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

Optimal. Leaf size=51 \[ -\frac{a^2}{2 x^2}+\frac{1}{2} x^2 \left (2 a c+b^2\right )+2 a b \log (x)+\frac{1}{2} b c x^4+\frac{c^2 x^6}{6} \]

[Out]

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

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Rubi [A]  time = 0.0408121, antiderivative size = 51, normalized size of antiderivative = 1., 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} \[ -\frac{a^2}{2 x^2}+\frac{1}{2} x^2 \left (2 a c+b^2\right )+2 a b \log (x)+\frac{1}{2} b c x^4+\frac{c^2 x^6}{6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)^2/x^3,x]

[Out]

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

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]

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]))

Rubi steps

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

Mathematica [A]  time = 0.0144078, size = 46, normalized size = 0.9 \[ \frac{1}{6} \left (-\frac{3 a^2}{x^2}+3 x^2 \left (2 a c+b^2\right )+12 a b \log (x)+3 b c x^4+c^2 x^6\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-3*a^2)/x^2 + 3*(b^2 + 2*a*c)*x^2 + 3*b*c*x^4 + c^2*x^6 + 12*a*b*Log[x])/6

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Maple [A]  time = 0.048, size = 45, normalized size = 0.9 \begin{align*}{\frac{{c}^{2}{x}^{6}}{6}}+{\frac{bc{x}^{4}}{2}}+ca{x}^{2}+{\frac{{b}^{2}{x}^{2}}{2}}+2\,ab\ln \left ( x \right ) -{\frac{{a}^{2}}{2\,{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.955824, size = 59, normalized size = 1.16 \begin{align*} \frac{1}{6} \, c^{2} x^{6} + \frac{1}{2} \, b c x^{4} + \frac{1}{2} \,{\left (b^{2} + 2 \, a c\right )} x^{2} + a b \log \left (x^{2}\right ) - \frac{a^{2}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.43173, size = 109, normalized size = 2.14 \begin{align*} \frac{c^{2} x^{8} + 3 \, b c x^{6} + 3 \,{\left (b^{2} + 2 \, a c\right )} x^{4} + 12 \, a b x^{2} \log \left (x\right ) - 3 \, a^{2}}{6 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.345443, size = 44, normalized size = 0.86 \begin{align*} - \frac{a^{2}}{2 x^{2}} + 2 a b \log{\left (x \right )} + \frac{b c x^{4}}{2} + \frac{c^{2} x^{6}}{6} + x^{2} \left (a c + \frac{b^{2}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.14875, size = 72, normalized size = 1.41 \begin{align*} \frac{1}{6} \, c^{2} x^{6} + \frac{1}{2} \, b c x^{4} + \frac{1}{2} \, b^{2} x^{2} + a c x^{2} + a b \log \left (x^{2}\right ) - \frac{2 \, a b x^{2} + a^{2}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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