3.5.38 \(\int \frac {(a^2+2 a b x^2+b^2 x^4)^2}{x^{11}} \, dx\) [438]

Optimal. Leaf size=19 \[ -\frac {\left (a+b x^2\right )^5}{10 a x^{10}} \]

[Out]

-1/10*(b*x^2+a)^5/a/x^10

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {28, 270} \begin {gather*} -\frac {\left (a+b x^2\right )^5}{10 a x^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/10*(a + b*x^2)^5/(a*x^10)

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 270

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^2+2 a b x^2+b^2 x^4\right )^2}{x^{11}} \, dx &=\frac {\int \frac {\left (a b+b^2 x^2\right )^4}{x^{11}} \, dx}{b^4}\\ &=-\frac {\left (a+b x^2\right )^5}{10 a x^{10}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(52\) vs. \(2(19)=38\).
time = 0.00, size = 52, normalized size = 2.74 \begin {gather*} -\frac {a^4}{10 x^{10}}-\frac {a^3 b}{2 x^8}-\frac {a^2 b^2}{x^6}-\frac {a b^3}{x^4}-\frac {b^4}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(46\) vs. \(2(17)=34\).
time = 0.01, size = 47, normalized size = 2.47

method	result	size
gosper	\(-\frac {5 b^{4} x^{8}+10 a \,b^{3} x^{6}+10 a^{2} b^{2} x^{4}+5 a^{3} b \,x^{2}+a^{4}}{10 x^{10}}\)	\(47\)
default	\(-\frac {a \,b^{3}}{x^{4}}-\frac {b^{4}}{2 x^{2}}-\frac {a^{3} b}{2 x^{8}}-\frac {a^{4}}{10 x^{10}}-\frac {a^{2} b^{2}}{x^{6}}\)	\(47\)
norman	\(\frac {-\frac {1}{2} b^{4} x^{8}-a \,b^{3} x^{6}-a^{2} b^{2} x^{4}-\frac {1}{2} a^{3} b \,x^{2}-\frac {1}{10} a^{4}}{x^{10}}\)	\(48\)
risch	\(\frac {-\frac {1}{2} b^{4} x^{8}-a \,b^{3} x^{6}-a^{2} b^{2} x^{4}-\frac {1}{2} a^{3} b \,x^{2}-\frac {1}{10} a^{4}}{x^{10}}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^4+2*a*b*x^2+a^2)^2/x^11,x,method=_RETURNVERBOSE)

[Out]

-a*b^3/x^4-1/2*b^4/x^2-1/2*a^3*b/x^8-1/10*a^4/x^10-a^2*b^2/x^6

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (17) = 34\).
time = 0.38, size = 46, normalized size = 2.42 \begin {gather*} -\frac {5 \, b^{4} x^{8} + 10 \, a b^{3} x^{6} + 10 \, a^{2} b^{2} x^{4} + 5 \, a^{3} b x^{2} + a^{4}}{10 \, 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)^2/x^11,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (17) = 34\).
time = 0.37, size = 46, normalized size = 2.42 \begin {gather*} -\frac {5 \, b^{4} x^{8} + 10 \, a b^{3} x^{6} + 10 \, a^{2} b^{2} x^{4} + 5 \, a^{3} b x^{2} + a^{4}}{10 \, 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)^2/x^11,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (15) = 30\).
time = 0.16, size = 49, normalized size = 2.58 \begin {gather*} \frac {- a^{4} - 5 a^{3} b x^{2} - 10 a^{2} b^{2} x^{4} - 10 a b^{3} x^{6} - 5 b^{4} x^{8}}{10 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)**2/x**11,x)

[Out]

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

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (17) = 34\).
time = 3.62, size = 46, normalized size = 2.42 \begin {gather*} -\frac {5 \, b^{4} x^{8} + 10 \, a b^{3} x^{6} + 10 \, a^{2} b^{2} x^{4} + 5 \, a^{3} b x^{2} + a^{4}}{10 \, 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)^2/x^11,x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 0.03, size = 46, normalized size = 2.42 \begin {gather*} -\frac {\frac {a^4}{10}+\frac {a^3\,b\,x^2}{2}+a^2\,b^2\,x^4+a\,b^3\,x^6+\frac {b^4\,x^8}{2}}{x^{10}} \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)^2/x^11,x)

[Out]

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

________________________________________________________________________________________