3.2380 \(\int \frac{1}{(a+b \sqrt [3]{x})^3 x^4} \, dx\)

Optimal. Leaf size=183 \[ \frac{54 b^7}{a^{10} x^{2/3}}+\frac{63 b^5}{4 a^8 x^{4/3}}-\frac{9 b^4}{a^7 x^{5/3}}+\frac{5 b^3}{a^6 x^2}-\frac{18 b^2}{7 a^5 x^{7/3}}-\frac{30 b^9}{a^{11} \left (a+b \sqrt [3]{x}\right )}-\frac{3 b^9}{2 a^{10} \left (a+b \sqrt [3]{x}\right )^2}-\frac{135 b^8}{a^{11} \sqrt [3]{x}}-\frac{28 b^6}{a^9 x}+\frac{165 b^9 \log \left (a+b \sqrt [3]{x}\right )}{a^{12}}-\frac{55 b^9 \log (x)}{a^{12}}+\frac{9 b}{8 a^4 x^{8/3}}-\frac{1}{3 a^3 x^3} \]

[Out]

(-3*b^9)/(2*a^10*(a + b*x^(1/3))^2) - (30*b^9)/(a^11*(a + b*x^(1/3))) - 1/(3*a^3*x^3) + (9*b)/(8*a^4*x^(8/3))
- (18*b^2)/(7*a^5*x^(7/3)) + (5*b^3)/(a^6*x^2) - (9*b^4)/(a^7*x^(5/3)) + (63*b^5)/(4*a^8*x^(4/3)) - (28*b^6)/(
a^9*x) + (54*b^7)/(a^10*x^(2/3)) - (135*b^8)/(a^11*x^(1/3)) + (165*b^9*Log[a + b*x^(1/3)])/a^12 - (55*b^9*Log[
x])/a^12

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Rubi [A]  time = 0.139836, antiderivative size = 183, 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} \[ \frac{54 b^7}{a^{10} x^{2/3}}+\frac{63 b^5}{4 a^8 x^{4/3}}-\frac{9 b^4}{a^7 x^{5/3}}+\frac{5 b^3}{a^6 x^2}-\frac{18 b^2}{7 a^5 x^{7/3}}-\frac{30 b^9}{a^{11} \left (a+b \sqrt [3]{x}\right )}-\frac{3 b^9}{2 a^{10} \left (a+b \sqrt [3]{x}\right )^2}-\frac{135 b^8}{a^{11} \sqrt [3]{x}}-\frac{28 b^6}{a^9 x}+\frac{165 b^9 \log \left (a+b \sqrt [3]{x}\right )}{a^{12}}-\frac{55 b^9 \log (x)}{a^{12}}+\frac{9 b}{8 a^4 x^{8/3}}-\frac{1}{3 a^3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x^(1/3))^3*x^4),x]

[Out]

(-3*b^9)/(2*a^10*(a + b*x^(1/3))^2) - (30*b^9)/(a^11*(a + b*x^(1/3))) - 1/(3*a^3*x^3) + (9*b)/(8*a^4*x^(8/3))
- (18*b^2)/(7*a^5*x^(7/3)) + (5*b^3)/(a^6*x^2) - (9*b^4)/(a^7*x^(5/3)) + (63*b^5)/(4*a^8*x^(4/3)) - (28*b^6)/(
a^9*x) + (54*b^7)/(a^10*x^(2/3)) - (135*b^8)/(a^11*x^(1/3)) + (165*b^9*Log[a + b*x^(1/3)])/a^12 - (55*b^9*Log[
x])/a^12

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 )^3 x^4} \, dx &=3 \operatorname{Subst}\left (\int \frac{1}{x^{10} (a+b x)^3} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{1}{a^3 x^{10}}-\frac{3 b}{a^4 x^9}+\frac{6 b^2}{a^5 x^8}-\frac{10 b^3}{a^6 x^7}+\frac{15 b^4}{a^7 x^6}-\frac{21 b^5}{a^8 x^5}+\frac{28 b^6}{a^9 x^4}-\frac{36 b^7}{a^{10} x^3}+\frac{45 b^8}{a^{11} x^2}-\frac{55 b^9}{a^{12} x}+\frac{b^{10}}{a^{10} (a+b x)^3}+\frac{10 b^{10}}{a^{11} (a+b x)^2}+\frac{55 b^{10}}{a^{12} (a+b x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{3 b^9}{2 a^{10} \left (a+b \sqrt [3]{x}\right )^2}-\frac{30 b^9}{a^{11} \left (a+b \sqrt [3]{x}\right )}-\frac{1}{3 a^3 x^3}+\frac{9 b}{8 a^4 x^{8/3}}-\frac{18 b^2}{7 a^5 x^{7/3}}+\frac{5 b^3}{a^6 x^2}-\frac{9 b^4}{a^7 x^{5/3}}+\frac{63 b^5}{4 a^8 x^{4/3}}-\frac{28 b^6}{a^9 x}+\frac{54 b^7}{a^{10} x^{2/3}}-\frac{135 b^8}{a^{11} \sqrt [3]{x}}+\frac{165 b^9 \log \left (a+b \sqrt [3]{x}\right )}{a^{12}}-\frac{55 b^9 \log (x)}{a^{12}}\\ \end{align*}

Mathematica [A]  time = 0.250023, size = 167, normalized size = 0.91 \[ -\frac{\frac{a \left (110 a^8 b^2 x^{2/3}+264 a^6 b^4 x^{4/3}-462 a^5 b^5 x^{5/3}+924 a^4 b^6 x^2-2310 a^3 b^7 x^{7/3}+9240 a^2 b^8 x^{8/3}-165 a^7 b^3 x-77 a^9 b \sqrt [3]{x}+56 a^{10}+41580 a b^9 x^3+27720 b^{10} x^{10/3}\right )}{x^3 \left (a+b \sqrt [3]{x}\right )^2}-27720 b^9 \log \left (a+b \sqrt [3]{x}\right )+9240 b^9 \log (x)}{168 a^{12}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*(56*a^10 - 77*a^9*b*x^(1/3) + 110*a^8*b^2*x^(2/3) - 165*a^7*b^3*x + 264*a^6*b^4*x^(4/3) - 462*a^5*b^5*x^(
5/3) + 924*a^4*b^6*x^2 - 2310*a^3*b^7*x^(7/3) + 9240*a^2*b^8*x^(8/3) + 41580*a*b^9*x^3 + 27720*b^10*x^(10/3)))
/((a + b*x^(1/3))^2*x^3) - 27720*b^9*Log[a + b*x^(1/3)] + 9240*b^9*Log[x])/(168*a^12)

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Maple [A]  time = 0.014, size = 156, normalized size = 0.9 \begin{align*} -{\frac{3\,{b}^{9}}{2\,{a}^{10}} \left ( a+b\sqrt [3]{x} \right ) ^{-2}}-30\,{\frac{{b}^{9}}{{a}^{11} \left ( a+b\sqrt [3]{x} \right ) }}-{\frac{1}{3\,{a}^{3}{x}^{3}}}+{\frac{9\,b}{8\,{a}^{4}}{x}^{-{\frac{8}{3}}}}-{\frac{18\,{b}^{2}}{7\,{a}^{5}}{x}^{-{\frac{7}{3}}}}+5\,{\frac{{b}^{3}}{{a}^{6}{x}^{2}}}-9\,{\frac{{b}^{4}}{{a}^{7}{x}^{5/3}}}+{\frac{63\,{b}^{5}}{4\,{a}^{8}}{x}^{-{\frac{4}{3}}}}-28\,{\frac{{b}^{6}}{{a}^{9}x}}+54\,{\frac{{b}^{7}}{{a}^{10}{x}^{2/3}}}-135\,{\frac{{b}^{8}}{{a}^{11}\sqrt [3]{x}}}+165\,{\frac{{b}^{9}\ln \left ( a+b\sqrt [3]{x} \right ) }{{a}^{12}}}-55\,{\frac{{b}^{9}\ln \left ( x \right ) }{{a}^{12}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/2*b^9/a^10/(a+b*x^(1/3))^2-30*b^9/a^11/(a+b*x^(1/3))-1/3/a^3/x^3+9/8*b/a^4/x^(8/3)-18/7*b^2/a^5/x^(7/3)+5*b
^3/a^6/x^2-9*b^4/a^7/x^(5/3)+63/4*b^5/a^8/x^(4/3)-28*b^6/a^9/x+54*b^7/a^10/x^(2/3)-135*b^8/a^11/x^(1/3)+165*b^
9*ln(a+b*x^(1/3))/a^12-55*b^9*ln(x)/a^12

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Maxima [A]  time = 1.02338, size = 223, normalized size = 1.22 \begin{align*} -\frac{27720 \, b^{10} x^{\frac{10}{3}} + 41580 \, a b^{9} x^{3} + 9240 \, a^{2} b^{8} x^{\frac{8}{3}} - 2310 \, a^{3} b^{7} x^{\frac{7}{3}} + 924 \, a^{4} b^{6} x^{2} - 462 \, a^{5} b^{5} x^{\frac{5}{3}} + 264 \, a^{6} b^{4} x^{\frac{4}{3}} - 165 \, a^{7} b^{3} x + 110 \, a^{8} b^{2} x^{\frac{2}{3}} - 77 \, a^{9} b x^{\frac{1}{3}} + 56 \, a^{10}}{168 \,{\left (a^{11} b^{2} x^{\frac{11}{3}} + 2 \, a^{12} b x^{\frac{10}{3}} + a^{13} x^{3}\right )}} + \frac{165 \, b^{9} \log \left (b x^{\frac{1}{3}} + a\right )}{a^{12}} - \frac{55 \, b^{9} \log \left (x\right )}{a^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/168*(27720*b^10*x^(10/3) + 41580*a*b^9*x^3 + 9240*a^2*b^8*x^(8/3) - 2310*a^3*b^7*x^(7/3) + 924*a^4*b^6*x^2
- 462*a^5*b^5*x^(5/3) + 264*a^6*b^4*x^(4/3) - 165*a^7*b^3*x + 110*a^8*b^2*x^(2/3) - 77*a^9*b*x^(1/3) + 56*a^10
)/(a^11*b^2*x^(11/3) + 2*a^12*b*x^(10/3) + a^13*x^3) + 165*b^9*log(b*x^(1/3) + a)/a^12 - 55*b^9*log(x)/a^12

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Fricas [A]  time = 1.56198, size = 625, normalized size = 3.42 \begin{align*} -\frac{9240 \, a^{3} b^{12} x^{4} + 13860 \, a^{6} b^{9} x^{3} + 3080 \, a^{9} b^{6} x^{2} - 728 \, a^{12} b^{3} x + 56 \, a^{15} - 27720 \,{\left (b^{15} x^{5} + 2 \, a^{3} b^{12} x^{4} + a^{6} b^{9} x^{3}\right )} \log \left (b x^{\frac{1}{3}} + a\right ) + 27720 \,{\left (b^{15} x^{5} + 2 \, a^{3} b^{12} x^{4} + a^{6} b^{9} x^{3}\right )} \log \left (x^{\frac{1}{3}}\right ) + 18 \,{\left (1540 \, a b^{14} x^{4} + 2695 \, a^{4} b^{11} x^{3} + 990 \, a^{7} b^{8} x^{2} - 99 \, a^{10} b^{5} x + 24 \, a^{13} b^{2}\right )} x^{\frac{2}{3}} - 63 \,{\left (220 \, a^{2} b^{13} x^{4} + 352 \, a^{5} b^{10} x^{3} + 99 \, a^{8} b^{7} x^{2} - 18 \, a^{11} b^{4} x + 3 \, a^{14} b\right )} x^{\frac{1}{3}}}{168 \,{\left (a^{12} b^{6} x^{5} + 2 \, a^{15} b^{3} x^{4} + a^{18} x^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/168*(9240*a^3*b^12*x^4 + 13860*a^6*b^9*x^3 + 3080*a^9*b^6*x^2 - 728*a^12*b^3*x + 56*a^15 - 27720*(b^15*x^5
+ 2*a^3*b^12*x^4 + a^6*b^9*x^3)*log(b*x^(1/3) + a) + 27720*(b^15*x^5 + 2*a^3*b^12*x^4 + a^6*b^9*x^3)*log(x^(1/
3)) + 18*(1540*a*b^14*x^4 + 2695*a^4*b^11*x^3 + 990*a^7*b^8*x^2 - 99*a^10*b^5*x + 24*a^13*b^2)*x^(2/3) - 63*(2
20*a^2*b^13*x^4 + 352*a^5*b^10*x^3 + 99*a^8*b^7*x^2 - 18*a^11*b^4*x + 3*a^14*b)*x^(1/3))/(a^12*b^6*x^5 + 2*a^1
5*b^3*x^4 + a^18*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))**3/x**4,x)

[Out]

Timed out

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Giac [A]  time = 1.17611, size = 211, normalized size = 1.15 \begin{align*} \frac{165 \, b^{9} \log \left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{a^{12}} - \frac{55 \, b^{9} \log \left ({\left | x \right |}\right )}{a^{12}} - \frac{27720 \, a b^{10} x^{\frac{10}{3}} + 41580 \, a^{2} b^{9} x^{3} + 9240 \, a^{3} b^{8} x^{\frac{8}{3}} - 2310 \, a^{4} b^{7} x^{\frac{7}{3}} + 924 \, a^{5} b^{6} x^{2} - 462 \, a^{6} b^{5} x^{\frac{5}{3}} + 264 \, a^{7} b^{4} x^{\frac{4}{3}} - 165 \, a^{8} b^{3} x + 110 \, a^{9} b^{2} x^{\frac{2}{3}} - 77 \, a^{10} b x^{\frac{1}{3}} + 56 \, a^{11}}{168 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} a^{12} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

165*b^9*log(abs(b*x^(1/3) + a))/a^12 - 55*b^9*log(abs(x))/a^12 - 1/168*(27720*a*b^10*x^(10/3) + 41580*a^2*b^9*
x^3 + 9240*a^3*b^8*x^(8/3) - 2310*a^4*b^7*x^(7/3) + 924*a^5*b^6*x^2 - 462*a^6*b^5*x^(5/3) + 264*a^7*b^4*x^(4/3
) - 165*a^8*b^3*x + 110*a^9*b^2*x^(2/3) - 77*a^10*b*x^(1/3) + 56*a^11)/((b*x^(1/3) + a)^2*a^12*x^3)