3.2443 \(\int \frac{1}{(a+\frac{b}{\sqrt [3]{x}})^3 x^4} \, dx\)

Optimal. Leaf size=146 \[ -\frac{45 a^4}{2 b^7 x^{2/3}}-\frac{9 a^2}{2 b^5 x^{4/3}}+\frac{21 a^6}{b^8 \left (a \sqrt [3]{x}+b\right )}+\frac{3 a^6}{2 b^7 \left (a \sqrt [3]{x}+b\right )^2}+\frac{63 a^5}{b^8 \sqrt [3]{x}}+\frac{10 a^3}{b^6 x}-\frac{84 a^6 \log \left (a \sqrt [3]{x}+b\right )}{b^9}+\frac{28 a^6 \log (x)}{b^9}+\frac{9 a}{5 b^4 x^{5/3}}-\frac{1}{2 b^3 x^2} \]

[Out]

(3*a^6)/(2*b^7*(b + a*x^(1/3))^2) + (21*a^6)/(b^8*(b + a*x^(1/3))) - 1/(2*b^3*x^2) + (9*a)/(5*b^4*x^(5/3)) - (
9*a^2)/(2*b^5*x^(4/3)) + (10*a^3)/(b^6*x) - (45*a^4)/(2*b^7*x^(2/3)) + (63*a^5)/(b^8*x^(1/3)) - (84*a^6*Log[b
+ a*x^(1/3)])/b^9 + (28*a^6*Log[x])/b^9

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Rubi [A]  time = 0.106149, antiderivative size = 146, 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} \[ -\frac{45 a^4}{2 b^7 x^{2/3}}-\frac{9 a^2}{2 b^5 x^{4/3}}+\frac{21 a^6}{b^8 \left (a \sqrt [3]{x}+b\right )}+\frac{3 a^6}{2 b^7 \left (a \sqrt [3]{x}+b\right )^2}+\frac{63 a^5}{b^8 \sqrt [3]{x}}+\frac{10 a^3}{b^6 x}-\frac{84 a^6 \log \left (a \sqrt [3]{x}+b\right )}{b^9}+\frac{28 a^6 \log (x)}{b^9}+\frac{9 a}{5 b^4 x^{5/3}}-\frac{1}{2 b^3 x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^(1/3))^3*x^4),x]

[Out]

(3*a^6)/(2*b^7*(b + a*x^(1/3))^2) + (21*a^6)/(b^8*(b + a*x^(1/3))) - 1/(2*b^3*x^2) + (9*a)/(5*b^4*x^(5/3)) - (
9*a^2)/(2*b^5*x^(4/3)) + (10*a^3)/(b^6*x) - (45*a^4)/(2*b^7*x^(2/3)) + (63*a^5)/(b^8*x^(1/3)) - (84*a^6*Log[b
+ a*x^(1/3)])/b^9 + (28*a^6*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 )^3 x^4} \, dx &=\int \frac{1}{\left (b+a \sqrt [3]{x}\right )^3 x^3} \, dx\\ &=3 \operatorname{Subst}\left (\int \frac{1}{x^7 (b+a x)^3} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{1}{b^3 x^7}-\frac{3 a}{b^4 x^6}+\frac{6 a^2}{b^5 x^5}-\frac{10 a^3}{b^6 x^4}+\frac{15 a^4}{b^7 x^3}-\frac{21 a^5}{b^8 x^2}+\frac{28 a^6}{b^9 x}-\frac{a^7}{b^7 (b+a x)^3}-\frac{7 a^7}{b^8 (b+a x)^2}-\frac{28 a^7}{b^9 (b+a x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{3 a^6}{2 b^7 \left (b+a \sqrt [3]{x}\right )^2}+\frac{21 a^6}{b^8 \left (b+a \sqrt [3]{x}\right )}-\frac{1}{2 b^3 x^2}+\frac{9 a}{5 b^4 x^{5/3}}-\frac{9 a^2}{2 b^5 x^{4/3}}+\frac{10 a^3}{b^6 x}-\frac{45 a^4}{2 b^7 x^{2/3}}+\frac{63 a^5}{b^8 \sqrt [3]{x}}-\frac{84 a^6 \log \left (b+a \sqrt [3]{x}\right )}{b^9}+\frac{28 a^6 \log (x)}{b^9}\\ \end{align*}

Mathematica [A]  time = 0.162161, size = 130, normalized size = 0.89 \[ \frac{\frac{b \left (280 a^5 b^2 x^{5/3}-70 a^4 b^3 x^{4/3}-14 a^2 b^5 x^{2/3}+28 a^3 b^4 x+1260 a^6 b x^2+840 a^7 x^{7/3}+8 a b^6 \sqrt [3]{x}-5 b^7\right )}{x^2 \left (a \sqrt [3]{x}+b\right )^2}-840 a^6 \log \left (a \sqrt [3]{x}+b\right )+280 a^6 \log (x)}{10 b^9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b*(-5*b^7 + 8*a*b^6*x^(1/3) - 14*a^2*b^5*x^(2/3) + 28*a^3*b^4*x - 70*a^4*b^3*x^(4/3) + 280*a^5*b^2*x^(5/3) +
1260*a^6*b*x^2 + 840*a^7*x^(7/3)))/((b + a*x^(1/3))^2*x^2) - 840*a^6*Log[b + a*x^(1/3)] + 280*a^6*Log[x])/(10
*b^9)

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Maple [A]  time = 0.013, size = 123, normalized size = 0.8 \begin{align*}{\frac{3\,{a}^{6}}{2\,{b}^{7}} \left ( b+a\sqrt [3]{x} \right ) ^{-2}}+21\,{\frac{{a}^{6}}{{b}^{8} \left ( b+a\sqrt [3]{x} \right ) }}-{\frac{1}{2\,{b}^{3}{x}^{2}}}+{\frac{9\,a}{5\,{b}^{4}}{x}^{-{\frac{5}{3}}}}-{\frac{9\,{a}^{2}}{2\,{b}^{5}}{x}^{-{\frac{4}{3}}}}+10\,{\frac{{a}^{3}}{{b}^{6}x}}-{\frac{45\,{a}^{4}}{2\,{b}^{7}}{x}^{-{\frac{2}{3}}}}+63\,{\frac{{a}^{5}}{{b}^{8}\sqrt [3]{x}}}-84\,{\frac{{a}^{6}\ln \left ( b+a\sqrt [3]{x} \right ) }{{b}^{9}}}+28\,{\frac{{a}^{6}\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))^3/x^4,x)

[Out]

3/2*a^6/b^7/(b+a*x^(1/3))^2+21*a^6/b^8/(b+a*x^(1/3))-1/2/b^3/x^2+9/5*a/b^4/x^(5/3)-9/2*a^2/b^5/x^(4/3)+10*a^3/
b^6/x-45/2*a^4/b^7/x^(2/3)+63*a^5/b^8/x^(1/3)-84*a^6*ln(b+a*x^(1/3))/b^9+28*a^6*ln(x)/b^9

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Maxima [A]  time = 0.991442, size = 197, normalized size = 1.35 \begin{align*} -\frac{84 \, a^{6} \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{b^{9}} - \frac{{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{6}}{2 \, b^{9}} + \frac{24 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{5} a}{5 \, b^{9}} - \frac{21 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{4} a^{2}}{b^{9}} + \frac{56 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{3} a^{3}}{b^{9}} - \frac{105 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{2} a^{4}}{b^{9}} + \frac{168 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )} a^{5}}{b^{9}} - \frac{24 \, a^{7}}{{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )} b^{9}} + \frac{3 \, a^{8}}{2 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{2} b^{9}} \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]

-84*a^6*log(a + b/x^(1/3))/b^9 - 1/2*(a + b/x^(1/3))^6/b^9 + 24/5*(a + b/x^(1/3))^5*a/b^9 - 21*(a + b/x^(1/3))
^4*a^2/b^9 + 56*(a + b/x^(1/3))^3*a^3/b^9 - 105*(a + b/x^(1/3))^2*a^4/b^9 + 168*(a + b/x^(1/3))*a^5/b^9 - 24*a
^7/((a + b/x^(1/3))*b^9) + 3/2*a^8/((a + b/x^(1/3))^2*b^9)

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Fricas [A]  time = 1.5424, size = 517, normalized size = 3.54 \begin{align*} \frac{280 \, a^{9} b^{3} x^{3} + 420 \, a^{6} b^{6} x^{2} + 90 \, a^{3} b^{9} x - 5 \, b^{12} - 840 \,{\left (a^{12} x^{4} + 2 \, a^{9} b^{3} x^{3} + a^{6} b^{6} x^{2}\right )} \log \left (a x^{\frac{1}{3}} + b\right ) + 840 \,{\left (a^{12} x^{4} + 2 \, a^{9} b^{3} x^{3} + a^{6} b^{6} x^{2}\right )} \log \left (x^{\frac{1}{3}}\right ) + 15 \,{\left (56 \, a^{11} b x^{3} + 98 \, a^{8} b^{4} x^{2} + 36 \, a^{5} b^{7} x - 3 \, a^{2} b^{10}\right )} x^{\frac{2}{3}} - 3 \,{\left (140 \, a^{10} b^{2} x^{3} + 224 \, a^{7} b^{5} x^{2} + 63 \, a^{4} b^{8} x - 6 \, a b^{11}\right )} x^{\frac{1}{3}}}{10 \,{\left (a^{6} b^{9} x^{4} + 2 \, a^{3} b^{12} x^{3} + b^{15} x^{2}\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]

1/10*(280*a^9*b^3*x^3 + 420*a^6*b^6*x^2 + 90*a^3*b^9*x - 5*b^12 - 840*(a^12*x^4 + 2*a^9*b^3*x^3 + a^6*b^6*x^2)
*log(a*x^(1/3) + b) + 840*(a^12*x^4 + 2*a^9*b^3*x^3 + a^6*b^6*x^2)*log(x^(1/3)) + 15*(56*a^11*b*x^3 + 98*a^8*b
^4*x^2 + 36*a^5*b^7*x - 3*a^2*b^10)*x^(2/3) - 3*(140*a^10*b^2*x^3 + 224*a^7*b^5*x^2 + 63*a^4*b^8*x - 6*a*b^11)
*x^(1/3))/(a^6*b^9*x^4 + 2*a^3*b^12*x^3 + b^15*x^2)

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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 = 1.24383, size = 166, normalized size = 1.14 \begin{align*} -\frac{84 \, a^{6} \log \left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{b^{9}} + \frac{28 \, a^{6} \log \left ({\left | x \right |}\right )}{b^{9}} + \frac{840 \, a^{7} b x^{\frac{7}{3}} + 1260 \, a^{6} b^{2} x^{2} + 280 \, a^{5} b^{3} x^{\frac{5}{3}} - 70 \, a^{4} b^{4} x^{\frac{4}{3}} + 28 \, a^{3} b^{5} x - 14 \, a^{2} b^{6} x^{\frac{2}{3}} + 8 \, a b^{7} x^{\frac{1}{3}} - 5 \, b^{8}}{10 \,{\left (a x^{\frac{1}{3}} + b\right )}^{2} b^{9} x^{2}} \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]

-84*a^6*log(abs(a*x^(1/3) + b))/b^9 + 28*a^6*log(abs(x))/b^9 + 1/10*(840*a^7*b*x^(7/3) + 1260*a^6*b^2*x^2 + 28
0*a^5*b^3*x^(5/3) - 70*a^4*b^4*x^(4/3) + 28*a^3*b^5*x - 14*a^2*b^6*x^(2/3) + 8*a*b^7*x^(1/3) - 5*b^8)/((a*x^(1
/3) + b)^2*b^9*x^2)