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

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

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Rubi [A]  time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {\left (a+b x^2\right )^7}{14 a x^{14}} \end {gather*}

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*}

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Mathematica [B]  time = 0.01, size = 82, normalized size = 4.32 \begin {gather*} -\frac {a^6}{14 x^{14}}-\frac {a^5 b}{2 x^{12}}-\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 {3 a b^5}{2 x^4}-\frac {b^6}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/14*a^6/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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{15}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.81, size = 68, normalized size = 3.58 \begin {gather*} -\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 {gather*}

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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giac [B]  time = 0.16, size = 68, normalized size = 3.58 \begin {gather*} -\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 {gather*}

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

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maple [B]  time = 0.01, size = 69, normalized size = 3.63 \begin {gather*} -\frac {b^{6}}{2 x^{2}}-\frac {3 a \,b^{5}}{2 x^{4}}-\frac {5 a^{2} b^{4}}{2 x^{6}}-\frac {5 a^{3} b^{3}}{2 x^{8}}-\frac {3 a^{4} b^{2}}{2 x^{10}}-\frac {a^{5} b}{2 x^{12}}-\frac {a^{6}}{14 x^{14}} \end {gather*}

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

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maxima [B]  time = 1.38, size = 68, normalized size = 3.58 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 4.36, size = 70, normalized size = 3.68 \begin {gather*} -\frac {\frac {a^6}{14}+\frac {a^5\,b\,x^2}{2}+\frac {3\,a^4\,b^2\,x^4}{2}+\frac {5\,a^3\,b^3\,x^6}{2}+\frac {5\,a^2\,b^4\,x^8}{2}+\frac {3\,a\,b^5\,x^{10}}{2}+\frac {b^6\,x^{12}}{2}}{x^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 0.63, size = 73, normalized size = 3.84 \begin {gather*} \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 {gather*}

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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