3.40 \(\int \frac{(a+b x^2)^3}{x^{13}} \, dx\)

Optimal. Leaf size=43 \[ -\frac{3 a^2 b}{10 x^{10}}-\frac{a^3}{12 x^{12}}-\frac{3 a b^2}{8 x^8}-\frac{b^3}{6 x^6} \]

[Out]

-a^3/(12*x^12) - (3*a^2*b)/(10*x^10) - (3*a*b^2)/(8*x^8) - b^3/(6*x^6)

________________________________________________________________________________________

Rubi [A]  time = 0.0185851, 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} \[ -\frac{3 a^2 b}{10 x^{10}}-\frac{a^3}{12 x^{12}}-\frac{3 a b^2}{8 x^8}-\frac{b^3}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^3/x^13,x]

[Out]

-a^3/(12*x^12) - (3*a^2*b)/(10*x^10) - (3*a*b^2)/(8*x^8) - b^3/(6*x^6)

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^2\right )^3}{x^{13}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^3}{x^7} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a^3}{x^7}+\frac{3 a^2 b}{x^6}+\frac{3 a b^2}{x^5}+\frac{b^3}{x^4}\right ) \, dx,x,x^2\right )\\ &=-\frac{a^3}{12 x^{12}}-\frac{3 a^2 b}{10 x^{10}}-\frac{3 a b^2}{8 x^8}-\frac{b^3}{6 x^6}\\ \end{align*}

Mathematica [A]  time = 0.0037502, size = 43, normalized size = 1. \[ -\frac{3 a^2 b}{10 x^{10}}-\frac{a^3}{12 x^{12}}-\frac{3 a b^2}{8 x^8}-\frac{b^3}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^3/x^13,x]

[Out]

-a^3/(12*x^12) - (3*a^2*b)/(10*x^10) - (3*a*b^2)/(8*x^8) - b^3/(6*x^6)

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 36, normalized size = 0.8 \begin{align*} -{\frac{{a}^{3}}{12\,{x}^{12}}}-{\frac{3\,{a}^{2}b}{10\,{x}^{10}}}-{\frac{3\,a{b}^{2}}{8\,{x}^{8}}}-{\frac{{b}^{3}}{6\,{x}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^3/x^13,x)

[Out]

-1/12*a^3/x^12-3/10*a^2*b/x^10-3/8*a*b^2/x^8-1/6*b^3/x^6

________________________________________________________________________________________

Maxima [A]  time = 2.317, size = 50, normalized size = 1.16 \begin{align*} -\frac{20 \, b^{3} x^{6} + 45 \, a b^{2} x^{4} + 36 \, a^{2} b x^{2} + 10 \, a^{3}}{120 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/x^13,x, algorithm="maxima")

[Out]

-1/120*(20*b^3*x^6 + 45*a*b^2*x^4 + 36*a^2*b*x^2 + 10*a^3)/x^12

________________________________________________________________________________________

Fricas [A]  time = 1.31164, size = 88, normalized size = 2.05 \begin{align*} -\frac{20 \, b^{3} x^{6} + 45 \, a b^{2} x^{4} + 36 \, a^{2} b x^{2} + 10 \, a^{3}}{120 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/x^13,x, algorithm="fricas")

[Out]

-1/120*(20*b^3*x^6 + 45*a*b^2*x^4 + 36*a^2*b*x^2 + 10*a^3)/x^12

________________________________________________________________________________________

Sympy [A]  time = 0.459102, size = 39, normalized size = 0.91 \begin{align*} - \frac{10 a^{3} + 36 a^{2} b x^{2} + 45 a b^{2} x^{4} + 20 b^{3} x^{6}}{120 x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**3/x**13,x)

[Out]

-(10*a**3 + 36*a**2*b*x**2 + 45*a*b**2*x**4 + 20*b**3*x**6)/(120*x**12)

________________________________________________________________________________________

Giac [A]  time = 1.79705, size = 50, normalized size = 1.16 \begin{align*} -\frac{20 \, b^{3} x^{6} + 45 \, a b^{2} x^{4} + 36 \, a^{2} b x^{2} + 10 \, a^{3}}{120 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/x^13,x, algorithm="giac")

[Out]

-1/120*(20*b^3*x^6 + 45*a*b^2*x^4 + 36*a^2*b*x^2 + 10*a^3)/x^12