3.120 \(\int \frac{(a+b x^2)^8}{x^{14}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0377404, 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} \[ -\frac{28 a^6 b^2}{9 x^9}-\frac{8 a^5 b^3}{x^7}-\frac{14 a^4 b^4}{x^5}-\frac{56 a^3 b^5}{3 x^3}-\frac{28 a^2 b^6}{x}-\frac{8 a^7 b}{11 x^{11}}-\frac{a^8}{13 x^{13}}+8 a b^7 x+\frac{b^8 x^3}{3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^8/x^14,x]

[Out]

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

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

Mathematica [A]  time = 0.0097264, size = 98, normalized size = 1. \[ -\frac{28 a^6 b^2}{9 x^9}-\frac{8 a^5 b^3}{x^7}-\frac{14 a^4 b^4}{x^5}-\frac{56 a^3 b^5}{3 x^3}-\frac{28 a^2 b^6}{x}-\frac{8 a^7 b}{11 x^{11}}-\frac{a^8}{13 x^{13}}+8 a b^7 x+\frac{b^8 x^3}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^8/x^14,x]

[Out]

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

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Maple [A]  time = 0.008, size = 89, normalized size = 0.9 \begin{align*} -{\frac{{a}^{8}}{13\,{x}^{13}}}-{\frac{8\,{a}^{7}b}{11\,{x}^{11}}}-{\frac{28\,{a}^{6}{b}^{2}}{9\,{x}^{9}}}-8\,{\frac{{a}^{5}{b}^{3}}{{x}^{7}}}-14\,{\frac{{a}^{4}{b}^{4}}{{x}^{5}}}-{\frac{56\,{a}^{3}{b}^{5}}{3\,{x}^{3}}}-28\,{\frac{{a}^{2}{b}^{6}}{x}}+8\,a{b}^{7}x+{\frac{{b}^{8}{x}^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^8/x^14,x)

[Out]

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

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Maxima [A]  time = 2.71249, size = 123, normalized size = 1.26 \begin{align*} \frac{1}{3} \, b^{8} x^{3} + 8 \, a b^{7} x - \frac{36036 \, a^{2} b^{6} x^{12} + 24024 \, a^{3} b^{5} x^{10} + 18018 \, a^{4} b^{4} x^{8} + 10296 \, a^{5} b^{3} x^{6} + 4004 \, a^{6} b^{2} x^{4} + 936 \, a^{7} b x^{2} + 99 \, a^{8}}{1287 \, x^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^14,x, algorithm="maxima")

[Out]

1/3*b^8*x^3 + 8*a*b^7*x - 1/1287*(36036*a^2*b^6*x^12 + 24024*a^3*b^5*x^10 + 18018*a^4*b^4*x^8 + 10296*a^5*b^3*
x^6 + 4004*a^6*b^2*x^4 + 936*a^7*b*x^2 + 99*a^8)/x^13

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Fricas [A]  time = 1.12773, size = 234, normalized size = 2.39 \begin{align*} \frac{429 \, b^{8} x^{16} + 10296 \, a b^{7} x^{14} - 36036 \, a^{2} b^{6} x^{12} - 24024 \, a^{3} b^{5} x^{10} - 18018 \, a^{4} b^{4} x^{8} - 10296 \, a^{5} b^{3} x^{6} - 4004 \, a^{6} b^{2} x^{4} - 936 \, a^{7} b x^{2} - 99 \, a^{8}}{1287 \, x^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^14,x, algorithm="fricas")

[Out]

1/1287*(429*b^8*x^16 + 10296*a*b^7*x^14 - 36036*a^2*b^6*x^12 - 24024*a^3*b^5*x^10 - 18018*a^4*b^4*x^8 - 10296*
a^5*b^3*x^6 - 4004*a^6*b^2*x^4 - 936*a^7*b*x^2 - 99*a^8)/x^13

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Sympy [A]  time = 0.888839, size = 95, normalized size = 0.97 \begin{align*} 8 a b^{7} x + \frac{b^{8} x^{3}}{3} - \frac{99 a^{8} + 936 a^{7} b x^{2} + 4004 a^{6} b^{2} x^{4} + 10296 a^{5} b^{3} x^{6} + 18018 a^{4} b^{4} x^{8} + 24024 a^{3} b^{5} x^{10} + 36036 a^{2} b^{6} x^{12}}{1287 x^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**8/x**14,x)

[Out]

8*a*b**7*x + b**8*x**3/3 - (99*a**8 + 936*a**7*b*x**2 + 4004*a**6*b**2*x**4 + 10296*a**5*b**3*x**6 + 18018*a**
4*b**4*x**8 + 24024*a**3*b**5*x**10 + 36036*a**2*b**6*x**12)/(1287*x**13)

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Giac [A]  time = 2.13636, size = 123, normalized size = 1.26 \begin{align*} \frac{1}{3} \, b^{8} x^{3} + 8 \, a b^{7} x - \frac{36036 \, a^{2} b^{6} x^{12} + 24024 \, a^{3} b^{5} x^{10} + 18018 \, a^{4} b^{4} x^{8} + 10296 \, a^{5} b^{3} x^{6} + 4004 \, a^{6} b^{2} x^{4} + 936 \, a^{7} b x^{2} + 99 \, a^{8}}{1287 \, x^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^14,x, algorithm="giac")

[Out]

1/3*b^8*x^3 + 8*a*b^7*x - 1/1287*(36036*a^2*b^6*x^12 + 24024*a^3*b^5*x^10 + 18018*a^4*b^4*x^8 + 10296*a^5*b^3*
x^6 + 4004*a^6*b^2*x^4 + 936*a^7*b*x^2 + 99*a^8)/x^13