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

Optimal. Leaf size=74 \[ 3 a^2 b^4 x^5+\frac{20}{3} a^3 b^3 x^3+15 a^4 b^2 x-\frac{6 a^5 b}{x}-\frac{a^6}{3 x^3}+\frac{6}{7} a b^5 x^7+\frac{b^6 x^9}{9} \]

[Out]

-a^6/(3*x^3) - (6*a^5*b)/x + 15*a^4*b^2*x + (20*a^3*b^3*x^3)/3 + 3*a^2*b^4*x^5 + (6*a*b^5*x^7)/7 + (b^6*x^9)/9

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Rubi [A]  time = 0.0364336, antiderivative size = 74, 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} \[ 3 a^2 b^4 x^5+\frac{20}{3} a^3 b^3 x^3+15 a^4 b^2 x-\frac{6 a^5 b}{x}-\frac{a^6}{3 x^3}+\frac{6}{7} a b^5 x^7+\frac{b^6 x^9}{9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-a^6/(3*x^3) - (6*a^5*b)/x + 15*a^4*b^2*x + (20*a^3*b^3*x^3)/3 + 3*a^2*b^4*x^5 + (6*a*b^5*x^7)/7 + (b^6*x^9)/9

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 )^3}{x^4} \, dx &=\frac{\int \frac{\left (a b+b^2 x^2\right )^6}{x^4} \, dx}{b^6}\\ &=\frac{\int \left (15 a^4 b^8+\frac{a^6 b^6}{x^4}+\frac{6 a^5 b^7}{x^2}+20 a^3 b^9 x^2+15 a^2 b^{10} x^4+6 a b^{11} x^6+b^{12} x^8\right ) \, dx}{b^6}\\ &=-\frac{a^6}{3 x^3}-\frac{6 a^5 b}{x}+15 a^4 b^2 x+\frac{20}{3} a^3 b^3 x^3+3 a^2 b^4 x^5+\frac{6}{7} a b^5 x^7+\frac{b^6 x^9}{9}\\ \end{align*}

Mathematica [A]  time = 0.0089066, size = 74, normalized size = 1. \[ 3 a^2 b^4 x^5+\frac{20}{3} a^3 b^3 x^3+15 a^4 b^2 x-\frac{6 a^5 b}{x}-\frac{a^6}{3 x^3}+\frac{6}{7} a b^5 x^7+\frac{b^6 x^9}{9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-a^6/(3*x^3) - (6*a^5*b)/x + 15*a^4*b^2*x + (20*a^3*b^3*x^3)/3 + 3*a^2*b^4*x^5 + (6*a*b^5*x^7)/7 + (b^6*x^9)/9

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Maple [A]  time = 0.046, size = 67, normalized size = 0.9 \begin{align*} -{\frac{{a}^{6}}{3\,{x}^{3}}}-6\,{\frac{{a}^{5}b}{x}}+15\,{a}^{4}{b}^{2}x+{\frac{20\,{a}^{3}{b}^{3}{x}^{3}}{3}}+3\,{a}^{2}{b}^{4}{x}^{5}+{\frac{6\,a{b}^{5}{x}^{7}}{7}}+{\frac{{b}^{6}{x}^{9}}{9}} \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)^3/x^4,x)

[Out]

-1/3*a^6/x^3-6*a^5*b/x+15*a^4*b^2*x+20/3*a^3*b^3*x^3+3*a^2*b^4*x^5+6/7*a*b^5*x^7+1/9*b^6*x^9

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Maxima [A]  time = 1.01736, size = 90, normalized size = 1.22 \begin{align*} \frac{1}{9} \, b^{6} x^{9} + \frac{6}{7} \, a b^{5} x^{7} + 3 \, a^{2} b^{4} x^{5} + \frac{20}{3} \, a^{3} b^{3} x^{3} + 15 \, a^{4} b^{2} x - \frac{18 \, a^{5} b x^{2} + a^{6}}{3 \, x^{3}} \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)^3/x^4,x, algorithm="maxima")

[Out]

1/9*b^6*x^9 + 6/7*a*b^5*x^7 + 3*a^2*b^4*x^5 + 20/3*a^3*b^3*x^3 + 15*a^4*b^2*x - 1/3*(18*a^5*b*x^2 + a^6)/x^3

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Fricas [A]  time = 1.62616, size = 159, normalized size = 2.15 \begin{align*} \frac{7 \, b^{6} x^{12} + 54 \, a b^{5} x^{10} + 189 \, a^{2} b^{4} x^{8} + 420 \, a^{3} b^{3} x^{6} + 945 \, a^{4} b^{2} x^{4} - 378 \, a^{5} b x^{2} - 21 \, a^{6}}{63 \, x^{3}} \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)^3/x^4,x, algorithm="fricas")

[Out]

1/63*(7*b^6*x^12 + 54*a*b^5*x^10 + 189*a^2*b^4*x^8 + 420*a^3*b^3*x^6 + 945*a^4*b^2*x^4 - 378*a^5*b*x^2 - 21*a^
6)/x^3

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Sympy [A]  time = 0.34225, size = 73, normalized size = 0.99 \begin{align*} 15 a^{4} b^{2} x + \frac{20 a^{3} b^{3} x^{3}}{3} + 3 a^{2} b^{4} x^{5} + \frac{6 a b^{5} x^{7}}{7} + \frac{b^{6} x^{9}}{9} - \frac{a^{6} + 18 a^{5} b x^{2}}{3 x^{3}} \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)**3/x**4,x)

[Out]

15*a**4*b**2*x + 20*a**3*b**3*x**3/3 + 3*a**2*b**4*x**5 + 6*a*b**5*x**7/7 + b**6*x**9/9 - (a**6 + 18*a**5*b*x*
*2)/(3*x**3)

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Giac [A]  time = 1.12572, size = 90, normalized size = 1.22 \begin{align*} \frac{1}{9} \, b^{6} x^{9} + \frac{6}{7} \, a b^{5} x^{7} + 3 \, a^{2} b^{4} x^{5} + \frac{20}{3} \, a^{3} b^{3} x^{3} + 15 \, a^{4} b^{2} x - \frac{18 \, a^{5} b x^{2} + a^{6}}{3 \, x^{3}} \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)^3/x^4,x, algorithm="giac")

[Out]

1/9*b^6*x^9 + 6/7*a*b^5*x^7 + 3*a^2*b^4*x^5 + 20/3*a^3*b^3*x^3 + 15*a^4*b^2*x - 1/3*(18*a^5*b*x^2 + a^6)/x^3