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

Optimal. Leaf size=49 \[ 6 a^2 b^2 \log (x)-\frac{2 a^3 b}{x^2}-\frac{a^4}{4 x^4}+2 a b^3 x^2+\frac{b^4 x^4}{4} \]

[Out]

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

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Rubi [A]  time = 0.0356296, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {28, 266, 43} \[ 6 a^2 b^2 \log (x)-\frac{2 a^3 b}{x^2}-\frac{a^4}{4 x^4}+2 a b^3 x^2+\frac{b^4 x^4}{4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 266

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^2}{x^5} \, dx &=\frac{\int \frac{\left (a b+b^2 x^2\right )^4}{x^5} \, dx}{b^4}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (a b+b^2 x\right )^4}{x^3} \, dx,x,x^2\right )}{2 b^4}\\ &=\frac{\operatorname{Subst}\left (\int \left (4 a b^7+\frac{a^4 b^4}{x^3}+\frac{4 a^3 b^5}{x^2}+\frac{6 a^2 b^6}{x}+b^8 x\right ) \, dx,x,x^2\right )}{2 b^4}\\ &=-\frac{a^4}{4 x^4}-\frac{2 a^3 b}{x^2}+2 a b^3 x^2+\frac{b^4 x^4}{4}+6 a^2 b^2 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0045238, size = 49, normalized size = 1. \[ 6 a^2 b^2 \log (x)-\frac{2 a^3 b}{x^2}-\frac{a^4}{4 x^4}+2 a b^3 x^2+\frac{b^4 x^4}{4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.046, size = 46, normalized size = 0.9 \begin{align*} -{\frac{{a}^{4}}{4\,{x}^{4}}}-2\,{\frac{{a}^{3}b}{{x}^{2}}}+2\,a{b}^{3}{x}^{2}+{\frac{{b}^{4}{x}^{4}}{4}}+6\,{a}^{2}{b}^{2}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.996828, size = 65, normalized size = 1.33 \begin{align*} \frac{1}{4} \, b^{4} x^{4} + 2 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} \log \left (x^{2}\right ) - \frac{8 \, a^{3} b x^{2} + a^{4}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.70179, size = 104, normalized size = 2.12 \begin{align*} \frac{b^{4} x^{8} + 8 \, a b^{3} x^{6} + 24 \, a^{2} b^{2} x^{4} \log \left (x\right ) - 8 \, a^{3} b x^{2} - a^{4}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.360037, size = 48, normalized size = 0.98 \begin{align*} 6 a^{2} b^{2} \log{\left (x \right )} + 2 a b^{3} x^{2} + \frac{b^{4} x^{4}}{4} - \frac{a^{4} + 8 a^{3} b x^{2}}{4 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.18422, size = 80, normalized size = 1.63 \begin{align*} \frac{1}{4} \, b^{4} x^{4} + 2 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} \log \left (x^{2}\right ) - \frac{18 \, a^{2} b^{2} x^{4} + 8 \, a^{3} b x^{2} + a^{4}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b^4*x^4 + 2*a*b^3*x^2 + 3*a^2*b^2*log(x^2) - 1/4*(18*a^2*b^2*x^4 + 8*a^3*b*x^2 + a^4)/x^4