3.460 \(\int \frac {(a^2+2 a b x^2+b^2 x^4)^3}{x^8} \, dx\)

Optimal. Leaf size=72 \[ -\frac {a^6}{7 x^7}-\frac {6 a^5 b}{5 x^5}-\frac {5 a^4 b^2}{x^3}-\frac {20 a^3 b^3}{x}+15 a^2 b^4 x+2 a b^5 x^3+\frac {b^6 x^5}{5} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 72, 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} \[ -\frac {5 a^4 b^2}{x^3}-\frac {20 a^3 b^3}{x}+15 a^2 b^4 x-\frac {6 a^5 b}{5 x^5}-\frac {a^6}{7 x^7}+2 a b^5 x^3+\frac {b^6 x^5}{5} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^8,x]

[Out]

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

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Mathematica [A]  time = 0.01, size = 72, normalized size = 1.00 \[ -\frac {a^6}{7 x^7}-\frac {6 a^5 b}{5 x^5}-\frac {5 a^4 b^2}{x^3}-\frac {20 a^3 b^3}{x}+15 a^2 b^4 x+2 a b^5 x^3+\frac {b^6 x^5}{5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^8,x]

[Out]

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

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fricas [A]  time = 0.72, size = 70, normalized size = 0.97 \[ \frac {7 \, b^{6} x^{12} + 70 \, a b^{5} x^{10} + 525 \, a^{2} b^{4} x^{8} - 700 \, a^{3} b^{3} x^{6} - 175 \, a^{4} b^{2} x^{4} - 42 \, a^{5} b x^{2} - 5 \, a^{6}}{35 \, x^{7}} \]

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^8,x, algorithm="fricas")

[Out]

1/35*(7*b^6*x^12 + 70*a*b^5*x^10 + 525*a^2*b^4*x^8 - 700*a^3*b^3*x^6 - 175*a^4*b^2*x^4 - 42*a^5*b*x^2 - 5*a^6)
/x^7

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giac [A]  time = 0.15, size = 69, normalized size = 0.96 \[ \frac {1}{5} \, b^{6} x^{5} + 2 \, a b^{5} x^{3} + 15 \, a^{2} b^{4} x - \frac {700 \, a^{3} b^{3} x^{6} + 175 \, a^{4} b^{2} x^{4} + 42 \, a^{5} b x^{2} + 5 \, a^{6}}{35 \, x^{7}} \]

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^8,x, algorithm="giac")

[Out]

1/5*b^6*x^5 + 2*a*b^5*x^3 + 15*a^2*b^4*x - 1/35*(700*a^3*b^3*x^6 + 175*a^4*b^2*x^4 + 42*a^5*b*x^2 + 5*a^6)/x^7

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maple [A]  time = 0.01, size = 67, normalized size = 0.93 \[ \frac {b^{6} x^{5}}{5}+2 a \,b^{5} x^{3}+15 a^{2} b^{4} x -\frac {20 a^{3} b^{3}}{x}-\frac {5 a^{4} b^{2}}{x^{3}}-\frac {6 a^{5} b}{5 x^{5}}-\frac {a^{6}}{7 x^{7}} \]

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^8,x)

[Out]

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

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maxima [A]  time = 1.38, size = 69, normalized size = 0.96 \[ \frac {1}{5} \, b^{6} x^{5} + 2 \, a b^{5} x^{3} + 15 \, a^{2} b^{4} x - \frac {700 \, a^{3} b^{3} x^{6} + 175 \, a^{4} b^{2} x^{4} + 42 \, a^{5} b x^{2} + 5 \, a^{6}}{35 \, x^{7}} \]

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^8,x, algorithm="maxima")

[Out]

1/5*b^6*x^5 + 2*a*b^5*x^3 + 15*a^2*b^4*x - 1/35*(700*a^3*b^3*x^6 + 175*a^4*b^2*x^4 + 42*a^5*b*x^2 + 5*a^6)/x^7

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mupad [B]  time = 0.05, size = 69, normalized size = 0.96 \[ \frac {b^6\,x^5}{5}-\frac {\frac {a^6}{7}+\frac {6\,a^5\,b\,x^2}{5}+5\,a^4\,b^2\,x^4+20\,a^3\,b^3\,x^6}{x^7}+15\,a^2\,b^4\,x+2\,a\,b^5\,x^3 \]

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^8,x)

[Out]

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

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sympy [A]  time = 0.34, size = 73, normalized size = 1.01 \[ 15 a^{2} b^{4} x + 2 a b^{5} x^{3} + \frac {b^{6} x^{5}}{5} + \frac {- 5 a^{6} - 42 a^{5} b x^{2} - 175 a^{4} b^{2} x^{4} - 700 a^{3} b^{3} x^{6}}{35 x^{7}} \]

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**8,x)

[Out]

15*a**2*b**4*x + 2*a*b**5*x**3 + b**6*x**5/5 + (-5*a**6 - 42*a**5*b*x**2 - 175*a**4*b**2*x**4 - 700*a**3*b**3*
x**6)/(35*x**7)

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