3.1.97 \(\int \frac {(a+b x^2)^8}{x^{11}} \, dx\)

Optimal. Leaf size=95 \[ -\frac {a^8}{10 x^{10}}-\frac {a^7 b}{x^8}-\frac {14 a^6 b^2}{3 x^6}-\frac {14 a^5 b^3}{x^4}-\frac {35 a^4 b^4}{x^2}+56 a^3 b^5 \log (x)+14 a^2 b^6 x^2+2 a b^7 x^4+\frac {b^8 x^6}{6} \]

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Rubi [A]  time = 0.06, antiderivative size = 95, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {14 a^6 b^2}{3 x^6}-\frac {14 a^5 b^3}{x^4}-\frac {35 a^4 b^4}{x^2}+14 a^2 b^6 x^2+56 a^3 b^5 \log (x)-\frac {a^7 b}{x^8}-\frac {a^8}{10 x^{10}}+2 a b^7 x^4+\frac {b^8 x^6}{6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^8/x^11,x]

[Out]

-a^8/(10*x^10) - (a^7*b)/x^8 - (14*a^6*b^2)/(3*x^6) - (14*a^5*b^3)/x^4 - (35*a^4*b^4)/x^2 + 14*a^2*b^6*x^2 + 2
*a*b^7*x^4 + (b^8*x^6)/6 + 56*a^3*b^5*Log[x]

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])

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]]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^2\right )^8}{x^{11}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^8}{x^6} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (28 a^2 b^6+\frac {a^8}{x^6}+\frac {8 a^7 b}{x^5}+\frac {28 a^6 b^2}{x^4}+\frac {56 a^5 b^3}{x^3}+\frac {70 a^4 b^4}{x^2}+\frac {56 a^3 b^5}{x}+8 a b^7 x+b^8 x^2\right ) \, dx,x,x^2\right )\\ &=-\frac {a^8}{10 x^{10}}-\frac {a^7 b}{x^8}-\frac {14 a^6 b^2}{3 x^6}-\frac {14 a^5 b^3}{x^4}-\frac {35 a^4 b^4}{x^2}+14 a^2 b^6 x^2+2 a b^7 x^4+\frac {b^8 x^6}{6}+56 a^3 b^5 \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 95, normalized size = 1.00 \begin {gather*} -\frac {a^8}{10 x^{10}}-\frac {a^7 b}{x^8}-\frac {14 a^6 b^2}{3 x^6}-\frac {14 a^5 b^3}{x^4}-\frac {35 a^4 b^4}{x^2}+56 a^3 b^5 \log (x)+14 a^2 b^6 x^2+2 a b^7 x^4+\frac {b^8 x^6}{6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^8/x^11,x]

[Out]

-1/10*a^8/x^10 - (a^7*b)/x^8 - (14*a^6*b^2)/(3*x^6) - (14*a^5*b^3)/x^4 - (35*a^4*b^4)/x^2 + 14*a^2*b^6*x^2 + 2
*a*b^7*x^4 + (b^8*x^6)/6 + 56*a^3*b^5*Log[x]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b x^2\right )^8}{x^{11}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x^2)^8/x^11,x]

[Out]

IntegrateAlgebraic[(a + b*x^2)^8/x^11, x]

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fricas [A]  time = 1.38, size = 94, normalized size = 0.99 \begin {gather*} \frac {5 \, b^{8} x^{16} + 60 \, a b^{7} x^{14} + 420 \, a^{2} b^{6} x^{12} + 1680 \, a^{3} b^{5} x^{10} \log \relax (x) - 1050 \, a^{4} b^{4} x^{8} - 420 \, a^{5} b^{3} x^{6} - 140 \, a^{6} b^{2} x^{4} - 30 \, a^{7} b x^{2} - 3 \, a^{8}}{30 \, x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^11,x, algorithm="fricas")

[Out]

1/30*(5*b^8*x^16 + 60*a*b^7*x^14 + 420*a^2*b^6*x^12 + 1680*a^3*b^5*x^10*log(x) - 1050*a^4*b^4*x^8 - 420*a^5*b^
3*x^6 - 140*a^6*b^2*x^4 - 30*a^7*b*x^2 - 3*a^8)/x^10

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giac [A]  time = 1.13, size = 105, normalized size = 1.11 \begin {gather*} \frac {1}{6} \, b^{8} x^{6} + 2 \, a b^{7} x^{4} + 14 \, a^{2} b^{6} x^{2} + 28 \, a^{3} b^{5} \log \left (x^{2}\right ) - \frac {1918 \, a^{3} b^{5} x^{10} + 1050 \, a^{4} b^{4} x^{8} + 420 \, a^{5} b^{3} x^{6} + 140 \, a^{6} b^{2} x^{4} + 30 \, a^{7} b x^{2} + 3 \, a^{8}}{30 \, x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^11,x, algorithm="giac")

[Out]

1/6*b^8*x^6 + 2*a*b^7*x^4 + 14*a^2*b^6*x^2 + 28*a^3*b^5*log(x^2) - 1/30*(1918*a^3*b^5*x^10 + 1050*a^4*b^4*x^8
+ 420*a^5*b^3*x^6 + 140*a^6*b^2*x^4 + 30*a^7*b*x^2 + 3*a^8)/x^10

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maple [A]  time = 0.01, size = 90, normalized size = 0.95 \begin {gather*} \frac {b^{8} x^{6}}{6}+2 a \,b^{7} x^{4}+14 a^{2} b^{6} x^{2}+56 a^{3} b^{5} \ln \relax (x )-\frac {35 a^{4} b^{4}}{x^{2}}-\frac {14 a^{5} b^{3}}{x^{4}}-\frac {14 a^{6} b^{2}}{3 x^{6}}-\frac {a^{7} b}{x^{8}}-\frac {a^{8}}{10 x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^8/x^11,x)

[Out]

-1/10*a^8/x^10-a^7*b/x^8-14/3*a^6*b^2/x^6-14*a^5*b^3/x^4-35*a^4*b^4/x^2+14*a^2*b^6*x^2+2*a*b^7*x^4+1/6*b^8*x^6
+56*a^3*b^5*ln(x)

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maxima [A]  time = 1.33, size = 94, normalized size = 0.99 \begin {gather*} \frac {1}{6} \, b^{8} x^{6} + 2 \, a b^{7} x^{4} + 14 \, a^{2} b^{6} x^{2} + 28 \, a^{3} b^{5} \log \left (x^{2}\right ) - \frac {1050 \, a^{4} b^{4} x^{8} + 420 \, a^{5} b^{3} x^{6} + 140 \, a^{6} b^{2} x^{4} + 30 \, a^{7} b x^{2} + 3 \, a^{8}}{30 \, x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^11,x, algorithm="maxima")

[Out]

1/6*b^8*x^6 + 2*a*b^7*x^4 + 14*a^2*b^6*x^2 + 28*a^3*b^5*log(x^2) - 1/30*(1050*a^4*b^4*x^8 + 420*a^5*b^3*x^6 +
140*a^6*b^2*x^4 + 30*a^7*b*x^2 + 3*a^8)/x^10

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mupad [B]  time = 5.11, size = 91, normalized size = 0.96 \begin {gather*} \frac {b^8\,x^6}{6}-\frac {\frac {a^8}{10}+a^7\,b\,x^2+\frac {14\,a^6\,b^2\,x^4}{3}+14\,a^5\,b^3\,x^6+35\,a^4\,b^4\,x^8}{x^{10}}+2\,a\,b^7\,x^4+14\,a^2\,b^6\,x^2+56\,a^3\,b^5\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^8/x^11,x)

[Out]

(b^8*x^6)/6 - (a^8/10 + a^7*b*x^2 + (14*a^6*b^2*x^4)/3 + 14*a^5*b^3*x^6 + 35*a^4*b^4*x^8)/x^10 + 2*a*b^7*x^4 +
14*a^2*b^6*x^2 + 56*a^3*b^5*log(x)

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sympy [A]  time = 0.51, size = 99, normalized size = 1.04 \begin {gather*} 56 a^{3} b^{5} \log {\relax (x )} + 14 a^{2} b^{6} x^{2} + 2 a b^{7} x^{4} + \frac {b^{8} x^{6}}{6} + \frac {- 3 a^{8} - 30 a^{7} b x^{2} - 140 a^{6} b^{2} x^{4} - 420 a^{5} b^{3} x^{6} - 1050 a^{4} b^{4} x^{8}}{30 x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**8/x**11,x)

[Out]

56*a**3*b**5*log(x) + 14*a**2*b**6*x**2 + 2*a*b**7*x**4 + b**8*x**6/6 + (-3*a**8 - 30*a**7*b*x**2 - 140*a**6*b
**2*x**4 - 420*a**5*b**3*x**6 - 1050*a**4*b**4*x**8)/(30*x**10)

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