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

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

[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

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Rubi [A]  time = 0.0390975, antiderivative size = 100, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0093797, size = 100, normalized size = 1. \[ 4 a^2 b^6 x^7+\frac{56}{5} a^3 b^5 x^5+\frac{70}{3} a^4 b^4 x^3+56 a^5 b^3 x-\frac{28 a^6 b^2}{x}-\frac{8 a^7 b}{3 x^3}-\frac{a^8}{5 x^5}+\frac{8}{9} a b^7 x^9+\frac{b^8 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]

Integrate[(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

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Maple [A]  time = 0.005, size = 89, normalized size = 0.9 \begin{align*} -{\frac{{a}^{8}}{5\,{x}^{5}}}-{\frac{8\,{a}^{7}b}{3\,{x}^{3}}}-28\,{\frac{{a}^{6}{b}^{2}}{x}}+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}} \end{align*}

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 = 2.64936, size = 123, normalized size = 1.23 \begin{align*} \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}} \end{align*}

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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Fricas [A]  time = 1.17004, size = 227, normalized size = 2.27 \begin{align*} \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}} \end{align*}

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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Sympy [A]  time = 0.451726, size = 100, normalized size = 1. \begin{align*} 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}} \end{align*}

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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Giac [A]  time = 2.438, size = 123, normalized size = 1.23 \begin{align*} \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}} \end{align*}

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