3.2434 \(\int \frac{1}{(a+\frac{b}{\sqrt [3]{x}})^2 x^3} \, dx\)

Optimal. Leaf size=97 \[ -\frac{9 a^2}{2 b^4 x^{2/3}}+\frac{3 a^4}{b^5 \left (a \sqrt [3]{x}+b\right )}+\frac{12 a^3}{b^5 \sqrt [3]{x}}-\frac{15 a^4 \log \left (a \sqrt [3]{x}+b\right )}{b^6}+\frac{5 a^4 \log (x)}{b^6}+\frac{2 a}{b^3 x}-\frac{3}{4 b^2 x^{4/3}} \]

[Out]

(3*a^4)/(b^5*(b + a*x^(1/3))) - 3/(4*b^2*x^(4/3)) + (2*a)/(b^3*x) - (9*a^2)/(2*b^4*x^(2/3)) + (12*a^3)/(b^5*x^
(1/3)) - (15*a^4*Log[b + a*x^(1/3)])/b^6 + (5*a^4*Log[x])/b^6

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Rubi [A]  time = 0.0647476, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {263, 266, 44} \[ -\frac{9 a^2}{2 b^4 x^{2/3}}+\frac{3 a^4}{b^5 \left (a \sqrt [3]{x}+b\right )}+\frac{12 a^3}{b^5 \sqrt [3]{x}}-\frac{15 a^4 \log \left (a \sqrt [3]{x}+b\right )}{b^6}+\frac{5 a^4 \log (x)}{b^6}+\frac{2 a}{b^3 x}-\frac{3}{4 b^2 x^{4/3}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^(1/3))^2*x^3),x]

[Out]

(3*a^4)/(b^5*(b + a*x^(1/3))) - 3/(4*b^2*x^(4/3)) + (2*a)/(b^3*x) - (9*a^2)/(2*b^4*x^(2/3)) + (12*a^3)/(b^5*x^
(1/3)) - (15*a^4*Log[b + a*x^(1/3)])/b^6 + (5*a^4*Log[x])/b^6

Rule 263

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 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+\frac{b}{\sqrt [3]{x}}\right )^2 x^3} \, dx &=\int \frac{1}{\left (b+a \sqrt [3]{x}\right )^2 x^{7/3}} \, dx\\ &=3 \operatorname{Subst}\left (\int \frac{1}{x^5 (b+a x)^2} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{1}{b^2 x^5}-\frac{2 a}{b^3 x^4}+\frac{3 a^2}{b^4 x^3}-\frac{4 a^3}{b^5 x^2}+\frac{5 a^4}{b^6 x}-\frac{a^5}{b^5 (b+a x)^2}-\frac{5 a^5}{b^6 (b+a x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{3 a^4}{b^5 \left (b+a \sqrt [3]{x}\right )}-\frac{3}{4 b^2 x^{4/3}}+\frac{2 a}{b^3 x}-\frac{9 a^2}{2 b^4 x^{2/3}}+\frac{12 a^3}{b^5 \sqrt [3]{x}}-\frac{15 a^4 \log \left (b+a \sqrt [3]{x}\right )}{b^6}+\frac{5 a^4 \log (x)}{b^6}\\ \end{align*}

Mathematica [A]  time = 0.0962485, size = 82, normalized size = 0.85 \[ -\frac{\frac{18 a^2 b^2}{x^{2/3}}+\frac{12 a^5}{a+\frac{b}{\sqrt [3]{x}}}-\frac{48 a^3 b}{\sqrt [3]{x}}+60 a^4 \log \left (a+\frac{b}{\sqrt [3]{x}}\right )-\frac{8 a b^3}{x}+\frac{3 b^4}{x^{4/3}}}{4 b^6} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^(1/3))^2*x^3),x]

[Out]

-((12*a^5)/(a + b/x^(1/3)) + (3*b^4)/x^(4/3) - (8*a*b^3)/x + (18*a^2*b^2)/x^(2/3) - (48*a^3*b)/x^(1/3) + 60*a^
4*Log[a + b/x^(1/3)])/(4*b^6)

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Maple [A]  time = 0.01, size = 84, normalized size = 0.9 \begin{align*} 3\,{\frac{{a}^{4}}{{b}^{5} \left ( b+a\sqrt [3]{x} \right ) }}-{\frac{3}{4\,{b}^{2}}{x}^{-{\frac{4}{3}}}}+2\,{\frac{a}{{b}^{3}x}}-{\frac{9\,{a}^{2}}{2\,{b}^{4}}{x}^{-{\frac{2}{3}}}}+12\,{\frac{{a}^{3}}{{b}^{5}\sqrt [3]{x}}}-15\,{\frac{{a}^{4}\ln \left ( b+a\sqrt [3]{x} \right ) }{{b}^{6}}}+5\,{\frac{{a}^{4}\ln \left ( x \right ) }{{b}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x^(1/3))^2/x^3,x)

[Out]

3*a^4/b^5/(b+a*x^(1/3))-3/4/b^2/x^(4/3)+2*a/b^3/x-9/2*a^2/b^4/x^(2/3)+12*a^3/b^5/x^(1/3)-15*a^4*ln(b+a*x^(1/3)
)/b^6+5*a^4*ln(x)/b^6

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Maxima [A]  time = 0.978025, size = 128, normalized size = 1.32 \begin{align*} -\frac{15 \, a^{4} \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{b^{6}} - \frac{3 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{4}}{4 \, b^{6}} + \frac{5 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{3} a}{b^{6}} - \frac{15 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{2} a^{2}}{b^{6}} + \frac{30 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )} a^{3}}{b^{6}} - \frac{3 \, a^{5}}{{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )} b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^(1/3))^2/x^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.56435, size = 321, normalized size = 3.31 \begin{align*} \frac{20 \, a^{4} b^{3} x^{2} + 8 \, a b^{6} x - 60 \,{\left (a^{7} x^{3} + a^{4} b^{3} x^{2}\right )} \log \left (a x^{\frac{1}{3}} + b\right ) + 60 \,{\left (a^{7} x^{3} + a^{4} b^{3} x^{2}\right )} \log \left (x^{\frac{1}{3}}\right ) + 3 \,{\left (20 \, a^{6} b x^{2} + 15 \, a^{3} b^{4} x - b^{7}\right )} x^{\frac{2}{3}} - 6 \,{\left (5 \, a^{5} b^{2} x^{2} + 3 \, a^{2} b^{5} x\right )} x^{\frac{1}{3}}}{4 \,{\left (a^{3} b^{6} x^{3} + b^{9} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^(1/3))^2/x^3,x, algorithm="fricas")

[Out]

1/4*(20*a^4*b^3*x^2 + 8*a*b^6*x - 60*(a^7*x^3 + a^4*b^3*x^2)*log(a*x^(1/3) + b) + 60*(a^7*x^3 + a^4*b^3*x^2)*l
og(x^(1/3)) + 3*(20*a^6*b*x^2 + 15*a^3*b^4*x - b^7)*x^(2/3) - 6*(5*a^5*b^2*x^2 + 3*a^2*b^5*x)*x^(1/3))/(a^3*b^
6*x^3 + b^9*x^2)

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Sympy [A]  time = 19.9769, size = 340, normalized size = 3.51 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{\frac{4}{3}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{3}{4 b^{2} x^{\frac{4}{3}}} & \text{for}\: a = 0 \\- \frac{1}{2 a^{2} x^{2}} & \text{for}\: b = 0 \\\frac{20 a^{5} x^{3} \log{\left (x \right )}}{4 a b^{6} x^{3} + 4 b^{7} x^{\frac{8}{3}}} - \frac{60 a^{5} x^{3} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{4 a b^{6} x^{3} + 4 b^{7} x^{\frac{8}{3}}} + \frac{20 a^{4} b x^{\frac{8}{3}} \log{\left (x \right )}}{4 a b^{6} x^{3} + 4 b^{7} x^{\frac{8}{3}}} - \frac{60 a^{4} b x^{\frac{8}{3}} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{4 a b^{6} x^{3} + 4 b^{7} x^{\frac{8}{3}}} + \frac{60 a^{4} b x^{\frac{8}{3}}}{4 a b^{6} x^{3} + 4 b^{7} x^{\frac{8}{3}}} + \frac{30 a^{3} b^{2} x^{\frac{7}{3}}}{4 a b^{6} x^{3} + 4 b^{7} x^{\frac{8}{3}}} - \frac{10 a^{2} b^{3} x^{2}}{4 a b^{6} x^{3} + 4 b^{7} x^{\frac{8}{3}}} + \frac{5 a b^{4} x^{\frac{5}{3}}}{4 a b^{6} x^{3} + 4 b^{7} x^{\frac{8}{3}}} - \frac{3 b^{5} x^{\frac{4}{3}}}{4 a b^{6} x^{3} + 4 b^{7} x^{\frac{8}{3}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x**(1/3))**2/x**3,x)

[Out]

Piecewise((zoo/x**(4/3), Eq(a, 0) & Eq(b, 0)), (-3/(4*b**2*x**(4/3)), Eq(a, 0)), (-1/(2*a**2*x**2), Eq(b, 0)),
(20*a**5*x**3*log(x)/(4*a*b**6*x**3 + 4*b**7*x**(8/3)) - 60*a**5*x**3*log(x**(1/3) + b/a)/(4*a*b**6*x**3 + 4*
b**7*x**(8/3)) + 20*a**4*b*x**(8/3)*log(x)/(4*a*b**6*x**3 + 4*b**7*x**(8/3)) - 60*a**4*b*x**(8/3)*log(x**(1/3)
+ b/a)/(4*a*b**6*x**3 + 4*b**7*x**(8/3)) + 60*a**4*b*x**(8/3)/(4*a*b**6*x**3 + 4*b**7*x**(8/3)) + 30*a**3*b**
2*x**(7/3)/(4*a*b**6*x**3 + 4*b**7*x**(8/3)) - 10*a**2*b**3*x**2/(4*a*b**6*x**3 + 4*b**7*x**(8/3)) + 5*a*b**4*
x**(5/3)/(4*a*b**6*x**3 + 4*b**7*x**(8/3)) - 3*b**5*x**(4/3)/(4*a*b**6*x**3 + 4*b**7*x**(8/3)), True))

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Giac [A]  time = 1.16705, size = 122, normalized size = 1.26 \begin{align*} -\frac{15 \, a^{4} \log \left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{b^{6}} + \frac{5 \, a^{4} \log \left ({\left | x \right |}\right )}{b^{6}} + \frac{60 \, a^{4} b x^{\frac{4}{3}} + 30 \, a^{3} b^{2} x - 10 \, a^{2} b^{3} x^{\frac{2}{3}} + 5 \, a b^{4} x^{\frac{1}{3}} - 3 \, b^{5}}{4 \,{\left (a x^{\frac{1}{3}} + b\right )} b^{6} x^{\frac{4}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^(1/3))^2/x^3,x, algorithm="giac")

[Out]

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