3.2318 \(\int \frac{(a+b \sqrt [3]{x})^5}{x^2} \, dx\)

Optimal. Leaf size=68 \[ -\frac{30 a^3 b^2}{\sqrt [3]{x}}+10 a^2 b^3 \log (x)-\frac{15 a^4 b}{2 x^{2/3}}-\frac{a^5}{x}+15 a b^4 \sqrt [3]{x}+\frac{3}{2} b^5 x^{2/3} \]

[Out]

-(a^5/x) - (15*a^4*b)/(2*x^(2/3)) - (30*a^3*b^2)/x^(1/3) + 15*a*b^4*x^(1/3) + (3*b^5*x^(2/3))/2 + 10*a^2*b^3*L
og[x]

________________________________________________________________________________________

Rubi [A]  time = 0.0319101, antiderivative size = 68, 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} \[ -\frac{30 a^3 b^2}{\sqrt [3]{x}}+10 a^2 b^3 \log (x)-\frac{15 a^4 b}{2 x^{2/3}}-\frac{a^5}{x}+15 a b^4 \sqrt [3]{x}+\frac{3}{2} b^5 x^{2/3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^5/x^2,x]

[Out]

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

Mathematica [A]  time = 0.0240683, size = 68, normalized size = 1. \[ -\frac{30 a^3 b^2}{\sqrt [3]{x}}+10 a^2 b^3 \log (x)-\frac{15 a^4 b}{2 x^{2/3}}-\frac{a^5}{x}+15 a b^4 \sqrt [3]{x}+\frac{3}{2} b^5 x^{2/3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^5/x^2,x]

[Out]

-(a^5/x) - (15*a^4*b)/(2*x^(2/3)) - (30*a^3*b^2)/x^(1/3) + 15*a*b^4*x^(1/3) + (3*b^5*x^(2/3))/2 + 10*a^2*b^3*L
og[x]

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 57, normalized size = 0.8 \begin{align*} -{\frac{{a}^{5}}{x}}-{\frac{15\,{a}^{4}b}{2}{x}^{-{\frac{2}{3}}}}-30\,{\frac{{a}^{3}{b}^{2}}{\sqrt [3]{x}}}+15\,a{b}^{4}\sqrt [3]{x}+{\frac{3\,{b}^{5}}{2}{x}^{{\frac{2}{3}}}}+10\,{a}^{2}{b}^{3}\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^2,x)

[Out]

-a^5/x-15/2*a^4*b/x^(2/3)-30*a^3*b^2/x^(1/3)+15*a*b^4*x^(1/3)+3/2*b^5*x^(2/3)+10*a^2*b^3*ln(x)

________________________________________________________________________________________

Maxima [A]  time = 1.01243, size = 80, normalized size = 1.18 \begin{align*} 10 \, a^{2} b^{3} \log \left (x\right ) + \frac{3}{2} \, b^{5} x^{\frac{2}{3}} + 15 \, a b^{4} x^{\frac{1}{3}} - \frac{60 \, a^{3} b^{2} x^{\frac{2}{3}} + 15 \, a^{4} b x^{\frac{1}{3}} + 2 \, a^{5}}{2 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^5/x^2,x, algorithm="maxima")

[Out]

10*a^2*b^3*log(x) + 3/2*b^5*x^(2/3) + 15*a*b^4*x^(1/3) - 1/2*(60*a^3*b^2*x^(2/3) + 15*a^4*b*x^(1/3) + 2*a^5)/x

________________________________________________________________________________________

Fricas [A]  time = 1.5051, size = 147, normalized size = 2.16 \begin{align*} \frac{60 \, a^{2} b^{3} x \log \left (x^{\frac{1}{3}}\right ) - 2 \, a^{5} + 3 \,{\left (b^{5} x - 20 \, a^{3} b^{2}\right )} x^{\frac{2}{3}} + 15 \,{\left (2 \, a b^{4} x - a^{4} b\right )} x^{\frac{1}{3}}}{2 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^5/x^2,x, algorithm="fricas")

[Out]

1/2*(60*a^2*b^3*x*log(x^(1/3)) - 2*a^5 + 3*(b^5*x - 20*a^3*b^2)*x^(2/3) + 15*(2*a*b^4*x - a^4*b)*x^(1/3))/x

________________________________________________________________________________________

Sympy [A]  time = 0.891623, size = 66, normalized size = 0.97 \begin{align*} - \frac{a^{5}}{x} - \frac{15 a^{4} b}{2 x^{\frac{2}{3}}} - \frac{30 a^{3} b^{2}}{\sqrt [3]{x}} + 10 a^{2} b^{3} \log{\left (x \right )} + 15 a b^{4} \sqrt [3]{x} + \frac{3 b^{5} x^{\frac{2}{3}}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/3))**5/x**2,x)

[Out]

-a**5/x - 15*a**4*b/(2*x**(2/3)) - 30*a**3*b**2/x**(1/3) + 10*a**2*b**3*log(x) + 15*a*b**4*x**(1/3) + 3*b**5*x
**(2/3)/2

________________________________________________________________________________________

Giac [A]  time = 1.15233, size = 81, normalized size = 1.19 \begin{align*} 10 \, a^{2} b^{3} \log \left ({\left | x \right |}\right ) + \frac{3}{2} \, b^{5} x^{\frac{2}{3}} + 15 \, a b^{4} x^{\frac{1}{3}} - \frac{60 \, a^{3} b^{2} x^{\frac{2}{3}} + 15 \, a^{4} b x^{\frac{1}{3}} + 2 \, a^{5}}{2 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^5/x^2,x, algorithm="giac")

[Out]

10*a^2*b^3*log(abs(x)) + 3/2*b^5*x^(2/3) + 15*a*b^4*x^(1/3) - 1/2*(60*a^3*b^2*x^(2/3) + 15*a^4*b*x^(1/3) + 2*a
^5)/x