3.2357 \(\int \frac{x^2}{a+b \sqrt [3]{x}} \, dx\)

Optimal. Leaf size=124 \[ \frac{3 a^6 x^{2/3}}{2 b^7}+\frac{3 a^4 x^{4/3}}{4 b^5}-\frac{3 a^3 x^{5/3}}{5 b^4}+\frac{a^2 x^2}{2 b^3}-\frac{3 a^7 \sqrt [3]{x}}{b^8}-\frac{a^5 x}{b^6}+\frac{3 a^8 \log \left (a+b \sqrt [3]{x}\right )}{b^9}-\frac{3 a x^{7/3}}{7 b^2}+\frac{3 x^{8/3}}{8 b} \]

[Out]

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

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Rubi [A]  time = 0.0704568, antiderivative size = 124, 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} \[ \frac{3 a^6 x^{2/3}}{2 b^7}+\frac{3 a^4 x^{4/3}}{4 b^5}-\frac{3 a^3 x^{5/3}}{5 b^4}+\frac{a^2 x^2}{2 b^3}-\frac{3 a^7 \sqrt [3]{x}}{b^8}-\frac{a^5 x}{b^6}+\frac{3 a^8 \log \left (a+b \sqrt [3]{x}\right )}{b^9}-\frac{3 a x^{7/3}}{7 b^2}+\frac{3 x^{8/3}}{8 b} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*x^(1/3)),x]

[Out]

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

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

Mathematica [A]  time = 0.0741907, size = 124, normalized size = 1. \[ \frac{3 a^6 x^{2/3}}{2 b^7}+\frac{3 a^4 x^{4/3}}{4 b^5}-\frac{3 a^3 x^{5/3}}{5 b^4}+\frac{a^2 x^2}{2 b^3}-\frac{3 a^7 \sqrt [3]{x}}{b^8}-\frac{a^5 x}{b^6}+\frac{3 a^8 \log \left (a+b \sqrt [3]{x}\right )}{b^9}-\frac{3 a x^{7/3}}{7 b^2}+\frac{3 x^{8/3}}{8 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b*x^(1/3)),x]

[Out]

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

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Maple [A]  time = 0.003, size = 99, normalized size = 0.8 \begin{align*} -3\,{\frac{{a}^{7}\sqrt [3]{x}}{{b}^{8}}}+{\frac{3\,{a}^{6}}{2\,{b}^{7}}{x}^{{\frac{2}{3}}}}-{\frac{x{a}^{5}}{{b}^{6}}}+{\frac{3\,{a}^{4}}{4\,{b}^{5}}{x}^{{\frac{4}{3}}}}-{\frac{3\,{a}^{3}}{5\,{b}^{4}}{x}^{{\frac{5}{3}}}}+{\frac{{a}^{2}{x}^{2}}{2\,{b}^{3}}}-{\frac{3\,a}{7\,{b}^{2}}{x}^{{\frac{7}{3}}}}+{\frac{3}{8\,b}{x}^{{\frac{8}{3}}}}+3\,{\frac{{a}^{8}\ln \left ( a+b\sqrt [3]{x} \right ) }{{b}^{9}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.01248, size = 197, normalized size = 1.59 \begin{align*} \frac{3 \, a^{8} \log \left (b x^{\frac{1}{3}} + a\right )}{b^{9}} + \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{8}}{8 \, b^{9}} - \frac{24 \,{\left (b x^{\frac{1}{3}} + a\right )}^{7} a}{7 \, b^{9}} + \frac{14 \,{\left (b x^{\frac{1}{3}} + a\right )}^{6} a^{2}}{b^{9}} - \frac{168 \,{\left (b x^{\frac{1}{3}} + a\right )}^{5} a^{3}}{5 \, b^{9}} + \frac{105 \,{\left (b x^{\frac{1}{3}} + a\right )}^{4} a^{4}}{2 \, b^{9}} - \frac{56 \,{\left (b x^{\frac{1}{3}} + a\right )}^{3} a^{5}}{b^{9}} + \frac{42 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} a^{6}}{b^{9}} - \frac{24 \,{\left (b x^{\frac{1}{3}} + a\right )} a^{7}}{b^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*a^8*log(b*x^(1/3) + a)/b^9 + 3/8*(b*x^(1/3) + a)^8/b^9 - 24/7*(b*x^(1/3) + a)^7*a/b^9 + 14*(b*x^(1/3) + a)^6
*a^2/b^9 - 168/5*(b*x^(1/3) + a)^5*a^3/b^9 + 105/2*(b*x^(1/3) + a)^4*a^4/b^9 - 56*(b*x^(1/3) + a)^3*a^5/b^9 +
42*(b*x^(1/3) + a)^2*a^6/b^9 - 24*(b*x^(1/3) + a)*a^7/b^9

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Fricas [A]  time = 1.44679, size = 240, normalized size = 1.94 \begin{align*} \frac{140 \, a^{2} b^{6} x^{2} - 280 \, a^{5} b^{3} x + 840 \, a^{8} \log \left (b x^{\frac{1}{3}} + a\right ) + 21 \,{\left (5 \, b^{8} x^{2} - 8 \, a^{3} b^{5} x + 20 \, a^{6} b^{2}\right )} x^{\frac{2}{3}} - 30 \,{\left (4 \, a b^{7} x^{2} - 7 \, a^{4} b^{4} x + 28 \, a^{7} b\right )} x^{\frac{1}{3}}}{280 \, b^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/280*(140*a^2*b^6*x^2 - 280*a^5*b^3*x + 840*a^8*log(b*x^(1/3) + a) + 21*(5*b^8*x^2 - 8*a^3*b^5*x + 20*a^6*b^2
)*x^(2/3) - 30*(4*a*b^7*x^2 - 7*a^4*b^4*x + 28*a^7*b)*x^(1/3))/b^9

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Sympy [A]  time = 25.1611, size = 122, normalized size = 0.98 \begin{align*} \frac{3 a^{8} \log{\left (1 + \frac{b \sqrt [3]{x}}{a} \right )}}{b^{9}} - \frac{3 a^{7} \sqrt [3]{x}}{b^{8}} + \frac{3 a^{6} x^{\frac{2}{3}}}{2 b^{7}} - \frac{a^{5} x}{b^{6}} + \frac{3 a^{4} x^{\frac{4}{3}}}{4 b^{5}} - \frac{3 a^{3} x^{\frac{5}{3}}}{5 b^{4}} + \frac{a^{2} x^{2}}{2 b^{3}} - \frac{3 a x^{\frac{7}{3}}}{7 b^{2}} + \frac{3 x^{\frac{8}{3}}}{8 b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.16368, size = 135, normalized size = 1.09 \begin{align*} \frac{3 \, a^{8} \log \left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{b^{9}} + \frac{105 \, b^{7} x^{\frac{8}{3}} - 120 \, a b^{6} x^{\frac{7}{3}} + 140 \, a^{2} b^{5} x^{2} - 168 \, a^{3} b^{4} x^{\frac{5}{3}} + 210 \, a^{4} b^{3} x^{\frac{4}{3}} - 280 \, a^{5} b^{2} x + 420 \, a^{6} b x^{\frac{2}{3}} - 840 \, a^{7} x^{\frac{1}{3}}}{280 \, b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*a^8*log(abs(b*x^(1/3) + a))/b^9 + 1/280*(105*b^7*x^(8/3) - 120*a*b^6*x^(7/3) + 140*a^2*b^5*x^2 - 168*a^3*b^4
*x^(5/3) + 210*a^4*b^3*x^(4/3) - 280*a^5*b^2*x + 420*a^6*b*x^(2/3) - 840*a^7*x^(1/3))/b^8