3.3.91 \(\int \frac {(a^2+2 a b x^2+b^2 x^4)^3}{x^{11}} \, dx\)

Optimal. Leaf size=77 \[ -\frac {a^6}{10 x^{10}}-\frac {3 a^5 b}{4 x^8}-\frac {5 a^4 b^2}{2 x^6}-\frac {5 a^3 b^3}{x^4}-\frac {15 a^2 b^4}{2 x^2}+6 a b^5 \log (x)+\frac {b^6 x^2}{2} \]

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Rubi [A]  time = 0.05, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {28, 266, 43} \begin {gather*} -\frac {5 a^4 b^2}{2 x^6}-\frac {5 a^3 b^3}{x^4}-\frac {15 a^2 b^4}{2 x^2}-\frac {3 a^5 b}{4 x^8}-\frac {a^6}{10 x^{10}}+6 a b^5 \log (x)+\frac {b^6 x^2}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^11,x]

[Out]

-a^6/(10*x^10) - (3*a^5*b)/(4*x^8) - (5*a^4*b^2)/(2*x^6) - (5*a^3*b^3)/x^4 - (15*a^2*b^4)/(2*x^2) + (b^6*x^2)/
2 + 6*a*b^5*Log[x]

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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^2+2 a b x^2+b^2 x^4\right )^3}{x^{11}} \, dx &=\frac {\int \frac {\left (a b+b^2 x^2\right )^6}{x^{11}} \, dx}{b^6}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (a b+b^2 x\right )^6}{x^6} \, dx,x,x^2\right )}{2 b^6}\\ &=\frac {\operatorname {Subst}\left (\int \left (b^{12}+\frac {a^6 b^6}{x^6}+\frac {6 a^5 b^7}{x^5}+\frac {15 a^4 b^8}{x^4}+\frac {20 a^3 b^9}{x^3}+\frac {15 a^2 b^{10}}{x^2}+\frac {6 a b^{11}}{x}\right ) \, dx,x,x^2\right )}{2 b^6}\\ &=-\frac {a^6}{10 x^{10}}-\frac {3 a^5 b}{4 x^8}-\frac {5 a^4 b^2}{2 x^6}-\frac {5 a^3 b^3}{x^4}-\frac {15 a^2 b^4}{2 x^2}+\frac {b^6 x^2}{2}+6 a b^5 \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 77, normalized size = 1.00 \begin {gather*} -\frac {a^6}{10 x^{10}}-\frac {3 a^5 b}{4 x^8}-\frac {5 a^4 b^2}{2 x^6}-\frac {5 a^3 b^3}{x^4}-\frac {15 a^2 b^4}{2 x^2}+6 a b^5 \log (x)+\frac {b^6 x^2}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^11,x]

[Out]

-1/10*a^6/x^10 - (3*a^5*b)/(4*x^8) - (5*a^4*b^2)/(2*x^6) - (5*a^3*b^3)/x^4 - (15*a^2*b^4)/(2*x^2) + (b^6*x^2)/
2 + 6*a*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^2+2 a b x^2+b^2 x^4\right )^3}{x^{11}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^11,x]

[Out]

IntegrateAlgebraic[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^11, x]

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fricas [A]  time = 0.53, size = 72, normalized size = 0.94 \begin {gather*} \frac {10 \, b^{6} x^{12} + 120 \, a b^{5} x^{10} \log \relax (x) - 150 \, a^{2} b^{4} x^{8} - 100 \, a^{3} b^{3} x^{6} - 50 \, a^{4} b^{2} x^{4} - 15 \, a^{5} b x^{2} - 2 \, a^{6}}{20 \, x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^3/x^11,x, algorithm="fricas")

[Out]

1/20*(10*b^6*x^12 + 120*a*b^5*x^10*log(x) - 150*a^2*b^4*x^8 - 100*a^3*b^3*x^6 - 50*a^4*b^2*x^4 - 15*a^5*b*x^2
- 2*a^6)/x^10

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giac [A]  time = 0.15, size = 81, normalized size = 1.05 \begin {gather*} \frac {1}{2} \, b^{6} x^{2} + 3 \, a b^{5} \log \left (x^{2}\right ) - \frac {137 \, a b^{5} x^{10} + 150 \, a^{2} b^{4} x^{8} + 100 \, a^{3} b^{3} x^{6} + 50 \, a^{4} b^{2} x^{4} + 15 \, a^{5} b x^{2} + 2 \, a^{6}}{20 \, x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^3/x^11,x, algorithm="giac")

[Out]

1/2*b^6*x^2 + 3*a*b^5*log(x^2) - 1/20*(137*a*b^5*x^10 + 150*a^2*b^4*x^8 + 100*a^3*b^3*x^6 + 50*a^4*b^2*x^4 + 1
5*a^5*b*x^2 + 2*a^6)/x^10

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maple [A]  time = 0.01, size = 68, normalized size = 0.88 \begin {gather*} \frac {b^{6} x^{2}}{2}+6 a \,b^{5} \ln \relax (x )-\frac {15 a^{2} b^{4}}{2 x^{2}}-\frac {5 a^{3} b^{3}}{x^{4}}-\frac {5 a^{4} b^{2}}{2 x^{6}}-\frac {3 a^{5} b}{4 x^{8}}-\frac {a^{6}}{10 x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^4+2*a*b*x^2+a^2)^3/x^11,x)

[Out]

-1/10*a^6/x^10-3/4*a^5*b/x^8-5/2*a^4*b^2/x^6-5*a^3*b^3/x^4-15/2*a^2*b^4/x^2+1/2*b^6*x^2+6*a*b^5*ln(x)

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maxima [A]  time = 1.38, size = 72, normalized size = 0.94 \begin {gather*} \frac {1}{2} \, b^{6} x^{2} + 3 \, a b^{5} \log \left (x^{2}\right ) - \frac {150 \, a^{2} b^{4} x^{8} + 100 \, a^{3} b^{3} x^{6} + 50 \, a^{4} b^{2} x^{4} + 15 \, a^{5} b x^{2} + 2 \, a^{6}}{20 \, x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^3/x^11,x, algorithm="maxima")

[Out]

1/2*b^6*x^2 + 3*a*b^5*log(x^2) - 1/20*(150*a^2*b^4*x^8 + 100*a^3*b^3*x^6 + 50*a^4*b^2*x^4 + 15*a^5*b*x^2 + 2*a
^6)/x^10

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mupad [B]  time = 4.40, size = 70, normalized size = 0.91 \begin {gather*} \frac {b^6\,x^2}{2}-\frac {\frac {a^6}{10}+\frac {3\,a^5\,b\,x^2}{4}+\frac {5\,a^4\,b^2\,x^4}{2}+5\,a^3\,b^3\,x^6+\frac {15\,a^2\,b^4\,x^8}{2}}{x^{10}}+6\,a\,b^5\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2 + b^2*x^4 + 2*a*b*x^2)^3/x^11,x)

[Out]

(b^6*x^2)/2 - (a^6/10 + (3*a^5*b*x^2)/4 + (5*a^4*b^2*x^4)/2 + 5*a^3*b^3*x^6 + (15*a^2*b^4*x^8)/2)/x^10 + 6*a*b
^5*log(x)

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sympy [A]  time = 0.63, size = 75, normalized size = 0.97 \begin {gather*} 6 a b^{5} \log {\relax (x )} + \frac {b^{6} x^{2}}{2} + \frac {- 2 a^{6} - 15 a^{5} b x^{2} - 50 a^{4} b^{2} x^{4} - 100 a^{3} b^{3} x^{6} - 150 a^{2} b^{4} x^{8}}{20 x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**4+2*a*b*x**2+a**2)**3/x**11,x)

[Out]

6*a*b**5*log(x) + b**6*x**2/2 + (-2*a**6 - 15*a**5*b*x**2 - 50*a**4*b**2*x**4 - 100*a**3*b**3*x**6 - 150*a**2*
b**4*x**8)/(20*x**10)

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