3.2376 \(\int \frac {1}{(a+b \sqrt [3]{x})^3} \, dx\)

Optimal. Leaf size=54 \[ -\frac {3 a^2}{2 b^3 \left (a+b \sqrt [3]{x}\right )^2}+\frac {6 a}{b^3 \left (a+b \sqrt [3]{x}\right )}+\frac {3 \log \left (a+b \sqrt [3]{x}\right )}{b^3} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 54, 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 = {190, 43} \[ -\frac {3 a^2}{2 b^3 \left (a+b \sqrt [3]{x}\right )^2}+\frac {6 a}{b^3 \left (a+b \sqrt [3]{x}\right )}+\frac {3 \log \left (a+b \sqrt [3]{x}\right )}{b^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^(-3),x]

[Out]

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

Rule 43

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 190

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+b \sqrt [3]{x}\right )^3} \, dx &=3 \operatorname {Subst}\left (\int \frac {x^2}{(a+b x)^3} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname {Subst}\left (\int \left (\frac {a^2}{b^2 (a+b x)^3}-\frac {2 a}{b^2 (a+b x)^2}+\frac {1}{b^2 (a+b x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {3 a^2}{2 b^3 \left (a+b \sqrt [3]{x}\right )^2}+\frac {6 a}{b^3 \left (a+b \sqrt [3]{x}\right )}+\frac {3 \log \left (a+b \sqrt [3]{x}\right )}{b^3}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 45, normalized size = 0.83 \[ \frac {3 \left (\frac {a \left (3 a+4 b \sqrt [3]{x}\right )}{\left (a+b \sqrt [3]{x}\right )^2}+2 \log \left (a+b \sqrt [3]{x}\right )\right )}{2 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^(-3),x]

[Out]

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

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fricas [B]  time = 0.55, size = 113, normalized size = 2.09 \[ \frac {3 \, {\left (6 \, a^{3} b^{3} x + 3 \, a^{6} + 2 \, {\left (b^{6} x^{2} + 2 \, a^{3} b^{3} x + a^{6}\right )} \log \left (b x^{\frac {1}{3}} + a\right ) + {\left (4 \, a b^{5} x + a^{4} b^{2}\right )} x^{\frac {2}{3}} - {\left (5 \, a^{2} b^{4} x + 2 \, a^{5} b\right )} x^{\frac {1}{3}}\right )}}{2 \, {\left (b^{9} x^{2} + 2 \, a^{3} b^{6} x + a^{6} b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.16, size = 44, normalized size = 0.81 \[ \frac {3 \, \log \left ({\left | b x^{\frac {1}{3}} + a \right |}\right )}{b^{3}} + \frac {3 \, {\left (4 \, a x^{\frac {1}{3}} + \frac {3 \, a^{2}}{b}\right )}}{2 \, {\left (b x^{\frac {1}{3}} + a\right )}^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*log(abs(b*x^(1/3) + a))/b^3 + 3/2*(4*a*x^(1/3) + 3*a^2/b)/((b*x^(1/3) + a)^2*b^2)

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maple [B]  time = 0.08, size = 237, normalized size = 4.39 \[ -\frac {9 a^{6}}{2 \left (b^{3} x +a^{3}\right )^{2} b^{3}}+\frac {2 a x}{\left (b^{2} x^{\frac {2}{3}}-a b \,x^{\frac {1}{3}}+a^{2}\right )^{2}}-\frac {13 a^{2} x^{\frac {2}{3}}}{2 \left (b^{2} x^{\frac {2}{3}}-a b \,x^{\frac {1}{3}}+a^{2}\right )^{2} b}+\frac {5 a^{3} x^{\frac {1}{3}}}{\left (b^{2} x^{\frac {2}{3}}-a b \,x^{\frac {1}{3}}+a^{2}\right )^{2} b^{2}}-\frac {3 a^{4}}{\left (b^{2} x^{\frac {2}{3}}-a b \,x^{\frac {1}{3}}+a^{2}\right )^{2} b^{3}}+\frac {9 a^{3}}{\left (b^{3} x +a^{3}\right ) b^{3}}-\frac {a^{2}}{\left (b \,x^{\frac {1}{3}}+a \right )^{2} b^{3}}+\frac {4 a}{\left (b \,x^{\frac {1}{3}}+a \right ) b^{3}}+\frac {2 \ln \left (b \,x^{\frac {1}{3}}+a \right )}{b^{3}}-\frac {\ln \left (b^{2} x^{\frac {2}{3}}-a b \,x^{\frac {1}{3}}+a^{2}\right )}{b^{3}}+\frac {\ln \left (b^{3} x +a^{3}\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/2*a^6/(b^3*x+a^3)^2/b^3+9/b^3*a^3/(b^3*x+a^3)+ln(b^3*x+a^3)/b^3+2*ln(b*x^(1/3)+a)/b^3-a^2/b^3/(b*x^(1/3)+a)
^2-13/2*a^2/b/(b^2*x^(2/3)-a*b*x^(1/3)+a^2)^2*x^(2/3)+5*a^3/b^2/(b^2*x^(2/3)-a*b*x^(1/3)+a^2)^2*x^(1/3)-3*a^4/
b^3/(b^2*x^(2/3)-a*b*x^(1/3)+a^2)^2-1/b^3*ln(b^2*x^(2/3)-a*b*x^(1/3)+a^2)+4*a/b^3/(b*x^(1/3)+a)+2*a/(b^2*x^(2/
3)-a*b*x^(1/3)+a^2)^2*x

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maxima [A]  time = 0.54, size = 46, normalized size = 0.85 \[ \frac {3 \, \log \left (b x^{\frac {1}{3}} + a\right )}{b^{3}} + \frac {6 \, a}{{\left (b x^{\frac {1}{3}} + a\right )} b^{3}} - \frac {3 \, a^{2}}{2 \, {\left (b x^{\frac {1}{3}} + a\right )}^{2} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.14, size = 53, normalized size = 0.98 \[ \frac {\frac {9\,a^2}{2\,b^3}+\frac {6\,a\,x^{1/3}}{b^2}}{a^2+b^2\,x^{2/3}+2\,a\,b\,x^{1/3}}+\frac {3\,\ln \left (a+b\,x^{1/3}\right )}{b^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*x^(1/3))^3,x)

[Out]

((9*a^2)/(2*b^3) + (6*a*x^(1/3))/b^2)/(a^2 + b^2*x^(2/3) + 2*a*b*x^(1/3)) + (3*log(a + b*x^(1/3)))/b^3

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sympy [A]  time = 0.67, size = 228, normalized size = 4.22 \[ \begin {cases} \frac {6 a^{2} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{2 a^{2} b^{3} + 4 a b^{4} \sqrt [3]{x} + 2 b^{5} x^{\frac {2}{3}}} + \frac {9 a^{2}}{2 a^{2} b^{3} + 4 a b^{4} \sqrt [3]{x} + 2 b^{5} x^{\frac {2}{3}}} + \frac {12 a b \sqrt [3]{x} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{2 a^{2} b^{3} + 4 a b^{4} \sqrt [3]{x} + 2 b^{5} x^{\frac {2}{3}}} + \frac {12 a b \sqrt [3]{x}}{2 a^{2} b^{3} + 4 a b^{4} \sqrt [3]{x} + 2 b^{5} x^{\frac {2}{3}}} + \frac {6 b^{2} x^{\frac {2}{3}} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{2 a^{2} b^{3} + 4 a b^{4} \sqrt [3]{x} + 2 b^{5} x^{\frac {2}{3}}} & \text {for}\: b \neq 0 \\\frac {x}{a^{3}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((6*a**2*log(a/b + x**(1/3))/(2*a**2*b**3 + 4*a*b**4*x**(1/3) + 2*b**5*x**(2/3)) + 9*a**2/(2*a**2*b**
3 + 4*a*b**4*x**(1/3) + 2*b**5*x**(2/3)) + 12*a*b*x**(1/3)*log(a/b + x**(1/3))/(2*a**2*b**3 + 4*a*b**4*x**(1/3
) + 2*b**5*x**(2/3)) + 12*a*b*x**(1/3)/(2*a**2*b**3 + 4*a*b**4*x**(1/3) + 2*b**5*x**(2/3)) + 6*b**2*x**(2/3)*l
og(a/b + x**(1/3))/(2*a**2*b**3 + 4*a*b**4*x**(1/3) + 2*b**5*x**(2/3)), Ne(b, 0)), (x/a**3, True))

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