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

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

[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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Rubi [A]  time = 0.04, antiderivative size = 74, 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} \[ 3 a^2 b^4 x^5+\frac {20}{3} a^3 b^3 x^3+15 a^4 b^2 x-\frac {6 a^5 b}{x}-\frac {a^6}{3 x^3}+\frac {6}{7} a b^5 x^7+\frac {b^6 x^9}{9} \]

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*}

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Mathematica [A]  time = 0.01, size = 74, normalized size = 1.00 \[ -\frac {a^6}{3 x^3}-\frac {6 a^5 b}{x}+15 a^4 b^2 x+\frac {20}{3} a^3 b^3 x^3+3 a^2 b^4 x^5+\frac {6}{7} a b^5 x^7+\frac {b^6 x^9}{9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*a^6/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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fricas [A]  time = 0.66, size = 70, normalized size = 0.95 \[ \frac {7 \, b^{6} x^{12} + 54 \, a b^{5} x^{10} + 189 \, a^{2} b^{4} x^{8} + 420 \, a^{3} b^{3} x^{6} + 945 \, a^{4} b^{2} x^{4} - 378 \, a^{5} b x^{2} - 21 \, a^{6}}{63 \, x^{3}} \]

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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giac [A]  time = 0.18, size = 67, normalized size = 0.91 \[ \frac {1}{9} \, b^{6} x^{9} + \frac {6}{7} \, a b^{5} x^{7} + 3 \, a^{2} b^{4} x^{5} + \frac {20}{3} \, a^{3} b^{3} x^{3} + 15 \, a^{4} b^{2} x - \frac {18 \, a^{5} b x^{2} + a^{6}}{3 \, x^{3}} \]

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

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maple [A]  time = 0.00, size = 67, normalized size = 0.91 \[ \frac {b^{6} x^{9}}{9}+\frac {6 a \,b^{5} x^{7}}{7}+3 a^{2} b^{4} x^{5}+\frac {20 a^{3} b^{3} x^{3}}{3}+15 a^{4} b^{2} x -\frac {6 a^{5} b}{x}-\frac {a^{6}}{3 x^{3}} \]

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 = 1.29, size = 67, normalized size = 0.91 \[ \frac {1}{9} \, b^{6} x^{9} + \frac {6}{7} \, a b^{5} x^{7} + 3 \, a^{2} b^{4} x^{5} + \frac {20}{3} \, a^{3} b^{3} x^{3} + 15 \, a^{4} b^{2} x - \frac {18 \, a^{5} b x^{2} + a^{6}}{3 \, x^{3}} \]

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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mupad [B]  time = 0.03, size = 69, normalized size = 0.93 \[ \frac {b^6\,x^9}{9}-\frac {\frac {a^6}{3}+6\,b\,a^5\,x^2}{x^3}+15\,a^4\,b^2\,x+\frac {6\,a\,b^5\,x^7}{7}+\frac {20\,a^3\,b^3\,x^3}{3}+3\,a^2\,b^4\,x^5 \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2 + b^2*x^4 + 2*a*b*x^2)^3/x^4,x)

[Out]

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

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sympy [A]  time = 0.21, size = 75, normalized size = 1.01 \[ 15 a^{4} b^{2} x + \frac {20 a^{3} b^{3} x^{3}}{3} + 3 a^{2} b^{4} x^{5} + \frac {6 a b^{5} x^{7}}{7} + \frac {b^{6} x^{9}}{9} + \frac {- a^{6} - 18 a^{5} b x^{2}}{3 x^{3}} \]

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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