3.2438 \(\int \frac {x}{(a+\frac {b}{\sqrt [3]{x}})^3} \, dx\)

Optimal. Leaf size=134 \[ -\frac {3 b^8}{2 a^9 \left (a \sqrt [3]{x}+b\right )^2}+\frac {24 b^7}{a^9 \left (a \sqrt [3]{x}+b\right )}+\frac {84 b^6 \log \left (a \sqrt [3]{x}+b\right )}{a^9}-\frac {63 b^5 \sqrt [3]{x}}{a^8}+\frac {45 b^4 x^{2/3}}{2 a^7}-\frac {10 b^3 x}{a^6}+\frac {9 b^2 x^{4/3}}{2 a^5}-\frac {9 b x^{5/3}}{5 a^4}+\frac {x^2}{2 a^3} \]

[Out]

-3/2*b^8/a^9/(b+a*x^(1/3))^2+24*b^7/a^9/(b+a*x^(1/3))-63*b^5*x^(1/3)/a^8+45/2*b^4*x^(2/3)/a^7-10*b^3*x/a^6+9/2
*b^2*x^(4/3)/a^5-9/5*b*x^(5/3)/a^4+1/2*x^2/a^3+84*b^6*ln(b+a*x^(1/3))/a^9

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Rubi [A]  time = 0.11, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {263, 266, 43} \[ \frac {45 b^4 x^{2/3}}{2 a^7}+\frac {9 b^2 x^{4/3}}{2 a^5}-\frac {3 b^8}{2 a^9 \left (a \sqrt [3]{x}+b\right )^2}+\frac {24 b^7}{a^9 \left (a \sqrt [3]{x}+b\right )}-\frac {63 b^5 \sqrt [3]{x}}{a^8}-\frac {10 b^3 x}{a^6}+\frac {84 b^6 \log \left (a \sqrt [3]{x}+b\right )}{a^9}-\frac {9 b x^{5/3}}{5 a^4}+\frac {x^2}{2 a^3} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b/x^(1/3))^3,x]

[Out]

(-3*b^8)/(2*a^9*(b + a*x^(1/3))^2) + (24*b^7)/(a^9*(b + a*x^(1/3))) - (63*b^5*x^(1/3))/a^8 + (45*b^4*x^(2/3))/
(2*a^7) - (10*b^3*x)/a^6 + (9*b^2*x^(4/3))/(2*a^5) - (9*b*x^(5/3))/(5*a^4) + x^2/(2*a^3) + (84*b^6*Log[b + a*x
^(1/3)])/a^9

Rule 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[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 {x}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3} \, dx &=\int \frac {x^2}{\left (b+a \sqrt [3]{x}\right )^3} \, dx\\ &=3 \operatorname {Subst}\left (\int \frac {x^8}{(b+a x)^3} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname {Subst}\left (\int \left (-\frac {21 b^5}{a^8}+\frac {15 b^4 x}{a^7}-\frac {10 b^3 x^2}{a^6}+\frac {6 b^2 x^3}{a^5}-\frac {3 b x^4}{a^4}+\frac {x^5}{a^3}+\frac {b^8}{a^8 (b+a x)^3}-\frac {8 b^7}{a^8 (b+a x)^2}+\frac {28 b^6}{a^8 (b+a x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {3 b^8}{2 a^9 \left (b+a \sqrt [3]{x}\right )^2}+\frac {24 b^7}{a^9 \left (b+a \sqrt [3]{x}\right )}-\frac {63 b^5 \sqrt [3]{x}}{a^8}+\frac {45 b^4 x^{2/3}}{2 a^7}-\frac {10 b^3 x}{a^6}+\frac {9 b^2 x^{4/3}}{2 a^5}-\frac {9 b x^{5/3}}{5 a^4}+\frac {x^2}{2 a^3}+\frac {84 b^6 \log \left (b+a \sqrt [3]{x}\right )}{a^9}\\ \end {align*}

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Mathematica [A]  time = 0.16, size = 132, normalized size = 0.99 \[ \frac {\frac {a \sqrt [3]{x} \left (5 a^7 x^{7/3}-8 a^6 b x^2+14 a^5 b^2 x^{5/3}-28 a^4 b^3 x^{4/3}+70 a^3 b^4 x-280 a^2 b^5 x^{2/3}-1260 a b^6 \sqrt [3]{x}-840 b^7\right )}{\left (a \sqrt [3]{x}+b\right )^2}+840 b^6 \log \left (a+\frac {b}{\sqrt [3]{x}}\right )+280 b^6 \log (x)}{10 a^9} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b/x^(1/3))^3,x]

[Out]

((a*x^(1/3)*(-840*b^7 - 1260*a*b^6*x^(1/3) - 280*a^2*b^5*x^(2/3) + 70*a^3*b^4*x - 28*a^4*b^3*x^(4/3) + 14*a^5*
b^2*x^(5/3) - 8*a^6*b*x^2 + 5*a^7*x^(7/3)))/(b + a*x^(1/3))^2 + 840*b^6*Log[a + b/x^(1/3)] + 280*b^6*Log[x])/(
10*a^9)

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fricas [A]  time = 0.85, size = 192, normalized size = 1.43 \[ \frac {5 \, a^{12} x^{4} - 90 \, a^{9} b^{3} x^{3} - 195 \, a^{6} b^{6} x^{2} + 170 \, a^{3} b^{9} x + 225 \, b^{12} + 840 \, {\left (a^{6} b^{6} x^{2} + 2 \, a^{3} b^{9} x + b^{12}\right )} \log \left (a x^{\frac {1}{3}} + b\right ) - 3 \, {\left (6 \, a^{11} b x^{3} - 63 \, a^{8} b^{4} x^{2} - 224 \, a^{5} b^{7} x - 140 \, a^{2} b^{10}\right )} x^{\frac {2}{3}} + 15 \, {\left (3 \, a^{10} b^{2} x^{3} - 36 \, a^{7} b^{5} x^{2} - 98 \, a^{4} b^{8} x - 56 \, a b^{11}\right )} x^{\frac {1}{3}}}{10 \, {\left (a^{15} x^{2} + 2 \, a^{12} b^{3} x + a^{9} b^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(5*a^12*x^4 - 90*a^9*b^3*x^3 - 195*a^6*b^6*x^2 + 170*a^3*b^9*x + 225*b^12 + 840*(a^6*b^6*x^2 + 2*a^3*b^9*
x + b^12)*log(a*x^(1/3) + b) - 3*(6*a^11*b*x^3 - 63*a^8*b^4*x^2 - 224*a^5*b^7*x - 140*a^2*b^10)*x^(2/3) + 15*(
3*a^10*b^2*x^3 - 36*a^7*b^5*x^2 - 98*a^4*b^8*x - 56*a*b^11)*x^(1/3))/(a^15*x^2 + 2*a^12*b^3*x + a^9*b^6)

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giac [A]  time = 0.16, size = 112, normalized size = 0.84 \[ \frac {84 \, b^{6} \log \left ({\left | a x^{\frac {1}{3}} + b \right |}\right )}{a^{9}} + \frac {3 \, {\left (16 \, a b^{7} x^{\frac {1}{3}} + 15 \, b^{8}\right )}}{2 \, {\left (a x^{\frac {1}{3}} + b\right )}^{2} a^{9}} + \frac {5 \, a^{15} x^{2} - 18 \, a^{14} b x^{\frac {5}{3}} + 45 \, a^{13} b^{2} x^{\frac {4}{3}} - 100 \, a^{12} b^{3} x + 225 \, a^{11} b^{4} x^{\frac {2}{3}} - 630 \, a^{10} b^{5} x^{\frac {1}{3}}}{10 \, a^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

84*b^6*log(abs(a*x^(1/3) + b))/a^9 + 3/2*(16*a*b^7*x^(1/3) + 15*b^8)/((a*x^(1/3) + b)^2*a^9) + 1/10*(5*a^15*x^
2 - 18*a^14*b*x^(5/3) + 45*a^13*b^2*x^(4/3) - 100*a^12*b^3*x + 225*a^11*b^4*x^(2/3) - 630*a^10*b^5*x^(1/3))/a^
18

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maple [A]  time = 0.01, size = 111, normalized size = 0.83 \[ \frac {x^{2}}{2 a^{3}}-\frac {3 b^{8}}{2 \left (a \,x^{\frac {1}{3}}+b \right )^{2} a^{9}}-\frac {9 b \,x^{\frac {5}{3}}}{5 a^{4}}+\frac {9 b^{2} x^{\frac {4}{3}}}{2 a^{5}}-\frac {10 b^{3} x}{a^{6}}+\frac {24 b^{7}}{\left (a \,x^{\frac {1}{3}}+b \right ) a^{9}}+\frac {84 b^{6} \ln \left (a \,x^{\frac {1}{3}}+b \right )}{a^{9}}+\frac {45 b^{4} x^{\frac {2}{3}}}{2 a^{7}}-\frac {63 b^{5} x^{\frac {1}{3}}}{a^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/2*b^8/a^9/(a*x^(1/3)+b)^2+24*b^7/a^9/(a*x^(1/3)+b)-63*b^5*x^(1/3)/a^8+45/2*b^4*x^(2/3)/a^7-10/a^6*b^3*x+9/2
*b^2*x^(4/3)/a^5-9/5*b*x^(5/3)/a^4+1/2/a^3*x^2+84*b^6*ln(a*x^(1/3)+b)/a^9

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maxima [A]  time = 0.50, size = 134, normalized size = 1.00 \[ \frac {5 \, a^{7} - \frac {8 \, a^{6} b}{x^{\frac {1}{3}}} + \frac {14 \, a^{5} b^{2}}{x^{\frac {2}{3}}} - \frac {28 \, a^{4} b^{3}}{x} + \frac {70 \, a^{3} b^{4}}{x^{\frac {4}{3}}} - \frac {280 \, a^{2} b^{5}}{x^{\frac {5}{3}}} - \frac {1260 \, a b^{6}}{x^{2}} - \frac {840 \, b^{7}}{x^{\frac {7}{3}}}}{10 \, {\left (\frac {a^{10}}{x^{2}} + \frac {2 \, a^{9} b}{x^{\frac {7}{3}}} + \frac {a^{8} b^{2}}{x^{\frac {8}{3}}}\right )}} + \frac {84 \, b^{6} \log \left (a + \frac {b}{x^{\frac {1}{3}}}\right )}{a^{9}} + \frac {28 \, b^{6} \log \relax (x)}{a^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(5*a^7 - 8*a^6*b/x^(1/3) + 14*a^5*b^2/x^(2/3) - 28*a^4*b^3/x + 70*a^3*b^4/x^(4/3) - 280*a^2*b^5/x^(5/3) -
1260*a*b^6/x^2 - 840*b^7/x^(7/3))/(a^10/x^2 + 2*a^9*b/x^(7/3) + a^8*b^2/x^(8/3)) + 84*b^6*log(a + b/x^(1/3))/
a^9 + 28*b^6*log(x)/a^9

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mupad [B]  time = 0.05, size = 120, normalized size = 0.90 \[ \frac {\frac {45\,b^8}{2\,a}+24\,b^7\,x^{1/3}}{a^8\,b^2+a^{10}\,x^{2/3}+2\,a^9\,b\,x^{1/3}}+\frac {x^2}{2\,a^3}-\frac {10\,b^3\,x}{a^6}-\frac {9\,b\,x^{5/3}}{5\,a^4}+\frac {84\,b^6\,\ln \left (b+a\,x^{1/3}\right )}{a^9}+\frac {9\,b^2\,x^{4/3}}{2\,a^5}+\frac {45\,b^4\,x^{2/3}}{2\,a^7}-\frac {63\,b^5\,x^{1/3}}{a^8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b/x^(1/3))^3,x)

[Out]

((45*b^8)/(2*a) + 24*b^7*x^(1/3))/(a^8*b^2 + a^10*x^(2/3) + 2*a^9*b*x^(1/3)) + x^2/(2*a^3) - (10*b^3*x)/a^6 -
(9*b*x^(5/3))/(5*a^4) + (84*b^6*log(b + a*x^(1/3)))/a^9 + (9*b^2*x^(4/3))/(2*a^5) + (45*b^4*x^(2/3))/(2*a^7) -
(63*b^5*x^(1/3))/a^8

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sympy [A]  time = 2.23, size = 493, normalized size = 3.68 \[ \begin {cases} \frac {5 a^{8} x^{\frac {8}{3}}}{10 a^{11} x^{\frac {2}{3}} + 20 a^{10} b \sqrt [3]{x} + 10 a^{9} b^{2}} - \frac {8 a^{7} b x^{\frac {7}{3}}}{10 a^{11} x^{\frac {2}{3}} + 20 a^{10} b \sqrt [3]{x} + 10 a^{9} b^{2}} + \frac {14 a^{6} b^{2} x^{2}}{10 a^{11} x^{\frac {2}{3}} + 20 a^{10} b \sqrt [3]{x} + 10 a^{9} b^{2}} - \frac {28 a^{5} b^{3} x^{\frac {5}{3}}}{10 a^{11} x^{\frac {2}{3}} + 20 a^{10} b \sqrt [3]{x} + 10 a^{9} b^{2}} + \frac {70 a^{4} b^{4} x^{\frac {4}{3}}}{10 a^{11} x^{\frac {2}{3}} + 20 a^{10} b \sqrt [3]{x} + 10 a^{9} b^{2}} - \frac {280 a^{3} b^{5} x}{10 a^{11} x^{\frac {2}{3}} + 20 a^{10} b \sqrt [3]{x} + 10 a^{9} b^{2}} + \frac {840 a^{2} b^{6} x^{\frac {2}{3}} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{10 a^{11} x^{\frac {2}{3}} + 20 a^{10} b \sqrt [3]{x} + 10 a^{9} b^{2}} + \frac {1680 a b^{7} \sqrt [3]{x} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{10 a^{11} x^{\frac {2}{3}} + 20 a^{10} b \sqrt [3]{x} + 10 a^{9} b^{2}} + \frac {1680 a b^{7} \sqrt [3]{x}}{10 a^{11} x^{\frac {2}{3}} + 20 a^{10} b \sqrt [3]{x} + 10 a^{9} b^{2}} + \frac {840 b^{8} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{10 a^{11} x^{\frac {2}{3}} + 20 a^{10} b \sqrt [3]{x} + 10 a^{9} b^{2}} + \frac {1260 b^{8}}{10 a^{11} x^{\frac {2}{3}} + 20 a^{10} b \sqrt [3]{x} + 10 a^{9} b^{2}} & \text {for}\: a \neq 0 \\\frac {x^{3}}{3 b^{3}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((5*a**8*x**(8/3)/(10*a**11*x**(2/3) + 20*a**10*b*x**(1/3) + 10*a**9*b**2) - 8*a**7*b*x**(7/3)/(10*a*
*11*x**(2/3) + 20*a**10*b*x**(1/3) + 10*a**9*b**2) + 14*a**6*b**2*x**2/(10*a**11*x**(2/3) + 20*a**10*b*x**(1/3
) + 10*a**9*b**2) - 28*a**5*b**3*x**(5/3)/(10*a**11*x**(2/3) + 20*a**10*b*x**(1/3) + 10*a**9*b**2) + 70*a**4*b
**4*x**(4/3)/(10*a**11*x**(2/3) + 20*a**10*b*x**(1/3) + 10*a**9*b**2) - 280*a**3*b**5*x/(10*a**11*x**(2/3) + 2
0*a**10*b*x**(1/3) + 10*a**9*b**2) + 840*a**2*b**6*x**(2/3)*log(x**(1/3) + b/a)/(10*a**11*x**(2/3) + 20*a**10*
b*x**(1/3) + 10*a**9*b**2) + 1680*a*b**7*x**(1/3)*log(x**(1/3) + b/a)/(10*a**11*x**(2/3) + 20*a**10*b*x**(1/3)
+ 10*a**9*b**2) + 1680*a*b**7*x**(1/3)/(10*a**11*x**(2/3) + 20*a**10*b*x**(1/3) + 10*a**9*b**2) + 840*b**8*lo
g(x**(1/3) + b/a)/(10*a**11*x**(2/3) + 20*a**10*b*x**(1/3) + 10*a**9*b**2) + 1260*b**8/(10*a**11*x**(2/3) + 20
*a**10*b*x**(1/3) + 10*a**9*b**2), Ne(a, 0)), (x**3/(3*b**3), True))

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