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

Optimal. Leaf size=98 \[ \frac{28}{9} a^2 b^6 x^9+8 a^3 b^5 x^7+14 a^4 b^4 x^5+\frac{56}{3} a^5 b^3 x^3+28 a^6 b^2 x-\frac{8 a^7 b}{x}-\frac{a^8}{3 x^3}+\frac{8}{11} a b^7 x^{11}+\frac{b^8 x^{13}}{13} \]

[Out]

-a^8/(3*x^3) - (8*a^7*b)/x + 28*a^6*b^2*x + (56*a^5*b^3*x^3)/3 + 14*a^4*b^4*x^5 + 8*a^3*b^5*x^7 + (28*a^2*b^6*
x^9)/9 + (8*a*b^7*x^11)/11 + (b^8*x^13)/13

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Rubi [A]  time = 0.0379941, antiderivative size = 98, 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} \[ \frac{28}{9} a^2 b^6 x^9+8 a^3 b^5 x^7+14 a^4 b^4 x^5+\frac{56}{3} a^5 b^3 x^3+28 a^6 b^2 x-\frac{8 a^7 b}{x}-\frac{a^8}{3 x^3}+\frac{8}{11} a b^7 x^{11}+\frac{b^8 x^{13}}{13} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^8/x^4,x]

[Out]

-a^8/(3*x^3) - (8*a^7*b)/x + 28*a^6*b^2*x + (56*a^5*b^3*x^3)/3 + 14*a^4*b^4*x^5 + 8*a^3*b^5*x^7 + (28*a^2*b^6*
x^9)/9 + (8*a*b^7*x^11)/11 + (b^8*x^13)/13

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^4} \, dx &=\int \left (28 a^6 b^2+\frac{a^8}{x^4}+\frac{8 a^7 b}{x^2}+56 a^5 b^3 x^2+70 a^4 b^4 x^4+56 a^3 b^5 x^6+28 a^2 b^6 x^8+8 a b^7 x^{10}+b^8 x^{12}\right ) \, dx\\ &=-\frac{a^8}{3 x^3}-\frac{8 a^7 b}{x}+28 a^6 b^2 x+\frac{56}{3} a^5 b^3 x^3+14 a^4 b^4 x^5+8 a^3 b^5 x^7+\frac{28}{9} a^2 b^6 x^9+\frac{8}{11} a b^7 x^{11}+\frac{b^8 x^{13}}{13}\\ \end{align*}

Mathematica [A]  time = 0.0077405, size = 98, normalized size = 1. \[ \frac{28}{9} a^2 b^6 x^9+8 a^3 b^5 x^7+14 a^4 b^4 x^5+\frac{56}{3} a^5 b^3 x^3+28 a^6 b^2 x-\frac{8 a^7 b}{x}-\frac{a^8}{3 x^3}+\frac{8}{11} a b^7 x^{11}+\frac{b^8 x^{13}}{13} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^8/x^4,x]

[Out]

-a^8/(3*x^3) - (8*a^7*b)/x + 28*a^6*b^2*x + (56*a^5*b^3*x^3)/3 + 14*a^4*b^4*x^5 + 8*a^3*b^5*x^7 + (28*a^2*b^6*
x^9)/9 + (8*a*b^7*x^11)/11 + (b^8*x^13)/13

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Maple [A]  time = 0.005, size = 89, normalized size = 0.9 \begin{align*} -{\frac{{a}^{8}}{3\,{x}^{3}}}-8\,{\frac{{a}^{7}b}{x}}+28\,{a}^{6}{b}^{2}x+{\frac{56\,{a}^{5}{b}^{3}{x}^{3}}{3}}+14\,{a}^{4}{b}^{4}{x}^{5}+8\,{a}^{3}{b}^{5}{x}^{7}+{\frac{28\,{a}^{2}{b}^{6}{x}^{9}}{9}}+{\frac{8\,a{b}^{7}{x}^{11}}{11}}+{\frac{{b}^{8}{x}^{13}}{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^8/x^4,x)

[Out]

-1/3*a^8/x^3-8*a^7*b/x+28*a^6*b^2*x+56/3*a^5*b^3*x^3+14*a^4*b^4*x^5+8*a^3*b^5*x^7+28/9*a^2*b^6*x^9+8/11*a*b^7*
x^11+1/13*b^8*x^13

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Maxima [A]  time = 1.96558, size = 120, normalized size = 1.22 \begin{align*} \frac{1}{13} \, b^{8} x^{13} + \frac{8}{11} \, a b^{7} x^{11} + \frac{28}{9} \, a^{2} b^{6} x^{9} + 8 \, a^{3} b^{5} x^{7} + 14 \, a^{4} b^{4} x^{5} + \frac{56}{3} \, a^{5} b^{3} x^{3} + 28 \, a^{6} b^{2} x - \frac{24 \, a^{7} b x^{2} + a^{8}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^4,x, algorithm="maxima")

[Out]

1/13*b^8*x^13 + 8/11*a*b^7*x^11 + 28/9*a^2*b^6*x^9 + 8*a^3*b^5*x^7 + 14*a^4*b^4*x^5 + 56/3*a^5*b^3*x^3 + 28*a^
6*b^2*x - 1/3*(24*a^7*b*x^2 + a^8)/x^3

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Fricas [A]  time = 1.25563, size = 232, normalized size = 2.37 \begin{align*} \frac{99 \, b^{8} x^{16} + 936 \, a b^{7} x^{14} + 4004 \, a^{2} b^{6} x^{12} + 10296 \, a^{3} b^{5} x^{10} + 18018 \, a^{4} b^{4} x^{8} + 24024 \, a^{5} b^{3} x^{6} + 36036 \, a^{6} b^{2} x^{4} - 10296 \, a^{7} b x^{2} - 429 \, a^{8}}{1287 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^4,x, algorithm="fricas")

[Out]

1/1287*(99*b^8*x^16 + 936*a*b^7*x^14 + 4004*a^2*b^6*x^12 + 10296*a^3*b^5*x^10 + 18018*a^4*b^4*x^8 + 24024*a^5*
b^3*x^6 + 36036*a^6*b^2*x^4 - 10296*a^7*b*x^2 - 429*a^8)/x^3

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Sympy [A]  time = 0.385195, size = 99, normalized size = 1.01 \begin{align*} 28 a^{6} b^{2} x + \frac{56 a^{5} b^{3} x^{3}}{3} + 14 a^{4} b^{4} x^{5} + 8 a^{3} b^{5} x^{7} + \frac{28 a^{2} b^{6} x^{9}}{9} + \frac{8 a b^{7} x^{11}}{11} + \frac{b^{8} x^{13}}{13} - \frac{a^{8} + 24 a^{7} b x^{2}}{3 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**8/x**4,x)

[Out]

28*a**6*b**2*x + 56*a**5*b**3*x**3/3 + 14*a**4*b**4*x**5 + 8*a**3*b**5*x**7 + 28*a**2*b**6*x**9/9 + 8*a*b**7*x
**11/11 + b**8*x**13/13 - (a**8 + 24*a**7*b*x**2)/(3*x**3)

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Giac [A]  time = 1.91878, size = 120, normalized size = 1.22 \begin{align*} \frac{1}{13} \, b^{8} x^{13} + \frac{8}{11} \, a b^{7} x^{11} + \frac{28}{9} \, a^{2} b^{6} x^{9} + 8 \, a^{3} b^{5} x^{7} + 14 \, a^{4} b^{4} x^{5} + \frac{56}{3} \, a^{5} b^{3} x^{3} + 28 \, a^{6} b^{2} x - \frac{24 \, a^{7} b x^{2} + a^{8}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^4,x, algorithm="giac")

[Out]

1/13*b^8*x^13 + 8/11*a*b^7*x^11 + 28/9*a^2*b^6*x^9 + 8*a^3*b^5*x^7 + 14*a^4*b^4*x^5 + 56/3*a^5*b^3*x^3 + 28*a^
6*b^2*x - 1/3*(24*a^7*b*x^2 + a^8)/x^3