3.7.39 \(\int \frac {(a+b x^2+c x^4)^2}{x^9} \, dx\)

Optimal. Leaf size=48 \[ -\frac {a^2}{8 x^8}-\frac {2 a c+b^2}{4 x^4}-\frac {a b}{3 x^6}-\frac {b c}{x^2}+c^2 \log (x) \]

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Rubi [A]  time = 0.04, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1114, 698} \begin {gather*} -\frac {a^2}{8 x^8}-\frac {2 a c+b^2}{4 x^4}-\frac {a b}{3 x^6}-\frac {b c}{x^2}+c^2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)^2/x^9,x]

[Out]

-a^2/(8*x^8) - (a*b)/(3*x^6) - (b^2 + 2*a*c)/(4*x^4) - (b*c)/x^2 + c^2*Log[x]

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rule 1114

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*(a +
b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^2+c x^4\right )^2}{x^9} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\left (a+b x+c x^2\right )^2}{x^5} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {a^2}{x^5}+\frac {2 a b}{x^4}+\frac {b^2+2 a c}{x^3}+\frac {2 b c}{x^2}+\frac {c^2}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac {a^2}{8 x^8}-\frac {a b}{3 x^6}-\frac {b^2+2 a c}{4 x^4}-\frac {b c}{x^2}+c^2 \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 50, normalized size = 1.04 \begin {gather*} -\frac {a^2}{8 x^8}+\frac {-2 a c-b^2}{4 x^4}-\frac {a b}{3 x^6}-\frac {b c}{x^2}+c^2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)^2/x^9,x]

[Out]

-1/8*a^2/x^8 - (a*b)/(3*x^6) + (-b^2 - 2*a*c)/(4*x^4) - (b*c)/x^2 + c^2*Log[x]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b x^2+c x^4\right )^2}{x^9} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x^2 + c*x^4)^2/x^9,x]

[Out]

IntegrateAlgebraic[(a + b*x^2 + c*x^4)^2/x^9, x]

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fricas [A]  time = 1.23, size = 48, normalized size = 1.00 \begin {gather*} \frac {24 \, c^{2} x^{8} \log \relax (x) - 24 \, b c x^{6} - 6 \, {\left (b^{2} + 2 \, a c\right )} x^{4} - 8 \, a b x^{2} - 3 \, a^{2}}{24 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^2/x^9,x, algorithm="fricas")

[Out]

1/24*(24*c^2*x^8*log(x) - 24*b*c*x^6 - 6*(b^2 + 2*a*c)*x^4 - 8*a*b*x^2 - 3*a^2)/x^8

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giac [A]  time = 0.15, size = 58, normalized size = 1.21 \begin {gather*} \frac {1}{2} \, c^{2} \log \left (x^{2}\right ) - \frac {25 \, c^{2} x^{8} + 24 \, b c x^{6} + 6 \, b^{2} x^{4} + 12 \, a c x^{4} + 8 \, a b x^{2} + 3 \, a^{2}}{24 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^2/x^9,x, algorithm="giac")

[Out]

1/2*c^2*log(x^2) - 1/24*(25*c^2*x^8 + 24*b*c*x^6 + 6*b^2*x^4 + 12*a*c*x^4 + 8*a*b*x^2 + 3*a^2)/x^8

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maple [A]  time = 0.01, size = 45, normalized size = 0.94 \begin {gather*} c^{2} \ln \relax (x )-\frac {b c}{x^{2}}-\frac {a c}{2 x^{4}}-\frac {b^{2}}{4 x^{4}}-\frac {a b}{3 x^{6}}-\frac {a^{2}}{8 x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2+a)^2/x^9,x)

[Out]

-b*c/x^2-1/8*a^2/x^8-1/3*a*b/x^6-1/2/x^4*a*c-1/4*b^2/x^4+c^2*ln(x)

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maxima [A]  time = 1.36, size = 48, normalized size = 1.00 \begin {gather*} \frac {1}{2} \, c^{2} \log \left (x^{2}\right ) - \frac {24 \, b c x^{6} + 6 \, {\left (b^{2} + 2 \, a c\right )} x^{4} + 8 \, a b x^{2} + 3 \, a^{2}}{24 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^2/x^9,x, algorithm="maxima")

[Out]

1/2*c^2*log(x^2) - 1/24*(24*b*c*x^6 + 6*(b^2 + 2*a*c)*x^4 + 8*a*b*x^2 + 3*a^2)/x^8

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mupad [B]  time = 4.18, size = 45, normalized size = 0.94 \begin {gather*} c^2\,\ln \relax (x)-\frac {\frac {a^2}{8}+x^4\,\left (\frac {b^2}{4}+\frac {a\,c}{2}\right )+\frac {a\,b\,x^2}{3}+b\,c\,x^6}{x^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2 + c*x^4)^2/x^9,x)

[Out]

c^2*log(x) - (a^2/8 + x^4*((a*c)/2 + b^2/4) + (a*b*x^2)/3 + b*c*x^6)/x^8

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sympy [A]  time = 1.31, size = 48, normalized size = 1.00 \begin {gather*} c^{2} \log {\relax (x )} + \frac {- 3 a^{2} - 8 a b x^{2} - 24 b c x^{6} + x^{4} \left (- 12 a c - 6 b^{2}\right )}{24 x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2+a)**2/x**9,x)

[Out]

c**2*log(x) + (-3*a**2 - 8*a*b*x**2 - 24*b*c*x**6 + x**4*(-12*a*c - 6*b**2))/(24*x**8)

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