3.96 \(\int \frac{(a+b x^2)^8}{x^9} \, dx\)

Optimal. Leaf size=97 \[ -\frac{7 a^6 b^2}{x^4}-\frac{28 a^5 b^3}{x^2}+28 a^3 b^5 x^2+7 a^2 b^6 x^4+70 a^4 b^4 \log (x)-\frac{4 a^7 b}{3 x^6}-\frac{a^8}{8 x^8}+\frac{4}{3} a b^7 x^6+\frac{b^8 x^8}{8} \]

[Out]

-a^8/(8*x^8) - (4*a^7*b)/(3*x^6) - (7*a^6*b^2)/x^4 - (28*a^5*b^3)/x^2 + 28*a^3*b^5*x^2 + 7*a^2*b^6*x^4 + (4*a*
b^7*x^6)/3 + (b^8*x^8)/8 + 70*a^4*b^4*Log[x]

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Rubi [A]  time = 0.0568999, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ -\frac{7 a^6 b^2}{x^4}-\frac{28 a^5 b^3}{x^2}+28 a^3 b^5 x^2+7 a^2 b^6 x^4+70 a^4 b^4 \log (x)-\frac{4 a^7 b}{3 x^6}-\frac{a^8}{8 x^8}+\frac{4}{3} a b^7 x^6+\frac{b^8 x^8}{8} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^8/x^9,x]

[Out]

-a^8/(8*x^8) - (4*a^7*b)/(3*x^6) - (7*a^6*b^2)/x^4 - (28*a^5*b^3)/x^2 + 28*a^3*b^5*x^2 + 7*a^2*b^6*x^4 + (4*a*
b^7*x^6)/3 + (b^8*x^8)/8 + 70*a^4*b^4*Log[x]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2\right )^8}{x^9} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^5} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (56 a^3 b^5+\frac{a^8}{x^5}+\frac{8 a^7 b}{x^4}+\frac{28 a^6 b^2}{x^3}+\frac{56 a^5 b^3}{x^2}+\frac{70 a^4 b^4}{x}+28 a^2 b^6 x+8 a b^7 x^2+b^8 x^3\right ) \, dx,x,x^2\right )\\ &=-\frac{a^8}{8 x^8}-\frac{4 a^7 b}{3 x^6}-\frac{7 a^6 b^2}{x^4}-\frac{28 a^5 b^3}{x^2}+28 a^3 b^5 x^2+7 a^2 b^6 x^4+\frac{4}{3} a b^7 x^6+\frac{b^8 x^8}{8}+70 a^4 b^4 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0046682, size = 97, normalized size = 1. \[ -\frac{7 a^6 b^2}{x^4}-\frac{28 a^5 b^3}{x^2}+28 a^3 b^5 x^2+7 a^2 b^6 x^4+70 a^4 b^4 \log (x)-\frac{4 a^7 b}{3 x^6}-\frac{a^8}{8 x^8}+\frac{4}{3} a b^7 x^6+\frac{b^8 x^8}{8} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^8/x^9,x]

[Out]

-a^8/(8*x^8) - (4*a^7*b)/(3*x^6) - (7*a^6*b^2)/x^4 - (28*a^5*b^3)/x^2 + 28*a^3*b^5*x^2 + 7*a^2*b^6*x^4 + (4*a*
b^7*x^6)/3 + (b^8*x^8)/8 + 70*a^4*b^4*Log[x]

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Maple [A]  time = 0.007, size = 90, normalized size = 0.9 \begin{align*} -{\frac{{a}^{8}}{8\,{x}^{8}}}-{\frac{4\,{a}^{7}b}{3\,{x}^{6}}}-7\,{\frac{{a}^{6}{b}^{2}}{{x}^{4}}}-28\,{\frac{{a}^{5}{b}^{3}}{{x}^{2}}}+28\,{a}^{3}{b}^{5}{x}^{2}+7\,{a}^{2}{b}^{6}{x}^{4}+{\frac{4\,a{b}^{7}{x}^{6}}{3}}+{\frac{{b}^{8}{x}^{8}}{8}}+70\,{a}^{4}{b}^{4}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^8/x^9,x)

[Out]

-1/8*a^8/x^8-4/3*a^7*b/x^6-7*a^6*b^2/x^4-28*a^5*b^3/x^2+28*a^3*b^5*x^2+7*a^2*b^6*x^4+4/3*a*b^7*x^6+1/8*b^8*x^8
+70*a^4*b^4*ln(x)

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Maxima [A]  time = 2.33408, size = 127, normalized size = 1.31 \begin{align*} \frac{1}{8} \, b^{8} x^{8} + \frac{4}{3} \, a b^{7} x^{6} + 7 \, a^{2} b^{6} x^{4} + 28 \, a^{3} b^{5} x^{2} + 35 \, a^{4} b^{4} \log \left (x^{2}\right ) - \frac{672 \, a^{5} b^{3} x^{6} + 168 \, a^{6} b^{2} x^{4} + 32 \, a^{7} b x^{2} + 3 \, a^{8}}{24 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^9,x, algorithm="maxima")

[Out]

1/8*b^8*x^8 + 4/3*a*b^7*x^6 + 7*a^2*b^6*x^4 + 28*a^3*b^5*x^2 + 35*a^4*b^4*log(x^2) - 1/24*(672*a^5*b^3*x^6 + 1
68*a^6*b^2*x^4 + 32*a^7*b*x^2 + 3*a^8)/x^8

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Fricas [A]  time = 1.17767, size = 219, normalized size = 2.26 \begin{align*} \frac{3 \, b^{8} x^{16} + 32 \, a b^{7} x^{14} + 168 \, a^{2} b^{6} x^{12} + 672 \, a^{3} b^{5} x^{10} + 1680 \, a^{4} b^{4} x^{8} \log \left (x\right ) - 672 \, a^{5} b^{3} x^{6} - 168 \, a^{6} b^{2} x^{4} - 32 \, a^{7} b x^{2} - 3 \, a^{8}}{24 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^9,x, algorithm="fricas")

[Out]

1/24*(3*b^8*x^16 + 32*a*b^7*x^14 + 168*a^2*b^6*x^12 + 672*a^3*b^5*x^10 + 1680*a^4*b^4*x^8*log(x) - 672*a^5*b^3
*x^6 - 168*a^6*b^2*x^4 - 32*a^7*b*x^2 - 3*a^8)/x^8

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Sympy [A]  time = 0.596116, size = 99, normalized size = 1.02 \begin{align*} 70 a^{4} b^{4} \log{\left (x \right )} + 28 a^{3} b^{5} x^{2} + 7 a^{2} b^{6} x^{4} + \frac{4 a b^{7} x^{6}}{3} + \frac{b^{8} x^{8}}{8} - \frac{3 a^{8} + 32 a^{7} b x^{2} + 168 a^{6} b^{2} x^{4} + 672 a^{5} b^{3} x^{6}}{24 x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**8/x**9,x)

[Out]

70*a**4*b**4*log(x) + 28*a**3*b**5*x**2 + 7*a**2*b**6*x**4 + 4*a*b**7*x**6/3 + b**8*x**8/8 - (3*a**8 + 32*a**7
*b*x**2 + 168*a**6*b**2*x**4 + 672*a**5*b**3*x**6)/(24*x**8)

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Giac [A]  time = 2.47352, size = 142, normalized size = 1.46 \begin{align*} \frac{1}{8} \, b^{8} x^{8} + \frac{4}{3} \, a b^{7} x^{6} + 7 \, a^{2} b^{6} x^{4} + 28 \, a^{3} b^{5} x^{2} + 35 \, a^{4} b^{4} \log \left (x^{2}\right ) - \frac{1750 \, a^{4} b^{4} x^{8} + 672 \, a^{5} b^{3} x^{6} + 168 \, a^{6} b^{2} x^{4} + 32 \, a^{7} b x^{2} + 3 \, a^{8}}{24 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^9,x, algorithm="giac")

[Out]

1/8*b^8*x^8 + 4/3*a*b^7*x^6 + 7*a^2*b^6*x^4 + 28*a^3*b^5*x^2 + 35*a^4*b^4*log(x^2) - 1/24*(1750*a^4*b^4*x^8 +
672*a^5*b^3*x^6 + 168*a^6*b^2*x^4 + 32*a^7*b*x^2 + 3*a^8)/x^8