3.2435 \(\int \frac{1}{(a+\frac{b}{\sqrt [3]{x}})^2 x^4} \, dx\)

Optimal. Leaf size=133 \[ \frac{9 a^5}{b^7 x^{2/3}}+\frac{3 a^3}{b^5 x^{4/3}}-\frac{9 a^2}{5 b^4 x^{5/3}}-\frac{3 a^7}{b^8 \left (a \sqrt [3]{x}+b\right )}-\frac{21 a^6}{b^8 \sqrt [3]{x}}-\frac{5 a^4}{b^6 x}+\frac{24 a^7 \log \left (a \sqrt [3]{x}+b\right )}{b^9}-\frac{8 a^7 \log (x)}{b^9}+\frac{a}{b^3 x^2}-\frac{3}{7 b^2 x^{7/3}} \]

[Out]

(-3*a^7)/(b^8*(b + a*x^(1/3))) - 3/(7*b^2*x^(7/3)) + a/(b^3*x^2) - (9*a^2)/(5*b^4*x^(5/3)) + (3*a^3)/(b^5*x^(4
/3)) - (5*a^4)/(b^6*x) + (9*a^5)/(b^7*x^(2/3)) - (21*a^6)/(b^8*x^(1/3)) + (24*a^7*Log[b + a*x^(1/3)])/b^9 - (8
*a^7*Log[x])/b^9

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Rubi [A]  time = 0.103239, antiderivative size = 133, 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} \[ \frac{9 a^5}{b^7 x^{2/3}}+\frac{3 a^3}{b^5 x^{4/3}}-\frac{9 a^2}{5 b^4 x^{5/3}}-\frac{3 a^7}{b^8 \left (a \sqrt [3]{x}+b\right )}-\frac{21 a^6}{b^8 \sqrt [3]{x}}-\frac{5 a^4}{b^6 x}+\frac{24 a^7 \log \left (a \sqrt [3]{x}+b\right )}{b^9}-\frac{8 a^7 \log (x)}{b^9}+\frac{a}{b^3 x^2}-\frac{3}{7 b^2 x^{7/3}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^(1/3))^2*x^4),x]

[Out]

(-3*a^7)/(b^8*(b + a*x^(1/3))) - 3/(7*b^2*x^(7/3)) + a/(b^3*x^2) - (9*a^2)/(5*b^4*x^(5/3)) + (3*a^3)/(b^5*x^(4
/3)) - (5*a^4)/(b^6*x) + (9*a^5)/(b^7*x^(2/3)) - (21*a^6)/(b^8*x^(1/3)) + (24*a^7*Log[b + a*x^(1/3)])/b^9 - (8
*a^7*Log[x])/b^9

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^4} \, dx &=\int \frac{1}{\left (b+a \sqrt [3]{x}\right )^2 x^{10/3}} \, dx\\ &=3 \operatorname{Subst}\left (\int \frac{1}{x^8 (b+a x)^2} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{1}{b^2 x^8}-\frac{2 a}{b^3 x^7}+\frac{3 a^2}{b^4 x^6}-\frac{4 a^3}{b^5 x^5}+\frac{5 a^4}{b^6 x^4}-\frac{6 a^5}{b^7 x^3}+\frac{7 a^6}{b^8 x^2}-\frac{8 a^7}{b^9 x}+\frac{a^8}{b^8 (b+a x)^2}+\frac{8 a^8}{b^9 (b+a x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{3 a^7}{b^8 \left (b+a \sqrt [3]{x}\right )}-\frac{3}{7 b^2 x^{7/3}}+\frac{a}{b^3 x^2}-\frac{9 a^2}{5 b^4 x^{5/3}}+\frac{3 a^3}{b^5 x^{4/3}}-\frac{5 a^4}{b^6 x}+\frac{9 a^5}{b^7 x^{2/3}}-\frac{21 a^6}{b^8 \sqrt [3]{x}}+\frac{24 a^7 \log \left (b+a \sqrt [3]{x}\right )}{b^9}-\frac{8 a^7 \log (x)}{b^9}\\ \end{align*}

Mathematica [A]  time = 0.114606, size = 123, normalized size = 0.92 \[ \frac{9 a^5}{b^7 x^{2/3}}+\frac{3 a^3}{b^5 x^{4/3}}-\frac{9 a^2}{5 b^4 x^{5/3}}+\frac{3 a^8}{b^9 \left (a+\frac{b}{\sqrt [3]{x}}\right )}-\frac{21 a^6}{b^8 \sqrt [3]{x}}-\frac{5 a^4}{b^6 x}+\frac{24 a^7 \log \left (a+\frac{b}{\sqrt [3]{x}}\right )}{b^9}+\frac{a}{b^3 x^2}-\frac{3}{7 b^2 x^{7/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*a^8)/(b^9*(a + b/x^(1/3))) - 3/(7*b^2*x^(7/3)) + a/(b^3*x^2) - (9*a^2)/(5*b^4*x^(5/3)) + (3*a^3)/(b^5*x^(4/
3)) - (5*a^4)/(b^6*x) + (9*a^5)/(b^7*x^(2/3)) - (21*a^6)/(b^8*x^(1/3)) + (24*a^7*Log[a + b/x^(1/3)])/b^9

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Maple [A]  time = 0.012, size = 116, normalized size = 0.9 \begin{align*} -3\,{\frac{{a}^{7}}{{b}^{8} \left ( b+a\sqrt [3]{x} \right ) }}-{\frac{3}{7\,{b}^{2}}{x}^{-{\frac{7}{3}}}}+{\frac{a}{{b}^{3}{x}^{2}}}-{\frac{9\,{a}^{2}}{5\,{b}^{4}}{x}^{-{\frac{5}{3}}}}+3\,{\frac{{a}^{3}}{{b}^{5}{x}^{4/3}}}-5\,{\frac{{a}^{4}}{{b}^{6}x}}+9\,{\frac{{a}^{5}}{{b}^{7}{x}^{2/3}}}-21\,{\frac{{a}^{6}}{{b}^{8}\sqrt [3]{x}}}+24\,{\frac{{a}^{7}\ln \left ( b+a\sqrt [3]{x} \right ) }{{b}^{9}}}-8\,{\frac{{a}^{7}\ln \left ( x \right ) }{{b}^{9}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x^(1/3))^2/x^4,x)

[Out]

-3*a^7/b^8/(b+a*x^(1/3))-3/7/b^2/x^(7/3)+a/b^3/x^2-9/5*a^2/b^4/x^(5/3)+3*a^3/b^5/x^(4/3)-5*a^4/b^6/x+9*a^5/b^7
/x^(2/3)-21*a^6/b^8/x^(1/3)+24*a^7*ln(b+a*x^(1/3))/b^9-8*a^7*ln(x)/b^9

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Maxima [A]  time = 1.01337, size = 197, normalized size = 1.48 \begin{align*} \frac{24 \, a^{7} \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{b^{9}} - \frac{3 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{7}}{7 \, b^{9}} + \frac{4 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{6} a}{b^{9}} - \frac{84 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{5} a^{2}}{5 \, b^{9}} + \frac{42 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{4} a^{3}}{b^{9}} - \frac{70 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{3} a^{4}}{b^{9}} + \frac{84 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{2} a^{5}}{b^{9}} - \frac{84 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )} a^{6}}{b^{9}} + \frac{3 \, a^{8}}{{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )} b^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^(1/3))^2/x^4,x, algorithm="maxima")

[Out]

24*a^7*log(a + b/x^(1/3))/b^9 - 3/7*(a + b/x^(1/3))^7/b^9 + 4*(a + b/x^(1/3))^6*a/b^9 - 84/5*(a + b/x^(1/3))^5
*a^2/b^9 + 42*(a + b/x^(1/3))^4*a^3/b^9 - 70*(a + b/x^(1/3))^3*a^4/b^9 + 84*(a + b/x^(1/3))^2*a^5/b^9 - 84*(a
+ b/x^(1/3))*a^6/b^9 + 3*a^8/((a + b/x^(1/3))*b^9)

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Fricas [A]  time = 1.54227, size = 408, normalized size = 3.07 \begin{align*} -\frac{280 \, a^{7} b^{3} x^{3} + 140 \, a^{4} b^{6} x^{2} - 35 \, a b^{9} x - 840 \,{\left (a^{10} x^{4} + a^{7} b^{3} x^{3}\right )} \log \left (a x^{\frac{1}{3}} + b\right ) + 840 \,{\left (a^{10} x^{4} + a^{7} b^{3} x^{3}\right )} \log \left (x^{\frac{1}{3}}\right ) + 15 \,{\left (56 \, a^{9} b x^{3} + 42 \, a^{6} b^{4} x^{2} - 6 \, a^{3} b^{7} x + b^{10}\right )} x^{\frac{2}{3}} - 21 \,{\left (20 \, a^{8} b^{2} x^{3} + 12 \, a^{5} b^{5} x^{2} - 3 \, a^{2} b^{8} x\right )} x^{\frac{1}{3}}}{35 \,{\left (a^{3} b^{9} x^{4} + b^{12} x^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^(1/3))^2/x^4,x, algorithm="fricas")

[Out]

-1/35*(280*a^7*b^3*x^3 + 140*a^4*b^6*x^2 - 35*a*b^9*x - 840*(a^10*x^4 + a^7*b^3*x^3)*log(a*x^(1/3) + b) + 840*
(a^10*x^4 + a^7*b^3*x^3)*log(x^(1/3)) + 15*(56*a^9*b*x^3 + 42*a^6*b^4*x^2 - 6*a^3*b^7*x + b^10)*x^(2/3) - 21*(
20*a^8*b^2*x^3 + 12*a^5*b^5*x^2 - 3*a^2*b^8*x)*x^(1/3))/(a^3*b^9*x^4 + b^12*x^3)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x**(1/3))**2/x**4,x)

[Out]

Timed out

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Giac [A]  time = 1.20133, size = 166, normalized size = 1.25 \begin{align*} \frac{24 \, a^{7} \log \left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{b^{9}} - \frac{8 \, a^{7} \log \left ({\left | x \right |}\right )}{b^{9}} - \frac{840 \, a^{7} b x^{\frac{7}{3}} + 420 \, a^{6} b^{2} x^{2} - 140 \, a^{5} b^{3} x^{\frac{5}{3}} + 70 \, a^{4} b^{4} x^{\frac{4}{3}} - 42 \, a^{3} b^{5} x + 28 \, a^{2} b^{6} x^{\frac{2}{3}} - 20 \, a b^{7} x^{\frac{1}{3}} + 15 \, b^{8}}{35 \,{\left (a x^{\frac{1}{3}} + b\right )} b^{9} x^{\frac{7}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^(1/3))^2/x^4,x, algorithm="giac")

[Out]

24*a^7*log(abs(a*x^(1/3) + b))/b^9 - 8*a^7*log(abs(x))/b^9 - 1/35*(840*a^7*b*x^(7/3) + 420*a^6*b^2*x^2 - 140*a
^5*b^3*x^(5/3) + 70*a^4*b^4*x^(4/3) - 42*a^3*b^5*x + 28*a^2*b^6*x^(2/3) - 20*a*b^7*x^(1/3) + 15*b^8)/((a*x^(1/
3) + b)*b^9*x^(7/3))