3.65 \(\int \frac{(a+b x^2)^5}{x^{13}} \, dx\)

Optimal. Leaf size=19 \[ -\frac{\left (a+b x^2\right )^6}{12 a x^{12}} \]

[Out]

-(a + b*x^2)^6/(12*a*x^12)

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Rubi [A]  time = 0.0035702, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {264} \[ -\frac{\left (a+b x^2\right )^6}{12 a x^{12}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^5/x^13,x]

[Out]

-(a + b*x^2)^6/(12*a*x^12)

Rule 264

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

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2\right )^5}{x^{13}} \, dx &=-\frac{\left (a+b x^2\right )^6}{12 a x^{12}}\\ \end{align*}

Mathematica [B]  time = 0.0040731, size = 69, normalized size = 3.63 \[ -\frac{5 a^3 b^2}{4 x^8}-\frac{5 a^2 b^3}{3 x^6}-\frac{a^4 b}{2 x^{10}}-\frac{a^5}{12 x^{12}}-\frac{5 a b^4}{4 x^4}-\frac{b^5}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^5/x^13,x]

[Out]

-a^5/(12*x^12) - (a^4*b)/(2*x^10) - (5*a^3*b^2)/(4*x^8) - (5*a^2*b^3)/(3*x^6) - (5*a*b^4)/(4*x^4) - b^5/(2*x^2
)

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Maple [B]  time = 0.004, size = 58, normalized size = 3.1 \begin{align*} -{\frac{5\,{a}^{3}{b}^{2}}{4\,{x}^{8}}}-{\frac{{a}^{4}b}{2\,{x}^{10}}}-{\frac{{a}^{5}}{12\,{x}^{12}}}-{\frac{5\,a{b}^{4}}{4\,{x}^{4}}}-{\frac{{b}^{5}}{2\,{x}^{2}}}-{\frac{5\,{a}^{2}{b}^{3}}{3\,{x}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5/x^13,x)

[Out]

-5/4*a^3*b^2/x^8-1/2*a^4*b/x^10-1/12*a^5/x^12-5/4*a*b^4/x^4-1/2*b^5/x^2-5/3*a^2*b^3/x^6

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Maxima [B]  time = 1.40163, size = 77, normalized size = 4.05 \begin{align*} -\frac{6 \, b^{5} x^{10} + 15 \, a b^{4} x^{8} + 20 \, a^{2} b^{3} x^{6} + 15 \, a^{3} b^{2} x^{4} + 6 \, a^{4} b x^{2} + a^{5}}{12 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^13,x, algorithm="maxima")

[Out]

-1/12*(6*b^5*x^10 + 15*a*b^4*x^8 + 20*a^2*b^3*x^6 + 15*a^3*b^2*x^4 + 6*a^4*b*x^2 + a^5)/x^12

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Fricas [B]  time = 1.25478, size = 127, normalized size = 6.68 \begin{align*} -\frac{6 \, b^{5} x^{10} + 15 \, a b^{4} x^{8} + 20 \, a^{2} b^{3} x^{6} + 15 \, a^{3} b^{2} x^{4} + 6 \, a^{4} b x^{2} + a^{5}}{12 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^13,x, algorithm="fricas")

[Out]

-1/12*(6*b^5*x^10 + 15*a*b^4*x^8 + 20*a^2*b^3*x^6 + 15*a^3*b^2*x^4 + 6*a^4*b*x^2 + a^5)/x^12

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Sympy [B]  time = 0.6015, size = 61, normalized size = 3.21 \begin{align*} - \frac{a^{5} + 6 a^{4} b x^{2} + 15 a^{3} b^{2} x^{4} + 20 a^{2} b^{3} x^{6} + 15 a b^{4} x^{8} + 6 b^{5} x^{10}}{12 x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5/x**13,x)

[Out]

-(a**5 + 6*a**4*b*x**2 + 15*a**3*b**2*x**4 + 20*a**2*b**3*x**6 + 15*a*b**4*x**8 + 6*b**5*x**10)/(12*x**12)

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Giac [B]  time = 2.87239, size = 77, normalized size = 4.05 \begin{align*} -\frac{6 \, b^{5} x^{10} + 15 \, a b^{4} x^{8} + 20 \, a^{2} b^{3} x^{6} + 15 \, a^{3} b^{2} x^{4} + 6 \, a^{4} b x^{2} + a^{5}}{12 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^13,x, algorithm="giac")

[Out]

-1/12*(6*b^5*x^10 + 15*a*b^4*x^8 + 20*a^2*b^3*x^6 + 15*a^3*b^2*x^4 + 6*a^4*b*x^2 + a^5)/x^12