3.834 \(\int \frac{(a+b x^2+c x^4)^2}{x^7} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.036362, 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} \[ -\frac{a^2}{6 x^6}-\frac{2 a c+b^2}{2 x^2}-\frac{a b}{2 x^4}+2 b c \log (x)+\frac{c^2 x^2}{2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)^2/x^7,x]

[Out]

-a^2/(6*x^6) - (a*b)/(2*x^4) - (b^2 + 2*a*c)/(2*x^2) + (c^2*x^2)/2 + 2*b*c*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^7} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\left (a+b x+c x^2\right )^2}{x^4} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (c^2+\frac{a^2}{x^4}+\frac{2 a b}{x^3}+\frac{b^2+2 a c}{x^2}+\frac{2 b c}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac{a^2}{6 x^6}-\frac{a b}{2 x^4}-\frac{b^2+2 a c}{2 x^2}+\frac{c^2 x^2}{2}+2 b c \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0174762, size = 50, normalized size = 0.98 \[ -\frac{a^2+3 a b x^2+6 a c x^4+3 b^2 x^4-12 b c x^6 \log (x)-3 c^2 x^8}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)^2/x^7,x]

[Out]

-(a^2 + 3*a*b*x^2 + 3*b^2*x^4 + 6*a*c*x^4 - 3*c^2*x^8 - 12*b*c*x^6*Log[x])/(6*x^6)

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Maple [A]  time = 0.046, size = 46, normalized size = 0.9 \begin{align*}{\frac{{c}^{2}{x}^{2}}{2}}+2\,bc\ln \left ( x \right ) -{\frac{ab}{2\,{x}^{4}}}-{\frac{ac}{{x}^{2}}}-{\frac{{b}^{2}}{2\,{x}^{2}}}-{\frac{{a}^{2}}{6\,{x}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2+a)^2/x^7,x)

[Out]

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

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Maxima [A]  time = 0.971878, size = 61, normalized size = 1.2 \begin{align*} \frac{1}{2} \, c^{2} x^{2} + b c \log \left (x^{2}\right ) - \frac{3 \,{\left (b^{2} + 2 \, a c\right )} x^{4} + 3 \, a b x^{2} + a^{2}}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^2/x^7,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.43859, size = 109, normalized size = 2.14 \begin{align*} \frac{3 \, c^{2} x^{8} + 12 \, b c x^{6} \log \left (x\right ) - 3 \,{\left (b^{2} + 2 \, a c\right )} x^{4} - 3 \, a b x^{2} - a^{2}}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^2/x^7,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.908223, size = 46, normalized size = 0.9 \begin{align*} 2 b c \log{\left (x \right )} + \frac{c^{2} x^{2}}{2} - \frac{a^{2} + 3 a b x^{2} + x^{4} \left (6 a c + 3 b^{2}\right )}{6 x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2+a)**2/x**7,x)

[Out]

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

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Giac [A]  time = 1.11117, size = 73, normalized size = 1.43 \begin{align*} \frac{1}{2} \, c^{2} x^{2} + b c \log \left (x^{2}\right ) - \frac{11 \, b c x^{6} + 3 \, b^{2} x^{4} + 6 \, a c x^{4} + 3 \, a b x^{2} + a^{2}}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^2/x^7,x, algorithm="giac")

[Out]

1/2*c^2*x^2 + b*c*log(x^2) - 1/6*(11*b*c*x^6 + 3*b^2*x^4 + 6*a*c*x^4 + 3*a*b*x^2 + a^2)/x^6