3.2359 \(\int \frac {1}{a+b \sqrt [3]{x}} \, dx\)

Optimal. Leaf size=42 \[ \frac {3 a^2 \log \left (a+b \sqrt [3]{x}\right )}{b^3}-\frac {3 a \sqrt [3]{x}}{b^2}+\frac {3 x^{2/3}}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 42, 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} \[ \frac {3 a^2 \log \left (a+b \sqrt [3]{x}\right )}{b^3}-\frac {3 a \sqrt [3]{x}}{b^2}+\frac {3 x^{2/3}}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^(-1),x]

[Out]

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

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Mathematica [A]  time = 0.02, size = 42, normalized size = 1.00 \[ \frac {3 a^2 \log \left (a+b \sqrt [3]{x}\right )}{b^3}-\frac {3 a \sqrt [3]{x}}{b^2}+\frac {3 x^{2/3}}{2 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^(-1),x]

[Out]

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

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fricas [A]  time = 0.53, size = 33, normalized size = 0.79 \[ \frac {3 \, {\left (2 \, a^{2} \log \left (b x^{\frac {1}{3}} + a\right ) + b^{2} x^{\frac {2}{3}} - 2 \, a b x^{\frac {1}{3}}\right )}}{2 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.16, size = 35, normalized size = 0.83 \[ \frac {3 \, a^{2} \log \left ({\left | b x^{\frac {1}{3}} + a \right |}\right )}{b^{3}} + \frac {3 \, {\left (b x^{\frac {2}{3}} - 2 \, a x^{\frac {1}{3}}\right )}}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.02, size = 79, normalized size = 1.88 \[ \frac {2 a^{2} \ln \left (b \,x^{\frac {1}{3}}+a \right )}{b^{3}}-\frac {a^{2} \ln \left (b^{2} x^{\frac {2}{3}}-a b \,x^{\frac {1}{3}}+a^{2}\right )}{b^{3}}+\frac {a^{2} \ln \left (b^{3} x +a^{3}\right )}{b^{3}}+\frac {3 x^{\frac {2}{3}}}{2 b}-\frac {3 a \,x^{\frac {1}{3}}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.45, size = 44, normalized size = 1.05 \[ \frac {3 \, a^{2} \log \left (b x^{\frac {1}{3}} + a\right )}{b^{3}} + \frac {3 \, {\left (b x^{\frac {1}{3}} + a\right )}^{2}}{2 \, b^{3}} - \frac {6 \, {\left (b x^{\frac {1}{3}} + a\right )} a}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.04, size = 34, normalized size = 0.81 \[ \frac {6\,a^2\,\ln \left (a+b\,x^{1/3}\right )+3\,b^2\,x^{2/3}-6\,a\,b\,x^{1/3}}{2\,b^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*x^(1/3)),x)

[Out]

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

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sympy [A]  time = 0.19, size = 42, normalized size = 1.00 \[ \begin {cases} \frac {3 a^{2} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{b^{3}} - \frac {3 a \sqrt [3]{x}}{b^{2}} + \frac {3 x^{\frac {2}{3}}}{2 b} & \text {for}\: b \neq 0 \\\frac {x}{a} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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