3.63 \(\int \frac {(a+b x^2)^5}{x^9} \, dx\)

Optimal. Leaf size=64 \[ -\frac {a^5}{8 x^8}-\frac {5 a^4 b}{6 x^6}-\frac {5 a^3 b^2}{2 x^4}-\frac {5 a^2 b^3}{x^2}+5 a b^4 \log (x)+\frac {b^5 x^2}{2} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 64, 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} \[ -\frac {5 a^3 b^2}{2 x^4}-\frac {5 a^2 b^3}{x^2}-\frac {5 a^4 b}{6 x^6}-\frac {a^5}{8 x^8}+5 a b^4 \log (x)+\frac {b^5 x^2}{2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^5/x^9,x]

[Out]

-a^5/(8*x^8) - (5*a^4*b)/(6*x^6) - (5*a^3*b^2)/(2*x^4) - (5*a^2*b^3)/x^2 + (b^5*x^2)/2 + 5*a*b^4*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 )^5}{x^9} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^5}{x^5} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (b^5+\frac {a^5}{x^5}+\frac {5 a^4 b}{x^4}+\frac {10 a^3 b^2}{x^3}+\frac {10 a^2 b^3}{x^2}+\frac {5 a b^4}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac {a^5}{8 x^8}-\frac {5 a^4 b}{6 x^6}-\frac {5 a^3 b^2}{2 x^4}-\frac {5 a^2 b^3}{x^2}+\frac {b^5 x^2}{2}+5 a b^4 \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 64, normalized size = 1.00 \[ -\frac {a^5}{8 x^8}-\frac {5 a^4 b}{6 x^6}-\frac {5 a^3 b^2}{2 x^4}-\frac {5 a^2 b^3}{x^2}+5 a b^4 \log (x)+\frac {b^5 x^2}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^5/x^9,x]

[Out]

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

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fricas [A]  time = 0.87, size = 61, normalized size = 0.95 \[ \frac {12 \, b^{5} x^{10} + 120 \, a b^{4} x^{8} \log \relax (x) - 120 \, a^{2} b^{3} x^{6} - 60 \, a^{3} b^{2} x^{4} - 20 \, a^{4} b x^{2} - 3 \, a^{5}}{24 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^9,x, algorithm="fricas")

[Out]

1/24*(12*b^5*x^10 + 120*a*b^4*x^8*log(x) - 120*a^2*b^3*x^6 - 60*a^3*b^2*x^4 - 20*a^4*b*x^2 - 3*a^5)/x^8

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giac [A]  time = 1.05, size = 70, normalized size = 1.09 \[ \frac {1}{2} \, b^{5} x^{2} + \frac {5}{2} \, a b^{4} \log \left (x^{2}\right ) - \frac {125 \, a b^{4} x^{8} + 120 \, a^{2} b^{3} x^{6} + 60 \, a^{3} b^{2} x^{4} + 20 \, a^{4} b x^{2} + 3 \, a^{5}}{24 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^9,x, algorithm="giac")

[Out]

1/2*b^5*x^2 + 5/2*a*b^4*log(x^2) - 1/24*(125*a*b^4*x^8 + 120*a^2*b^3*x^6 + 60*a^3*b^2*x^4 + 20*a^4*b*x^2 + 3*a
^5)/x^8

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maple [A]  time = 0.01, size = 57, normalized size = 0.89 \[ \frac {b^{5} x^{2}}{2}+5 a \,b^{4} \ln \relax (x )-\frac {5 a^{2} b^{3}}{x^{2}}-\frac {5 a^{3} b^{2}}{2 x^{4}}-\frac {5 a^{4} b}{6 x^{6}}-\frac {a^{5}}{8 x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5/x^9,x)

[Out]

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

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maxima [A]  time = 1.41, size = 61, normalized size = 0.95 \[ \frac {1}{2} \, b^{5} x^{2} + \frac {5}{2} \, a b^{4} \log \left (x^{2}\right ) - \frac {120 \, a^{2} b^{3} x^{6} + 60 \, a^{3} b^{2} x^{4} + 20 \, a^{4} b x^{2} + 3 \, a^{5}}{24 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^9,x, algorithm="maxima")

[Out]

1/2*b^5*x^2 + 5/2*a*b^4*log(x^2) - 1/24*(120*a^2*b^3*x^6 + 60*a^3*b^2*x^4 + 20*a^4*b*x^2 + 3*a^5)/x^8

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mupad [B]  time = 0.04, size = 59, normalized size = 0.92 \[ \frac {b^5\,x^2}{2}-\frac {\frac {a^5}{8}+\frac {5\,a^4\,b\,x^2}{6}+\frac {5\,a^3\,b^2\,x^4}{2}+5\,a^2\,b^3\,x^6}{x^8}+5\,a\,b^4\,\ln \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^5/x^9,x)

[Out]

(b^5*x^2)/2 - (a^5/8 + (5*a^4*b*x^2)/6 + (5*a^3*b^2*x^4)/2 + 5*a^2*b^3*x^6)/x^8 + 5*a*b^4*log(x)

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sympy [A]  time = 0.36, size = 63, normalized size = 0.98 \[ 5 a b^{4} \log {\relax (x )} + \frac {b^{5} x^{2}}{2} + \frac {- 3 a^{5} - 20 a^{4} b x^{2} - 60 a^{3} b^{2} x^{4} - 120 a^{2} b^{3} x^{6}}{24 x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5/x**9,x)

[Out]

5*a*b**4*log(x) + b**5*x**2/2 + (-3*a**5 - 20*a**4*b*x**2 - 60*a**3*b**2*x**4 - 120*a**2*b**3*x**6)/(24*x**8)

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