3.2362 \(\int \frac {1}{(a+b \sqrt [3]{x}) x^3} \, dx\)

Optimal. Leaf size=104 \[ -\frac {3 b^6 \log \left (a+b \sqrt [3]{x}\right )}{a^7}+\frac {b^6 \log (x)}{a^7}+\frac {3 b^5}{a^6 \sqrt [3]{x}}-\frac {3 b^4}{2 a^5 x^{2/3}}+\frac {b^3}{a^4 x}-\frac {3 b^2}{4 a^3 x^{4/3}}+\frac {3 b}{5 a^2 x^{5/3}}-\frac {1}{2 a x^2} \]

[Out]

-1/2/a/x^2+3/5*b/a^2/x^(5/3)-3/4*b^2/a^3/x^(4/3)+b^3/a^4/x-3/2*b^4/a^5/x^(2/3)+3*b^5/a^6/x^(1/3)-3*b^6*ln(a+b*
x^(1/3))/a^7+b^6*ln(x)/a^7

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Rubi [A]  time = 0.05, antiderivative size = 104, normalized size of antiderivative = 1.00, 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} \[ -\frac {3 b^4}{2 a^5 x^{2/3}}-\frac {3 b^2}{4 a^3 x^{4/3}}+\frac {3 b^5}{a^6 \sqrt [3]{x}}+\frac {b^3}{a^4 x}-\frac {3 b^6 \log \left (a+b \sqrt [3]{x}\right )}{a^7}+\frac {b^6 \log (x)}{a^7}+\frac {3 b}{5 a^2 x^{5/3}}-\frac {1}{2 a x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x^(1/3))*x^3),x]

[Out]

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

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])

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]]

Rubi steps

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

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Mathematica [A]  time = 0.07, size = 95, normalized size = 0.91 \[ \frac {\frac {a \left (-10 a^5+12 a^4 b \sqrt [3]{x}-15 a^3 b^2 x^{2/3}+20 a^2 b^3 x-30 a b^4 x^{4/3}+60 b^5 x^{5/3}\right )}{x^2}-60 b^6 \log \left (a+b \sqrt [3]{x}\right )+20 b^6 \log (x)}{20 a^7} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x^(1/3))*x^3),x]

[Out]

((a*(-10*a^5 + 12*a^4*b*x^(1/3) - 15*a^3*b^2*x^(2/3) + 20*a^2*b^3*x - 30*a*b^4*x^(4/3) + 60*b^5*x^(5/3)))/x^2
- 60*b^6*Log[a + b*x^(1/3)] + 20*b^6*Log[x])/(20*a^7)

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fricas [A]  time = 0.91, size = 93, normalized size = 0.89 \[ -\frac {60 \, b^{6} x^{2} \log \left (b x^{\frac {1}{3}} + a\right ) - 60 \, b^{6} x^{2} \log \left (x^{\frac {1}{3}}\right ) - 20 \, a^{3} b^{3} x + 10 \, a^{6} - 15 \, {\left (4 \, a b^{5} x - a^{4} b^{2}\right )} x^{\frac {2}{3}} + 6 \, {\left (5 \, a^{2} b^{4} x - 2 \, a^{5} b\right )} x^{\frac {1}{3}}}{20 \, a^{7} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/20*(60*b^6*x^2*log(b*x^(1/3) + a) - 60*b^6*x^2*log(x^(1/3)) - 20*a^3*b^3*x + 10*a^6 - 15*(4*a*b^5*x - a^4*b
^2)*x^(2/3) + 6*(5*a^2*b^4*x - 2*a^5*b)*x^(1/3))/(a^7*x^2)

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giac [A]  time = 0.16, size = 91, normalized size = 0.88 \[ -\frac {3 \, b^{6} \log \left ({\left | b x^{\frac {1}{3}} + a \right |}\right )}{a^{7}} + \frac {b^{6} \log \left ({\left | x \right |}\right )}{a^{7}} + \frac {60 \, a b^{5} x^{\frac {5}{3}} - 30 \, a^{2} b^{4} x^{\frac {4}{3}} + 20 \, a^{3} b^{3} x - 15 \, a^{4} b^{2} x^{\frac {2}{3}} + 12 \, a^{5} b x^{\frac {1}{3}} - 10 \, a^{6}}{20 \, a^{7} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*b^6*log(abs(b*x^(1/3) + a))/a^7 + b^6*log(abs(x))/a^7 + 1/20*(60*a*b^5*x^(5/3) - 30*a^2*b^4*x^(4/3) + 20*a^
3*b^3*x - 15*a^4*b^2*x^(2/3) + 12*a^5*b*x^(1/3) - 10*a^6)/(a^7*x^2)

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maple [A]  time = 0.01, size = 87, normalized size = 0.84 \[ \frac {b^{6} \ln \relax (x )}{a^{7}}-\frac {3 b^{6} \ln \left (b \,x^{\frac {1}{3}}+a \right )}{a^{7}}+\frac {3 b^{5}}{a^{6} x^{\frac {1}{3}}}-\frac {3 b^{4}}{2 a^{5} x^{\frac {2}{3}}}+\frac {b^{3}}{a^{4} x}-\frac {3 b^{2}}{4 a^{3} x^{\frac {4}{3}}}+\frac {3 b}{5 a^{2} x^{\frac {5}{3}}}-\frac {1}{2 a \,x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/a/x^2+3/5*b/a^2/x^(5/3)-3/4*b^2/a^3/x^(4/3)+b^3/a^4/x-3/2*b^4/a^5/x^(2/3)+3*b^5/a^6/x^(1/3)-3*b^6*ln(b*x^
(1/3)+a)/a^7+1/a^7*b^6*ln(x)

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maxima [A]  time = 0.54, size = 86, normalized size = 0.83 \[ -\frac {3 \, b^{6} \log \left (b x^{\frac {1}{3}} + a\right )}{a^{7}} + \frac {b^{6} \log \relax (x)}{a^{7}} + \frac {60 \, b^{5} x^{\frac {5}{3}} - 30 \, a b^{4} x^{\frac {4}{3}} + 20 \, a^{2} b^{3} x - 15 \, a^{3} b^{2} x^{\frac {2}{3}} + 12 \, a^{4} b x^{\frac {1}{3}} - 10 \, a^{5}}{20 \, a^{6} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*b^6*log(b*x^(1/3) + a)/a^7 + b^6*log(x)/a^7 + 1/20*(60*b^5*x^(5/3) - 30*a*b^4*x^(4/3) + 20*a^2*b^3*x - 15*a
^3*b^2*x^(2/3) + 12*a^4*b*x^(1/3) - 10*a^5)/(a^6*x^2)

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mupad [B]  time = 0.06, size = 83, normalized size = 0.80 \[ -\frac {\frac {1}{2\,a}-\frac {3\,b\,x^{1/3}}{5\,a^2}-\frac {b^3\,x}{a^4}+\frac {3\,b^2\,x^{2/3}}{4\,a^3}+\frac {3\,b^4\,x^{4/3}}{2\,a^5}-\frac {3\,b^5\,x^{5/3}}{a^6}}{x^2}-\frac {6\,b^6\,\mathrm {atanh}\left (\frac {2\,b\,x^{1/3}}{a}+1\right )}{a^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(a + b*x^(1/3))),x)

[Out]

- (1/(2*a) - (3*b*x^(1/3))/(5*a^2) - (b^3*x)/a^4 + (3*b^2*x^(2/3))/(4*a^3) + (3*b^4*x^(4/3))/(2*a^5) - (3*b^5*
x^(5/3))/a^6)/x^2 - (6*b^6*atanh((2*b*x^(1/3))/a + 1))/a^7

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sympy [A]  time = 8.70, size = 129, normalized size = 1.24 \[ \begin {cases} \frac {\tilde {\infty }}{x^{\frac {7}{3}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {3}{7 b x^{\frac {7}{3}}} & \text {for}\: a = 0 \\- \frac {1}{2 a x^{2}} & \text {for}\: b = 0 \\- \frac {1}{2 a x^{2}} + \frac {3 b}{5 a^{2} x^{\frac {5}{3}}} - \frac {3 b^{2}}{4 a^{3} x^{\frac {4}{3}}} + \frac {b^{3}}{a^{4} x} - \frac {3 b^{4}}{2 a^{5} x^{\frac {2}{3}}} + \frac {3 b^{5}}{a^{6} \sqrt [3]{x}} + \frac {b^{6} \log {\relax (x )}}{a^{7}} - \frac {3 b^{6} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{a^{7}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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