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

Optimal. Leaf size=102 \[ \frac{28}{5} a^2 b^6 x^5+\frac{56}{3} a^3 b^5 x^3-\frac{28 a^6 b^2}{3 x^3}+70 a^4 b^4 x-\frac{56 a^5 b^3}{x}-\frac{8 a^7 b}{5 x^5}-\frac{a^8}{7 x^7}+\frac{8}{7} a b^7 x^7+\frac{b^8 x^9}{9} \]

[Out]

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

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Rubi [A]  time = 0.036847, antiderivative size = 102, 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} \[ \frac{28}{5} a^2 b^6 x^5+\frac{56}{3} a^3 b^5 x^3-\frac{28 a^6 b^2}{3 x^3}+70 a^4 b^4 x-\frac{56 a^5 b^3}{x}-\frac{8 a^7 b}{5 x^5}-\frac{a^8}{7 x^7}+\frac{8}{7} a b^7 x^7+\frac{b^8 x^9}{9} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^8/x^8,x]

[Out]

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

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

Mathematica [A]  time = 0.0058527, size = 102, normalized size = 1. \[ \frac{28}{5} a^2 b^6 x^5+\frac{56}{3} a^3 b^5 x^3-\frac{28 a^6 b^2}{3 x^3}+70 a^4 b^4 x-\frac{56 a^5 b^3}{x}-\frac{8 a^7 b}{5 x^5}-\frac{a^8}{7 x^7}+\frac{8}{7} a b^7 x^7+\frac{b^8 x^9}{9} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^8/x^8,x]

[Out]

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

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Maple [A]  time = 0.005, size = 89, normalized size = 0.9 \begin{align*} -{\frac{{a}^{8}}{7\,{x}^{7}}}-{\frac{8\,{a}^{7}b}{5\,{x}^{5}}}-{\frac{28\,{a}^{6}{b}^{2}}{3\,{x}^{3}}}-56\,{\frac{{a}^{5}{b}^{3}}{x}}+70\,{a}^{4}{b}^{4}x+{\frac{56\,{a}^{3}{b}^{5}{x}^{3}}{3}}+{\frac{28\,{a}^{2}{b}^{6}{x}^{5}}{5}}+{\frac{8\,a{b}^{7}{x}^{7}}{7}}+{\frac{{b}^{8}{x}^{9}}{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.42532, size = 123, normalized size = 1.21 \begin{align*} \frac{1}{9} \, b^{8} x^{9} + \frac{8}{7} \, a b^{7} x^{7} + \frac{28}{5} \, a^{2} b^{6} x^{5} + \frac{56}{3} \, a^{3} b^{5} x^{3} + 70 \, a^{4} b^{4} x - \frac{5880 \, a^{5} b^{3} x^{6} + 980 \, a^{6} b^{2} x^{4} + 168 \, a^{7} b x^{2} + 15 \, a^{8}}{105 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*b^8*x^9 + 8/7*a*b^7*x^7 + 28/5*a^2*b^6*x^5 + 56/3*a^3*b^5*x^3 + 70*a^4*b^4*x - 1/105*(5880*a^5*b^3*x^6 + 9
80*a^6*b^2*x^4 + 168*a^7*b*x^2 + 15*a^8)/x^7

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Fricas [A]  time = 1.2508, size = 224, normalized size = 2.2 \begin{align*} \frac{35 \, b^{8} x^{16} + 360 \, a b^{7} x^{14} + 1764 \, a^{2} b^{6} x^{12} + 5880 \, a^{3} b^{5} x^{10} + 22050 \, a^{4} b^{4} x^{8} - 17640 \, a^{5} b^{3} x^{6} - 2940 \, a^{6} b^{2} x^{4} - 504 \, a^{7} b x^{2} - 45 \, a^{8}}{315 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/315*(35*b^8*x^16 + 360*a*b^7*x^14 + 1764*a^2*b^6*x^12 + 5880*a^3*b^5*x^10 + 22050*a^4*b^4*x^8 - 17640*a^5*b^
3*x^6 - 2940*a^6*b^2*x^4 - 504*a^7*b*x^2 - 45*a^8)/x^7

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Sympy [A]  time = 0.511301, size = 100, normalized size = 0.98 \begin{align*} 70 a^{4} b^{4} x + \frac{56 a^{3} b^{5} x^{3}}{3} + \frac{28 a^{2} b^{6} x^{5}}{5} + \frac{8 a b^{7} x^{7}}{7} + \frac{b^{8} x^{9}}{9} - \frac{15 a^{8} + 168 a^{7} b x^{2} + 980 a^{6} b^{2} x^{4} + 5880 a^{5} b^{3} x^{6}}{105 x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

70*a**4*b**4*x + 56*a**3*b**5*x**3/3 + 28*a**2*b**6*x**5/5 + 8*a*b**7*x**7/7 + b**8*x**9/9 - (15*a**8 + 168*a*
*7*b*x**2 + 980*a**6*b**2*x**4 + 5880*a**5*b**3*x**6)/(105*x**7)

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Giac [A]  time = 2.62742, size = 123, normalized size = 1.21 \begin{align*} \frac{1}{9} \, b^{8} x^{9} + \frac{8}{7} \, a b^{7} x^{7} + \frac{28}{5} \, a^{2} b^{6} x^{5} + \frac{56}{3} \, a^{3} b^{5} x^{3} + 70 \, a^{4} b^{4} x - \frac{5880 \, a^{5} b^{3} x^{6} + 980 \, a^{6} b^{2} x^{4} + 168 \, a^{7} b x^{2} + 15 \, a^{8}}{105 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*b^8*x^9 + 8/7*a*b^7*x^7 + 28/5*a^2*b^6*x^5 + 56/3*a^3*b^5*x^3 + 70*a^4*b^4*x - 1/105*(5880*a^5*b^3*x^6 + 9
80*a^6*b^2*x^4 + 168*a^7*b*x^2 + 15*a^8)/x^7