3.2371 \(\int \frac{1}{(a+b \sqrt [3]{x})^2 x^3} \, dx\)

Optimal. Leaf size=125 \[ -\frac{15 b^4}{2 a^6 x^{2/3}}-\frac{9 b^2}{4 a^4 x^{4/3}}+\frac{3 b^6}{a^7 \left (a+b \sqrt [3]{x}\right )}+\frac{18 b^5}{a^7 \sqrt [3]{x}}+\frac{4 b^3}{a^5 x}-\frac{21 b^6 \log \left (a+b \sqrt [3]{x}\right )}{a^8}+\frac{7 b^6 \log (x)}{a^8}+\frac{6 b}{5 a^3 x^{5/3}}-\frac{1}{2 a^2 x^2} \]

[Out]

(3*b^6)/(a^7*(a + b*x^(1/3))) - 1/(2*a^2*x^2) + (6*b)/(5*a^3*x^(5/3)) - (9*b^2)/(4*a^4*x^(4/3)) + (4*b^3)/(a^5
*x) - (15*b^4)/(2*a^6*x^(2/3)) + (18*b^5)/(a^7*x^(1/3)) - (21*b^6*Log[a + b*x^(1/3)])/a^8 + (7*b^6*Log[x])/a^8

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Rubi [A]  time = 0.0837684, antiderivative size = 125, 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} \[ -\frac{15 b^4}{2 a^6 x^{2/3}}-\frac{9 b^2}{4 a^4 x^{4/3}}+\frac{3 b^6}{a^7 \left (a+b \sqrt [3]{x}\right )}+\frac{18 b^5}{a^7 \sqrt [3]{x}}+\frac{4 b^3}{a^5 x}-\frac{21 b^6 \log \left (a+b \sqrt [3]{x}\right )}{a^8}+\frac{7 b^6 \log (x)}{a^8}+\frac{6 b}{5 a^3 x^{5/3}}-\frac{1}{2 a^2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x^(1/3))^2*x^3),x]

[Out]

(3*b^6)/(a^7*(a + b*x^(1/3))) - 1/(2*a^2*x^2) + (6*b)/(5*a^3*x^(5/3)) - (9*b^2)/(4*a^4*x^(4/3)) + (4*b^3)/(a^5
*x) - (15*b^4)/(2*a^6*x^(2/3)) + (18*b^5)/(a^7*x^(1/3)) - (21*b^6*Log[a + b*x^(1/3)])/a^8 + (7*b^6*Log[x])/a^8

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^3} \, dx &=3 \operatorname{Subst}\left (\int \frac{1}{x^7 (a+b x)^2} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{1}{a^2 x^7}-\frac{2 b}{a^3 x^6}+\frac{3 b^2}{a^4 x^5}-\frac{4 b^3}{a^5 x^4}+\frac{5 b^4}{a^6 x^3}-\frac{6 b^5}{a^7 x^2}+\frac{7 b^6}{a^8 x}-\frac{b^7}{a^7 (a+b x)^2}-\frac{7 b^7}{a^8 (a+b x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{3 b^6}{a^7 \left (a+b \sqrt [3]{x}\right )}-\frac{1}{2 a^2 x^2}+\frac{6 b}{5 a^3 x^{5/3}}-\frac{9 b^2}{4 a^4 x^{4/3}}+\frac{4 b^3}{a^5 x}-\frac{15 b^4}{2 a^6 x^{2/3}}+\frac{18 b^5}{a^7 \sqrt [3]{x}}-\frac{21 b^6 \log \left (a+b \sqrt [3]{x}\right )}{a^8}+\frac{7 b^6 \log (x)}{a^8}\\ \end{align*}

Mathematica [A]  time = 0.128633, size = 117, normalized size = 0.94 \[ \frac{\frac{a \left (-21 a^4 b^2 x^{2/3}-70 a^2 b^4 x^{4/3}+35 a^3 b^3 x+14 a^5 b \sqrt [3]{x}-10 a^6+210 a b^5 x^{5/3}+420 b^6 x^2\right )}{x^2 \left (a+b \sqrt [3]{x}\right )}-420 b^6 \log \left (a+b \sqrt [3]{x}\right )+140 b^6 \log (x)}{20 a^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a*(-10*a^6 + 14*a^5*b*x^(1/3) - 21*a^4*b^2*x^(2/3) + 35*a^3*b^3*x - 70*a^2*b^4*x^(4/3) + 210*a*b^5*x^(5/3) +
420*b^6*x^2))/((a + b*x^(1/3))*x^2) - 420*b^6*Log[a + b*x^(1/3)] + 140*b^6*Log[x])/(20*a^8)

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Maple [A]  time = 0.01, size = 106, normalized size = 0.9 \begin{align*} 3\,{\frac{{b}^{6}}{{a}^{7} \left ( a+b\sqrt [3]{x} \right ) }}-{\frac{1}{2\,{a}^{2}{x}^{2}}}+{\frac{6\,b}{5\,{a}^{3}}{x}^{-{\frac{5}{3}}}}-{\frac{9\,{b}^{2}}{4\,{a}^{4}}{x}^{-{\frac{4}{3}}}}+4\,{\frac{{b}^{3}}{x{a}^{5}}}-{\frac{15\,{b}^{4}}{2\,{a}^{6}}{x}^{-{\frac{2}{3}}}}+18\,{\frac{{b}^{5}}{{a}^{7}\sqrt [3]{x}}}-21\,{\frac{{b}^{6}\ln \left ( a+b\sqrt [3]{x} \right ) }{{a}^{8}}}+7\,{\frac{{b}^{6}\ln \left ( x \right ) }{{a}^{8}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*b^6/a^7/(a+b*x^(1/3))-1/2/a^2/x^2+6/5*b/a^3/x^(5/3)-9/4*b^2/a^4/x^(4/3)+4*b^3/x/a^5-15/2*b^4/a^6/x^(2/3)+18*
b^5/a^7/x^(1/3)-21*b^6*ln(a+b*x^(1/3))/a^8+7*b^6*ln(x)/a^8

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Maxima [A]  time = 0.968091, size = 149, normalized size = 1.19 \begin{align*} \frac{420 \, b^{6} x^{2} + 210 \, a b^{5} x^{\frac{5}{3}} - 70 \, a^{2} b^{4} x^{\frac{4}{3}} + 35 \, a^{3} b^{3} x - 21 \, a^{4} b^{2} x^{\frac{2}{3}} + 14 \, a^{5} b x^{\frac{1}{3}} - 10 \, a^{6}}{20 \,{\left (a^{7} b x^{\frac{7}{3}} + a^{8} x^{2}\right )}} - \frac{21 \, b^{6} \log \left (b x^{\frac{1}{3}} + a\right )}{a^{8}} + \frac{7 \, b^{6} \log \left (x\right )}{a^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*(420*b^6*x^2 + 210*a*b^5*x^(5/3) - 70*a^2*b^4*x^(4/3) + 35*a^3*b^3*x - 21*a^4*b^2*x^(2/3) + 14*a^5*b*x^(1
/3) - 10*a^6)/(a^7*b*x^(7/3) + a^8*x^2) - 21*b^6*log(b*x^(1/3) + a)/a^8 + 7*b^6*log(x)/a^8

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Fricas [A]  time = 1.51023, size = 370, normalized size = 2.96 \begin{align*} \frac{140 \, a^{3} b^{6} x^{2} + 70 \, a^{6} b^{3} x - 10 \, a^{9} - 420 \,{\left (b^{9} x^{3} + a^{3} b^{6} x^{2}\right )} \log \left (b x^{\frac{1}{3}} + a\right ) + 420 \,{\left (b^{9} x^{3} + a^{3} b^{6} x^{2}\right )} \log \left (x^{\frac{1}{3}}\right ) + 15 \,{\left (28 \, a b^{8} x^{2} + 21 \, a^{4} b^{5} x - 3 \, a^{7} b^{2}\right )} x^{\frac{2}{3}} - 6 \,{\left (35 \, a^{2} b^{7} x^{2} + 21 \, a^{5} b^{4} x - 4 \, a^{8} b\right )} x^{\frac{1}{3}}}{20 \,{\left (a^{8} b^{3} x^{3} + a^{11} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*(140*a^3*b^6*x^2 + 70*a^6*b^3*x - 10*a^9 - 420*(b^9*x^3 + a^3*b^6*x^2)*log(b*x^(1/3) + a) + 420*(b^9*x^3
+ a^3*b^6*x^2)*log(x^(1/3)) + 15*(28*a*b^8*x^2 + 21*a^4*b^5*x - 3*a^7*b^2)*x^(2/3) - 6*(35*a^2*b^7*x^2 + 21*a^
5*b^4*x - 4*a^8*b)*x^(1/3))/(a^8*b^3*x^3 + a^11*x^2)

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Sympy [A]  time = 20.6383, size = 405, normalized size = 3.24 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{\frac{8}{3}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{3}{8 b^{2} x^{\frac{8}{3}}} & \text{for}\: a = 0 \\- \frac{1}{2 a^{2} x^{2}} & \text{for}\: b = 0 \\- \frac{10 a^{7} x^{\frac{2}{3}}}{20 a^{9} x^{\frac{8}{3}} + 20 a^{8} b x^{3}} + \frac{14 a^{6} b x}{20 a^{9} x^{\frac{8}{3}} + 20 a^{8} b x^{3}} - \frac{21 a^{5} b^{2} x^{\frac{4}{3}}}{20 a^{9} x^{\frac{8}{3}} + 20 a^{8} b x^{3}} + \frac{35 a^{4} b^{3} x^{\frac{5}{3}}}{20 a^{9} x^{\frac{8}{3}} + 20 a^{8} b x^{3}} - \frac{70 a^{3} b^{4} x^{2}}{20 a^{9} x^{\frac{8}{3}} + 20 a^{8} b x^{3}} + \frac{210 a^{2} b^{5} x^{\frac{7}{3}}}{20 a^{9} x^{\frac{8}{3}} + 20 a^{8} b x^{3}} + \frac{140 a b^{6} x^{\frac{8}{3}} \log{\left (x \right )}}{20 a^{9} x^{\frac{8}{3}} + 20 a^{8} b x^{3}} - \frac{420 a b^{6} x^{\frac{8}{3}} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{20 a^{9} x^{\frac{8}{3}} + 20 a^{8} b x^{3}} + \frac{420 a b^{6} x^{\frac{8}{3}}}{20 a^{9} x^{\frac{8}{3}} + 20 a^{8} b x^{3}} + \frac{140 b^{7} x^{3} \log{\left (x \right )}}{20 a^{9} x^{\frac{8}{3}} + 20 a^{8} b x^{3}} - \frac{420 b^{7} x^{3} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{20 a^{9} x^{\frac{8}{3}} + 20 a^{8} b x^{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))**2/x**3,x)

[Out]

Piecewise((zoo/x**(8/3), Eq(a, 0) & Eq(b, 0)), (-3/(8*b**2*x**(8/3)), Eq(a, 0)), (-1/(2*a**2*x**2), Eq(b, 0)),
(-10*a**7*x**(2/3)/(20*a**9*x**(8/3) + 20*a**8*b*x**3) + 14*a**6*b*x/(20*a**9*x**(8/3) + 20*a**8*b*x**3) - 21
*a**5*b**2*x**(4/3)/(20*a**9*x**(8/3) + 20*a**8*b*x**3) + 35*a**4*b**3*x**(5/3)/(20*a**9*x**(8/3) + 20*a**8*b*
x**3) - 70*a**3*b**4*x**2/(20*a**9*x**(8/3) + 20*a**8*b*x**3) + 210*a**2*b**5*x**(7/3)/(20*a**9*x**(8/3) + 20*
a**8*b*x**3) + 140*a*b**6*x**(8/3)*log(x)/(20*a**9*x**(8/3) + 20*a**8*b*x**3) - 420*a*b**6*x**(8/3)*log(a/b +
x**(1/3))/(20*a**9*x**(8/3) + 20*a**8*b*x**3) + 420*a*b**6*x**(8/3)/(20*a**9*x**(8/3) + 20*a**8*b*x**3) + 140*
b**7*x**3*log(x)/(20*a**9*x**(8/3) + 20*a**8*b*x**3) - 420*b**7*x**3*log(a/b + x**(1/3))/(20*a**9*x**(8/3) + 2
0*a**8*b*x**3), True))

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Giac [A]  time = 1.18309, size = 151, normalized size = 1.21 \begin{align*} -\frac{21 \, b^{6} \log \left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{a^{8}} + \frac{7 \, b^{6} \log \left ({\left | x \right |}\right )}{a^{8}} + \frac{420 \, a b^{6} x^{2} + 210 \, a^{2} b^{5} x^{\frac{5}{3}} - 70 \, a^{3} b^{4} x^{\frac{4}{3}} + 35 \, a^{4} b^{3} x - 21 \, a^{5} b^{2} x^{\frac{2}{3}} + 14 \, a^{6} b x^{\frac{1}{3}} - 10 \, a^{7}}{20 \,{\left (b x^{\frac{1}{3}} + a\right )} a^{8} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-21*b^6*log(abs(b*x^(1/3) + a))/a^8 + 7*b^6*log(abs(x))/a^8 + 1/20*(420*a*b^6*x^2 + 210*a^2*b^5*x^(5/3) - 70*a
^3*b^4*x^(4/3) + 35*a^4*b^3*x - 21*a^5*b^2*x^(2/3) + 14*a^6*b*x^(1/3) - 10*a^7)/((b*x^(1/3) + a)*a^8*x^2)