3.2362 \(\int \frac{1}{(a+b \sqrt [3]{x}) x^3} \, dx\)

Optimal. Leaf size=104 \[ -\frac{3 b^4}{2 a^5 x^{2/3}}-\frac{3 b^2}{4 a^3 x^{4/3}}+\frac{3 b^5}{a^6 \sqrt [3]{x}}+\frac{b^3}{a^4 x}-\frac{3 b^6 \log \left (a+b \sqrt [3]{x}\right )}{a^7}+\frac{b^6 \log (x)}{a^7}+\frac{3 b}{5 a^2 x^{5/3}}-\frac{1}{2 a x^2} \]

[Out]

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

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Rubi [A]  time = 0.0537924, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 44} \[ -\frac{3 b^4}{2 a^5 x^{2/3}}-\frac{3 b^2}{4 a^3 x^{4/3}}+\frac{3 b^5}{a^6 \sqrt [3]{x}}+\frac{b^3}{a^4 x}-\frac{3 b^6 \log \left (a+b \sqrt [3]{x}\right )}{a^7}+\frac{b^6 \log (x)}{a^7}+\frac{3 b}{5 a^2 x^{5/3}}-\frac{1}{2 a x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x^(1/3))*x^3),x]

[Out]

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

Rule 266

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]]

Rule 44

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] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{\left (a+b \sqrt [3]{x}\right ) x^3} \, dx &=3 \operatorname{Subst}\left (\int \frac{1}{x^7 (a+b x)} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{1}{a x^7}-\frac{b}{a^2 x^6}+\frac{b^2}{a^3 x^5}-\frac{b^3}{a^4 x^4}+\frac{b^4}{a^5 x^3}-\frac{b^5}{a^6 x^2}+\frac{b^6}{a^7 x}-\frac{b^7}{a^7 (a+b x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{1}{2 a x^2}+\frac{3 b}{5 a^2 x^{5/3}}-\frac{3 b^2}{4 a^3 x^{4/3}}+\frac{b^3}{a^4 x}-\frac{3 b^4}{2 a^5 x^{2/3}}+\frac{3 b^5}{a^6 \sqrt [3]{x}}-\frac{3 b^6 \log \left (a+b \sqrt [3]{x}\right )}{a^7}+\frac{b^6 \log (x)}{a^7}\\ \end{align*}

Mathematica [A]  time = 0.0619384, size = 95, normalized size = 0.91 \[ \frac{\frac{a \left (-15 a^3 b^2 x^{2/3}+20 a^2 b^3 x+12 a^4 b \sqrt [3]{x}-10 a^5-30 a b^4 x^{4/3}+60 b^5 x^{5/3}\right )}{x^2}-60 b^6 \log \left (a+b \sqrt [3]{x}\right )+20 b^6 \log (x)}{20 a^7} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x^(1/3))*x^3),x]

[Out]

((a*(-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
- 60*b^6*Log[a + b*x^(1/3)] + 20*b^6*Log[x])/(20*a^7)

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Maple [A]  time = 0.008, size = 87, normalized size = 0.8 \begin{align*} -{\frac{1}{2\,a{x}^{2}}}+{\frac{3\,b}{5\,{a}^{2}}{x}^{-{\frac{5}{3}}}}-{\frac{3\,{b}^{2}}{4\,{a}^{3}}{x}^{-{\frac{4}{3}}}}+{\frac{{b}^{3}}{{a}^{4}x}}-{\frac{3\,{b}^{4}}{2\,{a}^{5}}{x}^{-{\frac{2}{3}}}}+3\,{\frac{{b}^{5}}{{a}^{6}\sqrt [3]{x}}}-3\,{\frac{{b}^{6}\ln \left ( a+b\sqrt [3]{x} \right ) }{{a}^{7}}}+{\frac{{b}^{6}\ln \left ( x \right ) }{{a}^{7}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.99467, size = 116, normalized size = 1.12 \begin{align*} -\frac{3 \, b^{6} \log \left (b x^{\frac{1}{3}} + a\right )}{a^{7}} + \frac{b^{6} \log \left (x\right )}{a^{7}} + \frac{60 \, b^{5} x^{\frac{5}{3}} - 30 \, a b^{4} x^{\frac{4}{3}} + 20 \, a^{2} b^{3} x - 15 \, a^{3} b^{2} x^{\frac{2}{3}} + 12 \, a^{4} b x^{\frac{1}{3}} - 10 \, a^{5}}{20 \, a^{6} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.4483, size = 230, normalized size = 2.21 \begin{align*} -\frac{60 \, b^{6} x^{2} \log \left (b x^{\frac{1}{3}} + a\right ) - 60 \, b^{6} x^{2} \log \left (x^{\frac{1}{3}}\right ) - 20 \, a^{3} b^{3} x + 10 \, a^{6} - 15 \,{\left (4 \, a b^{5} x - a^{4} b^{2}\right )} x^{\frac{2}{3}} + 6 \,{\left (5 \, a^{2} b^{4} x - 2 \, a^{5} b\right )} x^{\frac{1}{3}}}{20 \, a^{7} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 10.5118, size = 129, normalized size = 1.24 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{\frac{7}{3}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{3}{7 b x^{\frac{7}{3}}} & \text{for}\: a = 0 \\- \frac{1}{2 a x^{2}} & \text{for}\: b = 0 \\- \frac{1}{2 a x^{2}} + \frac{3 b}{5 a^{2} x^{\frac{5}{3}}} - \frac{3 b^{2}}{4 a^{3} x^{\frac{4}{3}}} + \frac{b^{3}}{a^{4} x} - \frac{3 b^{4}}{2 a^{5} x^{\frac{2}{3}}} + \frac{3 b^{5}}{a^{6} \sqrt [3]{x}} + \frac{b^{6} \log{\left (x \right )}}{a^{7}} - \frac{3 b^{6} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{a^{7}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.17873, size = 123, normalized size = 1.18 \begin{align*} -\frac{3 \, b^{6} \log \left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{a^{7}} + \frac{b^{6} \log \left ({\left | x \right |}\right )}{a^{7}} + \frac{60 \, a b^{5} x^{\frac{5}{3}} - 30 \, a^{2} b^{4} x^{\frac{4}{3}} + 20 \, a^{3} b^{3} x - 15 \, a^{4} b^{2} x^{\frac{2}{3}} + 12 \, a^{5} b x^{\frac{1}{3}} - 10 \, a^{6}}{20 \, a^{7} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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