3.4.94 \(\int \frac {x}{\sqrt [3]{a+b x}} \, dx\) [394]

Optimal. Leaf size=34 \[ -\frac {3 a (a+b x)^{2/3}}{2 b^2}+\frac {3 (a+b x)^{5/3}}{5 b^2} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 34, 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 (a+b x)^{5/3}}{5 b^2}-\frac {3 a (a+b x)^{2/3}}{2 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(a + b*x)^(1/3),x]

[Out]

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

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Mathematica [A]
time = 0.01, size = 24, normalized size = 0.71 \begin {gather*} \frac {3 (a+b x)^{2/3} (-3 a+2 b x)}{10 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*x)^(1/3),x]

[Out]

(3*(a + b*x)^(2/3)*(-3*a + 2*b*x))/(10*b^2)

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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(84\) vs. \(2(34)=68\).
time = 3.19, size = 74, normalized size = 2.18 \begin {gather*} \frac {3 a^{\frac {2}{3}} \left (3 a^2 \left (1-\left (\frac {a+b x}{a}\right )^{\frac {2}{3}}\right )+a b x \left (3-\left (\frac {a+b x}{a}\right )^{\frac {2}{3}}\right )+2 b^2 x^2 \left (\frac {a+b x}{a}\right )^{\frac {2}{3}}\right )}{10 b^2 \left (a+b x\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

3 a ^ (2 / 3) (3 a ^ 2 (1 - ((a + b x) / a) ^ (2 / 3)) + a b x (3 - ((a + b x) / a) ^ (2 / 3)) + 2 b ^ 2 x ^ 2
((a + b x) / a) ^ (2 / 3)) / (10 b ^ 2 (a + b x))

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Maple [A]
time = 0.12, size = 26, normalized size = 0.76

method	result	size
gosper	\(-\frac {3 \left (b x +a \right )^{\frac {2}{3}} \left (-2 b x +3 a \right )}{10 b^{2}}\)	\(21\)
trager	\(-\frac {3 \left (b x +a \right )^{\frac {2}{3}} \left (-2 b x +3 a \right )}{10 b^{2}}\)	\(21\)
risch	\(-\frac {3 \left (b x +a \right )^{\frac {2}{3}} \left (-2 b x +3 a \right )}{10 b^{2}}\)	\(21\)
derivativedivides	\(\frac {\frac {3 \left (b x +a \right )^{\frac {5}{3}}}{5}-\frac {3 a \left (b x +a \right )^{\frac {2}{3}}}{2}}{b^{2}}\)	\(26\)
default	\(\frac {\frac {3 \left (b x +a \right )^{\frac {5}{3}}}{5}-\frac {3 a \left (b x +a \right )^{\frac {2}{3}}}{2}}{b^{2}}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/b^2*(1/5*(b*x+a)^(5/3)-1/2*a*(b*x+a)^(2/3))

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Maxima [A]
time = 0.27, size = 26, normalized size = 0.76 \begin {gather*} \frac {3 \, {\left (b x + a\right )}^{\frac {5}{3}}}{5 \, b^{2}} - \frac {3 \, {\left (b x + a\right )}^{\frac {2}{3}} a}{2 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.32, size = 20, normalized size = 0.59 \begin {gather*} \frac {3 \, {\left (2 \, b x - 3 \, a\right )} {\left (b x + a\right )}^{\frac {2}{3}}}{10 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/10*(2*b*x - 3*a)*(b*x + a)^(2/3)/b^2

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 162 vs. \(2 (31) = 62\).
time = 0.57, size = 162, normalized size = 4.76 \begin {gather*} - \frac {9 a^{\frac {11}{3}} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{10 a^{2} b^{2} + 10 a b^{3} x} + \frac {9 a^{\frac {11}{3}}}{10 a^{2} b^{2} + 10 a b^{3} x} - \frac {3 a^{\frac {8}{3}} b x \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{10 a^{2} b^{2} + 10 a b^{3} x} + \frac {9 a^{\frac {8}{3}} b x}{10 a^{2} b^{2} + 10 a b^{3} x} + \frac {6 a^{\frac {5}{3}} b^{2} x^{2} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{10 a^{2} b^{2} + 10 a b^{3} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9*a**(11/3)*(1 + b*x/a)**(2/3)/(10*a**2*b**2 + 10*a*b**3*x) + 9*a**(11/3)/(10*a**2*b**2 + 10*a*b**3*x) - 3*a*
*(8/3)*b*x*(1 + b*x/a)**(2/3)/(10*a**2*b**2 + 10*a*b**3*x) + 9*a**(8/3)*b*x/(10*a**2*b**2 + 10*a*b**3*x) + 6*a
**(5/3)*b**2*x**2*(1 + b*x/a)**(2/3)/(10*a**2*b**2 + 10*a*b**3*x)

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Giac [A]
time = 0.00, size = 42, normalized size = 1.24 \begin {gather*} \frac {3 \left (\frac {1}{5} \left (\left (a+b x\right )^{\frac {1}{3}}\right )^{2} \left (a+b x\right )-\frac {1}{2} \left (\left (a+b x\right )^{\frac {1}{3}}\right )^{2} a\right )}{b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/10*(2*(b*x + a)^(5/3) - 5*(b*x + a)^(2/3)*a)/b^2

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Mupad [B]
time = 0.03, size = 25, normalized size = 0.74 \begin {gather*} -\frac {15\,a\,{\left (a+b\,x\right )}^{2/3}-6\,{\left (a+b\,x\right )}^{5/3}}{10\,b^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b*x)^(1/3),x)

[Out]

-(15*a*(a + b*x)^(2/3) - 6*(a + b*x)^(5/3))/(10*b^2)

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