3.2437 \(\int \frac {x^2}{(a+\frac {b}{\sqrt [3]{x}})^3} \, dx\)

Optimal. Leaf size=171 \[ \frac {3 b^{11}}{2 a^{12} \left (a \sqrt [3]{x}+b\right )^2}-\frac {33 b^{10}}{a^{12} \left (a \sqrt [3]{x}+b\right )}-\frac {165 b^9 \log \left (a \sqrt [3]{x}+b\right )}{a^{12}}+\frac {135 b^8 \sqrt [3]{x}}{a^{11}}-\frac {54 b^7 x^{2/3}}{a^{10}}+\frac {28 b^6 x}{a^9}-\frac {63 b^5 x^{4/3}}{4 a^8}+\frac {9 b^4 x^{5/3}}{a^7}-\frac {5 b^3 x^2}{a^6}+\frac {18 b^2 x^{7/3}}{7 a^5}-\frac {9 b x^{8/3}}{8 a^4}+\frac {x^3}{3 a^3} \]

[Out]

3/2*b^11/a^12/(b+a*x^(1/3))^2-33*b^10/a^12/(b+a*x^(1/3))+135*b^8*x^(1/3)/a^11-54*b^7*x^(2/3)/a^10+28*b^6*x/a^9
-63/4*b^5*x^(4/3)/a^8+9*b^4*x^(5/3)/a^7-5*b^3*x^2/a^6+18/7*b^2*x^(7/3)/a^5-9/8*b*x^(8/3)/a^4+1/3*x^3/a^3-165*b
^9*ln(b+a*x^(1/3))/a^12

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Rubi [A]  time = 0.15, antiderivative size = 171, 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, 43} \[ -\frac {54 b^7 x^{2/3}}{a^{10}}-\frac {63 b^5 x^{4/3}}{4 a^8}+\frac {9 b^4 x^{5/3}}{a^7}-\frac {5 b^3 x^2}{a^6}+\frac {18 b^2 x^{7/3}}{7 a^5}+\frac {3 b^{11}}{2 a^{12} \left (a \sqrt [3]{x}+b\right )^2}-\frac {33 b^{10}}{a^{12} \left (a \sqrt [3]{x}+b\right )}+\frac {135 b^8 \sqrt [3]{x}}{a^{11}}+\frac {28 b^6 x}{a^9}-\frac {165 b^9 \log \left (a \sqrt [3]{x}+b\right )}{a^{12}}-\frac {9 b x^{8/3}}{8 a^4}+\frac {x^3}{3 a^3} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b/x^(1/3))^3,x]

[Out]

(3*b^11)/(2*a^12*(b + a*x^(1/3))^2) - (33*b^10)/(a^12*(b + a*x^(1/3))) + (135*b^8*x^(1/3))/a^11 - (54*b^7*x^(2
/3))/a^10 + (28*b^6*x)/a^9 - (63*b^5*x^(4/3))/(4*a^8) + (9*b^4*x^(5/3))/a^7 - (5*b^3*x^2)/a^6 + (18*b^2*x^(7/3
))/(7*a^5) - (9*b*x^(8/3))/(8*a^4) + x^3/(3*a^3) - (165*b^9*Log[b + a*x^(1/3)])/a^12

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^2}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3} \, dx &=\int \frac {x^3}{\left (b+a \sqrt [3]{x}\right )^3} \, dx\\ &=3 \operatorname {Subst}\left (\int \frac {x^{11}}{(b+a x)^3} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname {Subst}\left (\int \left (\frac {45 b^8}{a^{11}}-\frac {36 b^7 x}{a^{10}}+\frac {28 b^6 x^2}{a^9}-\frac {21 b^5 x^3}{a^8}+\frac {15 b^4 x^4}{a^7}-\frac {10 b^3 x^5}{a^6}+\frac {6 b^2 x^6}{a^5}-\frac {3 b x^7}{a^4}+\frac {x^8}{a^3}-\frac {b^{11}}{a^{11} (b+a x)^3}+\frac {11 b^{10}}{a^{11} (b+a x)^2}-\frac {55 b^9}{a^{11} (b+a x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {3 b^{11}}{2 a^{12} \left (b+a \sqrt [3]{x}\right )^2}-\frac {33 b^{10}}{a^{12} \left (b+a \sqrt [3]{x}\right )}+\frac {135 b^8 \sqrt [3]{x}}{a^{11}}-\frac {54 b^7 x^{2/3}}{a^{10}}+\frac {28 b^6 x}{a^9}-\frac {63 b^5 x^{4/3}}{4 a^8}+\frac {9 b^4 x^{5/3}}{a^7}-\frac {5 b^3 x^2}{a^6}+\frac {18 b^2 x^{7/3}}{7 a^5}-\frac {9 b x^{8/3}}{8 a^4}+\frac {x^3}{3 a^3}-\frac {165 b^9 \log \left (b+a \sqrt [3]{x}\right )}{a^{12}}\\ \end {align*}

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Mathematica [A]  time = 0.16, size = 157, normalized size = 0.92 \[ \frac {56 a^9 x^3-189 a^8 b x^{8/3}+432 a^7 b^2 x^{7/3}-840 a^6 b^3 x^2+1512 a^5 b^4 x^{5/3}-2646 a^4 b^5 x^{4/3}+4704 a^3 b^6 x-9072 a^2 b^7 x^{2/3}+\frac {252 b^{11}}{\left (a \sqrt [3]{x}+b\right )^2}-\frac {5544 b^{10}}{a \sqrt [3]{x}+b}-27720 b^9 \log \left (a \sqrt [3]{x}+b\right )+22680 a b^8 \sqrt [3]{x}}{168 a^{12}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b/x^(1/3))^3,x]

[Out]

((252*b^11)/(b + a*x^(1/3))^2 - (5544*b^10)/(b + a*x^(1/3)) + 22680*a*b^8*x^(1/3) - 9072*a^2*b^7*x^(2/3) + 470
4*a^3*b^6*x - 2646*a^4*b^5*x^(4/3) + 1512*a^5*b^4*x^(5/3) - 840*a^6*b^3*x^2 + 432*a^7*b^2*x^(7/3) - 189*a^8*b*
x^(8/3) + 56*a^9*x^3 - 27720*b^9*Log[b + a*x^(1/3)])/(168*a^12)

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fricas [A]  time = 0.85, size = 225, normalized size = 1.32 \[ \frac {56 \, a^{15} x^{5} - 728 \, a^{12} b^{3} x^{4} + 3080 \, a^{9} b^{6} x^{3} + 8568 \, a^{6} b^{9} x^{2} - 1344 \, a^{3} b^{12} x - 5292 \, b^{15} - 27720 \, {\left (a^{6} b^{9} x^{2} + 2 \, a^{3} b^{12} x + b^{15}\right )} \log \left (a x^{\frac {1}{3}} + b\right ) - 63 \, {\left (3 \, a^{14} b x^{4} - 18 \, a^{11} b^{4} x^{3} + 99 \, a^{8} b^{7} x^{2} + 352 \, a^{5} b^{10} x + 220 \, a^{2} b^{13}\right )} x^{\frac {2}{3}} + 18 \, {\left (24 \, a^{13} b^{2} x^{4} - 99 \, a^{10} b^{5} x^{3} + 990 \, a^{7} b^{8} x^{2} + 2695 \, a^{4} b^{11} x + 1540 \, a b^{14}\right )} x^{\frac {1}{3}}}{168 \, {\left (a^{18} x^{2} + 2 \, a^{15} b^{3} x + a^{12} b^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/168*(56*a^15*x^5 - 728*a^12*b^3*x^4 + 3080*a^9*b^6*x^3 + 8568*a^6*b^9*x^2 - 1344*a^3*b^12*x - 5292*b^15 - 27
720*(a^6*b^9*x^2 + 2*a^3*b^12*x + b^15)*log(a*x^(1/3) + b) - 63*(3*a^14*b*x^4 - 18*a^11*b^4*x^3 + 99*a^8*b^7*x
^2 + 352*a^5*b^10*x + 220*a^2*b^13)*x^(2/3) + 18*(24*a^13*b^2*x^4 - 99*a^10*b^5*x^3 + 990*a^7*b^8*x^2 + 2695*a
^4*b^11*x + 1540*a*b^14)*x^(1/3))/(a^18*x^2 + 2*a^15*b^3*x + a^12*b^6)

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giac [A]  time = 0.17, size = 145, normalized size = 0.85 \[ -\frac {165 \, b^{9} \log \left ({\left | a x^{\frac {1}{3}} + b \right |}\right )}{a^{12}} - \frac {3 \, {\left (22 \, a b^{10} x^{\frac {1}{3}} + 21 \, b^{11}\right )}}{2 \, {\left (a x^{\frac {1}{3}} + b\right )}^{2} a^{12}} + \frac {56 \, a^{24} x^{3} - 189 \, a^{23} b x^{\frac {8}{3}} + 432 \, a^{22} b^{2} x^{\frac {7}{3}} - 840 \, a^{21} b^{3} x^{2} + 1512 \, a^{20} b^{4} x^{\frac {5}{3}} - 2646 \, a^{19} b^{5} x^{\frac {4}{3}} + 4704 \, a^{18} b^{6} x - 9072 \, a^{17} b^{7} x^{\frac {2}{3}} + 22680 \, a^{16} b^{8} x^{\frac {1}{3}}}{168 \, a^{27}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-165*b^9*log(abs(a*x^(1/3) + b))/a^12 - 3/2*(22*a*b^10*x^(1/3) + 21*b^11)/((a*x^(1/3) + b)^2*a^12) + 1/168*(56
*a^24*x^3 - 189*a^23*b*x^(8/3) + 432*a^22*b^2*x^(7/3) - 840*a^21*b^3*x^2 + 1512*a^20*b^4*x^(5/3) - 2646*a^19*b
^5*x^(4/3) + 4704*a^18*b^6*x - 9072*a^17*b^7*x^(2/3) + 22680*a^16*b^8*x^(1/3))/a^27

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maple [A]  time = 0.01, size = 144, normalized size = 0.84 \[ \frac {x^{3}}{3 a^{3}}-\frac {9 b \,x^{\frac {8}{3}}}{8 a^{4}}+\frac {18 b^{2} x^{\frac {7}{3}}}{7 a^{5}}-\frac {5 b^{3} x^{2}}{a^{6}}+\frac {3 b^{11}}{2 \left (a \,x^{\frac {1}{3}}+b \right )^{2} a^{12}}+\frac {9 b^{4} x^{\frac {5}{3}}}{a^{7}}-\frac {63 b^{5} x^{\frac {4}{3}}}{4 a^{8}}+\frac {28 b^{6} x}{a^{9}}-\frac {33 b^{10}}{\left (a \,x^{\frac {1}{3}}+b \right ) a^{12}}-\frac {165 b^{9} \ln \left (a \,x^{\frac {1}{3}}+b \right )}{a^{12}}-\frac {54 b^{7} x^{\frac {2}{3}}}{a^{10}}+\frac {135 b^{8} x^{\frac {1}{3}}}{a^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*b^11/a^12/(a*x^(1/3)+b)^2-33*b^10/a^12/(a*x^(1/3)+b)+135*b^8*x^(1/3)/a^11-54*b^7*x^(2/3)/a^10+28*b^6*x/a^9
-63/4*b^5*x^(4/3)/a^8+9*b^4*x^(5/3)/a^7-5/a^6*b^3*x^2+18/7*b^2*x^(7/3)/a^5-9/8*b*x^(8/3)/a^4+1/3/a^3*x^3-165*b
^9*ln(a*x^(1/3)+b)/a^12

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maxima [A]  time = 0.57, size = 167, normalized size = 0.98 \[ \frac {56 \, a^{10} - \frac {77 \, a^{9} b}{x^{\frac {1}{3}}} + \frac {110 \, a^{8} b^{2}}{x^{\frac {2}{3}}} - \frac {165 \, a^{7} b^{3}}{x} + \frac {264 \, a^{6} b^{4}}{x^{\frac {4}{3}}} - \frac {462 \, a^{5} b^{5}}{x^{\frac {5}{3}}} + \frac {924 \, a^{4} b^{6}}{x^{2}} - \frac {2310 \, a^{3} b^{7}}{x^{\frac {7}{3}}} + \frac {9240 \, a^{2} b^{8}}{x^{\frac {8}{3}}} + \frac {41580 \, a b^{9}}{x^{3}} + \frac {27720 \, b^{10}}{x^{\frac {10}{3}}}}{168 \, {\left (\frac {a^{13}}{x^{3}} + \frac {2 \, a^{12} b}{x^{\frac {10}{3}}} + \frac {a^{11} b^{2}}{x^{\frac {11}{3}}}\right )}} - \frac {165 \, b^{9} \log \left (a + \frac {b}{x^{\frac {1}{3}}}\right )}{a^{12}} - \frac {55 \, b^{9} \log \relax (x)}{a^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/168*(56*a^10 - 77*a^9*b/x^(1/3) + 110*a^8*b^2/x^(2/3) - 165*a^7*b^3/x + 264*a^6*b^4/x^(4/3) - 462*a^5*b^5/x^
(5/3) + 924*a^4*b^6/x^2 - 2310*a^3*b^7/x^(7/3) + 9240*a^2*b^8/x^(8/3) + 41580*a*b^9/x^3 + 27720*b^10/x^(10/3))
/(a^13/x^3 + 2*a^12*b/x^(10/3) + a^11*b^2/x^(11/3)) - 165*b^9*log(a + b/x^(1/3))/a^12 - 55*b^9*log(x)/a^12

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mupad [B]  time = 1.15, size = 154, normalized size = 0.90 \[ \frac {x^3}{3\,a^3}-\frac {\frac {63\,b^{11}}{2\,a}+33\,b^{10}\,x^{1/3}}{a^{11}\,b^2+a^{13}\,x^{2/3}+2\,a^{12}\,b\,x^{1/3}}-\frac {9\,b\,x^{8/3}}{8\,a^4}+\frac {28\,b^6\,x}{a^9}-\frac {165\,b^9\,\ln \left (b+a\,x^{1/3}\right )}{a^{12}}-\frac {5\,b^3\,x^2}{a^6}+\frac {18\,b^2\,x^{7/3}}{7\,a^5}+\frac {9\,b^4\,x^{5/3}}{a^7}-\frac {63\,b^5\,x^{4/3}}{4\,a^8}-\frac {54\,b^7\,x^{2/3}}{a^{10}}+\frac {135\,b^8\,x^{1/3}}{a^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a + b/x^(1/3))^3,x)

[Out]

x^3/(3*a^3) - ((63*b^11)/(2*a) + 33*b^10*x^(1/3))/(a^11*b^2 + a^13*x^(2/3) + 2*a^12*b*x^(1/3)) - (9*b*x^(8/3))
/(8*a^4) + (28*b^6*x)/a^9 - (165*b^9*log(b + a*x^(1/3)))/a^12 - (5*b^3*x^2)/a^6 + (18*b^2*x^(7/3))/(7*a^5) + (
9*b^4*x^(5/3))/a^7 - (63*b^5*x^(4/3))/(4*a^8) - (54*b^7*x^(2/3))/a^10 + (135*b^8*x^(1/3))/a^11

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sympy [A]  time = 6.31, size = 624, normalized size = 3.65 \[ \begin {cases} \frac {56 a^{11} x^{\frac {11}{3}}}{168 a^{14} x^{\frac {2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac {77 a^{10} b x^{\frac {10}{3}}}{168 a^{14} x^{\frac {2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} + \frac {110 a^{9} b^{2} x^{3}}{168 a^{14} x^{\frac {2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac {165 a^{8} b^{3} x^{\frac {8}{3}}}{168 a^{14} x^{\frac {2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} + \frac {264 a^{7} b^{4} x^{\frac {7}{3}}}{168 a^{14} x^{\frac {2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac {462 a^{6} b^{5} x^{2}}{168 a^{14} x^{\frac {2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} + \frac {924 a^{5} b^{6} x^{\frac {5}{3}}}{168 a^{14} x^{\frac {2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac {2310 a^{4} b^{7} x^{\frac {4}{3}}}{168 a^{14} x^{\frac {2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} + \frac {9240 a^{3} b^{8} x}{168 a^{14} x^{\frac {2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac {27720 a^{2} b^{9} x^{\frac {2}{3}} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{168 a^{14} x^{\frac {2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac {55440 a b^{10} \sqrt [3]{x} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{168 a^{14} x^{\frac {2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac {55440 a b^{10} \sqrt [3]{x}}{168 a^{14} x^{\frac {2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac {27720 b^{11} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{168 a^{14} x^{\frac {2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac {41580 b^{11}}{168 a^{14} x^{\frac {2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} & \text {for}\: a \neq 0 \\\frac {x^{4}}{4 b^{3}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((56*a**11*x**(11/3)/(168*a**14*x**(2/3) + 336*a**13*b*x**(1/3) + 168*a**12*b**2) - 77*a**10*b*x**(10
/3)/(168*a**14*x**(2/3) + 336*a**13*b*x**(1/3) + 168*a**12*b**2) + 110*a**9*b**2*x**3/(168*a**14*x**(2/3) + 33
6*a**13*b*x**(1/3) + 168*a**12*b**2) - 165*a**8*b**3*x**(8/3)/(168*a**14*x**(2/3) + 336*a**13*b*x**(1/3) + 168
*a**12*b**2) + 264*a**7*b**4*x**(7/3)/(168*a**14*x**(2/3) + 336*a**13*b*x**(1/3) + 168*a**12*b**2) - 462*a**6*
b**5*x**2/(168*a**14*x**(2/3) + 336*a**13*b*x**(1/3) + 168*a**12*b**2) + 924*a**5*b**6*x**(5/3)/(168*a**14*x**
(2/3) + 336*a**13*b*x**(1/3) + 168*a**12*b**2) - 2310*a**4*b**7*x**(4/3)/(168*a**14*x**(2/3) + 336*a**13*b*x**
(1/3) + 168*a**12*b**2) + 9240*a**3*b**8*x/(168*a**14*x**(2/3) + 336*a**13*b*x**(1/3) + 168*a**12*b**2) - 2772
0*a**2*b**9*x**(2/3)*log(x**(1/3) + b/a)/(168*a**14*x**(2/3) + 336*a**13*b*x**(1/3) + 168*a**12*b**2) - 55440*
a*b**10*x**(1/3)*log(x**(1/3) + b/a)/(168*a**14*x**(2/3) + 336*a**13*b*x**(1/3) + 168*a**12*b**2) - 55440*a*b*
*10*x**(1/3)/(168*a**14*x**(2/3) + 336*a**13*b*x**(1/3) + 168*a**12*b**2) - 27720*b**11*log(x**(1/3) + b/a)/(1
68*a**14*x**(2/3) + 336*a**13*b*x**(1/3) + 168*a**12*b**2) - 41580*b**11/(168*a**14*x**(2/3) + 336*a**13*b*x**
(1/3) + 168*a**12*b**2), Ne(a, 0)), (x**4/(4*b**3), True))

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