3.118 \(\int \frac{(a+b x^2)^8}{x^{10}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0390688, 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} \[ -\frac{28 a^6 b^2}{5 x^5}-\frac{56 a^5 b^3}{3 x^3}+\frac{28}{3} a^2 b^6 x^3-\frac{70 a^4 b^4}{x}+56 a^3 b^5 x-\frac{8 a^7 b}{7 x^7}-\frac{a^8}{9 x^9}+\frac{8}{5} a b^7 x^5+\frac{b^8 x^7}{7} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^8/x^10,x]

[Out]

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

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

Mathematica [A]  time = 0.0096779, size = 102, normalized size = 1. \[ -\frac{28 a^6 b^2}{5 x^5}-\frac{56 a^5 b^3}{3 x^3}+\frac{28}{3} a^2 b^6 x^3-\frac{70 a^4 b^4}{x}+56 a^3 b^5 x-\frac{8 a^7 b}{7 x^7}-\frac{a^8}{9 x^9}+\frac{8}{5} a b^7 x^5+\frac{b^8 x^7}{7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^8/x^10,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^8/x^10,x)

[Out]

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

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Maxima [A]  time = 2.33814, size = 123, normalized size = 1.21 \begin{align*} \frac{1}{7} \, b^{8} x^{7} + \frac{8}{5} \, a b^{7} x^{5} + \frac{28}{3} \, a^{2} b^{6} x^{3} + 56 \, a^{3} b^{5} x - \frac{22050 \, a^{4} b^{4} x^{8} + 5880 \, a^{5} b^{3} x^{6} + 1764 \, a^{6} b^{2} x^{4} + 360 \, a^{7} b x^{2} + 35 \, a^{8}}{315 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^10,x, algorithm="maxima")

[Out]

1/7*b^8*x^7 + 8/5*a*b^7*x^5 + 28/3*a^2*b^6*x^3 + 56*a^3*b^5*x - 1/315*(22050*a^4*b^4*x^8 + 5880*a^5*b^3*x^6 +
1764*a^6*b^2*x^4 + 360*a^7*b*x^2 + 35*a^8)/x^9

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Fricas [A]  time = 1.19949, size = 224, normalized size = 2.2 \begin{align*} \frac{45 \, b^{8} x^{16} + 504 \, a b^{7} x^{14} + 2940 \, a^{2} b^{6} x^{12} + 17640 \, a^{3} b^{5} x^{10} - 22050 \, a^{4} b^{4} x^{8} - 5880 \, a^{5} b^{3} x^{6} - 1764 \, a^{6} b^{2} x^{4} - 360 \, a^{7} b x^{2} - 35 \, a^{8}}{315 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^10,x, algorithm="fricas")

[Out]

1/315*(45*b^8*x^16 + 504*a*b^7*x^14 + 2940*a^2*b^6*x^12 + 17640*a^3*b^5*x^10 - 22050*a^4*b^4*x^8 - 5880*a^5*b^
3*x^6 - 1764*a^6*b^2*x^4 - 360*a^7*b*x^2 - 35*a^8)/x^9

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Sympy [A]  time = 0.602902, size = 99, normalized size = 0.97 \begin{align*} 56 a^{3} b^{5} x + \frac{28 a^{2} b^{6} x^{3}}{3} + \frac{8 a b^{7} x^{5}}{5} + \frac{b^{8} x^{7}}{7} - \frac{35 a^{8} + 360 a^{7} b x^{2} + 1764 a^{6} b^{2} x^{4} + 5880 a^{5} b^{3} x^{6} + 22050 a^{4} b^{4} x^{8}}{315 x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**8/x**10,x)

[Out]

56*a**3*b**5*x + 28*a**2*b**6*x**3/3 + 8*a*b**7*x**5/5 + b**8*x**7/7 - (35*a**8 + 360*a**7*b*x**2 + 1764*a**6*
b**2*x**4 + 5880*a**5*b**3*x**6 + 22050*a**4*b**4*x**8)/(315*x**9)

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Giac [A]  time = 1.94556, size = 123, normalized size = 1.21 \begin{align*} \frac{1}{7} \, b^{8} x^{7} + \frac{8}{5} \, a b^{7} x^{5} + \frac{28}{3} \, a^{2} b^{6} x^{3} + 56 \, a^{3} b^{5} x - \frac{22050 \, a^{4} b^{4} x^{8} + 5880 \, a^{5} b^{3} x^{6} + 1764 \, a^{6} b^{2} x^{4} + 360 \, a^{7} b x^{2} + 35 \, a^{8}}{315 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^10,x, algorithm="giac")

[Out]

1/7*b^8*x^7 + 8/5*a*b^7*x^5 + 28/3*a^2*b^6*x^3 + 56*a^3*b^5*x - 1/315*(22050*a^4*b^4*x^8 + 5880*a^5*b^3*x^6 +
1764*a^6*b^2*x^4 + 360*a^7*b*x^2 + 35*a^8)/x^9