3.7.35 \(\int \frac {(a+b x^2+c x^4)^2}{x^5} \, dx\)

Optimal. Leaf size=45 \[ -\frac {a^2}{4 x^4}+\log (x) \left (2 a c+b^2\right )-\frac {a b}{x^2}+b c x^2+\frac {c^2 x^4}{4} \]

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Rubi [A]  time = 0.04, antiderivative size = 45, 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}{4 x^4}+\log (x) \left (2 a c+b^2\right )-\frac {a b}{x^2}+b c x^2+\frac {c^2 x^4}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)^2/x^5,x]

[Out]

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

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Mathematica [A]  time = 0.02, size = 41, normalized size = 0.91 \begin {gather*} \log (x) \left (2 a c+b^2\right )+\frac {\left (c x^4-a\right ) \left (a+4 b x^2+c x^4\right )}{4 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)^2/x^5,x]

[Out]

((-a + c*x^4)*(a + 4*b*x^2 + c*x^4))/(4*x^4) + (b^2 + 2*a*c)*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^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x^2 + c*x^4)^2/x^5,x]

[Out]

IntegrateAlgebraic[(a + b*x^2 + c*x^4)^2/x^5, x]

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fricas [A]  time = 1.01, size = 47, normalized size = 1.04 \begin {gather*} \frac {c^{2} x^{8} + 4 \, b c x^{6} + 4 \, {\left (b^{2} + 2 \, a c\right )} x^{4} \log \relax (x) - 4 \, a b x^{2} - a^{2}}{4 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^2/x^5,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.18, size = 60, normalized size = 1.33 \begin {gather*} \frac {1}{4} \, c^{2} x^{4} + b c x^{2} + \frac {1}{2} \, {\left (b^{2} + 2 \, a c\right )} \log \left (x^{2}\right ) - \frac {3 \, b^{2} x^{4} + 6 \, a c x^{4} + 4 \, a b x^{2} + a^{2}}{4 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^2/x^5,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.01, size = 43, normalized size = 0.96 \begin {gather*} \frac {c^{2} x^{4}}{4}+b c \,x^{2}+2 a c \ln \relax (x )+b^{2} \ln \relax (x )-\frac {a b}{x^{2}}-\frac {a^{2}}{4 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2+a)^2/x^5,x)

[Out]

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

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maxima [A]  time = 1.34, size = 45, normalized size = 1.00 \begin {gather*} \frac {1}{4} \, c^{2} x^{4} + b c x^{2} + \frac {1}{2} \, {\left (b^{2} + 2 \, a c\right )} \log \left (x^{2}\right ) - \frac {4 \, a b x^{2} + a^{2}}{4 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^2/x^5,x, algorithm="maxima")

[Out]

1/4*c^2*x^4 + b*c*x^2 + 1/2*(b^2 + 2*a*c)*log(x^2) - 1/4*(4*a*b*x^2 + a^2)/x^4

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mupad [B]  time = 0.04, size = 43, normalized size = 0.96 \begin {gather*} \ln \relax (x)\,\left (b^2+2\,a\,c\right )-\frac {\frac {a^2}{4}+b\,a\,x^2}{x^4}+\frac {c^2\,x^4}{4}+b\,c\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2 + c*x^4)^2/x^5,x)

[Out]

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

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sympy [A]  time = 0.37, size = 44, normalized size = 0.98 \begin {gather*} b c x^{2} + \frac {c^{2} x^{4}}{4} + \left (2 a c + b^{2}\right ) \log {\relax (x )} + \frac {- a^{2} - 4 a b x^{2}}{4 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2+a)**2/x**5,x)

[Out]

b*c*x**2 + c**2*x**4/4 + (2*a*c + b**2)*log(x) + (-a**2 - 4*a*b*x**2)/(4*x**4)

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