3.2367 \(\int \frac{x}{(a+b \sqrt [3]{x})^2} \, dx\)

Optimal. Leaf size=85 \[ \frac{9 a^2 x^{2/3}}{2 b^4}+\frac{3 a^5}{b^6 \left (a+b \sqrt [3]{x}\right )}-\frac{12 a^3 \sqrt [3]{x}}{b^5}+\frac{15 a^4 \log \left (a+b \sqrt [3]{x}\right )}{b^6}-\frac{2 a x}{b^3}+\frac{3 x^{4/3}}{4 b^2} \]

[Out]

(3*a^5)/(b^6*(a + b*x^(1/3))) - (12*a^3*x^(1/3))/b^5 + (9*a^2*x^(2/3))/(2*b^4) - (2*a*x)/b^3 + (3*x^(4/3))/(4*
b^2) + (15*a^4*Log[a + b*x^(1/3)])/b^6

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Rubi [A]  time = 0.0554201, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{9 a^2 x^{2/3}}{2 b^4}+\frac{3 a^5}{b^6 \left (a+b \sqrt [3]{x}\right )}-\frac{12 a^3 \sqrt [3]{x}}{b^5}+\frac{15 a^4 \log \left (a+b \sqrt [3]{x}\right )}{b^6}-\frac{2 a x}{b^3}+\frac{3 x^{4/3}}{4 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*x^(1/3))^2,x]

[Out]

(3*a^5)/(b^6*(a + b*x^(1/3))) - (12*a^3*x^(1/3))/b^5 + (9*a^2*x^(2/3))/(2*b^4) - (2*a*x)/b^3 + (3*x^(4/3))/(4*
b^2) + (15*a^4*Log[a + b*x^(1/3)])/b^6

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}{\left (a+b \sqrt [3]{x}\right )^2} \, dx &=3 \operatorname{Subst}\left (\int \frac{x^5}{(a+b x)^2} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (-\frac{4 a^3}{b^5}+\frac{3 a^2 x}{b^4}-\frac{2 a x^2}{b^3}+\frac{x^3}{b^2}-\frac{a^5}{b^5 (a+b x)^2}+\frac{5 a^4}{b^5 (a+b x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{3 a^5}{b^6 \left (a+b \sqrt [3]{x}\right )}-\frac{12 a^3 \sqrt [3]{x}}{b^5}+\frac{9 a^2 x^{2/3}}{2 b^4}-\frac{2 a x}{b^3}+\frac{3 x^{4/3}}{4 b^2}+\frac{15 a^4 \log \left (a+b \sqrt [3]{x}\right )}{b^6}\\ \end{align*}

Mathematica [A]  time = 0.0568362, size = 80, normalized size = 0.94 \[ \frac{18 a^2 b^2 x^{2/3}+\frac{12 a^5}{a+b \sqrt [3]{x}}-48 a^3 b \sqrt [3]{x}+60 a^4 \log \left (a+b \sqrt [3]{x}\right )-8 a b^3 x+3 b^4 x^{4/3}}{4 b^6} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*x^(1/3))^2,x]

[Out]

((12*a^5)/(a + b*x^(1/3)) - 48*a^3*b*x^(1/3) + 18*a^2*b^2*x^(2/3) - 8*a*b^3*x + 3*b^4*x^(4/3) + 60*a^4*Log[a +
b*x^(1/3)])/(4*b^6)

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Maple [A]  time = 0.007, size = 72, normalized size = 0.9 \begin{align*} 3\,{\frac{{a}^{5}}{{b}^{6} \left ( a+b\sqrt [3]{x} \right ) }}-12\,{\frac{{a}^{3}\sqrt [3]{x}}{{b}^{5}}}+{\frac{9\,{a}^{2}}{2\,{b}^{4}}{x}^{{\frac{2}{3}}}}-2\,{\frac{ax}{{b}^{3}}}+{\frac{3}{4\,{b}^{2}}{x}^{{\frac{4}{3}}}}+15\,{\frac{{a}^{4}\ln \left ( a+b\sqrt [3]{x} \right ) }{{b}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*a^5/b^6/(a+b*x^(1/3))-12*a^3*x^(1/3)/b^5+9/2*a^2*x^(2/3)/b^4-2*a*x/b^3+3/4*x^(4/3)/b^2+15*a^4*ln(a+b*x^(1/3)
)/b^6

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Maxima [A]  time = 0.972462, size = 128, normalized size = 1.51 \begin{align*} \frac{15 \, a^{4} \log \left (b x^{\frac{1}{3}} + a\right )}{b^{6}} + \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{4}}{4 \, b^{6}} - \frac{5 \,{\left (b x^{\frac{1}{3}} + a\right )}^{3} a}{b^{6}} + \frac{15 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} a^{2}}{b^{6}} - \frac{30 \,{\left (b x^{\frac{1}{3}} + a\right )} a^{3}}{b^{6}} + \frac{3 \, a^{5}}{{\left (b x^{\frac{1}{3}} + a\right )} b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

15*a^4*log(b*x^(1/3) + a)/b^6 + 3/4*(b*x^(1/3) + a)^4/b^6 - 5*(b*x^(1/3) + a)^3*a/b^6 + 15*(b*x^(1/3) + a)^2*a
^2/b^6 - 30*(b*x^(1/3) + a)*a^3/b^6 + 3*a^5/((b*x^(1/3) + a)*b^6)

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Fricas [A]  time = 1.5176, size = 255, normalized size = 3. \begin{align*} -\frac{8 \, a b^{6} x^{2} + 8 \, a^{4} b^{3} x - 12 \, a^{7} - 60 \,{\left (a^{4} b^{3} x + a^{7}\right )} \log \left (b x^{\frac{1}{3}} + a\right ) - 6 \,{\left (3 \, a^{2} b^{5} x + 5 \, a^{5} b^{2}\right )} x^{\frac{2}{3}} - 3 \,{\left (b^{7} x^{2} - 15 \, a^{3} b^{4} x - 20 \, a^{6} b\right )} x^{\frac{1}{3}}}{4 \,{\left (b^{9} x + a^{3} b^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(8*a*b^6*x^2 + 8*a^4*b^3*x - 12*a^7 - 60*(a^4*b^3*x + a^7)*log(b*x^(1/3) + a) - 6*(3*a^2*b^5*x + 5*a^5*b^
2)*x^(2/3) - 3*(b^7*x^2 - 15*a^3*b^4*x - 20*a^6*b)*x^(1/3))/(b^9*x + a^3*b^6)

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Sympy [B]  time = 7.11568, size = 243, normalized size = 2.86 \begin{align*} \frac{60 a^{5} x^{\frac{80}{3}} \log{\left (1 + \frac{b \sqrt [3]{x}}{a} \right )}}{4 a b^{6} x^{\frac{80}{3}} + 4 b^{7} x^{27}} + \frac{60 a^{4} b x^{27} \log{\left (1 + \frac{b \sqrt [3]{x}}{a} \right )}}{4 a b^{6} x^{\frac{80}{3}} + 4 b^{7} x^{27}} - \frac{60 a^{4} b x^{27}}{4 a b^{6} x^{\frac{80}{3}} + 4 b^{7} x^{27}} - \frac{30 a^{3} b^{2} x^{\frac{82}{3}}}{4 a b^{6} x^{\frac{80}{3}} + 4 b^{7} x^{27}} + \frac{10 a^{2} b^{3} x^{\frac{83}{3}}}{4 a b^{6} x^{\frac{80}{3}} + 4 b^{7} x^{27}} - \frac{5 a b^{4} x^{28}}{4 a b^{6} x^{\frac{80}{3}} + 4 b^{7} x^{27}} + \frac{3 b^{5} x^{\frac{85}{3}}}{4 a b^{6} x^{\frac{80}{3}} + 4 b^{7} x^{27}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

60*a**5*x**(80/3)*log(1 + b*x**(1/3)/a)/(4*a*b**6*x**(80/3) + 4*b**7*x**27) + 60*a**4*b*x**27*log(1 + b*x**(1/
3)/a)/(4*a*b**6*x**(80/3) + 4*b**7*x**27) - 60*a**4*b*x**27/(4*a*b**6*x**(80/3) + 4*b**7*x**27) - 30*a**3*b**2
*x**(82/3)/(4*a*b**6*x**(80/3) + 4*b**7*x**27) + 10*a**2*b**3*x**(83/3)/(4*a*b**6*x**(80/3) + 4*b**7*x**27) -
5*a*b**4*x**28/(4*a*b**6*x**(80/3) + 4*b**7*x**27) + 3*b**5*x**(85/3)/(4*a*b**6*x**(80/3) + 4*b**7*x**27)

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Giac [A]  time = 1.20181, size = 105, normalized size = 1.24 \begin{align*} \frac{15 \, a^{4} \log \left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{b^{6}} + \frac{3 \, a^{5}}{{\left (b x^{\frac{1}{3}} + a\right )} b^{6}} + \frac{3 \, b^{6} x^{\frac{4}{3}} - 8 \, a b^{5} x + 18 \, a^{2} b^{4} x^{\frac{2}{3}} - 48 \, a^{3} b^{3} x^{\frac{1}{3}}}{4 \, b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

15*a^4*log(abs(b*x^(1/3) + a))/b^6 + 3*a^5/((b*x^(1/3) + a)*b^6) + 1/4*(3*b^6*x^(4/3) - 8*a*b^5*x + 18*a^2*b^4
*x^(2/3) - 48*a^3*b^3*x^(1/3))/b^8