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

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

[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

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Rubi [A]  time = 0.148063, antiderivative size = 171, normalized size of antiderivative = 1., 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 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]]

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])

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*}

Mathematica [A]  time = 0.151713, size = 157, normalized size = 0.92 \[ \frac{-9072 a^7 b^2 x^{2/3}-2646 a^5 b^4 x^{4/3}+1512 a^4 b^5 x^{5/3}-840 a^3 b^6 x^2+432 a^2 b^7 x^{7/3}+4704 a^6 b^3 x+\frac{252 a^{11}}{\left (a+b \sqrt [3]{x}\right )^2}-\frac{5544 a^{10}}{a+b \sqrt [3]{x}}+22680 a^8 b \sqrt [3]{x}-27720 a^9 \log \left (a+b \sqrt [3]{x}\right )-189 a b^8 x^{8/3}+56 b^9 x^3}{168 b^{12}} \]

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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Maple [A]  time = 0.007, size = 144, normalized size = 0.8 \begin{align*}{\frac{3\,{a}^{11}}{2\,{b}^{12}} \left ( a+b\sqrt [3]{x} \right ) ^{-2}}-33\,{\frac{{a}^{10}}{{b}^{12} \left ( a+b\sqrt [3]{x} \right ) }}+135\,{\frac{{a}^{8}\sqrt [3]{x}}{{b}^{11}}}-54\,{\frac{{a}^{7}{x}^{2/3}}{{b}^{10}}}+28\,{\frac{{a}^{6}x}{{b}^{9}}}-{\frac{63\,{a}^{5}}{4\,{b}^{8}}{x}^{{\frac{4}{3}}}}+9\,{\frac{{a}^{4}{x}^{5/3}}{{b}^{7}}}-5\,{\frac{{x}^{2}{a}^{3}}{{b}^{6}}}+{\frac{18\,{a}^{2}}{7\,{b}^{5}}{x}^{{\frac{7}{3}}}}-{\frac{9\,a}{8\,{b}^{4}}{x}^{{\frac{8}{3}}}}+{\frac{{x}^{3}}{3\,{b}^{3}}}-165\,{\frac{{a}^{9}\ln \left ( a+b\sqrt [3]{x} \right ) }{{b}^{12}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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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Maxima [A]  time = 0.969329, size = 266, normalized size = 1.56 \begin{align*} -\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}} \end{align*}

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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Fricas [A]  time = 1.5674, size = 543, normalized size = 3.18 \begin{align*} \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 )}} \end{align*}

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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Sympy [A]  time = 7.25401, size = 624, normalized size = 3.65 \begin{align*} \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} \end{align*}

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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Giac [A]  time = 1.1914, size = 196, normalized size = 1.15 \begin{align*} -\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}} \end{align*}

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