3.2378 \(\int \frac{1}{(a+b \sqrt [3]{x})^3 x^2} \, dx\)

Optimal. Leaf size=103 \[ -\frac{12 b^3}{a^5 \left (a+b \sqrt [3]{x}\right )}-\frac{3 b^3}{2 a^4 \left (a+b \sqrt [3]{x}\right )^2}-\frac{18 b^2}{a^5 \sqrt [3]{x}}+\frac{30 b^3 \log \left (a+b \sqrt [3]{x}\right )}{a^6}-\frac{10 b^3 \log (x)}{a^6}+\frac{9 b}{2 a^4 x^{2/3}}-\frac{1}{a^3 x} \]

[Out]

(-3*b^3)/(2*a^4*(a + b*x^(1/3))^2) - (12*b^3)/(a^5*(a + b*x^(1/3))) - 1/(a^3*x) + (9*b)/(2*a^4*x^(2/3)) - (18*
b^2)/(a^5*x^(1/3)) + (30*b^3*Log[a + b*x^(1/3)])/a^6 - (10*b^3*Log[x])/a^6

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Rubi [A]  time = 0.065565, antiderivative size = 103, 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} \[ -\frac{12 b^3}{a^5 \left (a+b \sqrt [3]{x}\right )}-\frac{3 b^3}{2 a^4 \left (a+b \sqrt [3]{x}\right )^2}-\frac{18 b^2}{a^5 \sqrt [3]{x}}+\frac{30 b^3 \log \left (a+b \sqrt [3]{x}\right )}{a^6}-\frac{10 b^3 \log (x)}{a^6}+\frac{9 b}{2 a^4 x^{2/3}}-\frac{1}{a^3 x} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x^(1/3))^3*x^2),x]

[Out]

(-3*b^3)/(2*a^4*(a + b*x^(1/3))^2) - (12*b^3)/(a^5*(a + b*x^(1/3))) - 1/(a^3*x) + (9*b)/(2*a^4*x^(2/3)) - (18*
b^2)/(a^5*x^(1/3)) + (30*b^3*Log[a + b*x^(1/3)])/a^6 - (10*b^3*Log[x])/a^6

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^2} \, dx &=3 \operatorname{Subst}\left (\int \frac{1}{x^4 (a+b x)^3} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{1}{a^3 x^4}-\frac{3 b}{a^4 x^3}+\frac{6 b^2}{a^5 x^2}-\frac{10 b^3}{a^6 x}+\frac{b^4}{a^4 (a+b x)^3}+\frac{4 b^4}{a^5 (a+b x)^2}+\frac{10 b^4}{a^6 (a+b x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{3 b^3}{2 a^4 \left (a+b \sqrt [3]{x}\right )^2}-\frac{12 b^3}{a^5 \left (a+b \sqrt [3]{x}\right )}-\frac{1}{a^3 x}+\frac{9 b}{2 a^4 x^{2/3}}-\frac{18 b^2}{a^5 \sqrt [3]{x}}+\frac{30 b^3 \log \left (a+b \sqrt [3]{x}\right )}{a^6}-\frac{10 b^3 \log (x)}{a^6}\\ \end{align*}

Mathematica [A]  time = 0.138873, size = 93, normalized size = 0.9 \[ -\frac{\frac{a \left (20 a^2 b^2 x^{2/3}-5 a^3 b \sqrt [3]{x}+2 a^4+90 a b^3 x+60 b^4 x^{4/3}\right )}{x \left (a+b \sqrt [3]{x}\right )^2}-60 b^3 \log \left (a+b \sqrt [3]{x}\right )+20 b^3 \log (x)}{2 a^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*(2*a^4 - 5*a^3*b*x^(1/3) + 20*a^2*b^2*x^(2/3) + 90*a*b^3*x + 60*b^4*x^(4/3)))/((a + b*x^(1/3))^2*x) - 60*
b^3*Log[a + b*x^(1/3)] + 20*b^3*Log[x])/(2*a^6)

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Maple [A]  time = 0.01, size = 90, normalized size = 0.9 \begin{align*} -{\frac{3\,{b}^{3}}{2\,{a}^{4}} \left ( a+b\sqrt [3]{x} \right ) ^{-2}}-12\,{\frac{{b}^{3}}{{a}^{5} \left ( a+b\sqrt [3]{x} \right ) }}-{\frac{1}{{a}^{3}x}}+{\frac{9\,b}{2\,{a}^{4}}{x}^{-{\frac{2}{3}}}}-18\,{\frac{{b}^{2}}{{a}^{5}\sqrt [3]{x}}}+30\,{\frac{{b}^{3}\ln \left ( a+b\sqrt [3]{x} \right ) }{{a}^{6}}}-10\,{\frac{{b}^{3}\ln \left ( x \right ) }{{a}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/2*b^3/a^4/(a+b*x^(1/3))^2-12*b^3/a^5/(a+b*x^(1/3))-1/a^3/x+9/2*b/a^4/x^(2/3)-18*b^2/a^5/x^(1/3)+30*b^3*ln(a
+b*x^(1/3))/a^6-10*b^3*ln(x)/a^6

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Maxima [A]  time = 0.974684, size = 131, normalized size = 1.27 \begin{align*} -\frac{60 \, b^{4} x^{\frac{4}{3}} + 90 \, a b^{3} x + 20 \, a^{2} b^{2} x^{\frac{2}{3}} - 5 \, a^{3} b x^{\frac{1}{3}} + 2 \, a^{4}}{2 \,{\left (a^{5} b^{2} x^{\frac{5}{3}} + 2 \, a^{6} b x^{\frac{4}{3}} + a^{7} x\right )}} + \frac{30 \, b^{3} \log \left (b x^{\frac{1}{3}} + a\right )}{a^{6}} - \frac{10 \, b^{3} \log \left (x\right )}{a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(60*b^4*x^(4/3) + 90*a*b^3*x + 20*a^2*b^2*x^(2/3) - 5*a^3*b*x^(1/3) + 2*a^4)/(a^5*b^2*x^(5/3) + 2*a^6*b*x
^(4/3) + a^7*x) + 30*b^3*log(b*x^(1/3) + a)/a^6 - 10*b^3*log(x)/a^6

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Fricas [B]  time = 1.5533, size = 421, normalized size = 4.09 \begin{align*} -\frac{20 \, a^{3} b^{6} x^{2} + 31 \, a^{6} b^{3} x + 2 \, a^{9} - 60 \,{\left (b^{9} x^{3} + 2 \, a^{3} b^{6} x^{2} + a^{6} b^{3} x\right )} \log \left (b x^{\frac{1}{3}} + a\right ) + 60 \,{\left (b^{9} x^{3} + 2 \, a^{3} b^{6} x^{2} + a^{6} b^{3} x\right )} \log \left (x^{\frac{1}{3}}\right ) + 3 \,{\left (20 \, a b^{8} x^{2} + 35 \, a^{4} b^{5} x + 12 \, a^{7} b^{2}\right )} x^{\frac{2}{3}} - 3 \,{\left (10 \, a^{2} b^{7} x^{2} + 16 \, a^{5} b^{4} x + 3 \, a^{8} b\right )} x^{\frac{1}{3}}}{2 \,{\left (a^{6} b^{6} x^{3} + 2 \, a^{9} b^{3} x^{2} + a^{12} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(20*a^3*b^6*x^2 + 31*a^6*b^3*x + 2*a^9 - 60*(b^9*x^3 + 2*a^3*b^6*x^2 + a^6*b^3*x)*log(b*x^(1/3) + a) + 60
*(b^9*x^3 + 2*a^3*b^6*x^2 + a^6*b^3*x)*log(x^(1/3)) + 3*(20*a*b^8*x^2 + 35*a^4*b^5*x + 12*a^7*b^2)*x^(2/3) - 3
*(10*a^2*b^7*x^2 + 16*a^5*b^4*x + 3*a^8*b)*x^(1/3))/(a^6*b^6*x^3 + 2*a^9*b^3*x^2 + a^12*x)

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Sympy [A]  time = 5.88143, size = 558, normalized size = 5.42 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{2}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{1}{2 b^{3} x^{2}} & \text{for}\: a = 0 \\- \frac{1}{a^{3} x} & \text{for}\: b = 0 \\- \frac{2 a^{5} x^{\frac{2}{3}}}{2 a^{8} x^{\frac{5}{3}} + 4 a^{7} b x^{2} + 2 a^{6} b^{2} x^{\frac{7}{3}}} + \frac{5 a^{4} b x}{2 a^{8} x^{\frac{5}{3}} + 4 a^{7} b x^{2} + 2 a^{6} b^{2} x^{\frac{7}{3}}} - \frac{20 a^{3} b^{2} x^{\frac{4}{3}}}{2 a^{8} x^{\frac{5}{3}} + 4 a^{7} b x^{2} + 2 a^{6} b^{2} x^{\frac{7}{3}}} - \frac{20 a^{2} b^{3} x^{\frac{5}{3}} \log{\left (x \right )}}{2 a^{8} x^{\frac{5}{3}} + 4 a^{7} b x^{2} + 2 a^{6} b^{2} x^{\frac{7}{3}}} + \frac{60 a^{2} b^{3} x^{\frac{5}{3}} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{2 a^{8} x^{\frac{5}{3}} + 4 a^{7} b x^{2} + 2 a^{6} b^{2} x^{\frac{7}{3}}} - \frac{40 a b^{4} x^{2} \log{\left (x \right )}}{2 a^{8} x^{\frac{5}{3}} + 4 a^{7} b x^{2} + 2 a^{6} b^{2} x^{\frac{7}{3}}} + \frac{120 a b^{4} x^{2} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{2 a^{8} x^{\frac{5}{3}} + 4 a^{7} b x^{2} + 2 a^{6} b^{2} x^{\frac{7}{3}}} + \frac{120 a b^{4} x^{2}}{2 a^{8} x^{\frac{5}{3}} + 4 a^{7} b x^{2} + 2 a^{6} b^{2} x^{\frac{7}{3}}} - \frac{20 b^{5} x^{\frac{7}{3}} \log{\left (x \right )}}{2 a^{8} x^{\frac{5}{3}} + 4 a^{7} b x^{2} + 2 a^{6} b^{2} x^{\frac{7}{3}}} + \frac{60 b^{5} x^{\frac{7}{3}} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{2 a^{8} x^{\frac{5}{3}} + 4 a^{7} b x^{2} + 2 a^{6} b^{2} x^{\frac{7}{3}}} + \frac{90 b^{5} x^{\frac{7}{3}}}{2 a^{8} x^{\frac{5}{3}} + 4 a^{7} b x^{2} + 2 a^{6} b^{2} x^{\frac{7}{3}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo/x**2, Eq(a, 0) & Eq(b, 0)), (-1/(2*b**3*x**2), Eq(a, 0)), (-1/(a**3*x), Eq(b, 0)), (-2*a**5*x**
(2/3)/(2*a**8*x**(5/3) + 4*a**7*b*x**2 + 2*a**6*b**2*x**(7/3)) + 5*a**4*b*x/(2*a**8*x**(5/3) + 4*a**7*b*x**2 +
2*a**6*b**2*x**(7/3)) - 20*a**3*b**2*x**(4/3)/(2*a**8*x**(5/3) + 4*a**7*b*x**2 + 2*a**6*b**2*x**(7/3)) - 20*a
**2*b**3*x**(5/3)*log(x)/(2*a**8*x**(5/3) + 4*a**7*b*x**2 + 2*a**6*b**2*x**(7/3)) + 60*a**2*b**3*x**(5/3)*log(
a/b + x**(1/3))/(2*a**8*x**(5/3) + 4*a**7*b*x**2 + 2*a**6*b**2*x**(7/3)) - 40*a*b**4*x**2*log(x)/(2*a**8*x**(5
/3) + 4*a**7*b*x**2 + 2*a**6*b**2*x**(7/3)) + 120*a*b**4*x**2*log(a/b + x**(1/3))/(2*a**8*x**(5/3) + 4*a**7*b*
x**2 + 2*a**6*b**2*x**(7/3)) + 120*a*b**4*x**2/(2*a**8*x**(5/3) + 4*a**7*b*x**2 + 2*a**6*b**2*x**(7/3)) - 20*b
**5*x**(7/3)*log(x)/(2*a**8*x**(5/3) + 4*a**7*b*x**2 + 2*a**6*b**2*x**(7/3)) + 60*b**5*x**(7/3)*log(a/b + x**(
1/3))/(2*a**8*x**(5/3) + 4*a**7*b*x**2 + 2*a**6*b**2*x**(7/3)) + 90*b**5*x**(7/3)/(2*a**8*x**(5/3) + 4*a**7*b*
x**2 + 2*a**6*b**2*x**(7/3)), True))

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Giac [A]  time = 1.15767, size = 122, normalized size = 1.18 \begin{align*} \frac{30 \, b^{3} \log \left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{a^{6}} - \frac{10 \, b^{3} \log \left ({\left | x \right |}\right )}{a^{6}} - \frac{60 \, a b^{4} x^{\frac{4}{3}} + 90 \, a^{2} b^{3} x + 20 \, a^{3} b^{2} x^{\frac{2}{3}} - 5 \, a^{4} b x^{\frac{1}{3}} + 2 \, a^{5}}{2 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} a^{6} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

30*b^3*log(abs(b*x^(1/3) + a))/a^6 - 10*b^3*log(abs(x))/a^6 - 1/2*(60*a*b^4*x^(4/3) + 90*a^2*b^3*x + 20*a^3*b^
2*x^(2/3) - 5*a^4*b*x^(1/3) + 2*a^5)/((b*x^(1/3) + a)^2*a^6*x)