3.101 \(\int \frac{(a+b x^2)^8}{x^{19}} \, dx\)

Optimal. Leaf size=19 \[ -\frac{\left (a+b x^2\right )^9}{18 a x^{18}} \]

[Out]

-(a + b*x^2)^9/(18*a*x^18)

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Rubi [A]  time = 0.0031426, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {264} \[ -\frac{\left (a+b x^2\right )^9}{18 a x^{18}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^8/x^19,x]

[Out]

-(a + b*x^2)^9/(18*a*x^18)

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2\right )^8}{x^{19}} \, dx &=-\frac{\left (a+b x^2\right )^9}{18 a x^{18}}\\ \end{align*}

Mathematica [B]  time = 0.0045611, size = 100, normalized size = 5.26 \[ -\frac{2 a^6 b^2}{x^{14}}-\frac{14 a^5 b^3}{3 x^{12}}-\frac{7 a^4 b^4}{x^{10}}-\frac{7 a^3 b^5}{x^8}-\frac{14 a^2 b^6}{3 x^6}-\frac{a^7 b}{2 x^{16}}-\frac{a^8}{18 x^{18}}-\frac{2 a b^7}{x^4}-\frac{b^8}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^8/x^19,x]

[Out]

-a^8/(18*x^18) - (a^7*b)/(2*x^16) - (2*a^6*b^2)/x^14 - (14*a^5*b^3)/(3*x^12) - (7*a^4*b^4)/x^10 - (7*a^3*b^5)/
x^8 - (14*a^2*b^6)/(3*x^6) - (2*a*b^7)/x^4 - b^8/(2*x^2)

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Maple [B]  time = 0.006, size = 91, normalized size = 4.8 \begin{align*} -7\,{\frac{{a}^{3}{b}^{5}}{{x}^{8}}}-7\,{\frac{{a}^{4}{b}^{4}}{{x}^{10}}}-2\,{\frac{a{b}^{7}}{{x}^{4}}}-{\frac{{a}^{8}}{18\,{x}^{18}}}-{\frac{14\,{a}^{5}{b}^{3}}{3\,{x}^{12}}}-{\frac{{b}^{8}}{2\,{x}^{2}}}-{\frac{14\,{a}^{2}{b}^{6}}{3\,{x}^{6}}}-2\,{\frac{{a}^{6}{b}^{2}}{{x}^{14}}}-{\frac{{a}^{7}b}{2\,{x}^{16}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^8/x^19,x)

[Out]

-7*a^3*b^5/x^8-7*a^4*b^4/x^10-2*a*b^7/x^4-1/18*a^8/x^18-14/3*a^5*b^3/x^12-1/2*b^8/x^2-14/3*a^2*b^6/x^6-2*a^6*b
^2/x^14-1/2*a^7*b/x^16

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Maxima [B]  time = 2.8204, size = 122, normalized size = 6.42 \begin{align*} -\frac{9 \, b^{8} x^{16} + 36 \, a b^{7} x^{14} + 84 \, a^{2} b^{6} x^{12} + 126 \, a^{3} b^{5} x^{10} + 126 \, a^{4} b^{4} x^{8} + 84 \, a^{5} b^{3} x^{6} + 36 \, a^{6} b^{2} x^{4} + 9 \, a^{7} b x^{2} + a^{8}}{18 \, x^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^19,x, algorithm="maxima")

[Out]

-1/18*(9*b^8*x^16 + 36*a*b^7*x^14 + 84*a^2*b^6*x^12 + 126*a^3*b^5*x^10 + 126*a^4*b^4*x^8 + 84*a^5*b^3*x^6 + 36
*a^6*b^2*x^4 + 9*a^7*b*x^2 + a^8)/x^18

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Fricas [B]  time = 1.23415, size = 203, normalized size = 10.68 \begin{align*} -\frac{9 \, b^{8} x^{16} + 36 \, a b^{7} x^{14} + 84 \, a^{2} b^{6} x^{12} + 126 \, a^{3} b^{5} x^{10} + 126 \, a^{4} b^{4} x^{8} + 84 \, a^{5} b^{3} x^{6} + 36 \, a^{6} b^{2} x^{4} + 9 \, a^{7} b x^{2} + a^{8}}{18 \, x^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^19,x, algorithm="fricas")

[Out]

-1/18*(9*b^8*x^16 + 36*a*b^7*x^14 + 84*a^2*b^6*x^12 + 126*a^3*b^5*x^10 + 126*a^4*b^4*x^8 + 84*a^5*b^3*x^6 + 36
*a^6*b^2*x^4 + 9*a^7*b*x^2 + a^8)/x^18

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Sympy [B]  time = 1.13006, size = 97, normalized size = 5.11 \begin{align*} - \frac{a^{8} + 9 a^{7} b x^{2} + 36 a^{6} b^{2} x^{4} + 84 a^{5} b^{3} x^{6} + 126 a^{4} b^{4} x^{8} + 126 a^{3} b^{5} x^{10} + 84 a^{2} b^{6} x^{12} + 36 a b^{7} x^{14} + 9 b^{8} x^{16}}{18 x^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**8/x**19,x)

[Out]

-(a**8 + 9*a**7*b*x**2 + 36*a**6*b**2*x**4 + 84*a**5*b**3*x**6 + 126*a**4*b**4*x**8 + 126*a**3*b**5*x**10 + 84
*a**2*b**6*x**12 + 36*a*b**7*x**14 + 9*b**8*x**16)/(18*x**18)

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Giac [B]  time = 2.09674, size = 122, normalized size = 6.42 \begin{align*} -\frac{9 \, b^{8} x^{16} + 36 \, a b^{7} x^{14} + 84 \, a^{2} b^{6} x^{12} + 126 \, a^{3} b^{5} x^{10} + 126 \, a^{4} b^{4} x^{8} + 84 \, a^{5} b^{3} x^{6} + 36 \, a^{6} b^{2} x^{4} + 9 \, a^{7} b x^{2} + a^{8}}{18 \, x^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^19,x, algorithm="giac")

[Out]

-1/18*(9*b^8*x^16 + 36*a*b^7*x^14 + 84*a^2*b^6*x^12 + 126*a^3*b^5*x^10 + 126*a^4*b^4*x^8 + 84*a^5*b^3*x^6 + 36
*a^6*b^2*x^4 + 9*a^7*b*x^2 + a^8)/x^18