3.202 \(\int \frac {x^5}{(a+b x^2)^{10}} \, dx\)

Optimal. Leaf size=53 \[ -\frac {a^2}{18 b^3 \left (a+b x^2\right )^9}+\frac {a}{8 b^3 \left (a+b x^2\right )^8}-\frac {1}{14 b^3 \left (a+b x^2\right )^7} \]

[Out]

-1/18*a^2/b^3/(b*x^2+a)^9+1/8*a/b^3/(b*x^2+a)^8-1/14/b^3/(b*x^2+a)^7

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Rubi [A]  time = 0.04, antiderivative size = 53, 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} \[ -\frac {a^2}{18 b^3 \left (a+b x^2\right )^9}+\frac {a}{8 b^3 \left (a+b x^2\right )^8}-\frac {1}{14 b^3 \left (a+b x^2\right )^7} \]

Antiderivative was successfully verified.

[In]

Int[x^5/(a + b*x^2)^10,x]

[Out]

-a^2/(18*b^3*(a + b*x^2)^9) + a/(8*b^3*(a + b*x^2)^8) - 1/(14*b^3*(a + b*x^2)^7)

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^5}{\left (a+b x^2\right )^{10}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2}{(a+b x)^{10}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {a^2}{b^2 (a+b x)^{10}}-\frac {2 a}{b^2 (a+b x)^9}+\frac {1}{b^2 (a+b x)^8}\right ) \, dx,x,x^2\right )\\ &=-\frac {a^2}{18 b^3 \left (a+b x^2\right )^9}+\frac {a}{8 b^3 \left (a+b x^2\right )^8}-\frac {1}{14 b^3 \left (a+b x^2\right )^7}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 35, normalized size = 0.66 \[ -\frac {a^2+9 a b x^2+36 b^2 x^4}{504 b^3 \left (a+b x^2\right )^9} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(a + b*x^2)^10,x]

[Out]

-1/504*(a^2 + 9*a*b*x^2 + 36*b^2*x^4)/(b^3*(a + b*x^2)^9)

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fricas [B]  time = 0.87, size = 124, normalized size = 2.34 \[ -\frac {36 \, b^{2} x^{4} + 9 \, a b x^{2} + a^{2}}{504 \, {\left (b^{12} x^{18} + 9 \, a b^{11} x^{16} + 36 \, a^{2} b^{10} x^{14} + 84 \, a^{3} b^{9} x^{12} + 126 \, a^{4} b^{8} x^{10} + 126 \, a^{5} b^{7} x^{8} + 84 \, a^{6} b^{6} x^{6} + 36 \, a^{7} b^{5} x^{4} + 9 \, a^{8} b^{4} x^{2} + a^{9} b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^2+a)^10,x, algorithm="fricas")

[Out]

-1/504*(36*b^2*x^4 + 9*a*b*x^2 + a^2)/(b^12*x^18 + 9*a*b^11*x^16 + 36*a^2*b^10*x^14 + 84*a^3*b^9*x^12 + 126*a^
4*b^8*x^10 + 126*a^5*b^7*x^8 + 84*a^6*b^6*x^6 + 36*a^7*b^5*x^4 + 9*a^8*b^4*x^2 + a^9*b^3)

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giac [A]  time = 0.64, size = 33, normalized size = 0.62 \[ -\frac {36 \, b^{2} x^{4} + 9 \, a b x^{2} + a^{2}}{504 \, {\left (b x^{2} + a\right )}^{9} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^2+a)^10,x, algorithm="giac")

[Out]

-1/504*(36*b^2*x^4 + 9*a*b*x^2 + a^2)/((b*x^2 + a)^9*b^3)

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maple [A]  time = 0.01, size = 48, normalized size = 0.91 \[ -\frac {a^{2}}{18 \left (b \,x^{2}+a \right )^{9} b^{3}}+\frac {a}{8 \left (b \,x^{2}+a \right )^{8} b^{3}}-\frac {1}{14 \left (b \,x^{2}+a \right )^{7} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(b*x^2+a)^10,x)

[Out]

-1/18*a^2/b^3/(b*x^2+a)^9+1/8*a/b^3/(b*x^2+a)^8-1/14/b^3/(b*x^2+a)^7

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maxima [B]  time = 1.51, size = 124, normalized size = 2.34 \[ -\frac {36 \, b^{2} x^{4} + 9 \, a b x^{2} + a^{2}}{504 \, {\left (b^{12} x^{18} + 9 \, a b^{11} x^{16} + 36 \, a^{2} b^{10} x^{14} + 84 \, a^{3} b^{9} x^{12} + 126 \, a^{4} b^{8} x^{10} + 126 \, a^{5} b^{7} x^{8} + 84 \, a^{6} b^{6} x^{6} + 36 \, a^{7} b^{5} x^{4} + 9 \, a^{8} b^{4} x^{2} + a^{9} b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^2+a)^10,x, algorithm="maxima")

[Out]

-1/504*(36*b^2*x^4 + 9*a*b*x^2 + a^2)/(b^12*x^18 + 9*a*b^11*x^16 + 36*a^2*b^10*x^14 + 84*a^3*b^9*x^12 + 126*a^
4*b^8*x^10 + 126*a^5*b^7*x^8 + 84*a^6*b^6*x^6 + 36*a^7*b^5*x^4 + 9*a^8*b^4*x^2 + a^9*b^3)

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mupad [B]  time = 4.83, size = 125, normalized size = 2.36 \[ -\frac {\frac {a^2}{504\,b^3}+\frac {x^4}{14\,b}+\frac {a\,x^2}{56\,b^2}}{a^9+9\,a^8\,b\,x^2+36\,a^7\,b^2\,x^4+84\,a^6\,b^3\,x^6+126\,a^5\,b^4\,x^8+126\,a^4\,b^5\,x^{10}+84\,a^3\,b^6\,x^{12}+36\,a^2\,b^7\,x^{14}+9\,a\,b^8\,x^{16}+b^9\,x^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(a + b*x^2)^10,x)

[Out]

-(a^2/(504*b^3) + x^4/(14*b) + (a*x^2)/(56*b^2))/(a^9 + b^9*x^18 + 9*a^8*b*x^2 + 9*a*b^8*x^16 + 36*a^7*b^2*x^4
+ 84*a^6*b^3*x^6 + 126*a^5*b^4*x^8 + 126*a^4*b^5*x^10 + 84*a^3*b^6*x^12 + 36*a^2*b^7*x^14)

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sympy [B]  time = 0.95, size = 131, normalized size = 2.47 \[ \frac {- a^{2} - 9 a b x^{2} - 36 b^{2} x^{4}}{504 a^{9} b^{3} + 4536 a^{8} b^{4} x^{2} + 18144 a^{7} b^{5} x^{4} + 42336 a^{6} b^{6} x^{6} + 63504 a^{5} b^{7} x^{8} + 63504 a^{4} b^{8} x^{10} + 42336 a^{3} b^{9} x^{12} + 18144 a^{2} b^{10} x^{14} + 4536 a b^{11} x^{16} + 504 b^{12} x^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(b*x**2+a)**10,x)

[Out]

(-a**2 - 9*a*b*x**2 - 36*b**2*x**4)/(504*a**9*b**3 + 4536*a**8*b**4*x**2 + 18144*a**7*b**5*x**4 + 42336*a**6*b
**6*x**6 + 63504*a**5*b**7*x**8 + 63504*a**4*b**8*x**10 + 42336*a**3*b**9*x**12 + 18144*a**2*b**10*x**14 + 453
6*a*b**11*x**16 + 504*b**12*x**18)

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