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

Optimal. Leaf size=48 \[ 2 a^2 b^2 x^3+4 a^3 b x-\frac{a^4}{x}+\frac{4}{5} a b^3 x^5+\frac{b^4 x^7}{7} \]

[Out]

-(a^4/x) + 4*a^3*b*x + 2*a^2*b^2*x^3 + (4*a*b^3*x^5)/5 + (b^4*x^7)/7

________________________________________________________________________________________

Rubi [A]  time = 0.0255003, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {28, 270} \[ 2 a^2 b^2 x^3+4 a^3 b x-\frac{a^4}{x}+\frac{4}{5} a b^3 x^5+\frac{b^4 x^7}{7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^4/x) + 4*a^3*b*x + 2*a^2*b^2*x^3 + (4*a*b^3*x^5)/5 + (b^4*x^7)/7

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 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

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

Mathematica [A]  time = 0.0081709, size = 48, normalized size = 1. \[ 2 a^2 b^2 x^3+4 a^3 b x-\frac{a^4}{x}+\frac{4}{5} a b^3 x^5+\frac{b^4 x^7}{7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^4/x) + 4*a^3*b*x + 2*a^2*b^2*x^3 + (4*a*b^3*x^5)/5 + (b^4*x^7)/7

________________________________________________________________________________________

Maple [A]  time = 0.046, size = 45, normalized size = 0.9 \begin{align*} -{\frac{{a}^{4}}{x}}+4\,x{a}^{3}b+2\,{a}^{2}{b}^{2}{x}^{3}+{\frac{4\,a{b}^{3}{x}^{5}}{5}}+{\frac{{b}^{4}{x}^{7}}{7}} \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^2,x)

[Out]

-a^4/x+4*x*a^3*b+2*a^2*b^2*x^3+4/5*a*b^3*x^5+1/7*b^4*x^7

________________________________________________________________________________________

Maxima [A]  time = 0.958235, size = 59, normalized size = 1.23 \begin{align*} \frac{1}{7} \, b^{4} x^{7} + \frac{4}{5} \, a b^{3} x^{5} + 2 \, a^{2} b^{2} x^{3} + 4 \, a^{3} b x - \frac{a^{4}}{x} \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^2,x, algorithm="maxima")

[Out]

1/7*b^4*x^7 + 4/5*a*b^3*x^5 + 2*a^2*b^2*x^3 + 4*a^3*b*x - a^4/x

________________________________________________________________________________________

Fricas [A]  time = 1.64799, size = 104, normalized size = 2.17 \begin{align*} \frac{5 \, b^{4} x^{8} + 28 \, a b^{3} x^{6} + 70 \, a^{2} b^{2} x^{4} + 140 \, a^{3} b x^{2} - 35 \, a^{4}}{35 \, x} \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^2,x, algorithm="fricas")

[Out]

1/35*(5*b^4*x^8 + 28*a*b^3*x^6 + 70*a^2*b^2*x^4 + 140*a^3*b*x^2 - 35*a^4)/x

________________________________________________________________________________________

Sympy [A]  time = 0.273524, size = 44, normalized size = 0.92 \begin{align*} - \frac{a^{4}}{x} + 4 a^{3} b x + 2 a^{2} b^{2} x^{3} + \frac{4 a b^{3} x^{5}}{5} + \frac{b^{4} x^{7}}{7} \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**2,x)

[Out]

-a**4/x + 4*a**3*b*x + 2*a**2*b**2*x**3 + 4*a*b**3*x**5/5 + b**4*x**7/7

________________________________________________________________________________________

Giac [A]  time = 1.11292, size = 59, normalized size = 1.23 \begin{align*} \frac{1}{7} \, b^{4} x^{7} + \frac{4}{5} \, a b^{3} x^{5} + 2 \, a^{2} b^{2} x^{3} + 4 \, a^{3} b x - \frac{a^{4}}{x} \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^2,x, algorithm="giac")

[Out]

1/7*b^4*x^7 + 4/5*a*b^3*x^5 + 2*a^2*b^2*x^3 + 4*a^3*b*x - a^4/x