3.2375 \(\int \frac{x}{(a+b \sqrt [3]{x})^3} \, dx\)

Optimal. Leaf size=90 \[ \frac{3 a^5}{2 b^6 \left (a+b \sqrt [3]{x}\right )^2}-\frac{15 a^4}{b^6 \left (a+b \sqrt [3]{x}\right )}+\frac{18 a^2 \sqrt [3]{x}}{b^5}-\frac{30 a^3 \log \left (a+b \sqrt [3]{x}\right )}{b^6}-\frac{9 a x^{2/3}}{2 b^4}+\frac{x}{b^3} \]

[Out]

(3*a^5)/(2*b^6*(a + b*x^(1/3))^2) - (15*a^4)/(b^6*(a + b*x^(1/3))) + (18*a^2*x^(1/3))/b^5 - (9*a*x^(2/3))/(2*b
^4) + x/b^3 - (30*a^3*Log[a + b*x^(1/3)])/b^6

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Rubi [A]  time = 0.0611164, antiderivative size = 90, 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} \[ \frac{3 a^5}{2 b^6 \left (a+b \sqrt [3]{x}\right )^2}-\frac{15 a^4}{b^6 \left (a+b \sqrt [3]{x}\right )}+\frac{18 a^2 \sqrt [3]{x}}{b^5}-\frac{30 a^3 \log \left (a+b \sqrt [3]{x}\right )}{b^6}-\frac{9 a x^{2/3}}{2 b^4}+\frac{x}{b^3} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*x^(1/3))^3,x]

[Out]

(3*a^5)/(2*b^6*(a + b*x^(1/3))^2) - (15*a^4)/(b^6*(a + b*x^(1/3))) + (18*a^2*x^(1/3))/b^5 - (9*a*x^(2/3))/(2*b
^4) + x/b^3 - (30*a^3*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 )^3} \, dx &=3 \operatorname{Subst}\left (\int \frac{x^5}{(a+b x)^3} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{6 a^2}{b^5}-\frac{3 a x}{b^4}+\frac{x^2}{b^3}-\frac{a^5}{b^5 (a+b x)^3}+\frac{5 a^4}{b^5 (a+b x)^2}-\frac{10 a^3}{b^5 (a+b x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{3 a^5}{2 b^6 \left (a+b \sqrt [3]{x}\right )^2}-\frac{15 a^4}{b^6 \left (a+b \sqrt [3]{x}\right )}+\frac{18 a^2 \sqrt [3]{x}}{b^5}-\frac{9 a x^{2/3}}{2 b^4}+\frac{x}{b^3}-\frac{30 a^3 \log \left (a+b \sqrt [3]{x}\right )}{b^6}\\ \end{align*}

Mathematica [A]  time = 0.0687078, size = 83, normalized size = 0.92 \[ \frac{\frac{3 a^5}{\left (a+b \sqrt [3]{x}\right )^2}-\frac{30 a^4}{a+b \sqrt [3]{x}}+36 a^2 b \sqrt [3]{x}-60 a^3 \log \left (a+b \sqrt [3]{x}\right )-9 a b^2 x^{2/3}+2 b^3 x}{2 b^6} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*x^(1/3))^3,x]

[Out]

((3*a^5)/(a + b*x^(1/3))^2 - (30*a^4)/(a + b*x^(1/3)) + 36*a^2*b*x^(1/3) - 9*a*b^2*x^(2/3) + 2*b^3*x - 60*a^3*
Log[a + b*x^(1/3)])/(2*b^6)

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Maple [A]  time = 0., size = 77, normalized size = 0.9 \begin{align*}{\frac{3\,{a}^{5}}{2\,{b}^{6}} \left ( a+b\sqrt [3]{x} \right ) ^{-2}}-15\,{\frac{{a}^{4}}{{b}^{6} \left ( a+b\sqrt [3]{x} \right ) }}+18\,{\frac{{a}^{2}\sqrt [3]{x}}{{b}^{5}}}-{\frac{9\,a}{2\,{b}^{4}}{x}^{{\frac{2}{3}}}}+{\frac{x}{{b}^{3}}}-30\,{\frac{{a}^{3}\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))^3,x)

[Out]

3/2*a^5/b^6/(a+b*x^(1/3))^2-15*a^4/b^6/(a+b*x^(1/3))+18*a^2*x^(1/3)/b^5-9/2*a*x^(2/3)/b^4+x/b^3-30*a^3*ln(a+b*
x^(1/3))/b^6

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Maxima [A]  time = 0.961063, size = 127, normalized size = 1.41 \begin{align*} -\frac{30 \, a^{3} \log \left (b x^{\frac{1}{3}} + a\right )}{b^{6}} + \frac{{\left (b x^{\frac{1}{3}} + a\right )}^{3}}{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^{2}}{b^{6}} - \frac{15 \, a^{4}}{{\left (b x^{\frac{1}{3}} + a\right )} b^{6}} + \frac{3 \, a^{5}}{2 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.52542, size = 350, normalized size = 3.89 \begin{align*} \frac{2 \, b^{9} x^{3} + 4 \, a^{3} b^{6} x^{2} - 34 \, a^{6} b^{3} x - 27 \, a^{9} - 60 \,{\left (a^{3} b^{6} x^{2} + 2 \, a^{6} b^{3} x + a^{9}\right )} \log \left (b x^{\frac{1}{3}} + a\right ) - 3 \,{\left (3 \, a b^{8} x^{2} + 16 \, a^{4} b^{5} x + 10 \, a^{7} b^{2}\right )} x^{\frac{2}{3}} + 3 \,{\left (12 \, a^{2} b^{7} x^{2} + 35 \, a^{5} b^{4} x + 20 \, a^{8} b\right )} x^{\frac{1}{3}}}{2 \,{\left (b^{12} x^{2} + 2 \, a^{3} b^{9} x + a^{6} b^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*b^9*x^3 + 4*a^3*b^6*x^2 - 34*a^6*b^3*x - 27*a^9 - 60*(a^3*b^6*x^2 + 2*a^6*b^3*x + a^9)*log(b*x^(1/3) +
a) - 3*(3*a*b^8*x^2 + 16*a^4*b^5*x + 10*a^7*b^2)*x^(2/3) + 3*(12*a^2*b^7*x^2 + 35*a^5*b^4*x + 20*a^8*b)*x^(1/3
))/(b^12*x^2 + 2*a^3*b^9*x + a^6*b^6)

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Sympy [A]  time = 1.13809, size = 362, normalized size = 4.02 \begin{align*} \begin{cases} - \frac{60 a^{5} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{2 a^{2} b^{6} + 4 a b^{7} \sqrt [3]{x} + 2 b^{8} x^{\frac{2}{3}}} - \frac{90 a^{5}}{2 a^{2} b^{6} + 4 a b^{7} \sqrt [3]{x} + 2 b^{8} x^{\frac{2}{3}}} - \frac{120 a^{4} b \sqrt [3]{x} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{2 a^{2} b^{6} + 4 a b^{7} \sqrt [3]{x} + 2 b^{8} x^{\frac{2}{3}}} - \frac{120 a^{4} b \sqrt [3]{x}}{2 a^{2} b^{6} + 4 a b^{7} \sqrt [3]{x} + 2 b^{8} x^{\frac{2}{3}}} - \frac{60 a^{3} b^{2} x^{\frac{2}{3}} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{2 a^{2} b^{6} + 4 a b^{7} \sqrt [3]{x} + 2 b^{8} x^{\frac{2}{3}}} + \frac{20 a^{2} b^{3} x}{2 a^{2} b^{6} + 4 a b^{7} \sqrt [3]{x} + 2 b^{8} x^{\frac{2}{3}}} - \frac{5 a b^{4} x^{\frac{4}{3}}}{2 a^{2} b^{6} + 4 a b^{7} \sqrt [3]{x} + 2 b^{8} x^{\frac{2}{3}}} + \frac{2 b^{5} x^{\frac{5}{3}}}{2 a^{2} b^{6} + 4 a b^{7} \sqrt [3]{x} + 2 b^{8} x^{\frac{2}{3}}} & \text{for}\: b \neq 0 \\\frac{x^{2}}{2 a^{3}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-60*a**5*log(a/b + x**(1/3))/(2*a**2*b**6 + 4*a*b**7*x**(1/3) + 2*b**8*x**(2/3)) - 90*a**5/(2*a**2*
b**6 + 4*a*b**7*x**(1/3) + 2*b**8*x**(2/3)) - 120*a**4*b*x**(1/3)*log(a/b + x**(1/3))/(2*a**2*b**6 + 4*a*b**7*
x**(1/3) + 2*b**8*x**(2/3)) - 120*a**4*b*x**(1/3)/(2*a**2*b**6 + 4*a*b**7*x**(1/3) + 2*b**8*x**(2/3)) - 60*a**
3*b**2*x**(2/3)*log(a/b + x**(1/3))/(2*a**2*b**6 + 4*a*b**7*x**(1/3) + 2*b**8*x**(2/3)) + 20*a**2*b**3*x/(2*a*
*2*b**6 + 4*a*b**7*x**(1/3) + 2*b**8*x**(2/3)) - 5*a*b**4*x**(4/3)/(2*a**2*b**6 + 4*a*b**7*x**(1/3) + 2*b**8*x
**(2/3)) + 2*b**5*x**(5/3)/(2*a**2*b**6 + 4*a*b**7*x**(1/3) + 2*b**8*x**(2/3)), Ne(b, 0)), (x**2/(2*a**3), Tru
e))

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Giac [A]  time = 1.16289, size = 107, normalized size = 1.19 \begin{align*} -\frac{30 \, a^{3} \log \left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{b^{6}} - \frac{3 \,{\left (10 \, a^{4} b x^{\frac{1}{3}} + 9 \, a^{5}\right )}}{2 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} b^{6}} + \frac{2 \, b^{6} x - 9 \, a b^{5} x^{\frac{2}{3}} + 36 \, a^{2} b^{4} x^{\frac{1}{3}}}{2 \, b^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-30*a^3*log(abs(b*x^(1/3) + a))/b^6 - 3/2*(10*a^4*b*x^(1/3) + 9*a^5)/((b*x^(1/3) + a)^2*b^6) + 1/2*(2*b^6*x -
9*a*b^5*x^(2/3) + 36*a^2*b^4*x^(1/3))/b^9