3.2373 \(\int \frac {x^3}{(a+b \sqrt [3]{x})^3} \, dx\)

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

[Out]

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

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

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*x^(1/3))^3,x]

[Out]

(3*a^11)/(2*b^12*(a + b*x^(1/3))^2) - (33*a^10)/(b^12*(a + b*x^(1/3))) + (135*a^8*x^(1/3))/b^11 - (54*a^7*x^(2
/3))/b^10 + (28*a^6*x)/b^9 - (63*a^5*x^(4/3))/(4*b^8) + (9*a^4*x^(5/3))/b^7 - (5*a^3*x^2)/b^6 + (18*a^2*x^(7/3
))/(7*b^5) - (9*a*x^(8/3))/(8*b^4) + x^3/(3*b^3) - (165*a^9*Log[a + b*x^(1/3)])/b^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 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^3}{\left (a+b \sqrt [3]{x}\right )^3} \, dx &=3 \operatorname {Subst}\left (\int \frac {x^{11}}{(a+b x)^3} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname {Subst}\left (\int \left (\frac {45 a^8}{b^{11}}-\frac {36 a^7 x}{b^{10}}+\frac {28 a^6 x^2}{b^9}-\frac {21 a^5 x^3}{b^8}+\frac {15 a^4 x^4}{b^7}-\frac {10 a^3 x^5}{b^6}+\frac {6 a^2 x^6}{b^5}-\frac {3 a x^7}{b^4}+\frac {x^8}{b^3}-\frac {a^{11}}{b^{11} (a+b x)^3}+\frac {11 a^{10}}{b^{11} (a+b x)^2}-\frac {55 a^9}{b^{11} (a+b x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {3 a^{11}}{2 b^{12} \left (a+b \sqrt [3]{x}\right )^2}-\frac {33 a^{10}}{b^{12} \left (a+b \sqrt [3]{x}\right )}+\frac {135 a^8 \sqrt [3]{x}}{b^{11}}-\frac {54 a^7 x^{2/3}}{b^{10}}+\frac {28 a^6 x}{b^9}-\frac {63 a^5 x^{4/3}}{4 b^8}+\frac {9 a^4 x^{5/3}}{b^7}-\frac {5 a^3 x^2}{b^6}+\frac {18 a^2 x^{7/3}}{7 b^5}-\frac {9 a x^{8/3}}{8 b^4}+\frac {x^3}{3 b^3}-\frac {165 a^9 \log \left (a+b \sqrt [3]{x}\right )}{b^{12}}\\ \end {align*}

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Mathematica [A]  time = 0.17, size = 157, normalized size = 0.92 \[ \frac {\frac {252 a^{11}}{\left (a+b \sqrt [3]{x}\right )^2}-\frac {5544 a^{10}}{a+b \sqrt [3]{x}}-27720 a^9 \log \left (a+b \sqrt [3]{x}\right )+22680 a^8 b \sqrt [3]{x}-9072 a^7 b^2 x^{2/3}+4704 a^6 b^3 x-2646 a^5 b^4 x^{4/3}+1512 a^4 b^5 x^{5/3}-840 a^3 b^6 x^2+432 a^2 b^7 x^{7/3}-189 a b^8 x^{8/3}+56 b^9 x^3}{168 b^{12}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*x^(1/3))^3,x]

[Out]

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

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fricas [A]  time = 0.76, size = 225, normalized size = 1.32 \[ \frac {56 \, b^{15} x^{5} - 728 \, a^{3} b^{12} x^{4} + 3080 \, a^{6} b^{9} x^{3} + 8568 \, a^{9} b^{6} x^{2} - 1344 \, a^{12} b^{3} x - 5292 \, a^{15} - 27720 \, {\left (a^{9} b^{6} x^{2} + 2 \, a^{12} b^{3} x + a^{15}\right )} \log \left (b x^{\frac {1}{3}} + a\right ) - 63 \, {\left (3 \, a b^{14} x^{4} - 18 \, a^{4} b^{11} x^{3} + 99 \, a^{7} b^{8} x^{2} + 352 \, a^{10} b^{5} x + 220 \, a^{13} b^{2}\right )} x^{\frac {2}{3}} + 18 \, {\left (24 \, a^{2} b^{13} x^{4} - 99 \, a^{5} b^{10} x^{3} + 990 \, a^{8} b^{7} x^{2} + 2695 \, a^{11} b^{4} x + 1540 \, a^{14} b\right )} x^{\frac {1}{3}}}{168 \, {\left (b^{18} x^{2} + 2 \, a^{3} b^{15} x + a^{6} b^{12}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.51, size = 197, normalized size = 1.15 \[ -\frac {165 \, a^{9} \log \left (b x^{\frac {1}{3}} + a\right )}{b^{12}} + \frac {{\left (b x^{\frac {1}{3}} + a\right )}^{9}}{3 \, b^{12}} - \frac {33 \, {\left (b x^{\frac {1}{3}} + a\right )}^{8} a}{8 \, b^{12}} + \frac {165 \, {\left (b x^{\frac {1}{3}} + a\right )}^{7} a^{2}}{7 \, b^{12}} - \frac {165 \, {\left (b x^{\frac {1}{3}} + a\right )}^{6} a^{3}}{2 \, b^{12}} + \frac {198 \, {\left (b x^{\frac {1}{3}} + a\right )}^{5} a^{4}}{b^{12}} - \frac {693 \, {\left (b x^{\frac {1}{3}} + a\right )}^{4} a^{5}}{2 \, b^{12}} + \frac {462 \, {\left (b x^{\frac {1}{3}} + a\right )}^{3} a^{6}}{b^{12}} - \frac {495 \, {\left (b x^{\frac {1}{3}} + a\right )}^{2} a^{7}}{b^{12}} + \frac {495 \, {\left (b x^{\frac {1}{3}} + a\right )} a^{8}}{b^{12}} - \frac {33 \, a^{10}}{{\left (b x^{\frac {1}{3}} + a\right )} b^{12}} + \frac {3 \, a^{11}}{2 \, {\left (b x^{\frac {1}{3}} + a\right )}^{2} b^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-165*a^9*log(b*x^(1/3) + a)/b^12 + 1/3*(b*x^(1/3) + a)^9/b^12 - 33/8*(b*x^(1/3) + a)^8*a/b^12 + 165/7*(b*x^(1/
3) + a)^7*a^2/b^12 - 165/2*(b*x^(1/3) + a)^6*a^3/b^12 + 198*(b*x^(1/3) + a)^5*a^4/b^12 - 693/2*(b*x^(1/3) + a)
^4*a^5/b^12 + 462*(b*x^(1/3) + a)^3*a^6/b^12 - 495*(b*x^(1/3) + a)^2*a^7/b^12 + 495*(b*x^(1/3) + a)*a^8/b^12 -
33*a^10/((b*x^(1/3) + a)*b^12) + 3/2*a^11/((b*x^(1/3) + a)^2*b^12)

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mupad [B]  time = 0.07, size = 154, normalized size = 0.90 \[ \frac {x^3}{3\,b^3}-\frac {\frac {63\,a^{11}}{2\,b}+33\,a^{10}\,x^{1/3}}{a^2\,b^{11}+b^{13}\,x^{2/3}+2\,a\,b^{12}\,x^{1/3}}-\frac {9\,a\,x^{8/3}}{8\,b^4}+\frac {28\,a^6\,x}{b^9}-\frac {165\,a^9\,\ln \left (a+b\,x^{1/3}\right )}{b^{12}}-\frac {5\,a^3\,x^2}{b^6}+\frac {18\,a^2\,x^{7/3}}{7\,b^5}+\frac {9\,a^4\,x^{5/3}}{b^7}-\frac {63\,a^5\,x^{4/3}}{4\,b^8}-\frac {54\,a^7\,x^{2/3}}{b^{10}}+\frac {135\,a^8\,x^{1/3}}{b^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a + b*x^(1/3))^3,x)

[Out]

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

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sympy [A]  time = 5.87, size = 624, normalized size = 3.65 \[ \begin {cases} - \frac {27720 a^{11} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{168 a^{2} b^{12} + 336 a b^{13} \sqrt [3]{x} + 168 b^{14} x^{\frac {2}{3}}} - \frac {41580 a^{11}}{168 a^{2} b^{12} + 336 a b^{13} \sqrt [3]{x} + 168 b^{14} x^{\frac {2}{3}}} - \frac {55440 a^{10} b \sqrt [3]{x} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{168 a^{2} b^{12} + 336 a b^{13} \sqrt [3]{x} + 168 b^{14} x^{\frac {2}{3}}} - \frac {55440 a^{10} b \sqrt [3]{x}}{168 a^{2} b^{12} + 336 a b^{13} \sqrt [3]{x} + 168 b^{14} x^{\frac {2}{3}}} - \frac {27720 a^{9} b^{2} x^{\frac {2}{3}} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{168 a^{2} b^{12} + 336 a b^{13} \sqrt [3]{x} + 168 b^{14} x^{\frac {2}{3}}} + \frac {9240 a^{8} b^{3} x}{168 a^{2} b^{12} + 336 a b^{13} \sqrt [3]{x} + 168 b^{14} x^{\frac {2}{3}}} - \frac {2310 a^{7} b^{4} x^{\frac {4}{3}}}{168 a^{2} b^{12} + 336 a b^{13} \sqrt [3]{x} + 168 b^{14} x^{\frac {2}{3}}} + \frac {924 a^{6} b^{5} x^{\frac {5}{3}}}{168 a^{2} b^{12} + 336 a b^{13} \sqrt [3]{x} + 168 b^{14} x^{\frac {2}{3}}} - \frac {462 a^{5} b^{6} x^{2}}{168 a^{2} b^{12} + 336 a b^{13} \sqrt [3]{x} + 168 b^{14} x^{\frac {2}{3}}} + \frac {264 a^{4} b^{7} x^{\frac {7}{3}}}{168 a^{2} b^{12} + 336 a b^{13} \sqrt [3]{x} + 168 b^{14} x^{\frac {2}{3}}} - \frac {165 a^{3} b^{8} x^{\frac {8}{3}}}{168 a^{2} b^{12} + 336 a b^{13} \sqrt [3]{x} + 168 b^{14} x^{\frac {2}{3}}} + \frac {110 a^{2} b^{9} x^{3}}{168 a^{2} b^{12} + 336 a b^{13} \sqrt [3]{x} + 168 b^{14} x^{\frac {2}{3}}} - \frac {77 a b^{10} x^{\frac {10}{3}}}{168 a^{2} b^{12} + 336 a b^{13} \sqrt [3]{x} + 168 b^{14} x^{\frac {2}{3}}} + \frac {56 b^{11} x^{\frac {11}{3}}}{168 a^{2} b^{12} + 336 a b^{13} \sqrt [3]{x} + 168 b^{14} x^{\frac {2}{3}}} & \text {for}\: b \neq 0 \\\frac {x^{4}}{4 a^{3}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-27720*a**11*log(a/b + x**(1/3))/(168*a**2*b**12 + 336*a*b**13*x**(1/3) + 168*b**14*x**(2/3)) - 415
80*a**11/(168*a**2*b**12 + 336*a*b**13*x**(1/3) + 168*b**14*x**(2/3)) - 55440*a**10*b*x**(1/3)*log(a/b + x**(1
/3))/(168*a**2*b**12 + 336*a*b**13*x**(1/3) + 168*b**14*x**(2/3)) - 55440*a**10*b*x**(1/3)/(168*a**2*b**12 + 3
36*a*b**13*x**(1/3) + 168*b**14*x**(2/3)) - 27720*a**9*b**2*x**(2/3)*log(a/b + x**(1/3))/(168*a**2*b**12 + 336
*a*b**13*x**(1/3) + 168*b**14*x**(2/3)) + 9240*a**8*b**3*x/(168*a**2*b**12 + 336*a*b**13*x**(1/3) + 168*b**14*
x**(2/3)) - 2310*a**7*b**4*x**(4/3)/(168*a**2*b**12 + 336*a*b**13*x**(1/3) + 168*b**14*x**(2/3)) + 924*a**6*b*
*5*x**(5/3)/(168*a**2*b**12 + 336*a*b**13*x**(1/3) + 168*b**14*x**(2/3)) - 462*a**5*b**6*x**2/(168*a**2*b**12
+ 336*a*b**13*x**(1/3) + 168*b**14*x**(2/3)) + 264*a**4*b**7*x**(7/3)/(168*a**2*b**12 + 336*a*b**13*x**(1/3) +
168*b**14*x**(2/3)) - 165*a**3*b**8*x**(8/3)/(168*a**2*b**12 + 336*a*b**13*x**(1/3) + 168*b**14*x**(2/3)) + 1
10*a**2*b**9*x**3/(168*a**2*b**12 + 336*a*b**13*x**(1/3) + 168*b**14*x**(2/3)) - 77*a*b**10*x**(10/3)/(168*a**
2*b**12 + 336*a*b**13*x**(1/3) + 168*b**14*x**(2/3)) + 56*b**11*x**(11/3)/(168*a**2*b**12 + 336*a*b**13*x**(1/
3) + 168*b**14*x**(2/3)), Ne(b, 0)), (x**4/(4*a**3), True))

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