3.2427 \(\int \frac {1}{(a+\frac {b}{\sqrt [3]{x}}) x^4} \, dx\)

Optimal. Leaf size=134 \[ -\frac {3 a^8 \log \left (a \sqrt [3]{x}+b\right )}{b^9}+\frac {a^8 \log (x)}{b^9}+\frac {3 a^7}{b^8 \sqrt [3]{x}}-\frac {3 a^6}{2 b^7 x^{2/3}}+\frac {a^5}{b^6 x}-\frac {3 a^4}{4 b^5 x^{4/3}}+\frac {3 a^3}{5 b^4 x^{5/3}}-\frac {a^2}{2 b^3 x^2}+\frac {3 a}{7 b^2 x^{7/3}}-\frac {3}{8 b x^{8/3}} \]

[Out]

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

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Rubi [A]  time = 0.08, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {263, 266, 44} \[ -\frac {3 a^6}{2 b^7 x^{2/3}}-\frac {3 a^4}{4 b^5 x^{4/3}}+\frac {3 a^3}{5 b^4 x^{5/3}}-\frac {a^2}{2 b^3 x^2}+\frac {3 a^7}{b^8 \sqrt [3]{x}}+\frac {a^5}{b^6 x}-\frac {3 a^8 \log \left (a \sqrt [3]{x}+b\right )}{b^9}+\frac {a^8 \log (x)}{b^9}+\frac {3 a}{7 b^2 x^{7/3}}-\frac {3}{8 b x^{8/3}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^(1/3))*x^4),x]

[Out]

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

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 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]]

Rubi steps

\begin {align*} \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right ) x^4} \, dx &=\int \frac {1}{\left (b+a \sqrt [3]{x}\right ) x^{11/3}} \, dx\\ &=3 \operatorname {Subst}\left (\int \frac {1}{x^9 (b+a x)} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname {Subst}\left (\int \left (\frac {1}{b x^9}-\frac {a}{b^2 x^8}+\frac {a^2}{b^3 x^7}-\frac {a^3}{b^4 x^6}+\frac {a^4}{b^5 x^5}-\frac {a^5}{b^6 x^4}+\frac {a^6}{b^7 x^3}-\frac {a^7}{b^8 x^2}+\frac {a^8}{b^9 x}-\frac {a^9}{b^9 (b+a x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {3}{8 b x^{8/3}}+\frac {3 a}{7 b^2 x^{7/3}}-\frac {a^2}{2 b^3 x^2}+\frac {3 a^3}{5 b^4 x^{5/3}}-\frac {3 a^4}{4 b^5 x^{4/3}}+\frac {a^5}{b^6 x}-\frac {3 a^6}{2 b^7 x^{2/3}}+\frac {3 a^7}{b^8 \sqrt [3]{x}}-\frac {3 a^8 \log \left (b+a \sqrt [3]{x}\right )}{b^9}+\frac {a^8 \log (x)}{b^9}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 121, normalized size = 0.90 \[ \frac {-840 a^8 \log \left (a \sqrt [3]{x}+b\right )+280 a^8 \log (x)+\frac {b \left (840 a^7 x^{7/3}-420 a^6 b x^2+280 a^5 b^2 x^{5/3}-210 a^4 b^3 x^{4/3}+168 a^3 b^4 x-140 a^2 b^5 x^{2/3}+120 a b^6 \sqrt [3]{x}-105 b^7\right )}{x^{8/3}}}{280 b^9} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^(1/3))*x^4),x]

[Out]

((b*(-105*b^7 + 120*a*b^6*x^(1/3) - 140*a^2*b^5*x^(2/3) + 168*a^3*b^4*x - 210*a^4*b^3*x^(4/3) + 280*a^5*b^2*x^
(5/3) - 420*a^6*b*x^2 + 840*a^7*x^(7/3)))/x^(8/3) - 840*a^8*Log[b + a*x^(1/3)] + 280*a^8*Log[x])/(280*b^9)

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fricas [A]  time = 0.60, size = 118, normalized size = 0.88 \[ -\frac {840 \, a^{8} x^{3} \log \left (a x^{\frac {1}{3}} + b\right ) - 840 \, a^{8} x^{3} \log \left (x^{\frac {1}{3}}\right ) - 280 \, a^{5} b^{3} x^{2} + 140 \, a^{2} b^{6} x - 30 \, {\left (28 \, a^{7} b x^{2} - 7 \, a^{4} b^{4} x + 4 \, a b^{7}\right )} x^{\frac {2}{3}} + 21 \, {\left (20 \, a^{6} b^{2} x^{2} - 8 \, a^{3} b^{5} x + 5 \, b^{8}\right )} x^{\frac {1}{3}}}{280 \, b^{9} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^(1/3))/x^4,x, algorithm="fricas")

[Out]

-1/280*(840*a^8*x^3*log(a*x^(1/3) + b) - 840*a^8*x^3*log(x^(1/3)) - 280*a^5*b^3*x^2 + 140*a^2*b^6*x - 30*(28*a
^7*b*x^2 - 7*a^4*b^4*x + 4*a*b^7)*x^(2/3) + 21*(20*a^6*b^2*x^2 - 8*a^3*b^5*x + 5*b^8)*x^(1/3))/(b^9*x^3)

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giac [A]  time = 0.18, size = 113, normalized size = 0.84 \[ -\frac {3 \, a^{8} \log \left ({\left | a x^{\frac {1}{3}} + b \right |}\right )}{b^{9}} + \frac {a^{8} \log \left ({\left | x \right |}\right )}{b^{9}} + \frac {840 \, a^{7} b x^{\frac {7}{3}} - 420 \, a^{6} b^{2} x^{2} + 280 \, a^{5} b^{3} x^{\frac {5}{3}} - 210 \, a^{4} b^{4} x^{\frac {4}{3}} + 168 \, a^{3} b^{5} x - 140 \, a^{2} b^{6} x^{\frac {2}{3}} + 120 \, a b^{7} x^{\frac {1}{3}} - 105 \, b^{8}}{280 \, b^{9} x^{\frac {8}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^(1/3))/x^4,x, algorithm="giac")

[Out]

-3*a^8*log(abs(a*x^(1/3) + b))/b^9 + a^8*log(abs(x))/b^9 + 1/280*(840*a^7*b*x^(7/3) - 420*a^6*b^2*x^2 + 280*a^
5*b^3*x^(5/3) - 210*a^4*b^4*x^(4/3) + 168*a^3*b^5*x - 140*a^2*b^6*x^(2/3) + 120*a*b^7*x^(1/3) - 105*b^8)/(b^9*
x^(8/3))

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maple [A]  time = 0.01, size = 109, normalized size = 0.81 \[ \frac {a^{8} \ln \relax (x )}{b^{9}}-\frac {3 a^{8} \ln \left (a \,x^{\frac {1}{3}}+b \right )}{b^{9}}+\frac {3 a^{7}}{b^{8} x^{\frac {1}{3}}}-\frac {3 a^{6}}{2 b^{7} x^{\frac {2}{3}}}+\frac {a^{5}}{b^{6} x}-\frac {3 a^{4}}{4 b^{5} x^{\frac {4}{3}}}+\frac {3 a^{3}}{5 b^{4} x^{\frac {5}{3}}}-\frac {a^{2}}{2 b^{3} x^{2}}+\frac {3 a}{7 b^{2} x^{\frac {7}{3}}}-\frac {3}{8 b \,x^{\frac {8}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x^(1/3))/x^4,x)

[Out]

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

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maxima [A]  time = 0.64, size = 146, normalized size = 1.09 \[ -\frac {3 \, a^{8} \log \left (a + \frac {b}{x^{\frac {1}{3}}}\right )}{b^{9}} - \frac {3 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{8}}{8 \, b^{9}} + \frac {24 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{7} a}{7 \, b^{9}} - \frac {14 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{6} a^{2}}{b^{9}} + \frac {168 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{5} a^{3}}{5 \, b^{9}} - \frac {105 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{4} a^{4}}{2 \, b^{9}} + \frac {56 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{3} a^{5}}{b^{9}} - \frac {42 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{2} a^{6}}{b^{9}} + \frac {24 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )} a^{7}}{b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^(1/3))/x^4,x, algorithm="maxima")

[Out]

-3*a^8*log(a + b/x^(1/3))/b^9 - 3/8*(a + b/x^(1/3))^8/b^9 + 24/7*(a + b/x^(1/3))^7*a/b^9 - 14*(a + b/x^(1/3))^
6*a^2/b^9 + 168/5*(a + b/x^(1/3))^5*a^3/b^9 - 105/2*(a + b/x^(1/3))^4*a^4/b^9 + 56*(a + b/x^(1/3))^3*a^5/b^9 -
42*(a + b/x^(1/3))^2*a^6/b^9 + 24*(a + b/x^(1/3))*a^7/b^9

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mupad [B]  time = 0.07, size = 105, normalized size = 0.78 \[ -\frac {\frac {3}{8\,b}-\frac {3\,a\,x^{1/3}}{7\,b^2}-\frac {3\,a^3\,x}{5\,b^4}+\frac {a^2\,x^{2/3}}{2\,b^3}+\frac {3\,a^6\,x^2}{2\,b^7}+\frac {3\,a^4\,x^{4/3}}{4\,b^5}-\frac {a^5\,x^{5/3}}{b^6}-\frac {3\,a^7\,x^{7/3}}{b^8}}{x^{8/3}}-\frac {6\,a^8\,\mathrm {atanh}\left (\frac {2\,a\,x^{1/3}}{b}+1\right )}{b^9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(a + b/x^(1/3))),x)

[Out]

- (3/(8*b) - (3*a*x^(1/3))/(7*b^2) - (3*a^3*x)/(5*b^4) + (a^2*x^(2/3))/(2*b^3) + (3*a^6*x^2)/(2*b^7) + (3*a^4*
x^(4/3))/(4*b^5) - (a^5*x^(5/3))/b^6 - (3*a^7*x^(7/3))/b^8)/x^(8/3) - (6*a^8*atanh((2*a*x^(1/3))/b + 1))/b^9

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sympy [A]  time = 13.98, size = 158, normalized size = 1.18 \[ \begin {cases} \frac {\tilde {\infty }}{x^{\frac {8}{3}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {1}{3 a x^{3}} & \text {for}\: b = 0 \\- \frac {3}{8 b x^{\frac {8}{3}}} & \text {for}\: a = 0 \\\frac {a^{8} \log {\relax (x )}}{b^{9}} - \frac {3 a^{8} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{b^{9}} + \frac {3 a^{7}}{b^{8} \sqrt [3]{x}} - \frac {3 a^{6}}{2 b^{7} x^{\frac {2}{3}}} + \frac {a^{5}}{b^{6} x} - \frac {3 a^{4}}{4 b^{5} x^{\frac {4}{3}}} + \frac {3 a^{3}}{5 b^{4} x^{\frac {5}{3}}} - \frac {a^{2}}{2 b^{3} x^{2}} + \frac {3 a}{7 b^{2} x^{\frac {7}{3}}} - \frac {3}{8 b x^{\frac {8}{3}}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x**(1/3))/x**4,x)

[Out]

Piecewise((zoo/x**(8/3), Eq(a, 0) & Eq(b, 0)), (-1/(3*a*x**3), Eq(b, 0)), (-3/(8*b*x**(8/3)), Eq(a, 0)), (a**8
*log(x)/b**9 - 3*a**8*log(x**(1/3) + b/a)/b**9 + 3*a**7/(b**8*x**(1/3)) - 3*a**6/(2*b**7*x**(2/3)) + a**5/(b**
6*x) - 3*a**4/(4*b**5*x**(4/3)) + 3*a**3/(5*b**4*x**(5/3)) - a**2/(2*b**3*x**2) + 3*a/(7*b**2*x**(7/3)) - 3/(8
*b*x**(8/3)), True))

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