3.64 \(\int \frac {(a+b x^2)^5}{x^{11}} \, dx\)

Optimal. Leaf size=65 \[ -\frac {a^5}{10 x^{10}}-\frac {5 a^4 b}{8 x^8}-\frac {5 a^3 b^2}{3 x^6}-\frac {5 a^2 b^3}{2 x^4}-\frac {5 a b^4}{2 x^2}+b^5 \log (x) \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 65, 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} \[ -\frac {5 a^3 b^2}{3 x^6}-\frac {5 a^2 b^3}{2 x^4}-\frac {5 a^4 b}{8 x^8}-\frac {a^5}{10 x^{10}}-\frac {5 a b^4}{2 x^2}+b^5 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^5/x^11,x]

[Out]

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

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Mathematica [A]  time = 0.00, size = 65, normalized size = 1.00 \[ -\frac {a^5}{10 x^{10}}-\frac {5 a^4 b}{8 x^8}-\frac {5 a^3 b^2}{3 x^6}-\frac {5 a^2 b^3}{2 x^4}-\frac {5 a b^4}{2 x^2}+b^5 \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^5/x^11,x]

[Out]

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

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fricas [A]  time = 0.85, size = 61, normalized size = 0.94 \[ \frac {120 \, b^{5} x^{10} \log \relax (x) - 300 \, a b^{4} x^{8} - 300 \, a^{2} b^{3} x^{6} - 200 \, a^{3} b^{2} x^{4} - 75 \, a^{4} b x^{2} - 12 \, a^{5}}{120 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^11,x, algorithm="fricas")

[Out]

1/120*(120*b^5*x^10*log(x) - 300*a*b^4*x^8 - 300*a^2*b^3*x^6 - 200*a^3*b^2*x^4 - 75*a^4*b*x^2 - 12*a^5)/x^10

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giac [A]  time = 1.06, size = 69, normalized size = 1.06 \[ \frac {1}{2} \, b^{5} \log \left (x^{2}\right ) - \frac {137 \, b^{5} x^{10} + 300 \, a b^{4} x^{8} + 300 \, a^{2} b^{3} x^{6} + 200 \, a^{3} b^{2} x^{4} + 75 \, a^{4} b x^{2} + 12 \, a^{5}}{120 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^11,x, algorithm="giac")

[Out]

1/2*b^5*log(x^2) - 1/120*(137*b^5*x^10 + 300*a*b^4*x^8 + 300*a^2*b^3*x^6 + 200*a^3*b^2*x^4 + 75*a^4*b*x^2 + 12
*a^5)/x^10

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maple [A]  time = 0.01, size = 56, normalized size = 0.86 \[ b^{5} \ln \relax (x )-\frac {5 a \,b^{4}}{2 x^{2}}-\frac {5 a^{2} b^{3}}{2 x^{4}}-\frac {5 a^{3} b^{2}}{3 x^{6}}-\frac {5 a^{4} b}{8 x^{8}}-\frac {a^{5}}{10 x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5/x^11,x)

[Out]

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

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maxima [A]  time = 1.35, size = 61, normalized size = 0.94 \[ \frac {1}{2} \, b^{5} \log \left (x^{2}\right ) - \frac {300 \, a b^{4} x^{8} + 300 \, a^{2} b^{3} x^{6} + 200 \, a^{3} b^{2} x^{4} + 75 \, a^{4} b x^{2} + 12 \, a^{5}}{120 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^11,x, algorithm="maxima")

[Out]

1/2*b^5*log(x^2) - 1/120*(300*a*b^4*x^8 + 300*a^2*b^3*x^6 + 200*a^3*b^2*x^4 + 75*a^4*b*x^2 + 12*a^5)/x^10

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mupad [B]  time = 4.77, size = 58, normalized size = 0.89 \[ b^5\,\ln \relax (x)-\frac {\frac {a^5}{10}+\frac {5\,a^4\,b\,x^2}{8}+\frac {5\,a^3\,b^2\,x^4}{3}+\frac {5\,a^2\,b^3\,x^6}{2}+\frac {5\,a\,b^4\,x^8}{2}}{x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^5/x^11,x)

[Out]

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

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sympy [A]  time = 0.44, size = 61, normalized size = 0.94 \[ b^{5} \log {\relax (x )} + \frac {- 12 a^{5} - 75 a^{4} b x^{2} - 200 a^{3} b^{2} x^{4} - 300 a^{2} b^{3} x^{6} - 300 a b^{4} x^{8}}{120 x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5/x**11,x)

[Out]

b**5*log(x) + (-12*a**5 - 75*a**4*b*x**2 - 200*a**3*b**2*x**4 - 300*a**2*b**3*x**6 - 300*a*b**4*x**8)/(120*x**
10)

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