3.244 \(\int \frac{(a+b x^3)^3}{x^{10}} \, dx\)

Optimal. Leaf size=37 \[ -\frac{a^2 b}{2 x^6}-\frac{a^3}{9 x^9}-\frac{a b^2}{x^3}+b^3 \log (x) \]

[Out]

-a^3/(9*x^9) - (a^2*b)/(2*x^6) - (a*b^2)/x^3 + b^3*Log[x]

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Rubi [A]  time = 0.017969, antiderivative size = 37, 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} \[ -\frac{a^2 b}{2 x^6}-\frac{a^3}{9 x^9}-\frac{a b^2}{x^3}+b^3 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^3/x^10,x]

[Out]

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

Mathematica [A]  time = 0.0042846, size = 37, normalized size = 1. \[ -\frac{a^2 b}{2 x^6}-\frac{a^3}{9 x^9}-\frac{a b^2}{x^3}+b^3 \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^3/x^10,x]

[Out]

-a^3/(9*x^9) - (a^2*b)/(2*x^6) - (a*b^2)/x^3 + b^3*Log[x]

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Maple [A]  time = 0.005, size = 34, normalized size = 0.9 \begin{align*} -{\frac{{a}^{3}}{9\,{x}^{9}}}-{\frac{{a}^{2}b}{2\,{x}^{6}}}-{\frac{a{b}^{2}}{{x}^{3}}}+{b}^{3}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^3/x^10,x)

[Out]

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

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Maxima [A]  time = 0.958666, size = 53, normalized size = 1.43 \begin{align*} \frac{1}{3} \, b^{3} \log \left (x^{3}\right ) - \frac{18 \, a b^{2} x^{6} + 9 \, a^{2} b x^{3} + 2 \, a^{3}}{18 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^3/x^10,x, algorithm="maxima")

[Out]

1/3*b^3*log(x^3) - 1/18*(18*a*b^2*x^6 + 9*a^2*b*x^3 + 2*a^3)/x^9

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Fricas [A]  time = 1.66096, size = 90, normalized size = 2.43 \begin{align*} \frac{18 \, b^{3} x^{9} \log \left (x\right ) - 18 \, a b^{2} x^{6} - 9 \, a^{2} b x^{3} - 2 \, a^{3}}{18 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^3/x^10,x, algorithm="fricas")

[Out]

1/18*(18*b^3*x^9*log(x) - 18*a*b^2*x^6 - 9*a^2*b*x^3 - 2*a^3)/x^9

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Sympy [A]  time = 0.561111, size = 36, normalized size = 0.97 \begin{align*} b^{3} \log{\left (x \right )} - \frac{2 a^{3} + 9 a^{2} b x^{3} + 18 a b^{2} x^{6}}{18 x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**3/x**10,x)

[Out]

b**3*log(x) - (2*a**3 + 9*a**2*b*x**3 + 18*a*b**2*x**6)/(18*x**9)

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Giac [A]  time = 1.10043, size = 61, normalized size = 1.65 \begin{align*} b^{3} \log \left ({\left | x \right |}\right ) - \frac{11 \, b^{3} x^{9} + 18 \, a b^{2} x^{6} + 9 \, a^{2} b x^{3} + 2 \, a^{3}}{18 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^3/x^10,x, algorithm="giac")

[Out]

b^3*log(abs(x)) - 1/18*(11*b^3*x^9 + 18*a*b^2*x^6 + 9*a^2*b*x^3 + 2*a^3)/x^9