3.39 \(\int \frac{(a+b x^2)^3}{x^{11}} \, dx\)

Optimal. Leaf size=40 \[ \frac{b \left (a+b x^2\right )^4}{40 a^2 x^8}-\frac{\left (a+b x^2\right )^4}{10 a x^{10}} \]

[Out]

-(a + b*x^2)^4/(10*a*x^10) + (b*(a + b*x^2)^4)/(40*a^2*x^8)

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Rubi [A]  time = 0.0193902, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {266, 45, 37} \[ \frac{b \left (a+b x^2\right )^4}{40 a^2 x^8}-\frac{\left (a+b x^2\right )^4}{10 a x^{10}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^3/x^11,x]

[Out]

-(a + b*x^2)^4/(10*a*x^10) + (b*(a + b*x^2)^4)/(40*a^2*x^8)

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 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2\right )^3}{x^{11}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^3}{x^6} \, dx,x,x^2\right )\\ &=-\frac{\left (a+b x^2\right )^4}{10 a x^{10}}-\frac{b \operatorname{Subst}\left (\int \frac{(a+b x)^3}{x^5} \, dx,x,x^2\right )}{10 a}\\ &=-\frac{\left (a+b x^2\right )^4}{10 a x^{10}}+\frac{b \left (a+b x^2\right )^4}{40 a^2 x^8}\\ \end{align*}

Mathematica [A]  time = 0.0035637, size = 43, normalized size = 1.08 \[ -\frac{3 a^2 b}{8 x^8}-\frac{a^3}{10 x^{10}}-\frac{a b^2}{2 x^6}-\frac{b^3}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^3/x^11,x]

[Out]

-a^3/(10*x^10) - (3*a^2*b)/(8*x^8) - (a*b^2)/(2*x^6) - b^3/(4*x^4)

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Maple [A]  time = 0.004, size = 36, normalized size = 0.9 \begin{align*} -{\frac{3\,{a}^{2}b}{8\,{x}^{8}}}-{\frac{{a}^{3}}{10\,{x}^{10}}}-{\frac{{b}^{3}}{4\,{x}^{4}}}-{\frac{a{b}^{2}}{2\,{x}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^3/x^11,x)

[Out]

-3/8*a^2*b/x^8-1/10*a^3/x^10-1/4*b^3/x^4-1/2*a*b^2/x^6

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Maxima [A]  time = 2.088, size = 50, normalized size = 1.25 \begin{align*} -\frac{10 \, b^{3} x^{6} + 20 \, a b^{2} x^{4} + 15 \, a^{2} b x^{2} + 4 \, a^{3}}{40 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/x^11,x, algorithm="maxima")

[Out]

-1/40*(10*b^3*x^6 + 20*a*b^2*x^4 + 15*a^2*b*x^2 + 4*a^3)/x^10

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Fricas [A]  time = 1.28533, size = 85, normalized size = 2.12 \begin{align*} -\frac{10 \, b^{3} x^{6} + 20 \, a b^{2} x^{4} + 15 \, a^{2} b x^{2} + 4 \, a^{3}}{40 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/x^11,x, algorithm="fricas")

[Out]

-1/40*(10*b^3*x^6 + 20*a*b^2*x^4 + 15*a^2*b*x^2 + 4*a^3)/x^10

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Sympy [A]  time = 0.439568, size = 39, normalized size = 0.98 \begin{align*} - \frac{4 a^{3} + 15 a^{2} b x^{2} + 20 a b^{2} x^{4} + 10 b^{3} x^{6}}{40 x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**3/x**11,x)

[Out]

-(4*a**3 + 15*a**2*b*x**2 + 20*a*b**2*x**4 + 10*b**3*x**6)/(40*x**10)

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Giac [A]  time = 2.52558, size = 50, normalized size = 1.25 \begin{align*} -\frac{10 \, b^{3} x^{6} + 20 \, a b^{2} x^{4} + 15 \, a^{2} b x^{2} + 4 \, a^{3}}{40 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/x^11,x, algorithm="giac")

[Out]

-1/40*(10*b^3*x^6 + 20*a*b^2*x^4 + 15*a^2*b*x^2 + 4*a^3)/x^10