3.2379 \(\int \frac{1}{(a+b \sqrt [3]{x})^3 x^3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0995701, antiderivative size = 146, 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} \[ -\frac{45 b^4}{2 a^7 x^{2/3}}-\frac{9 b^2}{2 a^5 x^{4/3}}+\frac{21 b^6}{a^8 \left (a+b \sqrt [3]{x}\right )}+\frac{3 b^6}{2 a^7 \left (a+b \sqrt [3]{x}\right )^2}+\frac{63 b^5}{a^8 \sqrt [3]{x}}+\frac{10 b^3}{a^6 x}-\frac{84 b^6 \log \left (a+b \sqrt [3]{x}\right )}{a^9}+\frac{28 b^6 \log (x)}{a^9}+\frac{9 b}{5 a^4 x^{5/3}}-\frac{1}{2 a^3 x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x^(1/3))^3*x^3),x]

[Out]

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

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

Mathematica [A]  time = 0.158862, size = 130, normalized size = 0.89 \[ \frac{\frac{a \left (-14 a^5 b^2 x^{2/3}-70 a^3 b^4 x^{4/3}+280 a^2 b^5 x^{5/3}+28 a^4 b^3 x+8 a^6 b \sqrt [3]{x}-5 a^7+1260 a b^6 x^2+840 b^7 x^{7/3}\right )}{x^2 \left (a+b \sqrt [3]{x}\right )^2}-840 b^6 \log \left (a+b \sqrt [3]{x}\right )+280 b^6 \log (x)}{10 a^9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.011, size = 123, normalized size = 0.8 \begin{align*}{\frac{3\,{b}^{6}}{2\,{a}^{7}} \left ( a+b\sqrt [3]{x} \right ) ^{-2}}+21\,{\frac{{b}^{6}}{{a}^{8} \left ( a+b\sqrt [3]{x} \right ) }}-{\frac{1}{2\,{x}^{2}{a}^{3}}}+{\frac{9\,b}{5\,{a}^{4}}{x}^{-{\frac{5}{3}}}}-{\frac{9\,{b}^{2}}{2\,{a}^{5}}{x}^{-{\frac{4}{3}}}}+10\,{\frac{{b}^{3}}{{a}^{6}x}}-{\frac{45\,{b}^{4}}{2\,{a}^{7}}{x}^{-{\frac{2}{3}}}}+63\,{\frac{{b}^{5}}{{a}^{8}\sqrt [3]{x}}}-84\,{\frac{{b}^{6}\ln \left ( a+b\sqrt [3]{x} \right ) }{{a}^{9}}}+28\,{\frac{{b}^{6}\ln \left ( x \right ) }{{a}^{9}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.01303, size = 178, normalized size = 1.22 \begin{align*} \frac{840 \, b^{7} x^{\frac{7}{3}} + 1260 \, a b^{6} x^{2} + 280 \, a^{2} b^{5} x^{\frac{5}{3}} - 70 \, a^{3} b^{4} x^{\frac{4}{3}} + 28 \, a^{4} b^{3} x - 14 \, a^{5} b^{2} x^{\frac{2}{3}} + 8 \, a^{6} b x^{\frac{1}{3}} - 5 \, a^{7}}{10 \,{\left (a^{8} b^{2} x^{\frac{8}{3}} + 2 \, a^{9} b x^{\frac{7}{3}} + a^{10} x^{2}\right )}} - \frac{84 \, b^{6} \log \left (b x^{\frac{1}{3}} + a\right )}{a^{9}} + \frac{28 \, b^{6} \log \left (x\right )}{a^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(840*b^7*x^(7/3) + 1260*a*b^6*x^2 + 280*a^2*b^5*x^(5/3) - 70*a^3*b^4*x^(4/3) + 28*a^4*b^3*x - 14*a^5*b^2*
x^(2/3) + 8*a^6*b*x^(1/3) - 5*a^7)/(a^8*b^2*x^(8/3) + 2*a^9*b*x^(7/3) + a^10*x^2) - 84*b^6*log(b*x^(1/3) + a)/
a^9 + 28*b^6*log(x)/a^9

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Fricas [A]  time = 1.56512, size = 517, normalized size = 3.54 \begin{align*} \frac{280 \, a^{3} b^{9} x^{3} + 420 \, a^{6} b^{6} x^{2} + 90 \, a^{9} b^{3} x - 5 \, a^{12} - 840 \,{\left (b^{12} x^{4} + 2 \, a^{3} b^{9} x^{3} + a^{6} b^{6} x^{2}\right )} \log \left (b x^{\frac{1}{3}} + a\right ) + 840 \,{\left (b^{12} x^{4} + 2 \, a^{3} b^{9} x^{3} + a^{6} b^{6} x^{2}\right )} \log \left (x^{\frac{1}{3}}\right ) + 15 \,{\left (56 \, a b^{11} x^{3} + 98 \, a^{4} b^{8} x^{2} + 36 \, a^{7} b^{5} x - 3 \, a^{10} b^{2}\right )} x^{\frac{2}{3}} - 3 \,{\left (140 \, a^{2} b^{10} x^{3} + 224 \, a^{5} b^{7} x^{2} + 63 \, a^{8} b^{4} x - 6 \, a^{11} b\right )} x^{\frac{1}{3}}}{10 \,{\left (a^{9} b^{6} x^{4} + 2 \, a^{12} b^{3} x^{3} + a^{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^3,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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Giac [A]  time = 1.21161, size = 166, normalized size = 1.14 \begin{align*} -\frac{84 \, b^{6} \log \left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{a^{9}} + \frac{28 \, b^{6} \log \left ({\left | x \right |}\right )}{a^{9}} + \frac{840 \, a b^{7} x^{\frac{7}{3}} + 1260 \, a^{2} b^{6} x^{2} + 280 \, a^{3} b^{5} x^{\frac{5}{3}} - 70 \, a^{4} b^{4} x^{\frac{4}{3}} + 28 \, a^{5} b^{3} x - 14 \, a^{6} b^{2} x^{\frac{2}{3}} + 8 \, a^{7} b x^{\frac{1}{3}} - 5 \, a^{8}}{10 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} a^{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^3,x, algorithm="giac")

[Out]

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