3.467 \(\int \frac{(a^2+2 a b x^2+b^2 x^4)^3}{x^{15}} \, dx\)

Optimal. Leaf size=19 \[ -\frac{\left (a+b x^2\right )^7}{14 a x^{14}} \]

[Out]

-(a + b*x^2)^7/(14*a*x^14)

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Rubi [A]  time = 0.0066284, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {28, 264} \[ -\frac{\left (a+b x^2\right )^7}{14 a x^{14}} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^15,x]

[Out]

-(a + b*x^2)^7/(14*a*x^14)

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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^2+2 a b x^2+b^2 x^4\right )^3}{x^{15}} \, dx &=\frac{\int \frac{\left (a b+b^2 x^2\right )^6}{x^{15}} \, dx}{b^6}\\ &=-\frac{\left (a+b x^2\right )^7}{14 a x^{14}}\\ \end{align*}

Mathematica [B]  time = 0.007406, size = 82, normalized size = 4.32 \[ -\frac{3 a^4 b^2}{2 x^{10}}-\frac{5 a^3 b^3}{2 x^8}-\frac{5 a^2 b^4}{2 x^6}-\frac{a^5 b}{2 x^{12}}-\frac{a^6}{14 x^{14}}-\frac{3 a b^5}{2 x^4}-\frac{b^6}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^15,x]

[Out]

-a^6/(14*x^14) - (a^5*b)/(2*x^12) - (3*a^4*b^2)/(2*x^10) - (5*a^3*b^3)/(2*x^8) - (5*a^2*b^4)/(2*x^6) - (3*a*b^
5)/(2*x^4) - b^6/(2*x^2)

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Maple [B]  time = 0.048, size = 69, normalized size = 3.6 \begin{align*} -{\frac{{a}^{5}b}{2\,{x}^{12}}}-{\frac{3\,{a}^{4}{b}^{2}}{2\,{x}^{10}}}-{\frac{3\,a{b}^{5}}{2\,{x}^{4}}}-{\frac{{a}^{6}}{14\,{x}^{14}}}-{\frac{{b}^{6}}{2\,{x}^{2}}}-{\frac{5\,{a}^{3}{b}^{3}}{2\,{x}^{8}}}-{\frac{5\,{a}^{2}{b}^{4}}{2\,{x}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^4+2*a*b*x^2+a^2)^3/x^15,x)

[Out]

-1/2*a^5*b/x^12-3/2*a^4*b^2/x^10-3/2*a*b^5/x^4-1/14*a^6/x^14-1/2*b^6/x^2-5/2*a^3*b^3/x^8-5/2*a^2*b^4/x^6

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Maxima [B]  time = 0.9972, size = 92, normalized size = 4.84 \begin{align*} -\frac{7 \, b^{6} x^{12} + 21 \, a b^{5} x^{10} + 35 \, a^{2} b^{4} x^{8} + 35 \, a^{3} b^{3} x^{6} + 21 \, a^{4} b^{2} x^{4} + 7 \, a^{5} b x^{2} + a^{6}}{14 \, x^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^3/x^15,x, algorithm="maxima")

[Out]

-1/14*(7*b^6*x^12 + 21*a*b^5*x^10 + 35*a^2*b^4*x^8 + 35*a^3*b^3*x^6 + 21*a^4*b^2*x^4 + 7*a^5*b*x^2 + a^6)/x^14

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Fricas [B]  time = 1.68576, size = 151, normalized size = 7.95 \begin{align*} -\frac{7 \, b^{6} x^{12} + 21 \, a b^{5} x^{10} + 35 \, a^{2} b^{4} x^{8} + 35 \, a^{3} b^{3} x^{6} + 21 \, a^{4} b^{2} x^{4} + 7 \, a^{5} b x^{2} + a^{6}}{14 \, x^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^3/x^15,x, algorithm="fricas")

[Out]

-1/14*(7*b^6*x^12 + 21*a*b^5*x^10 + 35*a^2*b^4*x^8 + 35*a^3*b^3*x^6 + 21*a^4*b^2*x^4 + 7*a^5*b*x^2 + a^6)/x^14

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Sympy [B]  time = 0.757519, size = 73, normalized size = 3.84 \begin{align*} - \frac{a^{6} + 7 a^{5} b x^{2} + 21 a^{4} b^{2} x^{4} + 35 a^{3} b^{3} x^{6} + 35 a^{2} b^{4} x^{8} + 21 a b^{5} x^{10} + 7 b^{6} x^{12}}{14 x^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**4+2*a*b*x**2+a**2)**3/x**15,x)

[Out]

-(a**6 + 7*a**5*b*x**2 + 21*a**4*b**2*x**4 + 35*a**3*b**3*x**6 + 35*a**2*b**4*x**8 + 21*a*b**5*x**10 + 7*b**6*
x**12)/(14*x**14)

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Giac [B]  time = 1.17481, size = 92, normalized size = 4.84 \begin{align*} -\frac{7 \, b^{6} x^{12} + 21 \, a b^{5} x^{10} + 35 \, a^{2} b^{4} x^{8} + 35 \, a^{3} b^{3} x^{6} + 21 \, a^{4} b^{2} x^{4} + 7 \, a^{5} b x^{2} + a^{6}}{14 \, x^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^3/x^15,x, algorithm="giac")

[Out]

-1/14*(7*b^6*x^12 + 21*a*b^5*x^10 + 35*a^2*b^4*x^8 + 35*a^3*b^3*x^6 + 21*a^4*b^2*x^4 + 7*a^5*b*x^2 + a^6)/x^14