3.2317 \(\int \frac{(a+b \sqrt [3]{x})^5}{x} \, dx\)

Optimal. Leaf size=65 \[ 15 a^3 b^2 x^{2/3}+10 a^2 b^3 x+15 a^4 b \sqrt [3]{x}+a^5 \log (x)+\frac{15}{4} a b^4 x^{4/3}+\frac{3}{5} b^5 x^{5/3} \]

[Out]

15*a^4*b*x^(1/3) + 15*a^3*b^2*x^(2/3) + 10*a^2*b^3*x + (15*a*b^4*x^(4/3))/4 + (3*b^5*x^(5/3))/5 + a^5*Log[x]

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Rubi [A]  time = 0.0287267, antiderivative size = 65, 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} \[ 15 a^3 b^2 x^{2/3}+10 a^2 b^3 x+15 a^4 b \sqrt [3]{x}+a^5 \log (x)+\frac{15}{4} a b^4 x^{4/3}+\frac{3}{5} b^5 x^{5/3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^5/x,x]

[Out]

15*a^4*b*x^(1/3) + 15*a^3*b^2*x^(2/3) + 10*a^2*b^3*x + (15*a*b^4*x^(4/3))/4 + (3*b^5*x^(5/3))/5 + a^5*Log[x]

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

Mathematica [A]  time = 0.0211744, size = 65, normalized size = 1. \[ 15 a^3 b^2 x^{2/3}+10 a^2 b^3 x+15 a^4 b \sqrt [3]{x}+a^5 \log (x)+\frac{15}{4} a b^4 x^{4/3}+\frac{3}{5} b^5 x^{5/3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^5/x,x]

[Out]

15*a^4*b*x^(1/3) + 15*a^3*b^2*x^(2/3) + 10*a^2*b^3*x + (15*a*b^4*x^(4/3))/4 + (3*b^5*x^(5/3))/5 + a^5*Log[x]

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Maple [A]  time = 0.002, size = 54, normalized size = 0.8 \begin{align*} 15\,{a}^{4}b\sqrt [3]{x}+15\,{a}^{3}{b}^{2}{x}^{2/3}+10\,{a}^{2}{b}^{3}x+{\frac{15\,a{b}^{4}}{4}{x}^{{\frac{4}{3}}}}+{\frac{3\,{b}^{5}}{5}{x}^{{\frac{5}{3}}}}+{a}^{5}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/3))^5/x,x)

[Out]

15*a^4*b*x^(1/3)+15*a^3*b^2*x^(2/3)+10*a^2*b^3*x+15/4*a*b^4*x^(4/3)+3/5*b^5*x^(5/3)+a^5*ln(x)

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Maxima [A]  time = 0.976911, size = 72, normalized size = 1.11 \begin{align*} \frac{3}{5} \, b^{5} x^{\frac{5}{3}} + \frac{15}{4} \, a b^{4} x^{\frac{4}{3}} + 10 \, a^{2} b^{3} x + a^{5} \log \left (x\right ) + 15 \, a^{3} b^{2} x^{\frac{2}{3}} + 15 \, a^{4} b x^{\frac{1}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^5/x,x, algorithm="maxima")

[Out]

3/5*b^5*x^(5/3) + 15/4*a*b^4*x^(4/3) + 10*a^2*b^3*x + a^5*log(x) + 15*a^3*b^2*x^(2/3) + 15*a^4*b*x^(1/3)

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Fricas [A]  time = 1.48146, size = 142, normalized size = 2.18 \begin{align*} 10 \, a^{2} b^{3} x + 3 \, a^{5} \log \left (x^{\frac{1}{3}}\right ) + \frac{3}{5} \,{\left (b^{5} x + 25 \, a^{3} b^{2}\right )} x^{\frac{2}{3}} + \frac{15}{4} \,{\left (a b^{4} x + 4 \, a^{4} b\right )} x^{\frac{1}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^5/x,x, algorithm="fricas")

[Out]

10*a^2*b^3*x + 3*a^5*log(x^(1/3)) + 3/5*(b^5*x + 25*a^3*b^2)*x^(2/3) + 15/4*(a*b^4*x + 4*a^4*b)*x^(1/3)

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Sympy [A]  time = 19.843, size = 66, normalized size = 1.02 \begin{align*} a^{5} \log{\left (x \right )} + 15 a^{4} b \sqrt [3]{x} + 15 a^{3} b^{2} x^{\frac{2}{3}} + 10 a^{2} b^{3} x + \frac{15 a b^{4} x^{\frac{4}{3}}}{4} + \frac{3 b^{5} x^{\frac{5}{3}}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/3))**5/x,x)

[Out]

a**5*log(x) + 15*a**4*b*x**(1/3) + 15*a**3*b**2*x**(2/3) + 10*a**2*b**3*x + 15*a*b**4*x**(4/3)/4 + 3*b**5*x**(
5/3)/5

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Giac [A]  time = 1.19133, size = 73, normalized size = 1.12 \begin{align*} \frac{3}{5} \, b^{5} x^{\frac{5}{3}} + \frac{15}{4} \, a b^{4} x^{\frac{4}{3}} + 10 \, a^{2} b^{3} x + a^{5} \log \left ({\left | x \right |}\right ) + 15 \, a^{3} b^{2} x^{\frac{2}{3}} + 15 \, a^{4} b x^{\frac{1}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^5/x,x, algorithm="giac")

[Out]

3/5*b^5*x^(5/3) + 15/4*a*b^4*x^(4/3) + 10*a^2*b^3*x + a^5*log(abs(x)) + 15*a^3*b^2*x^(2/3) + 15*a^4*b*x^(1/3)