3.25.22 \(\int \frac {x}{a+\frac {b}{\sqrt [3]{x}}} \, dx\) [2422]

Optimal. Leaf size=94 \[ -\frac {3 b^5 \sqrt [3]{x}}{a^6}+\frac {3 b^4 x^{2/3}}{2 a^5}-\frac {b^3 x}{a^4}+\frac {3 b^2 x^{4/3}}{4 a^3}-\frac {3 b x^{5/3}}{5 a^2}+\frac {x^2}{2 a}+\frac {3 b^6 \log \left (b+a \sqrt [3]{x}\right )}{a^7} \]

[Out]

-3*b^5*x^(1/3)/a^6+3/2*b^4*x^(2/3)/a^5-b^3*x/a^4+3/4*b^2*x^(4/3)/a^3-3/5*b*x^(5/3)/a^2+1/2*x^2/a+3*b^6*ln(b+a*
x^(1/3))/a^7

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Rubi [A]
time = 0.04, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {269, 272, 45} \begin {gather*} \frac {3 b^6 \log \left (a \sqrt [3]{x}+b\right )}{a^7}-\frac {3 b^5 \sqrt [3]{x}}{a^6}+\frac {3 b^4 x^{2/3}}{2 a^5}-\frac {b^3 x}{a^4}+\frac {3 b^2 x^{4/3}}{4 a^3}-\frac {3 b x^{5/3}}{5 a^2}+\frac {x^2}{2 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(a + b/x^(1/3)),x]

[Out]

(-3*b^5*x^(1/3))/a^6 + (3*b^4*x^(2/3))/(2*a^5) - (b^3*x)/a^4 + (3*b^2*x^(4/3))/(4*a^3) - (3*b*x^(5/3))/(5*a^2)
+ x^2/(2*a) + (3*b^6*Log[b + a*x^(1/3)])/a^7

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])

Rule 269

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {x}{a+\frac {b}{\sqrt [3]{x}}} \, dx &=\int \frac {x^{4/3}}{b+a \sqrt [3]{x}} \, dx\\ &=3 \text {Subst}\left (\int \frac {x^6}{b+a x} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \text {Subst}\left (\int \left (-\frac {b^5}{a^6}+\frac {b^4 x}{a^5}-\frac {b^3 x^2}{a^4}+\frac {b^2 x^3}{a^3}-\frac {b x^4}{a^2}+\frac {x^5}{a}+\frac {b^6}{a^6 (b+a x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {3 b^5 \sqrt [3]{x}}{a^6}+\frac {3 b^4 x^{2/3}}{2 a^5}-\frac {b^3 x}{a^4}+\frac {3 b^2 x^{4/3}}{4 a^3}-\frac {3 b x^{5/3}}{5 a^2}+\frac {x^2}{2 a}+\frac {3 b^6 \log \left (b+a \sqrt [3]{x}\right )}{a^7}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 88, normalized size = 0.94 \begin {gather*} \frac {-60 a b^5 \sqrt [3]{x}+30 a^2 b^4 x^{2/3}-20 a^3 b^3 x+15 a^4 b^2 x^{4/3}-12 a^5 b x^{5/3}+10 a^6 x^2+60 b^6 \log \left (b+a \sqrt [3]{x}\right )}{20 a^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b/x^(1/3)),x]

[Out]

(-60*a*b^5*x^(1/3) + 30*a^2*b^4*x^(2/3) - 20*a^3*b^3*x + 15*a^4*b^2*x^(4/3) - 12*a^5*b*x^(5/3) + 10*a^6*x^2 +
60*b^6*Log[b + a*x^(1/3)])/(20*a^7)

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Maple [A]
time = 0.20, size = 78, normalized size = 0.83

method	result	size
derivativedivides	\(\frac {\frac {a^{5} x^{2}}{2}-\frac {3 a^{4} b \,x^{\frac {5}{3}}}{5}+\frac {3 a^{3} b^{2} x^{\frac {4}{3}}}{4}-a^{2} b^{3} x +\frac {3 a \,b^{4} x^{\frac {2}{3}}}{2}-3 b^{5} x^{\frac {1}{3}}}{a^{6}}+\frac {3 b^{6} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{a^{7}}\)	\(78\)
default	\(\frac {\frac {a^{5} x^{2}}{2}-\frac {3 a^{4} b \,x^{\frac {5}{3}}}{5}+\frac {3 a^{3} b^{2} x^{\frac {4}{3}}}{4}-a^{2} b^{3} x +\frac {3 a \,b^{4} x^{\frac {2}{3}}}{2}-3 b^{5} x^{\frac {1}{3}}}{a^{6}}+\frac {3 b^{6} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{a^{7}}\)	\(78\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/a^6*(1/6*a^5*x^2-1/5*a^4*b*x^(5/3)+1/4*a^3*b^2*x^(4/3)-1/3*a^2*b^3*x+1/2*a*b^4*x^(2/3)-b^5*x^(1/3))+3*b^6*ln
(b+a*x^(1/3))/a^7

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Maxima [A]
time = 0.30, size = 88, normalized size = 0.94 \begin {gather*} \frac {3 \, b^{6} \log \left (a + \frac {b}{x^{\frac {1}{3}}}\right )}{a^{7}} + \frac {b^{6} \log \left (x\right )}{a^{7}} + \frac {{\left (10 \, a^{5} - \frac {12 \, a^{4} b}{x^{\frac {1}{3}}} + \frac {15 \, a^{3} b^{2}}{x^{\frac {2}{3}}} - \frac {20 \, a^{2} b^{3}}{x} + \frac {30 \, a b^{4}}{x^{\frac {4}{3}}} - \frac {60 \, b^{5}}{x^{\frac {5}{3}}}\right )} x^{2}}{20 \, a^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*b^6*log(a + b/x^(1/3))/a^7 + b^6*log(x)/a^7 + 1/20*(10*a^5 - 12*a^4*b/x^(1/3) + 15*a^3*b^2/x^(2/3) - 20*a^2*
b^3/x + 30*a*b^4/x^(4/3) - 60*b^5/x^(5/3))*x^2/a^6

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Fricas [A]
time = 0.37, size = 77, normalized size = 0.82 \begin {gather*} \frac {10 \, a^{6} x^{2} - 20 \, a^{3} b^{3} x + 60 \, b^{6} \log \left (a x^{\frac {1}{3}} + b\right ) - 6 \, {\left (2 \, a^{5} b x - 5 \, a^{2} b^{4}\right )} x^{\frac {2}{3}} + 15 \, {\left (a^{4} b^{2} x - 4 \, a b^{5}\right )} x^{\frac {1}{3}}}{20 \, a^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*(10*a^6*x^2 - 20*a^3*b^3*x + 60*b^6*log(a*x^(1/3) + b) - 6*(2*a^5*b*x - 5*a^2*b^4)*x^(2/3) + 15*(a^4*b^2*
x - 4*a*b^5)*x^(1/3))/a^7

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Sympy [A]
time = 0.30, size = 100, normalized size = 1.06 \begin {gather*} \begin {cases} \frac {x^{2}}{2 a} - \frac {3 b x^{\frac {5}{3}}}{5 a^{2}} + \frac {3 b^{2} x^{\frac {4}{3}}}{4 a^{3}} - \frac {b^{3} x}{a^{4}} + \frac {3 b^{4} x^{\frac {2}{3}}}{2 a^{5}} - \frac {3 b^{5} \sqrt [3]{x}}{a^{6}} + \frac {3 b^{6} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{a^{7}} & \text {for}\: a \neq 0 \\\frac {3 x^{\frac {7}{3}}}{7 b} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x**2/(2*a) - 3*b*x**(5/3)/(5*a**2) + 3*b**2*x**(4/3)/(4*a**3) - b**3*x/a**4 + 3*b**4*x**(2/3)/(2*a*
*5) - 3*b**5*x**(1/3)/a**6 + 3*b**6*log(x**(1/3) + b/a)/a**7, Ne(a, 0)), (3*x**(7/3)/(7*b), True))

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Giac [A]
time = 0.53, size = 78, normalized size = 0.83 \begin {gather*} \frac {3 \, b^{6} \log \left ({\left | a x^{\frac {1}{3}} + b \right |}\right )}{a^{7}} + \frac {10 \, a^{5} x^{2} - 12 \, a^{4} b x^{\frac {5}{3}} + 15 \, a^{3} b^{2} x^{\frac {4}{3}} - 20 \, a^{2} b^{3} x + 30 \, a b^{4} x^{\frac {2}{3}} - 60 \, b^{5} x^{\frac {1}{3}}}{20 \, a^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*b^6*log(abs(a*x^(1/3) + b))/a^7 + 1/20*(10*a^5*x^2 - 12*a^4*b*x^(5/3) + 15*a^3*b^2*x^(4/3) - 20*a^2*b^3*x +
30*a*b^4*x^(2/3) - 60*b^5*x^(1/3))/a^6

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Mupad [B]
time = 0.03, size = 76, normalized size = 0.81 \begin {gather*} \frac {x^2}{2\,a}-\frac {b^3\,x}{a^4}-\frac {3\,b\,x^{5/3}}{5\,a^2}+\frac {3\,b^6\,\ln \left (b+a\,x^{1/3}\right )}{a^7}+\frac {3\,b^2\,x^{4/3}}{4\,a^3}+\frac {3\,b^4\,x^{2/3}}{2\,a^5}-\frac {3\,b^5\,x^{1/3}}{a^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b/x^(1/3)),x)

[Out]

x^2/(2*a) - (b^3*x)/a^4 - (3*b*x^(5/3))/(5*a^2) + (3*b^6*log(b + a*x^(1/3)))/a^7 + (3*b^2*x^(4/3))/(4*a^3) + (
3*b^4*x^(2/3))/(2*a^5) - (3*b^5*x^(1/3))/a^6

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