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

Optimal. Leaf size=50 \[ -\frac {a^4}{3 x^3}-\frac {4 a^3 b}{x}+6 a^2 b^2 x+\frac {4}{3} a b^3 x^3+\frac {b^4 x^5}{5} \]

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Rubi [A]  time = 0.03, antiderivative size = 50, normalized size of antiderivative = 1.00, 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} \begin {gather*} 6 a^2 b^2 x-\frac {4 a^3 b}{x}-\frac {a^4}{3 x^3}+\frac {4}{3} a b^3 x^3+\frac {b^4 x^5}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-a^4/(3*x^3) - (4*a^3*b)/x + 6*a^2*b^2*x + (4*a*b^3*x^3)/3 + (b^4*x^5)/5

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

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Mathematica [A]  time = 0.01, size = 50, normalized size = 1.00 \begin {gather*} -\frac {a^4}{3 x^3}-\frac {4 a^3 b}{x}+6 a^2 b^2 x+\frac {4}{3} a b^3 x^3+\frac {b^4 x^5}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*a^4/x^3 - (4*a^3*b)/x + 6*a^2*b^2*x + (4*a*b^3*x^3)/3 + (b^4*x^5)/5

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.84, size = 48, normalized size = 0.96 \begin {gather*} \frac {3 \, b^{4} x^{8} + 20 \, a b^{3} x^{6} + 90 \, a^{2} b^{2} x^{4} - 60 \, a^{3} b x^{2} - 5 \, a^{4}}{15 \, x^{3}} \end {gather*}

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^4,x, algorithm="fricas")

[Out]

1/15*(3*b^4*x^8 + 20*a*b^3*x^6 + 90*a^2*b^2*x^4 - 60*a^3*b*x^2 - 5*a^4)/x^3

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giac [A]  time = 0.17, size = 45, normalized size = 0.90 \begin {gather*} \frac {1}{5} \, b^{4} x^{5} + \frac {4}{3} \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x - \frac {12 \, a^{3} b x^{2} + a^{4}}{3 \, x^{3}} \end {gather*}

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^4,x, algorithm="giac")

[Out]

1/5*b^4*x^5 + 4/3*a*b^3*x^3 + 6*a^2*b^2*x - 1/3*(12*a^3*b*x^2 + a^4)/x^3

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maple [A]  time = 0.00, size = 45, normalized size = 0.90 \begin {gather*} \frac {b^{4} x^{5}}{5}+\frac {4 a \,b^{3} x^{3}}{3}+6 a^{2} b^{2} x -\frac {4 a^{3} b}{x}-\frac {a^{4}}{3 x^{3}} \end {gather*}

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^4,x)

[Out]

-1/3*a^4/x^3-4*a^3*b/x+6*a^2*b^2*x+4/3*a*b^3*x^3+1/5*b^4*x^5

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maxima [A]  time = 1.44, size = 45, normalized size = 0.90 \begin {gather*} \frac {1}{5} \, b^{4} x^{5} + \frac {4}{3} \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x - \frac {12 \, a^{3} b x^{2} + a^{4}}{3 \, x^{3}} \end {gather*}

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^4,x, algorithm="maxima")

[Out]

1/5*b^4*x^5 + 4/3*a*b^3*x^3 + 6*a^2*b^2*x - 1/3*(12*a^3*b*x^2 + a^4)/x^3

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mupad [B]  time = 0.04, size = 47, normalized size = 0.94 \begin {gather*} \frac {b^4\,x^5}{5}-\frac {\frac {a^4}{3}+4\,b\,a^3\,x^2}{x^3}+6\,a^2\,b^2\,x+\frac {4\,a\,b^3\,x^3}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^4*x^5)/5 - (a^4/3 + 4*a^3*b*x^2)/x^3 + 6*a^2*b^2*x + (4*a*b^3*x^3)/3

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sympy [A]  time = 0.17, size = 49, normalized size = 0.98 \begin {gather*} 6 a^{2} b^{2} x + \frac {4 a b^{3} x^{3}}{3} + \frac {b^{4} x^{5}}{5} + \frac {- a^{4} - 12 a^{3} b x^{2}}{3 x^{3}} \end {gather*}

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**4,x)

[Out]

6*a**2*b**2*x + 4*a*b**3*x**3/3 + b**4*x**5/5 + (-a**4 - 12*a**3*b*x**2)/(3*x**3)

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