3.248 \(\int \frac{(a+b x^3)^3}{x^{22}} \, dx\)

Optimal. Leaf size=43 \[ -\frac{a^2 b}{6 x^{18}}-\frac{a^3}{21 x^{21}}-\frac{a b^2}{5 x^{15}}-\frac{b^3}{12 x^{12}} \]

[Out]

-a^3/(21*x^21) - (a^2*b)/(6*x^18) - (a*b^2)/(5*x^15) - b^3/(12*x^12)

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Rubi [A]  time = 0.0185465, antiderivative size = 43, 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} \[ -\frac{a^2 b}{6 x^{18}}-\frac{a^3}{21 x^{21}}-\frac{a b^2}{5 x^{15}}-\frac{b^3}{12 x^{12}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^3/x^22,x]

[Out]

-a^3/(21*x^21) - (a^2*b)/(6*x^18) - (a*b^2)/(5*x^15) - b^3/(12*x^12)

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^{22}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(a+b x)^3}{x^8} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{a^3}{x^8}+\frac{3 a^2 b}{x^7}+\frac{3 a b^2}{x^6}+\frac{b^3}{x^5}\right ) \, dx,x,x^3\right )\\ &=-\frac{a^3}{21 x^{21}}-\frac{a^2 b}{6 x^{18}}-\frac{a b^2}{5 x^{15}}-\frac{b^3}{12 x^{12}}\\ \end{align*}

Mathematica [A]  time = 0.0037663, size = 43, normalized size = 1. \[ -\frac{a^2 b}{6 x^{18}}-\frac{a^3}{21 x^{21}}-\frac{a b^2}{5 x^{15}}-\frac{b^3}{12 x^{12}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^3/x^22,x]

[Out]

-a^3/(21*x^21) - (a^2*b)/(6*x^18) - (a*b^2)/(5*x^15) - b^3/(12*x^12)

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Maple [A]  time = 0.005, size = 36, normalized size = 0.8 \begin{align*} -{\frac{{a}^{3}}{21\,{x}^{21}}}-{\frac{{a}^{2}b}{6\,{x}^{18}}}-{\frac{a{b}^{2}}{5\,{x}^{15}}}-{\frac{{b}^{3}}{12\,{x}^{12}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^3/x^22,x)

[Out]

-1/21*a^3/x^21-1/6*a^2*b/x^18-1/5*a*b^2/x^15-1/12*b^3/x^12

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Maxima [A]  time = 0.99636, size = 50, normalized size = 1.16 \begin{align*} -\frac{35 \, b^{3} x^{9} + 84 \, a b^{2} x^{6} + 70 \, a^{2} b x^{3} + 20 \, a^{3}}{420 \, x^{21}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^3/x^22,x, algorithm="maxima")

[Out]

-1/420*(35*b^3*x^9 + 84*a*b^2*x^6 + 70*a^2*b*x^3 + 20*a^3)/x^21

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Fricas [A]  time = 1.72042, size = 88, normalized size = 2.05 \begin{align*} -\frac{35 \, b^{3} x^{9} + 84 \, a b^{2} x^{6} + 70 \, a^{2} b x^{3} + 20 \, a^{3}}{420 \, x^{21}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^3/x^22,x, algorithm="fricas")

[Out]

-1/420*(35*b^3*x^9 + 84*a*b^2*x^6 + 70*a^2*b*x^3 + 20*a^3)/x^21

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Sympy [A]  time = 0.775907, size = 39, normalized size = 0.91 \begin{align*} - \frac{20 a^{3} + 70 a^{2} b x^{3} + 84 a b^{2} x^{6} + 35 b^{3} x^{9}}{420 x^{21}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**3/x**22,x)

[Out]

-(20*a**3 + 70*a**2*b*x**3 + 84*a*b**2*x**6 + 35*b**3*x**9)/(420*x**21)

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Giac [A]  time = 1.13253, size = 50, normalized size = 1.16 \begin{align*} -\frac{35 \, b^{3} x^{9} + 84 \, a b^{2} x^{6} + 70 \, a^{2} b x^{3} + 20 \, a^{3}}{420 \, x^{21}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^3/x^22,x, algorithm="giac")

[Out]

-1/420*(35*b^3*x^9 + 84*a*b^2*x^6 + 70*a^2*b*x^3 + 20*a^3)/x^21