3.116 \(\int \frac {(a+b x^2)^8}{x^6} \, dx\)

Optimal. Leaf size=100 \[ -\frac {a^8}{5 x^5}-\frac {8 a^7 b}{3 x^3}-\frac {28 a^6 b^2}{x}+56 a^5 b^3 x+\frac {70}{3} a^4 b^4 x^3+\frac {56}{5} a^3 b^5 x^5+4 a^2 b^6 x^7+\frac {8}{9} a b^7 x^9+\frac {b^8 x^{11}}{11} \]

[Out]

-1/5*a^8/x^5-8/3*a^7*b/x^3-28*a^6*b^2/x+56*a^5*b^3*x+70/3*a^4*b^4*x^3+56/5*a^3*b^5*x^5+4*a^2*b^6*x^7+8/9*a*b^7
*x^9+1/11*b^8*x^11

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Rubi [A]  time = 0.04, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {270} \[ 4 a^2 b^6 x^7+\frac {56}{5} a^3 b^5 x^5+\frac {70}{3} a^4 b^4 x^3+56 a^5 b^3 x-\frac {28 a^6 b^2}{x}-\frac {8 a^7 b}{3 x^3}-\frac {a^8}{5 x^5}+\frac {8}{9} a b^7 x^9+\frac {b^8 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^8/x^6,x]

[Out]

-a^8/(5*x^5) - (8*a^7*b)/(3*x^3) - (28*a^6*b^2)/x + 56*a^5*b^3*x + (70*a^4*b^4*x^3)/3 + (56*a^3*b^5*x^5)/5 + 4
*a^2*b^6*x^7 + (8*a*b^7*x^9)/9 + (b^8*x^11)/11

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+b x^2\right )^8}{x^6} \, dx &=\int \left (56 a^5 b^3+\frac {a^8}{x^6}+\frac {8 a^7 b}{x^4}+\frac {28 a^6 b^2}{x^2}+70 a^4 b^4 x^2+56 a^3 b^5 x^4+28 a^2 b^6 x^6+8 a b^7 x^8+b^8 x^{10}\right ) \, dx\\ &=-\frac {a^8}{5 x^5}-\frac {8 a^7 b}{3 x^3}-\frac {28 a^6 b^2}{x}+56 a^5 b^3 x+\frac {70}{3} a^4 b^4 x^3+\frac {56}{5} a^3 b^5 x^5+4 a^2 b^6 x^7+\frac {8}{9} a b^7 x^9+\frac {b^8 x^{11}}{11}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 100, normalized size = 1.00 \[ -\frac {a^8}{5 x^5}-\frac {8 a^7 b}{3 x^3}-\frac {28 a^6 b^2}{x}+56 a^5 b^3 x+\frac {70}{3} a^4 b^4 x^3+\frac {56}{5} a^3 b^5 x^5+4 a^2 b^6 x^7+\frac {8}{9} a b^7 x^9+\frac {b^8 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^8/x^6,x]

[Out]

-1/5*a^8/x^5 - (8*a^7*b)/(3*x^3) - (28*a^6*b^2)/x + 56*a^5*b^3*x + (70*a^4*b^4*x^3)/3 + (56*a^3*b^5*x^5)/5 + 4
*a^2*b^6*x^7 + (8*a*b^7*x^9)/9 + (b^8*x^11)/11

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fricas [A]  time = 0.62, size = 92, normalized size = 0.92 \[ \frac {45 \, b^{8} x^{16} + 440 \, a b^{7} x^{14} + 1980 \, a^{2} b^{6} x^{12} + 5544 \, a^{3} b^{5} x^{10} + 11550 \, a^{4} b^{4} x^{8} + 27720 \, a^{5} b^{3} x^{6} - 13860 \, a^{6} b^{2} x^{4} - 1320 \, a^{7} b x^{2} - 99 \, a^{8}}{495 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^6,x, algorithm="fricas")

[Out]

1/495*(45*b^8*x^16 + 440*a*b^7*x^14 + 1980*a^2*b^6*x^12 + 5544*a^3*b^5*x^10 + 11550*a^4*b^4*x^8 + 27720*a^5*b^
3*x^6 - 13860*a^6*b^2*x^4 - 1320*a^7*b*x^2 - 99*a^8)/x^5

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giac [A]  time = 0.97, size = 91, normalized size = 0.91 \[ \frac {1}{11} \, b^{8} x^{11} + \frac {8}{9} \, a b^{7} x^{9} + 4 \, a^{2} b^{6} x^{7} + \frac {56}{5} \, a^{3} b^{5} x^{5} + \frac {70}{3} \, a^{4} b^{4} x^{3} + 56 \, a^{5} b^{3} x - \frac {420 \, a^{6} b^{2} x^{4} + 40 \, a^{7} b x^{2} + 3 \, a^{8}}{15 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^6,x, algorithm="giac")

[Out]

1/11*b^8*x^11 + 8/9*a*b^7*x^9 + 4*a^2*b^6*x^7 + 56/5*a^3*b^5*x^5 + 70/3*a^4*b^4*x^3 + 56*a^5*b^3*x - 1/15*(420
*a^6*b^2*x^4 + 40*a^7*b*x^2 + 3*a^8)/x^5

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maple [A]  time = 0.01, size = 89, normalized size = 0.89 \[ \frac {b^{8} x^{11}}{11}+\frac {8 a \,b^{7} x^{9}}{9}+4 a^{2} b^{6} x^{7}+\frac {56 a^{3} b^{5} x^{5}}{5}+\frac {70 a^{4} b^{4} x^{3}}{3}+56 a^{5} b^{3} x -\frac {28 a^{6} b^{2}}{x}-\frac {8 a^{7} b}{3 x^{3}}-\frac {a^{8}}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^8/x^6,x)

[Out]

-1/5*a^8/x^5-8/3*a^7*b/x^3-28*a^6*b^2/x+56*a^5*b^3*x+70/3*a^4*b^4*x^3+56/5*a^3*b^5*x^5+4*a^2*b^6*x^7+8/9*a*b^7
*x^9+1/11*b^8*x^11

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maxima [A]  time = 1.21, size = 91, normalized size = 0.91 \[ \frac {1}{11} \, b^{8} x^{11} + \frac {8}{9} \, a b^{7} x^{9} + 4 \, a^{2} b^{6} x^{7} + \frac {56}{5} \, a^{3} b^{5} x^{5} + \frac {70}{3} \, a^{4} b^{4} x^{3} + 56 \, a^{5} b^{3} x - \frac {420 \, a^{6} b^{2} x^{4} + 40 \, a^{7} b x^{2} + 3 \, a^{8}}{15 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^6,x, algorithm="maxima")

[Out]

1/11*b^8*x^11 + 8/9*a*b^7*x^9 + 4*a^2*b^6*x^7 + 56/5*a^3*b^5*x^5 + 70/3*a^4*b^4*x^3 + 56*a^5*b^3*x - 1/15*(420
*a^6*b^2*x^4 + 40*a^7*b*x^2 + 3*a^8)/x^5

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mupad [B]  time = 4.96, size = 91, normalized size = 0.91 \[ \frac {b^8\,x^{11}}{11}-\frac {\frac {a^8}{5}+\frac {8\,a^7\,b\,x^2}{3}+28\,a^6\,b^2\,x^4}{x^5}+56\,a^5\,b^3\,x+\frac {8\,a\,b^7\,x^9}{9}+\frac {70\,a^4\,b^4\,x^3}{3}+\frac {56\,a^3\,b^5\,x^5}{5}+4\,a^2\,b^6\,x^7 \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^8/x^6,x)

[Out]

(b^8*x^11)/11 - (a^8/5 + (8*a^7*b*x^2)/3 + 28*a^6*b^2*x^4)/x^5 + 56*a^5*b^3*x + (8*a*b^7*x^9)/9 + (70*a^4*b^4*
x^3)/3 + (56*a^3*b^5*x^5)/5 + 4*a^2*b^6*x^7

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sympy [A]  time = 0.28, size = 102, normalized size = 1.02 \[ 56 a^{5} b^{3} x + \frac {70 a^{4} b^{4} x^{3}}{3} + \frac {56 a^{3} b^{5} x^{5}}{5} + 4 a^{2} b^{6} x^{7} + \frac {8 a b^{7} x^{9}}{9} + \frac {b^{8} x^{11}}{11} + \frac {- 3 a^{8} - 40 a^{7} b x^{2} - 420 a^{6} b^{2} x^{4}}{15 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**8/x**6,x)

[Out]

56*a**5*b**3*x + 70*a**4*b**4*x**3/3 + 56*a**3*b**5*x**5/5 + 4*a**2*b**6*x**7 + 8*a*b**7*x**9/9 + b**8*x**11/1
1 + (-3*a**8 - 40*a**7*b*x**2 - 420*a**6*b**2*x**4)/(15*x**5)

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