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]

-1/18*(b*x^2+a)^9/a/x^18

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Rubi [A]  time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, 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*}

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Mathematica [B]  time = 0.00, size = 100, normalized size = 5.26 \[ -\frac {a^8}{18 x^{18}}-\frac {a^7 b}{2 x^{16}}-\frac {2 a^6 b^2}{x^{14}}-\frac {14 a^5 b^3}{3 x^{12}}-\frac {7 a^4 b^4}{x^{10}}-\frac {7 a^3 b^5}{x^8}-\frac {14 a^2 b^6}{3 x^6}-\frac {2 a b^7}{x^4}-\frac {b^8}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^8/x^19,x]

[Out]

-1/18*a^8/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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fricas [B]  time = 0.79, size = 90, normalized size = 4.74 \[ -\frac {9 \, b^{8} x^{16} + 36 \, a b^{7} x^{14} + 84 \, a^{2} b^{6} x^{12} + 126 \, a^{3} b^{5} x^{10} + 126 \, a^{4} b^{4} x^{8} + 84 \, a^{5} b^{3} x^{6} + 36 \, a^{6} b^{2} x^{4} + 9 \, a^{7} b x^{2} + a^{8}}{18 \, x^{18}} \]

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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giac [B]  time = 1.07, size = 90, normalized size = 4.74 \[ -\frac {9 \, b^{8} x^{16} + 36 \, a b^{7} x^{14} + 84 \, a^{2} b^{6} x^{12} + 126 \, a^{3} b^{5} x^{10} + 126 \, a^{4} b^{4} x^{8} + 84 \, a^{5} b^{3} x^{6} + 36 \, a^{6} b^{2} x^{4} + 9 \, a^{7} b x^{2} + a^{8}}{18 \, x^{18}} \]

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

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maple [B]  time = 0.01, size = 91, normalized size = 4.79 \[ -\frac {b^{8}}{2 x^{2}}-\frac {2 a \,b^{7}}{x^{4}}-\frac {14 a^{2} b^{6}}{3 x^{6}}-\frac {7 a^{3} b^{5}}{x^{8}}-\frac {7 a^{4} b^{4}}{x^{10}}-\frac {14 a^{5} b^{3}}{3 x^{12}}-\frac {2 a^{6} b^{2}}{x^{14}}-\frac {a^{7} b}{2 x^{16}}-\frac {a^{8}}{18 x^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.32, size = 90, normalized size = 4.74 \[ -\frac {9 \, b^{8} x^{16} + 36 \, a b^{7} x^{14} + 84 \, a^{2} b^{6} x^{12} + 126 \, a^{3} b^{5} x^{10} + 126 \, a^{4} b^{4} x^{8} + 84 \, a^{5} b^{3} x^{6} + 36 \, a^{6} b^{2} x^{4} + 9 \, a^{7} b x^{2} + a^{8}}{18 \, x^{18}} \]

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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mupad [B]  time = 0.08, size = 92, normalized size = 4.84 \[ -\frac {\frac {a^8}{18}+\frac {a^7\,b\,x^2}{2}+2\,a^6\,b^2\,x^4+\frac {14\,a^5\,b^3\,x^6}{3}+7\,a^4\,b^4\,x^8+7\,a^3\,b^5\,x^{10}+\frac {14\,a^2\,b^6\,x^{12}}{3}+2\,a\,b^7\,x^{14}+\frac {b^8\,x^{16}}{2}}{x^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^8/x^19,x)

[Out]

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

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sympy [B]  time = 0.90, size = 97, normalized size = 5.11 \[ \frac {- a^{8} - 9 a^{7} b x^{2} - 36 a^{6} b^{2} x^{4} - 84 a^{5} b^{3} x^{6} - 126 a^{4} b^{4} x^{8} - 126 a^{3} b^{5} x^{10} - 84 a^{2} b^{6} x^{12} - 36 a b^{7} x^{14} - 9 b^{8} x^{16}}{18 x^{18}} \]

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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