3.2356 \(\int \frac{x^3}{a+b \sqrt [3]{x}} \, dx\)

Optimal. Leaf size=166 \[ -\frac{3 a^9 x^{2/3}}{2 b^{10}}-\frac{3 a^7 x^{4/3}}{4 b^8}+\frac{3 a^6 x^{5/3}}{5 b^7}-\frac{a^5 x^2}{2 b^6}+\frac{3 a^4 x^{7/3}}{7 b^5}-\frac{3 a^3 x^{8/3}}{8 b^4}+\frac{a^2 x^3}{3 b^3}+\frac{3 a^{10} \sqrt [3]{x}}{b^{11}}+\frac{a^8 x}{b^9}-\frac{3 a^{11} \log \left (a+b \sqrt [3]{x}\right )}{b^{12}}-\frac{3 a x^{10/3}}{10 b^2}+\frac{3 x^{11/3}}{11 b} \]

[Out]

(3*a^10*x^(1/3))/b^11 - (3*a^9*x^(2/3))/(2*b^10) + (a^8*x)/b^9 - (3*a^7*x^(4/3))/(4*b^8) + (3*a^6*x^(5/3))/(5*
b^7) - (a^5*x^2)/(2*b^6) + (3*a^4*x^(7/3))/(7*b^5) - (3*a^3*x^(8/3))/(8*b^4) + (a^2*x^3)/(3*b^3) - (3*a*x^(10/
3))/(10*b^2) + (3*x^(11/3))/(11*b) - (3*a^11*Log[a + b*x^(1/3)])/b^12

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Rubi [A]  time = 0.108825, antiderivative size = 166, 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} \[ -\frac{3 a^9 x^{2/3}}{2 b^{10}}-\frac{3 a^7 x^{4/3}}{4 b^8}+\frac{3 a^6 x^{5/3}}{5 b^7}-\frac{a^5 x^2}{2 b^6}+\frac{3 a^4 x^{7/3}}{7 b^5}-\frac{3 a^3 x^{8/3}}{8 b^4}+\frac{a^2 x^3}{3 b^3}+\frac{3 a^{10} \sqrt [3]{x}}{b^{11}}+\frac{a^8 x}{b^9}-\frac{3 a^{11} \log \left (a+b \sqrt [3]{x}\right )}{b^{12}}-\frac{3 a x^{10/3}}{10 b^2}+\frac{3 x^{11/3}}{11 b} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*x^(1/3)),x]

[Out]

(3*a^10*x^(1/3))/b^11 - (3*a^9*x^(2/3))/(2*b^10) + (a^8*x)/b^9 - (3*a^7*x^(4/3))/(4*b^8) + (3*a^6*x^(5/3))/(5*
b^7) - (a^5*x^2)/(2*b^6) + (3*a^4*x^(7/3))/(7*b^5) - (3*a^3*x^(8/3))/(8*b^4) + (a^2*x^3)/(3*b^3) - (3*a*x^(10/
3))/(10*b^2) + (3*x^(11/3))/(11*b) - (3*a^11*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}{a+b \sqrt [3]{x}} \, dx &=3 \operatorname{Subst}\left (\int \frac{x^{11}}{a+b x} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{a^{10}}{b^{11}}-\frac{a^9 x}{b^{10}}+\frac{a^8 x^2}{b^9}-\frac{a^7 x^3}{b^8}+\frac{a^6 x^4}{b^7}-\frac{a^5 x^5}{b^6}+\frac{a^4 x^6}{b^5}-\frac{a^3 x^7}{b^4}+\frac{a^2 x^8}{b^3}-\frac{a x^9}{b^2}+\frac{x^{10}}{b}-\frac{a^{11}}{b^{11} (a+b x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{3 a^{10} \sqrt [3]{x}}{b^{11}}-\frac{3 a^9 x^{2/3}}{2 b^{10}}+\frac{a^8 x}{b^9}-\frac{3 a^7 x^{4/3}}{4 b^8}+\frac{3 a^6 x^{5/3}}{5 b^7}-\frac{a^5 x^2}{2 b^6}+\frac{3 a^4 x^{7/3}}{7 b^5}-\frac{3 a^3 x^{8/3}}{8 b^4}+\frac{a^2 x^3}{3 b^3}-\frac{3 a x^{10/3}}{10 b^2}+\frac{3 x^{11/3}}{11 b}-\frac{3 a^{11} \log \left (a+b \sqrt [3]{x}\right )}{b^{12}}\\ \end{align*}

Mathematica [A]  time = 0.123036, size = 166, normalized size = 1. \[ -\frac{3 a^9 x^{2/3}}{2 b^{10}}-\frac{3 a^7 x^{4/3}}{4 b^8}+\frac{3 a^6 x^{5/3}}{5 b^7}-\frac{a^5 x^2}{2 b^6}+\frac{3 a^4 x^{7/3}}{7 b^5}-\frac{3 a^3 x^{8/3}}{8 b^4}+\frac{a^2 x^3}{3 b^3}+\frac{3 a^{10} \sqrt [3]{x}}{b^{11}}+\frac{a^8 x}{b^9}-\frac{3 a^{11} \log \left (a+b \sqrt [3]{x}\right )}{b^{12}}-\frac{3 a x^{10/3}}{10 b^2}+\frac{3 x^{11/3}}{11 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*x^(1/3)),x]

[Out]

(3*a^10*x^(1/3))/b^11 - (3*a^9*x^(2/3))/(2*b^10) + (a^8*x)/b^9 - (3*a^7*x^(4/3))/(4*b^8) + (3*a^6*x^(5/3))/(5*
b^7) - (a^5*x^2)/(2*b^6) + (3*a^4*x^(7/3))/(7*b^5) - (3*a^3*x^(8/3))/(8*b^4) + (a^2*x^3)/(3*b^3) - (3*a*x^(10/
3))/(10*b^2) + (3*x^(11/3))/(11*b) - (3*a^11*Log[a + b*x^(1/3)])/b^12

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Maple [A]  time = 0.006, size = 131, normalized size = 0.8 \begin{align*} 3\,{\frac{{a}^{10}\sqrt [3]{x}}{{b}^{11}}}-{\frac{3\,{a}^{9}}{2\,{b}^{10}}{x}^{{\frac{2}{3}}}}+{\frac{{a}^{8}x}{{b}^{9}}}-{\frac{3\,{a}^{7}}{4\,{b}^{8}}{x}^{{\frac{4}{3}}}}+{\frac{3\,{a}^{6}}{5\,{b}^{7}}{x}^{{\frac{5}{3}}}}-{\frac{{a}^{5}{x}^{2}}{2\,{b}^{6}}}+{\frac{3\,{a}^{4}}{7\,{b}^{5}}{x}^{{\frac{7}{3}}}}-{\frac{3\,{a}^{3}}{8\,{b}^{4}}{x}^{{\frac{8}{3}}}}+{\frac{{x}^{3}{a}^{2}}{3\,{b}^{3}}}-{\frac{3\,a}{10\,{b}^{2}}{x}^{{\frac{10}{3}}}}+{\frac{3}{11\,b}{x}^{{\frac{11}{3}}}}-3\,{\frac{{a}^{11}\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)),x)

[Out]

3*a^10*x^(1/3)/b^11-3/2*a^9*x^(2/3)/b^10+a^8*x/b^9-3/4*a^7*x^(4/3)/b^8+3/5*a^6*x^(5/3)/b^7-1/2*a^5*x^2/b^6+3/7
*a^4*x^(7/3)/b^5-3/8*a^3*x^(8/3)/b^4+1/3*a^2*x^3/b^3-3/10*a*x^(10/3)/b^2+3/11*x^(11/3)/b-3*a^11*ln(a+b*x^(1/3)
)/b^12

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Maxima [A]  time = 0.980407, size = 266, normalized size = 1.6 \begin{align*} -\frac{3 \, a^{11} \log \left (b x^{\frac{1}{3}} + a\right )}{b^{12}} + \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{11}}{11 \, b^{12}} - \frac{33 \,{\left (b x^{\frac{1}{3}} + a\right )}^{10} a}{10 \, b^{12}} + \frac{55 \,{\left (b x^{\frac{1}{3}} + a\right )}^{9} a^{2}}{3 \, b^{12}} - \frac{495 \,{\left (b x^{\frac{1}{3}} + a\right )}^{8} a^{3}}{8 \, b^{12}} + \frac{990 \,{\left (b x^{\frac{1}{3}} + a\right )}^{7} a^{4}}{7 \, b^{12}} - \frac{231 \,{\left (b x^{\frac{1}{3}} + a\right )}^{6} a^{5}}{b^{12}} + \frac{1386 \,{\left (b x^{\frac{1}{3}} + a\right )}^{5} a^{6}}{5 \, b^{12}} - \frac{495 \,{\left (b x^{\frac{1}{3}} + a\right )}^{4} a^{7}}{2 \, b^{12}} + \frac{165 \,{\left (b x^{\frac{1}{3}} + a\right )}^{3} a^{8}}{b^{12}} - \frac{165 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} a^{9}}{2 \, b^{12}} + \frac{33 \,{\left (b x^{\frac{1}{3}} + a\right )} a^{10}}{b^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*a^11*log(b*x^(1/3) + a)/b^12 + 3/11*(b*x^(1/3) + a)^11/b^12 - 33/10*(b*x^(1/3) + a)^10*a/b^12 + 55/3*(b*x^(
1/3) + a)^9*a^2/b^12 - 495/8*(b*x^(1/3) + a)^8*a^3/b^12 + 990/7*(b*x^(1/3) + a)^7*a^4/b^12 - 231*(b*x^(1/3) +
a)^6*a^5/b^12 + 1386/5*(b*x^(1/3) + a)^5*a^6/b^12 - 495/2*(b*x^(1/3) + a)^4*a^7/b^12 + 165*(b*x^(1/3) + a)^3*a
^8/b^12 - 165/2*(b*x^(1/3) + a)^2*a^9/b^12 + 33*(b*x^(1/3) + a)*a^10/b^12

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Fricas [A]  time = 1.5228, size = 335, normalized size = 2.02 \begin{align*} \frac{3080 \, a^{2} b^{9} x^{3} - 4620 \, a^{5} b^{6} x^{2} + 9240 \, a^{8} b^{3} x - 27720 \, a^{11} \log \left (b x^{\frac{1}{3}} + a\right ) + 63 \,{\left (40 \, b^{11} x^{3} - 55 \, a^{3} b^{8} x^{2} + 88 \, a^{6} b^{5} x - 220 \, a^{9} b^{2}\right )} x^{\frac{2}{3}} - 198 \,{\left (14 \, a b^{10} x^{3} - 20 \, a^{4} b^{7} x^{2} + 35 \, a^{7} b^{4} x - 140 \, a^{10} b\right )} x^{\frac{1}{3}}}{9240 \, b^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9240*(3080*a^2*b^9*x^3 - 4620*a^5*b^6*x^2 + 9240*a^8*b^3*x - 27720*a^11*log(b*x^(1/3) + a) + 63*(40*b^11*x^3
- 55*a^3*b^8*x^2 + 88*a^6*b^5*x - 220*a^9*b^2)*x^(2/3) - 198*(14*a*b^10*x^3 - 20*a^4*b^7*x^2 + 35*a^7*b^4*x -
140*a^10*b)*x^(1/3))/b^12

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Sympy [A]  time = 40.7558, size = 165, normalized size = 0.99 \begin{align*} - \frac{3 a^{11} \log{\left (1 + \frac{b \sqrt [3]{x}}{a} \right )}}{b^{12}} + \frac{3 a^{10} \sqrt [3]{x}}{b^{11}} - \frac{3 a^{9} x^{\frac{2}{3}}}{2 b^{10}} + \frac{a^{8} x}{b^{9}} - \frac{3 a^{7} x^{\frac{4}{3}}}{4 b^{8}} + \frac{3 a^{6} x^{\frac{5}{3}}}{5 b^{7}} - \frac{a^{5} x^{2}}{2 b^{6}} + \frac{3 a^{4} x^{\frac{7}{3}}}{7 b^{5}} - \frac{3 a^{3} x^{\frac{8}{3}}}{8 b^{4}} + \frac{a^{2} x^{3}}{3 b^{3}} - \frac{3 a x^{\frac{10}{3}}}{10 b^{2}} + \frac{3 x^{\frac{11}{3}}}{11 b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*a**11*log(1 + b*x**(1/3)/a)/b**12 + 3*a**10*x**(1/3)/b**11 - 3*a**9*x**(2/3)/(2*b**10) + a**8*x/b**9 - 3*a*
*7*x**(4/3)/(4*b**8) + 3*a**6*x**(5/3)/(5*b**7) - a**5*x**2/(2*b**6) + 3*a**4*x**(7/3)/(7*b**5) - 3*a**3*x**(8
/3)/(8*b**4) + a**2*x**3/(3*b**3) - 3*a*x**(10/3)/(10*b**2) + 3*x**(11/3)/(11*b)

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Giac [A]  time = 1.21503, size = 180, normalized size = 1.08 \begin{align*} -\frac{3 \, a^{11} \log \left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{b^{12}} + \frac{2520 \, b^{10} x^{\frac{11}{3}} - 2772 \, a b^{9} x^{\frac{10}{3}} + 3080 \, a^{2} b^{8} x^{3} - 3465 \, a^{3} b^{7} x^{\frac{8}{3}} + 3960 \, a^{4} b^{6} x^{\frac{7}{3}} - 4620 \, a^{5} b^{5} x^{2} + 5544 \, a^{6} b^{4} x^{\frac{5}{3}} - 6930 \, a^{7} b^{3} x^{\frac{4}{3}} + 9240 \, a^{8} b^{2} x - 13860 \, a^{9} b x^{\frac{2}{3}} + 27720 \, a^{10} x^{\frac{1}{3}}}{9240 \, b^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*a^11*log(abs(b*x^(1/3) + a))/b^12 + 1/9240*(2520*b^10*x^(11/3) - 2772*a*b^9*x^(10/3) + 3080*a^2*b^8*x^3 - 3
465*a^3*b^7*x^(8/3) + 3960*a^4*b^6*x^(7/3) - 4620*a^5*b^5*x^2 + 5544*a^6*b^4*x^(5/3) - 6930*a^7*b^3*x^(4/3) +
9240*a^8*b^2*x - 13860*a^9*b*x^(2/3) + 27720*a^10*x^(1/3))/b^11