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]

-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)

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Rubi [A]  time = 0.0388855, antiderivative size = 53, normalized size of antiderivative = 1., 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 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]]

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])

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*}

Mathematica [A]  time = 0.0126977, 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]

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

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Maple [A]  time = 0.007, size = 48, normalized size = 0.9 \begin{align*} -{\frac{{a}^{2}}{18\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{a}{8\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{8}}}-{\frac{1}{14\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{7}}} \end{align*}

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 = 2.34149, size = 167, normalized size = 3.15 \begin{align*} -\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 )}} \end{align*}

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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Fricas [B]  time = 1.18631, size = 271, normalized size = 5.11 \begin{align*} -\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 )}} \end{align*}

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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Sympy [B]  time = 7.05524, size = 131, normalized size = 2.47 \begin{align*} - \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}} \end{align*}

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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Giac [A]  time = 2.32654, size = 45, normalized size = 0.85 \begin{align*} -\frac{36 \, b^{2} x^{4} + 9 \, a b x^{2} + a^{2}}{504 \,{\left (b x^{2} + a\right )}^{9} b^{3}} \end{align*}

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)