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

Optimal. Leaf size=87 \[ \frac {a^5}{4 b^6 \left (a+b x^2\right )^2}-\frac {5 a^4}{2 b^6 \left (a+b x^2\right )}-\frac {5 a^3 \log \left (a+b x^2\right )}{b^6}+\frac {3 a^2 x^2}{b^5}-\frac {3 a x^4}{4 b^4}+\frac {x^6}{6 b^3} \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 87, 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} \[ \frac {3 a^2 x^2}{b^5}-\frac {5 a^4}{2 b^6 \left (a+b x^2\right )}+\frac {a^5}{4 b^6 \left (a+b x^2\right )^2}-\frac {5 a^3 \log \left (a+b x^2\right )}{b^6}-\frac {3 a x^4}{4 b^4}+\frac {x^6}{6 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^11/(a + b*x^2)^3,x]

[Out]

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

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

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Mathematica [A]  time = 0.03, size = 75, normalized size = 0.86 \[ \frac {\frac {3 a^5}{\left (a+b x^2\right )^2}-\frac {30 a^4}{a+b x^2}-60 a^3 \log \left (a+b x^2\right )+36 a^2 b x^2-9 a b^2 x^4+2 b^3 x^6}{12 b^6} \]

Antiderivative was successfully verified.

[In]

Integrate[x^11/(a + b*x^2)^3,x]

[Out]

(36*a^2*b*x^2 - 9*a*b^2*x^4 + 2*b^3*x^6 + (3*a^5)/(a + b*x^2)^2 - (30*a^4)/(a + b*x^2) - 60*a^3*Log[a + b*x^2]
)/(12*b^6)

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fricas [A]  time = 0.78, size = 115, normalized size = 1.32 \[ \frac {2 \, b^{5} x^{10} - 5 \, a b^{4} x^{8} + 20 \, a^{2} b^{3} x^{6} + 63 \, a^{3} b^{2} x^{4} + 6 \, a^{4} b x^{2} - 27 \, a^{5} - 60 \, {\left (a^{3} b^{2} x^{4} + 2 \, a^{4} b x^{2} + a^{5}\right )} \log \left (b x^{2} + a\right )}{12 \, {\left (b^{8} x^{4} + 2 \, a b^{7} x^{2} + a^{2} b^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(2*b^5*x^10 - 5*a*b^4*x^8 + 20*a^2*b^3*x^6 + 63*a^3*b^2*x^4 + 6*a^4*b*x^2 - 27*a^5 - 60*(a^3*b^2*x^4 + 2*
a^4*b*x^2 + a^5)*log(b*x^2 + a))/(b^8*x^4 + 2*a*b^7*x^2 + a^2*b^6)

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giac [A]  time = 0.63, size = 92, normalized size = 1.06 \[ -\frac {5 \, a^{3} \log \left ({\left | b x^{2} + a \right |}\right )}{b^{6}} + \frac {30 \, a^{3} b^{2} x^{4} + 50 \, a^{4} b x^{2} + 21 \, a^{5}}{4 \, {\left (b x^{2} + a\right )}^{2} b^{6}} + \frac {2 \, b^{6} x^{6} - 9 \, a b^{5} x^{4} + 36 \, a^{2} b^{4} x^{2}}{12 \, b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5*a^3*log(abs(b*x^2 + a))/b^6 + 1/4*(30*a^3*b^2*x^4 + 50*a^4*b*x^2 + 21*a^5)/((b*x^2 + a)^2*b^6) + 1/12*(2*b^
6*x^6 - 9*a*b^5*x^4 + 36*a^2*b^4*x^2)/b^9

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maple [A]  time = 0.01, size = 80, normalized size = 0.92 \[ \frac {x^{6}}{6 b^{3}}-\frac {3 a \,x^{4}}{4 b^{4}}+\frac {a^{5}}{4 \left (b \,x^{2}+a \right )^{2} b^{6}}+\frac {3 a^{2} x^{2}}{b^{5}}-\frac {5 a^{4}}{2 \left (b \,x^{2}+a \right ) b^{6}}-\frac {5 a^{3} \ln \left (b \,x^{2}+a \right )}{b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.34, size = 89, normalized size = 1.02 \[ -\frac {10 \, a^{4} b x^{2} + 9 \, a^{5}}{4 \, {\left (b^{8} x^{4} + 2 \, a b^{7} x^{2} + a^{2} b^{6}\right )}} - \frac {5 \, a^{3} \log \left (b x^{2} + a\right )}{b^{6}} + \frac {2 \, b^{2} x^{6} - 9 \, a b x^{4} + 36 \, a^{2} x^{2}}{12 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(10*a^4*b*x^2 + 9*a^5)/(b^8*x^4 + 2*a*b^7*x^2 + a^2*b^6) - 5*a^3*log(b*x^2 + a)/b^6 + 1/12*(2*b^2*x^6 - 9
*a*b*x^4 + 36*a^2*x^2)/b^5

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mupad [B]  time = 4.49, size = 90, normalized size = 1.03 \[ \frac {x^6}{6\,b^3}-\frac {\frac {9\,a^5}{4\,b}+\frac {5\,a^4\,x^2}{2}}{a^2\,b^5+2\,a\,b^6\,x^2+b^7\,x^4}-\frac {3\,a\,x^4}{4\,b^4}-\frac {5\,a^3\,\ln \left (b\,x^2+a\right )}{b^6}+\frac {3\,a^2\,x^2}{b^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11/(a + b*x^2)^3,x)

[Out]

x^6/(6*b^3) - ((9*a^5)/(4*b) + (5*a^4*x^2)/2)/(a^2*b^5 + b^7*x^4 + 2*a*b^6*x^2) - (3*a*x^4)/(4*b^4) - (5*a^3*l
og(a + b*x^2))/b^6 + (3*a^2*x^2)/b^5

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sympy [A]  time = 0.44, size = 92, normalized size = 1.06 \[ - \frac {5 a^{3} \log {\left (a + b x^{2} \right )}}{b^{6}} + \frac {3 a^{2} x^{2}}{b^{5}} - \frac {3 a x^{4}}{4 b^{4}} + \frac {- 9 a^{5} - 10 a^{4} b x^{2}}{4 a^{2} b^{6} + 8 a b^{7} x^{2} + 4 b^{8} x^{4}} + \frac {x^{6}}{6 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5*a**3*log(a + b*x**2)/b**6 + 3*a**2*x**2/b**5 - 3*a*x**4/(4*b**4) + (-9*a**5 - 10*a**4*b*x**2)/(4*a**2*b**6
+ 8*a*b**7*x**2 + 4*b**8*x**4) + x**6/(6*b**3)

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