3.2365 \(\int \frac{x^3}{(a+b \sqrt [3]{x})^2} \, dx\)

Optimal. Leaf size=171 \[ \frac{27 a^8 x^{2/3}}{2 b^{10}}+\frac{21 a^6 x^{4/3}}{4 b^8}-\frac{18 a^5 x^{5/3}}{5 b^7}+\frac{5 a^4 x^2}{2 b^6}-\frac{12 a^3 x^{7/3}}{7 b^5}+\frac{9 a^2 x^{8/3}}{8 b^4}+\frac{3 a^{11}}{b^{12} \left (a+b \sqrt [3]{x}\right )}-\frac{30 a^9 \sqrt [3]{x}}{b^{11}}-\frac{8 a^7 x}{b^9}+\frac{33 a^{10} \log \left (a+b \sqrt [3]{x}\right )}{b^{12}}-\frac{2 a x^3}{3 b^3}+\frac{3 x^{10/3}}{10 b^2} \]

[Out]

(3*a^11)/(b^12*(a + b*x^(1/3))) - (30*a^9*x^(1/3))/b^11 + (27*a^8*x^(2/3))/(2*b^10) - (8*a^7*x)/b^9 + (21*a^6*
x^(4/3))/(4*b^8) - (18*a^5*x^(5/3))/(5*b^7) + (5*a^4*x^2)/(2*b^6) - (12*a^3*x^(7/3))/(7*b^5) + (9*a^2*x^(8/3))
/(8*b^4) - (2*a*x^3)/(3*b^3) + (3*x^(10/3))/(10*b^2) + (33*a^10*Log[a + b*x^(1/3)])/b^12

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Rubi [A]  time = 0.135019, 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} \[ \frac{27 a^8 x^{2/3}}{2 b^{10}}+\frac{21 a^6 x^{4/3}}{4 b^8}-\frac{18 a^5 x^{5/3}}{5 b^7}+\frac{5 a^4 x^2}{2 b^6}-\frac{12 a^3 x^{7/3}}{7 b^5}+\frac{9 a^2 x^{8/3}}{8 b^4}+\frac{3 a^{11}}{b^{12} \left (a+b \sqrt [3]{x}\right )}-\frac{30 a^9 \sqrt [3]{x}}{b^{11}}-\frac{8 a^7 x}{b^9}+\frac{33 a^{10} \log \left (a+b \sqrt [3]{x}\right )}{b^{12}}-\frac{2 a x^3}{3 b^3}+\frac{3 x^{10/3}}{10 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*x^(1/3))^2,x]

[Out]

(3*a^11)/(b^12*(a + b*x^(1/3))) - (30*a^9*x^(1/3))/b^11 + (27*a^8*x^(2/3))/(2*b^10) - (8*a^7*x)/b^9 + (21*a^6*
x^(4/3))/(4*b^8) - (18*a^5*x^(5/3))/(5*b^7) + (5*a^4*x^2)/(2*b^6) - (12*a^3*x^(7/3))/(7*b^5) + (9*a^2*x^(8/3))
/(8*b^4) - (2*a*x^3)/(3*b^3) + (3*x^(10/3))/(10*b^2) + (33*a^10*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 )^2} \, dx &=3 \operatorname{Subst}\left (\int \frac{x^{11}}{(a+b x)^2} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (-\frac{10 a^9}{b^{11}}+\frac{9 a^8 x}{b^{10}}-\frac{8 a^7 x^2}{b^9}+\frac{7 a^6 x^3}{b^8}-\frac{6 a^5 x^4}{b^7}+\frac{5 a^4 x^5}{b^6}-\frac{4 a^3 x^6}{b^5}+\frac{3 a^2 x^7}{b^4}-\frac{2 a x^8}{b^3}+\frac{x^9}{b^2}-\frac{a^{11}}{b^{11} (a+b x)^2}+\frac{11 a^{10}}{b^{11} (a+b x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{3 a^{11}}{b^{12} \left (a+b \sqrt [3]{x}\right )}-\frac{30 a^9 \sqrt [3]{x}}{b^{11}}+\frac{27 a^8 x^{2/3}}{2 b^{10}}-\frac{8 a^7 x}{b^9}+\frac{21 a^6 x^{4/3}}{4 b^8}-\frac{18 a^5 x^{5/3}}{5 b^7}+\frac{5 a^4 x^2}{2 b^6}-\frac{12 a^3 x^{7/3}}{7 b^5}+\frac{9 a^2 x^{8/3}}{8 b^4}-\frac{2 a x^3}{3 b^3}+\frac{3 x^{10/3}}{10 b^2}+\frac{33 a^{10} \log \left (a+b \sqrt [3]{x}\right )}{b^{12}}\\ \end{align*}

Mathematica [A]  time = 0.148035, size = 171, normalized size = 1. \[ \frac{27 a^8 x^{2/3}}{2 b^{10}}+\frac{21 a^6 x^{4/3}}{4 b^8}-\frac{18 a^5 x^{5/3}}{5 b^7}+\frac{5 a^4 x^2}{2 b^6}-\frac{12 a^3 x^{7/3}}{7 b^5}+\frac{9 a^2 x^{8/3}}{8 b^4}+\frac{3 a^{11}}{b^{12} \left (a+b \sqrt [3]{x}\right )}-\frac{30 a^9 \sqrt [3]{x}}{b^{11}}-\frac{8 a^7 x}{b^9}+\frac{33 a^{10} \log \left (a+b \sqrt [3]{x}\right )}{b^{12}}-\frac{2 a x^3}{3 b^3}+\frac{3 x^{10/3}}{10 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*x^(1/3))^2,x]

[Out]

(3*a^11)/(b^12*(a + b*x^(1/3))) - (30*a^9*x^(1/3))/b^11 + (27*a^8*x^(2/3))/(2*b^10) - (8*a^7*x)/b^9 + (21*a^6*
x^(4/3))/(4*b^8) - (18*a^5*x^(5/3))/(5*b^7) + (5*a^4*x^2)/(2*b^6) - (12*a^3*x^(7/3))/(7*b^5) + (9*a^2*x^(8/3))
/(8*b^4) - (2*a*x^3)/(3*b^3) + (3*x^(10/3))/(10*b^2) + (33*a^10*Log[a + b*x^(1/3)])/b^12

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Maple [A]  time = 0.008, size = 138, normalized size = 0.8 \begin{align*} 3\,{\frac{{a}^{11}}{{b}^{12} \left ( a+b\sqrt [3]{x} \right ) }}-30\,{\frac{{a}^{9}\sqrt [3]{x}}{{b}^{11}}}+{\frac{27\,{a}^{8}}{2\,{b}^{10}}{x}^{{\frac{2}{3}}}}-8\,{\frac{{a}^{7}x}{{b}^{9}}}+{\frac{21\,{a}^{6}}{4\,{b}^{8}}{x}^{{\frac{4}{3}}}}-{\frac{18\,{a}^{5}}{5\,{b}^{7}}{x}^{{\frac{5}{3}}}}+{\frac{5\,{a}^{4}{x}^{2}}{2\,{b}^{6}}}-{\frac{12\,{a}^{3}}{7\,{b}^{5}}{x}^{{\frac{7}{3}}}}+{\frac{9\,{a}^{2}}{8\,{b}^{4}}{x}^{{\frac{8}{3}}}}-{\frac{2\,a{x}^{3}}{3\,{b}^{3}}}+{\frac{3}{10\,{b}^{2}}{x}^{{\frac{10}{3}}}}+33\,{\frac{{a}^{10}\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))^2,x)

[Out]

3*a^11/b^12/(a+b*x^(1/3))-30*a^9*x^(1/3)/b^11+27/2*a^8*x^(2/3)/b^10-8*a^7*x/b^9+21/4*a^6*x^(4/3)/b^8-18/5*a^5*
x^(5/3)/b^7+5/2*a^4*x^2/b^6-12/7*a^3*x^(7/3)/b^5+9/8*a^2*x^(8/3)/b^4-2/3*a*x^3/b^3+3/10*x^(10/3)/b^2+33*a^10*l
n(a+b*x^(1/3))/b^12

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Maxima [A]  time = 0.957459, size = 266, normalized size = 1.56 \begin{align*} \frac{33 \, a^{10} \log \left (b x^{\frac{1}{3}} + a\right )}{b^{12}} + \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{10}}{10 \, b^{12}} - \frac{11 \,{\left (b x^{\frac{1}{3}} + a\right )}^{9} a}{3 \, b^{12}} + \frac{165 \,{\left (b x^{\frac{1}{3}} + a\right )}^{8} a^{2}}{8 \, b^{12}} - \frac{495 \,{\left (b x^{\frac{1}{3}} + a\right )}^{7} a^{3}}{7 \, b^{12}} + \frac{165 \,{\left (b x^{\frac{1}{3}} + a\right )}^{6} a^{4}}{b^{12}} - \frac{1386 \,{\left (b x^{\frac{1}{3}} + a\right )}^{5} a^{5}}{5 \, b^{12}} + \frac{693 \,{\left (b x^{\frac{1}{3}} + a\right )}^{4} a^{6}}{2 \, b^{12}} - \frac{330 \,{\left (b x^{\frac{1}{3}} + a\right )}^{3} a^{7}}{b^{12}} + \frac{495 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} a^{8}}{2 \, b^{12}} - \frac{165 \,{\left (b x^{\frac{1}{3}} + a\right )} a^{9}}{b^{12}} + \frac{3 \, a^{11}}{{\left (b x^{\frac{1}{3}} + a\right )} b^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

33*a^10*log(b*x^(1/3) + a)/b^12 + 3/10*(b*x^(1/3) + a)^10/b^12 - 11/3*(b*x^(1/3) + a)^9*a/b^12 + 165/8*(b*x^(1
/3) + a)^8*a^2/b^12 - 495/7*(b*x^(1/3) + a)^7*a^3/b^12 + 165*(b*x^(1/3) + a)^6*a^4/b^12 - 1386/5*(b*x^(1/3) +
a)^5*a^5/b^12 + 693/2*(b*x^(1/3) + a)^4*a^6/b^12 - 330*(b*x^(1/3) + a)^3*a^7/b^12 + 495/2*(b*x^(1/3) + a)^2*a^
8/b^12 - 165*(b*x^(1/3) + a)*a^9/b^12 + 3*a^11/((b*x^(1/3) + a)*b^12)

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Fricas [A]  time = 1.52166, size = 450, normalized size = 2.63 \begin{align*} -\frac{560 \, a b^{12} x^{4} - 1540 \, a^{4} b^{9} x^{3} + 4620 \, a^{7} b^{6} x^{2} + 6720 \, a^{10} b^{3} x - 2520 \, a^{13} - 27720 \,{\left (a^{10} b^{3} x + a^{13}\right )} \log \left (b x^{\frac{1}{3}} + a\right ) - 63 \,{\left (15 \, a^{2} b^{11} x^{3} - 33 \, a^{5} b^{8} x^{2} + 132 \, a^{8} b^{5} x + 220 \, a^{11} b^{2}\right )} x^{\frac{2}{3}} - 18 \,{\left (14 \, b^{13} x^{4} - 66 \, a^{3} b^{10} x^{3} + 165 \, a^{6} b^{7} x^{2} - 1155 \, a^{9} b^{4} x - 1540 \, a^{12} b\right )} x^{\frac{1}{3}}}{840 \,{\left (b^{15} x + a^{3} b^{12}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/840*(560*a*b^12*x^4 - 1540*a^4*b^9*x^3 + 4620*a^7*b^6*x^2 + 6720*a^10*b^3*x - 2520*a^13 - 27720*(a^10*b^3*x
+ a^13)*log(b*x^(1/3) + a) - 63*(15*a^2*b^11*x^3 - 33*a^5*b^8*x^2 + 132*a^8*b^5*x + 220*a^11*b^2)*x^(2/3) - 1
8*(14*b^13*x^4 - 66*a^3*b^10*x^3 + 165*a^6*b^7*x^2 - 1155*a^9*b^4*x - 1540*a^12*b)*x^(1/3))/(b^15*x + a^3*b^12
)

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Sympy [B]  time = 52.2702, size = 444, normalized size = 2.6 \begin{align*} \frac{27720 a^{11} x^{\frac{308}{3}} \log{\left (1 + \frac{b \sqrt [3]{x}}{a} \right )}}{840 a b^{12} x^{\frac{308}{3}} + 840 b^{13} x^{103}} + \frac{27720 a^{10} b x^{103} \log{\left (1 + \frac{b \sqrt [3]{x}}{a} \right )}}{840 a b^{12} x^{\frac{308}{3}} + 840 b^{13} x^{103}} - \frac{27720 a^{10} b x^{103}}{840 a b^{12} x^{\frac{308}{3}} + 840 b^{13} x^{103}} - \frac{13860 a^{9} b^{2} x^{\frac{310}{3}}}{840 a b^{12} x^{\frac{308}{3}} + 840 b^{13} x^{103}} + \frac{4620 a^{8} b^{3} x^{\frac{311}{3}}}{840 a b^{12} x^{\frac{308}{3}} + 840 b^{13} x^{103}} - \frac{2310 a^{7} b^{4} x^{104}}{840 a b^{12} x^{\frac{308}{3}} + 840 b^{13} x^{103}} + \frac{1386 a^{6} b^{5} x^{\frac{313}{3}}}{840 a b^{12} x^{\frac{308}{3}} + 840 b^{13} x^{103}} - \frac{924 a^{5} b^{6} x^{\frac{314}{3}}}{840 a b^{12} x^{\frac{308}{3}} + 840 b^{13} x^{103}} + \frac{660 a^{4} b^{7} x^{105}}{840 a b^{12} x^{\frac{308}{3}} + 840 b^{13} x^{103}} - \frac{495 a^{3} b^{8} x^{\frac{316}{3}}}{840 a b^{12} x^{\frac{308}{3}} + 840 b^{13} x^{103}} + \frac{385 a^{2} b^{9} x^{\frac{317}{3}}}{840 a b^{12} x^{\frac{308}{3}} + 840 b^{13} x^{103}} - \frac{308 a b^{10} x^{106}}{840 a b^{12} x^{\frac{308}{3}} + 840 b^{13} x^{103}} + \frac{252 b^{11} x^{\frac{319}{3}}}{840 a b^{12} x^{\frac{308}{3}} + 840 b^{13} x^{103}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

27720*a**11*x**(308/3)*log(1 + b*x**(1/3)/a)/(840*a*b**12*x**(308/3) + 840*b**13*x**103) + 27720*a**10*b*x**10
3*log(1 + b*x**(1/3)/a)/(840*a*b**12*x**(308/3) + 840*b**13*x**103) - 27720*a**10*b*x**103/(840*a*b**12*x**(30
8/3) + 840*b**13*x**103) - 13860*a**9*b**2*x**(310/3)/(840*a*b**12*x**(308/3) + 840*b**13*x**103) + 4620*a**8*
b**3*x**(311/3)/(840*a*b**12*x**(308/3) + 840*b**13*x**103) - 2310*a**7*b**4*x**104/(840*a*b**12*x**(308/3) +
840*b**13*x**103) + 1386*a**6*b**5*x**(313/3)/(840*a*b**12*x**(308/3) + 840*b**13*x**103) - 924*a**5*b**6*x**(
314/3)/(840*a*b**12*x**(308/3) + 840*b**13*x**103) + 660*a**4*b**7*x**105/(840*a*b**12*x**(308/3) + 840*b**13*
x**103) - 495*a**3*b**8*x**(316/3)/(840*a*b**12*x**(308/3) + 840*b**13*x**103) + 385*a**2*b**9*x**(317/3)/(840
*a*b**12*x**(308/3) + 840*b**13*x**103) - 308*a*b**10*x**106/(840*a*b**12*x**(308/3) + 840*b**13*x**103) + 252
*b**11*x**(319/3)/(840*a*b**12*x**(308/3) + 840*b**13*x**103)

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Giac [A]  time = 1.22763, size = 194, normalized size = 1.13 \begin{align*} \frac{33 \, a^{10} \log \left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{b^{12}} + \frac{3 \, a^{11}}{{\left (b x^{\frac{1}{3}} + a\right )} b^{12}} + \frac{252 \, b^{18} x^{\frac{10}{3}} - 560 \, a b^{17} x^{3} + 945 \, a^{2} b^{16} x^{\frac{8}{3}} - 1440 \, a^{3} b^{15} x^{\frac{7}{3}} + 2100 \, a^{4} b^{14} x^{2} - 3024 \, a^{5} b^{13} x^{\frac{5}{3}} + 4410 \, a^{6} b^{12} x^{\frac{4}{3}} - 6720 \, a^{7} b^{11} x + 11340 \, a^{8} b^{10} x^{\frac{2}{3}} - 25200 \, a^{9} b^{9} x^{\frac{1}{3}}}{840 \, b^{20}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

33*a^10*log(abs(b*x^(1/3) + a))/b^12 + 3*a^11/((b*x^(1/3) + a)*b^12) + 1/840*(252*b^18*x^(10/3) - 560*a*b^17*x
^3 + 945*a^2*b^16*x^(8/3) - 1440*a^3*b^15*x^(7/3) + 2100*a^4*b^14*x^2 - 3024*a^5*b^13*x^(5/3) + 4410*a^6*b^12*
x^(4/3) - 6720*a^7*b^11*x + 11340*a^8*b^10*x^(2/3) - 25200*a^9*b^9*x^(1/3))/b^20