3.2372 \(\int \frac{1}{(a+b \sqrt [3]{x})^2 x^4} \, dx\)

Optimal. Leaf size=162 \[ \frac{12 b^7}{a^9 x^{2/3}}+\frac{9 b^5}{2 a^7 x^{4/3}}-\frac{3 b^4}{a^6 x^{5/3}}+\frac{2 b^3}{a^5 x^2}-\frac{9 b^2}{7 a^4 x^{7/3}}-\frac{3 b^9}{a^{10} \left (a+b \sqrt [3]{x}\right )}-\frac{27 b^8}{a^{10} \sqrt [3]{x}}-\frac{7 b^6}{a^8 x}+\frac{30 b^9 \log \left (a+b \sqrt [3]{x}\right )}{a^{11}}-\frac{10 b^9 \log (x)}{a^{11}}+\frac{3 b}{4 a^3 x^{8/3}}-\frac{1}{3 a^2 x^3} \]

[Out]

(-3*b^9)/(a^10*(a + b*x^(1/3))) - 1/(3*a^2*x^3) + (3*b)/(4*a^3*x^(8/3)) - (9*b^2)/(7*a^4*x^(7/3)) + (2*b^3)/(a
^5*x^2) - (3*b^4)/(a^6*x^(5/3)) + (9*b^5)/(2*a^7*x^(4/3)) - (7*b^6)/(a^8*x) + (12*b^7)/(a^9*x^(2/3)) - (27*b^8
)/(a^10*x^(1/3)) + (30*b^9*Log[a + b*x^(1/3)])/a^11 - (10*b^9*Log[x])/a^11

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Rubi [A]  time = 0.114986, antiderivative size = 162, 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} \[ \frac{12 b^7}{a^9 x^{2/3}}+\frac{9 b^5}{2 a^7 x^{4/3}}-\frac{3 b^4}{a^6 x^{5/3}}+\frac{2 b^3}{a^5 x^2}-\frac{9 b^2}{7 a^4 x^{7/3}}-\frac{3 b^9}{a^{10} \left (a+b \sqrt [3]{x}\right )}-\frac{27 b^8}{a^{10} \sqrt [3]{x}}-\frac{7 b^6}{a^8 x}+\frac{30 b^9 \log \left (a+b \sqrt [3]{x}\right )}{a^{11}}-\frac{10 b^9 \log (x)}{a^{11}}+\frac{3 b}{4 a^3 x^{8/3}}-\frac{1}{3 a^2 x^3} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x^(1/3))^2*x^4),x]

[Out]

(-3*b^9)/(a^10*(a + b*x^(1/3))) - 1/(3*a^2*x^3) + (3*b)/(4*a^3*x^(8/3)) - (9*b^2)/(7*a^4*x^(7/3)) + (2*b^3)/(a
^5*x^2) - (3*b^4)/(a^6*x^(5/3)) + (9*b^5)/(2*a^7*x^(4/3)) - (7*b^6)/(a^8*x) + (12*b^7)/(a^9*x^(2/3)) - (27*b^8
)/(a^10*x^(1/3)) + (30*b^9*Log[a + b*x^(1/3)])/a^11 - (10*b^9*Log[x])/a^11

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 )^2 x^4} \, dx &=3 \operatorname{Subst}\left (\int \frac{1}{x^{10} (a+b x)^2} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{1}{a^2 x^{10}}-\frac{2 b}{a^3 x^9}+\frac{3 b^2}{a^4 x^8}-\frac{4 b^3}{a^5 x^7}+\frac{5 b^4}{a^6 x^6}-\frac{6 b^5}{a^7 x^5}+\frac{7 b^6}{a^8 x^4}-\frac{8 b^7}{a^9 x^3}+\frac{9 b^8}{a^{10} x^2}-\frac{10 b^9}{a^{11} x}+\frac{b^{10}}{a^{10} (a+b x)^2}+\frac{10 b^{10}}{a^{11} (a+b x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{3 b^9}{a^{10} \left (a+b \sqrt [3]{x}\right )}-\frac{1}{3 a^2 x^3}+\frac{3 b}{4 a^3 x^{8/3}}-\frac{9 b^2}{7 a^4 x^{7/3}}+\frac{2 b^3}{a^5 x^2}-\frac{3 b^4}{a^6 x^{5/3}}+\frac{9 b^5}{2 a^7 x^{4/3}}-\frac{7 b^6}{a^8 x}+\frac{12 b^7}{a^9 x^{2/3}}-\frac{27 b^8}{a^{10} \sqrt [3]{x}}+\frac{30 b^9 \log \left (a+b \sqrt [3]{x}\right )}{a^{11}}-\frac{10 b^9 \log (x)}{a^{11}}\\ \end{align*}

Mathematica [A]  time = 0.20575, size = 154, normalized size = 0.95 \[ -\frac{\frac{a \left (45 a^7 b^2 x^{2/3}+84 a^5 b^4 x^{4/3}-126 a^4 b^5 x^{5/3}+210 a^3 b^6 x^2-420 a^2 b^7 x^{7/3}-60 a^6 b^3 x-35 a^8 b \sqrt [3]{x}+28 a^9+1260 a b^8 x^{8/3}+2520 b^9 x^3\right )}{x^3 \left (a+b \sqrt [3]{x}\right )}-2520 b^9 \log \left (a+b \sqrt [3]{x}\right )+840 b^9 \log (x)}{84 a^{11}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*(28*a^9 - 35*a^8*b*x^(1/3) + 45*a^7*b^2*x^(2/3) - 60*a^6*b^3*x + 84*a^5*b^4*x^(4/3) - 126*a^4*b^5*x^(5/3)
+ 210*a^3*b^6*x^2 - 420*a^2*b^7*x^(7/3) + 1260*a*b^8*x^(8/3) + 2520*b^9*x^3))/((a + b*x^(1/3))*x^3) - 2520*b^
9*Log[a + b*x^(1/3)] + 840*b^9*Log[x])/(84*a^11)

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Maple [A]  time = 0.013, size = 139, normalized size = 0.9 \begin{align*} -3\,{\frac{{b}^{9}}{{a}^{10} \left ( a+b\sqrt [3]{x} \right ) }}-{\frac{1}{3\,{x}^{3}{a}^{2}}}+{\frac{3\,b}{4\,{a}^{3}}{x}^{-{\frac{8}{3}}}}-{\frac{9\,{b}^{2}}{7\,{a}^{4}}{x}^{-{\frac{7}{3}}}}+2\,{\frac{{b}^{3}}{{a}^{5}{x}^{2}}}-3\,{\frac{{b}^{4}}{{a}^{6}{x}^{5/3}}}+{\frac{9\,{b}^{5}}{2\,{a}^{7}}{x}^{-{\frac{4}{3}}}}-7\,{\frac{{b}^{6}}{{a}^{8}x}}+12\,{\frac{{b}^{7}}{{a}^{9}{x}^{2/3}}}-27\,{\frac{{b}^{8}}{{a}^{10}\sqrt [3]{x}}}+30\,{\frac{{b}^{9}\ln \left ( a+b\sqrt [3]{x} \right ) }{{a}^{11}}}-10\,{\frac{{b}^{9}\ln \left ( x \right ) }{{a}^{11}}} \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*b^9/a^10/(a+b*x^(1/3))-1/3/x^3/a^2+3/4*b/a^3/x^(8/3)-9/7*b^2/a^4/x^(7/3)+2*b^3/a^5/x^2-3*b^4/a^6/x^(5/3)+9/
2*b^5/a^7/x^(4/3)-7*b^6/a^8/x+12*b^7/a^9/x^(2/3)-27*b^8/a^10/x^(1/3)+30*b^9*ln(a+b*x^(1/3))/a^11-10*b^9*ln(x)/
a^11

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Maxima [A]  time = 0.99505, size = 193, normalized size = 1.19 \begin{align*} -\frac{2520 \, b^{9} x^{3} + 1260 \, a b^{8} x^{\frac{8}{3}} - 420 \, a^{2} b^{7} x^{\frac{7}{3}} + 210 \, a^{3} b^{6} x^{2} - 126 \, a^{4} b^{5} x^{\frac{5}{3}} + 84 \, a^{5} b^{4} x^{\frac{4}{3}} - 60 \, a^{6} b^{3} x + 45 \, a^{7} b^{2} x^{\frac{2}{3}} - 35 \, a^{8} b x^{\frac{1}{3}} + 28 \, a^{9}}{84 \,{\left (a^{10} b x^{\frac{10}{3}} + a^{11} x^{3}\right )}} + \frac{30 \, b^{9} \log \left (b x^{\frac{1}{3}} + a\right )}{a^{11}} - \frac{10 \, b^{9} \log \left (x\right )}{a^{11}} \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]

-1/84*(2520*b^9*x^3 + 1260*a*b^8*x^(8/3) - 420*a^2*b^7*x^(7/3) + 210*a^3*b^6*x^2 - 126*a^4*b^5*x^(5/3) + 84*a^
5*b^4*x^(4/3) - 60*a^6*b^3*x + 45*a^7*b^2*x^(2/3) - 35*a^8*b*x^(1/3) + 28*a^9)/(a^10*b*x^(10/3) + a^11*x^3) +
30*b^9*log(b*x^(1/3) + a)/a^11 - 10*b^9*log(x)/a^11

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Fricas [A]  time = 1.54737, size = 456, normalized size = 2.81 \begin{align*} -\frac{840 \, a^{3} b^{9} x^{3} + 420 \, a^{6} b^{6} x^{2} - 140 \, a^{9} b^{3} x + 28 \, a^{12} - 2520 \,{\left (b^{12} x^{4} + a^{3} b^{9} x^{3}\right )} \log \left (b x^{\frac{1}{3}} + a\right ) + 2520 \,{\left (b^{12} x^{4} + a^{3} b^{9} x^{3}\right )} \log \left (x^{\frac{1}{3}}\right ) + 18 \,{\left (140 \, a b^{11} x^{3} + 105 \, a^{4} b^{8} x^{2} - 15 \, a^{7} b^{5} x + 6 \, a^{10} b^{2}\right )} x^{\frac{2}{3}} - 63 \,{\left (20 \, a^{2} b^{10} x^{3} + 12 \, a^{5} b^{7} x^{2} - 3 \, a^{8} b^{4} x + a^{11} b\right )} x^{\frac{1}{3}}}{84 \,{\left (a^{11} b^{3} x^{4} + a^{14} 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]

-1/84*(840*a^3*b^9*x^3 + 420*a^6*b^6*x^2 - 140*a^9*b^3*x + 28*a^12 - 2520*(b^12*x^4 + a^3*b^9*x^3)*log(b*x^(1/
3) + a) + 2520*(b^12*x^4 + a^3*b^9*x^3)*log(x^(1/3)) + 18*(140*a*b^11*x^3 + 105*a^4*b^8*x^2 - 15*a^7*b^5*x + 6
*a^10*b^2)*x^(2/3) - 63*(20*a^2*b^10*x^3 + 12*a^5*b^7*x^2 - 3*a^8*b^4*x + a^11*b)*x^(1/3))/(a^11*b^3*x^4 + a^1
4*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 = 1.21403, size = 196, normalized size = 1.21 \begin{align*} \frac{30 \, b^{9} \log \left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{a^{11}} - \frac{10 \, b^{9} \log \left ({\left | x \right |}\right )}{a^{11}} - \frac{2520 \, a b^{9} x^{3} + 1260 \, a^{2} b^{8} x^{\frac{8}{3}} - 420 \, a^{3} b^{7} x^{\frac{7}{3}} + 210 \, a^{4} b^{6} x^{2} - 126 \, a^{5} b^{5} x^{\frac{5}{3}} + 84 \, a^{6} b^{4} x^{\frac{4}{3}} - 60 \, a^{7} b^{3} x + 45 \, a^{8} b^{2} x^{\frac{2}{3}} - 35 \, a^{9} b x^{\frac{1}{3}} + 28 \, a^{10}}{84 \,{\left (b x^{\frac{1}{3}} + a\right )} a^{11} x^{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]

30*b^9*log(abs(b*x^(1/3) + a))/a^11 - 10*b^9*log(abs(x))/a^11 - 1/84*(2520*a*b^9*x^3 + 1260*a^2*b^8*x^(8/3) -
420*a^3*b^7*x^(7/3) + 210*a^4*b^6*x^2 - 126*a^5*b^5*x^(5/3) + 84*a^6*b^4*x^(4/3) - 60*a^7*b^3*x + 45*a^8*b^2*x
^(2/3) - 35*a^9*b*x^(1/3) + 28*a^10)/((b*x^(1/3) + a)*a^11*x^3)