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

Optimal. Leaf size=47 \[ a^2 \log (x)+\frac {1}{4} x^4 \left (2 a c+b^2\right )+a b x^2+\frac {1}{3} b c x^6+\frac {c^2 x^8}{8} \]

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Rubi [A]  time = 0.04, antiderivative size = 47, 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*} a^2 \log (x)+\frac {1}{4} x^4 \left (2 a c+b^2\right )+a b x^2+\frac {1}{3} b c x^6+\frac {c^2 x^8}{8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)^2/x,x]

[Out]

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

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Mathematica [A]  time = 0.01, size = 47, normalized size = 1.00 \begin {gather*} a^2 \log (x)+\frac {1}{4} x^4 \left (2 a c+b^2\right )+a b x^2+\frac {1}{3} b c x^6+\frac {c^2 x^8}{8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)^2/x,x]

[Out]

a*b*x^2 + ((b^2 + 2*a*c)*x^4)/4 + (b*c*x^6)/3 + (c^2*x^8)/8 + a^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} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.57, size = 41, normalized size = 0.87 \begin {gather*} \frac {1}{8} \, c^{2} x^{8} + \frac {1}{3} \, b c x^{6} + \frac {1}{4} \, {\left (b^{2} + 2 \, a c\right )} x^{4} + a b x^{2} + a^{2} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.15, size = 46, normalized size = 0.98 \begin {gather*} \frac {1}{8} \, c^{2} x^{8} + \frac {1}{3} \, b c x^{6} + \frac {1}{4} \, b^{2} x^{4} + \frac {1}{2} \, a c x^{4} + a b x^{2} + \frac {1}{2} \, a^{2} \log \left (x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 44, normalized size = 0.94 \begin {gather*} \frac {c^{2} x^{8}}{8}+\frac {b c \,x^{6}}{3}+\frac {a c \,x^{4}}{2}+\frac {b^{2} x^{4}}{4}+a b \,x^{2}+a^{2} \ln \relax (x ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.34, size = 44, normalized size = 0.94 \begin {gather*} \frac {1}{8} \, c^{2} x^{8} + \frac {1}{3} \, b c x^{6} + \frac {1}{4} \, {\left (b^{2} + 2 \, a c\right )} x^{4} + a b x^{2} + \frac {1}{2} \, a^{2} \log \left (x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.02, size = 42, normalized size = 0.89 \begin {gather*} a^2\,\ln \relax (x)+x^4\,\left (\frac {b^2}{4}+\frac {a\,c}{2}\right )+\frac {c^2\,x^8}{8}+a\,b\,x^2+\frac {b\,c\,x^6}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2 + c*x^4)^2/x,x)

[Out]

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

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sympy [A]  time = 0.14, size = 42, normalized size = 0.89 \begin {gather*} a^{2} \log {\relax (x )} + a b x^{2} + \frac {b c x^{6}}{3} + \frac {c^{2} x^{8}}{8} + x^{4} \left (\frac {a c}{2} + \frac {b^{2}}{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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