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

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

[Out]

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

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Rubi [A]  time = 0.0326376, 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} \[ -\frac{3 a^2 b^2}{x^2}-\frac{a^3 b}{x^4}-\frac{a^4}{6 x^6}+4 a b^3 \log (x)+\frac{b^4 x^2}{2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0045966, size = 49, normalized size = 1. \[ -\frac{3 a^2 b^2}{x^2}-\frac{a^3 b}{x^4}-\frac{a^4}{6 x^6}+4 a b^3 \log (x)+\frac{b^4 x^2}{2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.046, size = 46, normalized size = 0.9 \begin{align*} -{\frac{{a}^{4}}{6\,{x}^{6}}}-{\frac{{a}^{3}b}{{x}^{4}}}-3\,{\frac{{b}^{2}{a}^{2}}{{x}^{2}}}+{\frac{{b}^{4}{x}^{2}}{2}}+4\,a{b}^{3}\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^7,x)

[Out]

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

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Maxima [A]  time = 0.9814, size = 65, normalized size = 1.33 \begin{align*} \frac{1}{2} \, b^{4} x^{2} + 2 \, a b^{3} \log \left (x^{2}\right ) - \frac{18 \, a^{2} b^{2} x^{4} + 6 \, a^{3} b x^{2} + a^{4}}{6 \, x^{6}} \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^7,x, algorithm="maxima")

[Out]

1/2*b^4*x^2 + 2*a*b^3*log(x^2) - 1/6*(18*a^2*b^2*x^4 + 6*a^3*b*x^2 + a^4)/x^6

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Fricas [A]  time = 1.6557, size = 108, normalized size = 2.2 \begin{align*} \frac{3 \, b^{4} x^{8} + 24 \, a b^{3} x^{6} \log \left (x\right ) - 18 \, a^{2} b^{2} x^{4} - 6 \, a^{3} b x^{2} - a^{4}}{6 \, x^{6}} \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^7,x, algorithm="fricas")

[Out]

1/6*(3*b^4*x^8 + 24*a*b^3*x^6*log(x) - 18*a^2*b^2*x^4 - 6*a^3*b*x^2 - a^4)/x^6

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Sympy [A]  time = 0.419184, size = 48, normalized size = 0.98 \begin{align*} 4 a b^{3} \log{\left (x \right )} + \frac{b^{4} x^{2}}{2} - \frac{a^{4} + 6 a^{3} b x^{2} + 18 a^{2} b^{2} x^{4}}{6 x^{6}} \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**7,x)

[Out]

4*a*b**3*log(x) + b**4*x**2/2 - (a**4 + 6*a**3*b*x**2 + 18*a**2*b**2*x**4)/(6*x**6)

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Giac [A]  time = 1.10848, size = 77, normalized size = 1.57 \begin{align*} \frac{1}{2} \, b^{4} x^{2} + 2 \, a b^{3} \log \left (x^{2}\right ) - \frac{22 \, a b^{3} x^{6} + 18 \, a^{2} b^{2} x^{4} + 6 \, a^{3} b x^{2} + a^{4}}{6 \, x^{6}} \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^7,x, algorithm="giac")

[Out]

1/2*b^4*x^2 + 2*a*b^3*log(x^2) - 1/6*(22*a*b^3*x^6 + 18*a^2*b^2*x^4 + 6*a^3*b*x^2 + a^4)/x^6