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

Optimal. Leaf size=48 \[ 3 a^2 b^2 x^2+4 a^3 b \log (x)-\frac{a^4}{2 x^2}+a b^3 x^4+\frac{b^4 x^6}{6} \]

[Out]

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

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Rubi [A]  time = 0.0372027, antiderivative size = 48, 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} \[ 3 a^2 b^2 x^2+4 a^3 b \log (x)-\frac{a^4}{2 x^2}+a b^3 x^4+\frac{b^4 x^6}{6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0047694, size = 48, normalized size = 1. \[ 3 a^2 b^2 x^2+4 a^3 b \log (x)-\frac{a^4}{2 x^2}+a b^3 x^4+\frac{b^4 x^6}{6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.046, size = 45, normalized size = 0.9 \begin{align*} -{\frac{{a}^{4}}{2\,{x}^{2}}}+3\,{a}^{2}{b}^{2}{x}^{2}+a{b}^{3}{x}^{4}+{\frac{{b}^{4}{x}^{6}}{6}}+4\,{a}^{3}b\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^3,x)

[Out]

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

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Maxima [A]  time = 0.996465, size = 62, normalized size = 1.29 \begin{align*} \frac{1}{6} \, b^{4} x^{6} + a b^{3} x^{4} + 3 \, a^{2} b^{2} x^{2} + 2 \, a^{3} b \log \left (x^{2}\right ) - \frac{a^{4}}{2 \, x^{2}} \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^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.75956, size = 108, normalized size = 2.25 \begin{align*} \frac{b^{4} x^{8} + 6 \, a b^{3} x^{6} + 18 \, a^{2} b^{2} x^{4} + 24 \, a^{3} b x^{2} \log \left (x\right ) - 3 \, a^{4}}{6 \, x^{2}} \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^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.311597, size = 46, normalized size = 0.96 \begin{align*} - \frac{a^{4}}{2 x^{2}} + 4 a^{3} b \log{\left (x \right )} + 3 a^{2} b^{2} x^{2} + a b^{3} x^{4} + \frac{b^{4} x^{6}}{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**3,x)

[Out]

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

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Giac [A]  time = 1.13619, size = 76, normalized size = 1.58 \begin{align*} \frac{1}{6} \, b^{4} x^{6} + a b^{3} x^{4} + 3 \, a^{2} b^{2} x^{2} + 2 \, a^{3} b \log \left (x^{2}\right ) - \frac{4 \, a^{3} b x^{2} + a^{4}}{2 \, x^{2}} \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^3,x, algorithm="giac")

[Out]

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