3.25.39 \(\int \frac {1}{(a+\frac {b}{\sqrt [3]{x}})^3} \, dx\) [2439]

Optimal. Leaf size=100 \[ \frac {3 b^3}{2 a^4 \left (a+\frac {b}{\sqrt [3]{x}}\right )^2}+\frac {12 b^3}{a^5 \left (a+\frac {b}{\sqrt [3]{x}}\right )}+\frac {18 b^2 \sqrt [3]{x}}{a^5}-\frac {9 b x^{2/3}}{2 a^4}+\frac {x}{a^3}-\frac {30 b^3 \log \left (a+\frac {b}{\sqrt [3]{x}}\right )}{a^6}-\frac {10 b^3 \log (x)}{a^6} \]

[Out]

3/2*b^3/a^4/(a+b/x^(1/3))^2+12*b^3/a^5/(a+b/x^(1/3))+18*b^2*x^(1/3)/a^5-9/2*b*x^(2/3)/a^4+x/a^3-30*b^3*ln(a+b/
x^(1/3))/a^6-10*b^3*ln(x)/a^6

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Rubi [A]
time = 0.05, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {196, 46} \begin {gather*} -\frac {30 b^3 \log \left (a+\frac {b}{\sqrt [3]{x}}\right )}{a^6}-\frac {10 b^3 \log (x)}{a^6}+\frac {12 b^3}{a^5 \left (a+\frac {b}{\sqrt [3]{x}}\right )}+\frac {18 b^2 \sqrt [3]{x}}{a^5}+\frac {3 b^3}{2 a^4 \left (a+\frac {b}{\sqrt [3]{x}}\right )^2}-\frac {9 b x^{2/3}}{2 a^4}+\frac {x}{a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b/x^(1/3))^(-3),x]

[Out]

(3*b^3)/(2*a^4*(a + b/x^(1/3))^2) + (12*b^3)/(a^5*(a + b/x^(1/3))) + (18*b^2*x^(1/3))/a^5 - (9*b*x^(2/3))/(2*a
^4) + x/a^3 - (30*b^3*Log[a + b/x^(1/3)])/a^6 - (10*b^3*Log[x])/a^6

Rule 46

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 196

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

Rubi steps

\begin {align*} \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3} \, dx &=-\left (3 \text {Subst}\left (\int \frac {1}{x^4 (a+b x)^3} \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\right )\\ &=-\left (3 \text {Subst}\left (\int \left (\frac {1}{a^3 x^4}-\frac {3 b}{a^4 x^3}+\frac {6 b^2}{a^5 x^2}-\frac {10 b^3}{a^6 x}+\frac {b^4}{a^4 (a+b x)^3}+\frac {4 b^4}{a^5 (a+b x)^2}+\frac {10 b^4}{a^6 (a+b x)}\right ) \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\right )\\ &=\frac {3 b^3}{2 a^4 \left (a+\frac {b}{\sqrt [3]{x}}\right )^2}+\frac {12 b^3}{a^5 \left (a+\frac {b}{\sqrt [3]{x}}\right )}+\frac {18 b^2 \sqrt [3]{x}}{a^5}-\frac {9 b x^{2/3}}{2 a^4}+\frac {x}{a^3}-\frac {30 b^3 \log \left (a+\frac {b}{\sqrt [3]{x}}\right )}{a^6}-\frac {10 b^3 \log (x)}{a^6}\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 97, normalized size = 0.97 \begin {gather*} \frac {-27 b^5+6 a b^4 \sqrt [3]{x}+63 a^2 b^3 x^{2/3}+20 a^3 b^2 x-5 a^4 b x^{4/3}+2 a^5 x^{5/3}}{2 a^6 \left (b+a \sqrt [3]{x}\right )^2}-\frac {30 b^3 \log \left (b+a \sqrt [3]{x}\right )}{a^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^(1/3))^(-3),x]

[Out]

(-27*b^5 + 6*a*b^4*x^(1/3) + 63*a^2*b^3*x^(2/3) + 20*a^3*b^2*x - 5*a^4*b*x^(4/3) + 2*a^5*x^(5/3))/(2*a^6*(b +
a*x^(1/3))^2) - (30*b^3*Log[b + a*x^(1/3)])/a^6

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Maple [A]
time = 0.21, size = 79, normalized size = 0.79

method	result	size
derivativedivides	\(\frac {a^{2} x -\frac {9 a b \,x^{\frac {2}{3}}}{2}+18 b^{2} x^{\frac {1}{3}}}{a^{5}}+\frac {3 b^{5}}{2 a^{6} \left (b +a \,x^{\frac {1}{3}}\right )^{2}}-\frac {15 b^{4}}{a^{6} \left (b +a \,x^{\frac {1}{3}}\right )}-\frac {30 b^{3} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{a^{6}}\)	\(79\)
default	\(\frac {a^{2} x -\frac {9 a b \,x^{\frac {2}{3}}}{2}+18 b^{2} x^{\frac {1}{3}}}{a^{5}}+\frac {3 b^{5}}{2 a^{6} \left (b +a \,x^{\frac {1}{3}}\right )^{2}}-\frac {15 b^{4}}{a^{6} \left (b +a \,x^{\frac {1}{3}}\right )}-\frac {30 b^{3} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{a^{6}}\)	\(79\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/a^5*(1/3*a^2*x-3/2*a*b*x^(2/3)+6*b^2*x^(1/3))+3/2*b^5/a^6/(b+a*x^(1/3))^2-15/a^6*b^4/(b+a*x^(1/3))-30/a^6*b^
3*ln(b+a*x^(1/3))

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Maxima [A]
time = 0.31, size = 101, normalized size = 1.01 \begin {gather*} \frac {2 \, a^{4} - \frac {5 \, a^{3} b}{x^{\frac {1}{3}}} + \frac {20 \, a^{2} b^{2}}{x^{\frac {2}{3}}} + \frac {90 \, a b^{3}}{x} + \frac {60 \, b^{4}}{x^{\frac {4}{3}}}}{2 \, {\left (\frac {a^{7}}{x} + \frac {2 \, a^{6} b}{x^{\frac {4}{3}}} + \frac {a^{5} b^{2}}{x^{\frac {5}{3}}}\right )}} - \frac {30 \, b^{3} \log \left (a + \frac {b}{x^{\frac {1}{3}}}\right )}{a^{6}} - \frac {10 \, b^{3} \log \left (x\right )}{a^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*a^4 - 5*a^3*b/x^(1/3) + 20*a^2*b^2/x^(2/3) + 90*a*b^3/x + 60*b^4/x^(4/3))/(a^7/x + 2*a^6*b/x^(4/3) + a^
5*b^2/x^(5/3)) - 30*b^3*log(a + b/x^(1/3))/a^6 - 10*b^3*log(x)/a^6

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Fricas [A]
time = 0.40, size = 159, normalized size = 1.59 \begin {gather*} \frac {2 \, a^{9} x^{3} + 4 \, a^{6} b^{3} x^{2} - 34 \, a^{3} b^{6} x - 27 \, b^{9} - 60 \, {\left (a^{6} b^{3} x^{2} + 2 \, a^{3} b^{6} x + b^{9}\right )} \log \left (a x^{\frac {1}{3}} + b\right ) - 3 \, {\left (3 \, a^{8} b x^{2} + 16 \, a^{5} b^{4} x + 10 \, a^{2} b^{7}\right )} x^{\frac {2}{3}} + 3 \, {\left (12 \, a^{7} b^{2} x^{2} + 35 \, a^{4} b^{5} x + 20 \, a b^{8}\right )} x^{\frac {1}{3}}}{2 \, {\left (a^{12} x^{2} + 2 \, a^{9} b^{3} x + a^{6} b^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*a^9*x^3 + 4*a^6*b^3*x^2 - 34*a^3*b^6*x - 27*b^9 - 60*(a^6*b^3*x^2 + 2*a^3*b^6*x + b^9)*log(a*x^(1/3) +
b) - 3*(3*a^8*b*x^2 + 16*a^5*b^4*x + 10*a^2*b^7)*x^(2/3) + 3*(12*a^7*b^2*x^2 + 35*a^4*b^5*x + 20*a*b^8)*x^(1/3
))/(a^12*x^2 + 2*a^9*b^3*x + a^6*b^6)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 362 vs. \(2 (99) = 198\).
time = 0.28, size = 362, normalized size = 3.62 \begin {gather*} \begin {cases} \frac {2 a^{5} x^{\frac {5}{3}}}{2 a^{8} x^{\frac {2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} - \frac {5 a^{4} b x^{\frac {4}{3}}}{2 a^{8} x^{\frac {2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} + \frac {20 a^{3} b^{2} x}{2 a^{8} x^{\frac {2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} - \frac {60 a^{2} b^{3} x^{\frac {2}{3}} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{2 a^{8} x^{\frac {2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} - \frac {120 a b^{4} \sqrt [3]{x} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{2 a^{8} x^{\frac {2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} - \frac {120 a b^{4} \sqrt [3]{x}}{2 a^{8} x^{\frac {2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} - \frac {60 b^{5} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{2 a^{8} x^{\frac {2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} - \frac {90 b^{5}}{2 a^{8} x^{\frac {2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} & \text {for}\: a \neq 0 \\\frac {x^{2}}{2 b^{3}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2*a**5*x**(5/3)/(2*a**8*x**(2/3) + 4*a**7*b*x**(1/3) + 2*a**6*b**2) - 5*a**4*b*x**(4/3)/(2*a**8*x**
(2/3) + 4*a**7*b*x**(1/3) + 2*a**6*b**2) + 20*a**3*b**2*x/(2*a**8*x**(2/3) + 4*a**7*b*x**(1/3) + 2*a**6*b**2)
- 60*a**2*b**3*x**(2/3)*log(x**(1/3) + b/a)/(2*a**8*x**(2/3) + 4*a**7*b*x**(1/3) + 2*a**6*b**2) - 120*a*b**4*x
**(1/3)*log(x**(1/3) + b/a)/(2*a**8*x**(2/3) + 4*a**7*b*x**(1/3) + 2*a**6*b**2) - 120*a*b**4*x**(1/3)/(2*a**8*
x**(2/3) + 4*a**7*b*x**(1/3) + 2*a**6*b**2) - 60*b**5*log(x**(1/3) + b/a)/(2*a**8*x**(2/3) + 4*a**7*b*x**(1/3)
+ 2*a**6*b**2) - 90*b**5/(2*a**8*x**(2/3) + 4*a**7*b*x**(1/3) + 2*a**6*b**2), Ne(a, 0)), (x**2/(2*b**3), True
))

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Giac [A]
time = 1.63, size = 79, normalized size = 0.79 \begin {gather*} -\frac {30 \, b^{3} \log \left ({\left | a x^{\frac {1}{3}} + b \right |}\right )}{a^{6}} - \frac {3 \, {\left (10 \, a b^{4} x^{\frac {1}{3}} + 9 \, b^{5}\right )}}{2 \, {\left (a x^{\frac {1}{3}} + b\right )}^{2} a^{6}} + \frac {2 \, a^{6} x - 9 \, a^{5} b x^{\frac {2}{3}} + 36 \, a^{4} b^{2} x^{\frac {1}{3}}}{2 \, a^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-30*b^3*log(abs(a*x^(1/3) + b))/a^6 - 3/2*(10*a*b^4*x^(1/3) + 9*b^5)/((a*x^(1/3) + b)^2*a^6) + 1/2*(2*a^6*x -
9*a^5*b*x^(2/3) + 36*a^4*b^2*x^(1/3))/a^9

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Mupad [B]
time = 0.04, size = 87, normalized size = 0.87 \begin {gather*} \frac {x}{a^3}-\frac {\frac {27\,b^5}{2\,a}+15\,b^4\,x^{1/3}}{a^5\,b^2+a^7\,x^{2/3}+2\,a^6\,b\,x^{1/3}}-\frac {9\,b\,x^{2/3}}{2\,a^4}-\frac {30\,b^3\,\ln \left (b+a\,x^{1/3}\right )}{a^6}+\frac {18\,b^2\,x^{1/3}}{a^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b/x^(1/3))^3,x)

[Out]

x/a^3 - ((27*b^5)/(2*a) + 15*b^4*x^(1/3))/(a^5*b^2 + a^7*x^(2/3) + 2*a^6*b*x^(1/3)) - (9*b*x^(2/3))/(2*a^4) -
(30*b^3*log(b + a*x^(1/3)))/a^6 + (18*b^2*x^(1/3))/a^5

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