) / 10

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Maple [A]
time = 0.02, size = 14, normalized size = 0.67

method	result	size
trager	\(\left (\frac {3 b x}{5}+\frac {3 a}{2}\right ) x^{\frac {2}{3}}\)	\(13\)
gosper	\(\frac {3 x^{\frac {2}{3}} \left (2 b x +5 a \right )}{10}\)	\(14\)
derivativedivides	\(\frac {3 a \,x^{\frac {2}{3}}}{2}+\frac {3 b \,x^{\frac {5}{3}}}{5}\)	\(14\)
default	\(\frac {3 a \,x^{\frac {2}{3}}}{2}+\frac {3 b \,x^{\frac {5}{3}}}{5}\)	\(14\)
risch	\(\frac {3 x^{\frac {2}{3}} \left (2 b x +5 a \right )}{10}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*a*x^(2/3)+3/5*b*x^(5/3)

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Maxima [A]
time = 0.28, size = 13, normalized size = 0.62 \begin {gather*} \frac {3}{5} \, b x^{\frac {5}{3}} + \frac {3}{2} \, a x^{\frac {2}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/5*b*x^(5/3) + 3/2*a*x^(2/3)

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Fricas [A]
time = 0.30, size = 13, normalized size = 0.62 \begin {gather*} \frac {3}{10} \, {\left (2 \, b x + 5 \, a\right )} x^{\frac {2}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/10*(2*b*x + 5*a)*x^(2/3)

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Sympy [A]
time = 0.78, size = 19, normalized size = 0.90 \begin {gather*} \frac {3 a x^{\frac {2}{3}}}{2} + \frac {3 b x^{\frac {5}{3}}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*a*x**(2/3)/2 + 3*b*x**(5/3)/5

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Giac [A]
time = 0.00, size = 26, normalized size = 1.24 \begin {gather*} \frac {3}{5} b \left (x^{\frac {1}{3}}\right )^{2} x+\frac {3}{2} a \left (x^{\frac {1}{3}}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/5*b*x^(5/3) + 3/2*a*x^(2/3)

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Mupad [B]
time = 0.02, size = 13, normalized size = 0.62 \begin {gather*} \frac {3\,x^{2/3}\,\left (5\,a+2\,b\,x\right )}{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)/x^(1/3),x)

[Out]

(3*x^(2/3)*(5*a + 2*b*x))/10

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