3.2363 \(\int \frac {1}{(a+b \sqrt [3]{x}) x^4} \, dx\)

Optimal. Leaf size=149 \[ \frac {3 b^9 \log \left (a+b \sqrt [3]{x}\right )}{a^{10}}-\frac {b^9 \log (x)}{a^{10}}-\frac {3 b^8}{a^9 \sqrt [3]{x}}+\frac {3 b^7}{2 a^8 x^{2/3}}-\frac {b^6}{a^7 x}+\frac {3 b^5}{4 a^6 x^{4/3}}-\frac {3 b^4}{5 a^5 x^{5/3}}+\frac {b^3}{2 a^4 x^2}-\frac {3 b^2}{7 a^3 x^{7/3}}+\frac {3 b}{8 a^2 x^{8/3}}-\frac {1}{3 a x^3} \]

[Out]

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

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Rubi [A]  time = 0.08, antiderivative size = 149, 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} \[ \frac {3 b^7}{2 a^8 x^{2/3}}+\frac {3 b^5}{4 a^6 x^{4/3}}-\frac {3 b^4}{5 a^5 x^{5/3}}+\frac {b^3}{2 a^4 x^2}-\frac {3 b^2}{7 a^3 x^{7/3}}-\frac {3 b^8}{a^9 \sqrt [3]{x}}-\frac {b^6}{a^7 x}+\frac {3 b^9 \log \left (a+b \sqrt [3]{x}\right )}{a^{10}}-\frac {b^9 \log (x)}{a^{10}}+\frac {3 b}{8 a^2 x^{8/3}}-\frac {1}{3 a x^3} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x^(1/3))*x^4),x]

[Out]

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

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

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Mathematica [A]  time = 0.07, size = 138, normalized size = 0.93 \[ -\frac {280 a^9-315 a^8 b \sqrt [3]{x}+360 a^7 b^2 x^{2/3}-420 a^6 b^3 x+504 a^5 b^4 x^{4/3}-630 a^4 b^5 x^{5/3}+840 a^3 b^6 x^2-1260 a^2 b^7 x^{7/3}-2520 b^9 x^3 \log \left (a+b \sqrt [3]{x}\right )+2520 a b^8 x^{8/3}+840 b^9 x^3 \log (x)}{840 a^{10} x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x^(1/3))*x^4),x]

[Out]

-1/840*(280*a^9 - 315*a^8*b*x^(1/3) + 360*a^7*b^2*x^(2/3) - 420*a^6*b^3*x + 504*a^5*b^4*x^(4/3) - 630*a^4*b^5*
x^(5/3) + 840*a^3*b^6*x^2 - 1260*a^2*b^7*x^(7/3) + 2520*a*b^8*x^(8/3) - 2520*b^9*x^3*Log[a + b*x^(1/3)] + 840*
b^9*x^3*Log[x])/(a^10*x^3)

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fricas [A]  time = 0.70, size = 126, normalized size = 0.85 \[ \frac {2520 \, b^{9} x^{3} \log \left (b x^{\frac {1}{3}} + a\right ) - 2520 \, b^{9} x^{3} \log \left (x^{\frac {1}{3}}\right ) - 840 \, a^{3} b^{6} x^{2} + 420 \, a^{6} b^{3} x - 280 \, a^{9} - 90 \, {\left (28 \, a b^{8} x^{2} - 7 \, a^{4} b^{5} x + 4 \, a^{7} b^{2}\right )} x^{\frac {2}{3}} + 63 \, {\left (20 \, a^{2} b^{7} x^{2} - 8 \, a^{5} b^{4} x + 5 \, a^{8} b\right )} x^{\frac {1}{3}}}{840 \, a^{10} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/840*(2520*b^9*x^3*log(b*x^(1/3) + a) - 2520*b^9*x^3*log(x^(1/3)) - 840*a^3*b^6*x^2 + 420*a^6*b^3*x - 280*a^9
- 90*(28*a*b^8*x^2 - 7*a^4*b^5*x + 4*a^7*b^2)*x^(2/3) + 63*(20*a^2*b^7*x^2 - 8*a^5*b^4*x + 5*a^8*b)*x^(1/3))/
(a^10*x^3)

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giac [A]  time = 0.17, size = 125, normalized size = 0.84 \[ \frac {3 \, b^{9} \log \left ({\left | b x^{\frac {1}{3}} + a \right |}\right )}{a^{10}} - \frac {b^{9} \log \left ({\left | x \right |}\right )}{a^{10}} - \frac {2520 \, a b^{8} x^{\frac {8}{3}} - 1260 \, a^{2} b^{7} x^{\frac {7}{3}} + 840 \, a^{3} b^{6} x^{2} - 630 \, a^{4} b^{5} x^{\frac {5}{3}} + 504 \, a^{5} b^{4} x^{\frac {4}{3}} - 420 \, a^{6} b^{3} x + 360 \, a^{7} b^{2} x^{\frac {2}{3}} - 315 \, a^{8} b x^{\frac {1}{3}} + 280 \, a^{9}}{840 \, a^{10} x^{3}} \]

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*b^9*log(abs(b*x^(1/3) + a))/a^10 - b^9*log(abs(x))/a^10 - 1/840*(2520*a*b^8*x^(8/3) - 1260*a^2*b^7*x^(7/3) +
840*a^3*b^6*x^2 - 630*a^4*b^5*x^(5/3) + 504*a^5*b^4*x^(4/3) - 420*a^6*b^3*x + 360*a^7*b^2*x^(2/3) - 315*a^8*b
*x^(1/3) + 280*a^9)/(a^10*x^3)

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maple [A]  time = 0.01, size = 122, normalized size = 0.82 \[ -\frac {b^{9} \ln \relax (x )}{a^{10}}+\frac {3 b^{9} \ln \left (b \,x^{\frac {1}{3}}+a \right )}{a^{10}}-\frac {3 b^{8}}{a^{9} x^{\frac {1}{3}}}+\frac {3 b^{7}}{2 a^{8} x^{\frac {2}{3}}}-\frac {b^{6}}{a^{7} x}+\frac {3 b^{5}}{4 a^{6} x^{\frac {4}{3}}}-\frac {3 b^{4}}{5 a^{5} x^{\frac {5}{3}}}+\frac {b^{3}}{2 a^{4} x^{2}}-\frac {3 b^{2}}{7 a^{3} x^{\frac {7}{3}}}+\frac {3 b}{8 a^{2} x^{\frac {8}{3}}}-\frac {1}{3 a \,x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.55, size = 120, normalized size = 0.81 \[ \frac {3 \, b^{9} \log \left (b x^{\frac {1}{3}} + a\right )}{a^{10}} - \frac {b^{9} \log \relax (x)}{a^{10}} - \frac {2520 \, b^{8} x^{\frac {8}{3}} - 1260 \, a b^{7} x^{\frac {7}{3}} + 840 \, a^{2} b^{6} x^{2} - 630 \, a^{3} b^{5} x^{\frac {5}{3}} + 504 \, a^{4} b^{4} x^{\frac {4}{3}} - 420 \, a^{5} b^{3} x + 360 \, a^{6} b^{2} x^{\frac {2}{3}} - 315 \, a^{7} b x^{\frac {1}{3}} + 280 \, a^{8}}{840 \, a^{9} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*b^9*log(b*x^(1/3) + a)/a^10 - b^9*log(x)/a^10 - 1/840*(2520*b^8*x^(8/3) - 1260*a*b^7*x^(7/3) + 840*a^2*b^6*x
^2 - 630*a^3*b^5*x^(5/3) + 504*a^4*b^4*x^(4/3) - 420*a^5*b^3*x + 360*a^6*b^2*x^(2/3) - 315*a^7*b*x^(1/3) + 280
*a^8)/(a^9*x^3)

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mupad [B]  time = 1.16, size = 116, normalized size = 0.78 \[ -\frac {280\,a^9-5040\,b^9\,x^3\,\mathrm {atanh}\left (\frac {2\,b\,x^{1/3}}{a}+1\right )-420\,a^6\,b^3\,x-315\,a^8\,b\,x^{1/3}+2520\,a\,b^8\,x^{8/3}+840\,a^3\,b^6\,x^2+360\,a^7\,b^2\,x^{2/3}+504\,a^5\,b^4\,x^{4/3}-630\,a^4\,b^5\,x^{5/3}-1260\,a^2\,b^7\,x^{7/3}}{840\,a^{10}\,x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(a + b*x^(1/3))),x)

[Out]

-(280*a^9 - 5040*b^9*x^3*atanh((2*b*x^(1/3))/a + 1) - 420*a^6*b^3*x - 315*a^8*b*x^(1/3) + 2520*a*b^8*x^(8/3) +
840*a^3*b^6*x^2 + 360*a^7*b^2*x^(2/3) + 504*a^5*b^4*x^(4/3) - 630*a^4*b^5*x^(5/3) - 1260*a^2*b^7*x^(7/3))/(84
0*a^10*x^3)

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sympy [A]  time = 22.37, size = 172, normalized size = 1.15 \[ \begin {cases} \frac {\tilde {\infty }}{x^{\frac {10}{3}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {3}{10 b x^{\frac {10}{3}}} & \text {for}\: a = 0 \\- \frac {1}{3 a x^{3}} & \text {for}\: b = 0 \\- \frac {1}{3 a x^{3}} + \frac {3 b}{8 a^{2} x^{\frac {8}{3}}} - \frac {3 b^{2}}{7 a^{3} x^{\frac {7}{3}}} + \frac {b^{3}}{2 a^{4} x^{2}} - \frac {3 b^{4}}{5 a^{5} x^{\frac {5}{3}}} + \frac {3 b^{5}}{4 a^{6} x^{\frac {4}{3}}} - \frac {b^{6}}{a^{7} x} + \frac {3 b^{7}}{2 a^{8} x^{\frac {2}{3}}} - \frac {3 b^{8}}{a^{9} \sqrt [3]{x}} - \frac {b^{9} \log {\relax (x )}}{a^{10}} + \frac {3 b^{9} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{a^{10}} & \text {otherwise} \end {cases} \]

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**(10/3), Eq(a, 0) & Eq(b, 0)), (-3/(10*b*x**(10/3)), Eq(a, 0)), (-1/(3*a*x**3), Eq(b, 0)), (-
1/(3*a*x**3) + 3*b/(8*a**2*x**(8/3)) - 3*b**2/(7*a**3*x**(7/3)) + b**3/(2*a**4*x**2) - 3*b**4/(5*a**5*x**(5/3)
) + 3*b**5/(4*a**6*x**(4/3)) - b**6/(a**7*x) + 3*b**7/(2*a**8*x**(2/3)) - 3*b**8/(a**9*x**(1/3)) - b**9*log(x)
/a**10 + 3*b**9*log(a/b + x**(1/3))/a**10, True))

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