3.203 \(\int \frac{x^3}{(a+b x^2)^{10}} \, dx\)

Optimal. Leaf size=34 \[ \frac{a}{18 b^2 \left (a+b x^2\right )^9}-\frac{1}{16 b^2 \left (a+b x^2\right )^8} \]

[Out]

a/(18*b^2*(a + b*x^2)^9) - 1/(16*b^2*(a + b*x^2)^8)

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Rubi [A]  time = 0.0251604, antiderivative size = 34, 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} \[ \frac{a}{18 b^2 \left (a+b x^2\right )^9}-\frac{1}{16 b^2 \left (a+b x^2\right )^8} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*x^2)^10,x]

[Out]

a/(18*b^2*(a + b*x^2)^9) - 1/(16*b^2*(a + b*x^2)^8)

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^3}{\left (a+b x^2\right )^{10}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{(a+b x)^{10}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a}{b (a+b x)^{10}}+\frac{1}{b (a+b x)^9}\right ) \, dx,x,x^2\right )\\ &=\frac{a}{18 b^2 \left (a+b x^2\right )^9}-\frac{1}{16 b^2 \left (a+b x^2\right )^8}\\ \end{align*}

Mathematica [A]  time = 0.0079257, size = 24, normalized size = 0.71 \[ -\frac{a+9 b x^2}{144 b^2 \left (a+b x^2\right )^9} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*x^2)^10,x]

[Out]

-(a + 9*b*x^2)/(144*b^2*(a + b*x^2)^9)

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Maple [A]  time = 0.007, size = 31, normalized size = 0.9 \begin{align*}{\frac{a}{18\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{1}{16\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{8}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(b*x^2+a)^10,x)

[Out]

1/18*a/b^2/(b*x^2+a)^9-1/16/b^2/(b*x^2+a)^8

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Maxima [B]  time = 2.60832, size = 153, normalized size = 4.5 \begin{align*} -\frac{9 \, b x^{2} + a}{144 \,{\left (b^{11} x^{18} + 9 \, a b^{10} x^{16} + 36 \, a^{2} b^{9} x^{14} + 84 \, a^{3} b^{8} x^{12} + 126 \, a^{4} b^{7} x^{10} + 126 \, a^{5} b^{6} x^{8} + 84 \, a^{6} b^{5} x^{6} + 36 \, a^{7} b^{4} x^{4} + 9 \, a^{8} b^{3} x^{2} + a^{9} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^2+a)^10,x, algorithm="maxima")

[Out]

-1/144*(9*b*x^2 + a)/(b^11*x^18 + 9*a*b^10*x^16 + 36*a^2*b^9*x^14 + 84*a^3*b^8*x^12 + 126*a^4*b^7*x^10 + 126*a
^5*b^6*x^8 + 84*a^6*b^5*x^6 + 36*a^7*b^4*x^4 + 9*a^8*b^3*x^2 + a^9*b^2)

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Fricas [B]  time = 1.19908, size = 247, normalized size = 7.26 \begin{align*} -\frac{9 \, b x^{2} + a}{144 \,{\left (b^{11} x^{18} + 9 \, a b^{10} x^{16} + 36 \, a^{2} b^{9} x^{14} + 84 \, a^{3} b^{8} x^{12} + 126 \, a^{4} b^{7} x^{10} + 126 \, a^{5} b^{6} x^{8} + 84 \, a^{6} b^{5} x^{6} + 36 \, a^{7} b^{4} x^{4} + 9 \, a^{8} b^{3} x^{2} + a^{9} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^2+a)^10,x, algorithm="fricas")

[Out]

-1/144*(9*b*x^2 + a)/(b^11*x^18 + 9*a*b^10*x^16 + 36*a^2*b^9*x^14 + 84*a^3*b^8*x^12 + 126*a^4*b^7*x^10 + 126*a
^5*b^6*x^8 + 84*a^6*b^5*x^6 + 36*a^7*b^4*x^4 + 9*a^8*b^3*x^2 + a^9*b^2)

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Sympy [B]  time = 6.96263, size = 119, normalized size = 3.5 \begin{align*} - \frac{a + 9 b x^{2}}{144 a^{9} b^{2} + 1296 a^{8} b^{3} x^{2} + 5184 a^{7} b^{4} x^{4} + 12096 a^{6} b^{5} x^{6} + 18144 a^{5} b^{6} x^{8} + 18144 a^{4} b^{7} x^{10} + 12096 a^{3} b^{8} x^{12} + 5184 a^{2} b^{9} x^{14} + 1296 a b^{10} x^{16} + 144 b^{11} x^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(b*x**2+a)**10,x)

[Out]

-(a + 9*b*x**2)/(144*a**9*b**2 + 1296*a**8*b**3*x**2 + 5184*a**7*b**4*x**4 + 12096*a**6*b**5*x**6 + 18144*a**5
*b**6*x**8 + 18144*a**4*b**7*x**10 + 12096*a**3*b**8*x**12 + 5184*a**2*b**9*x**14 + 1296*a*b**10*x**16 + 144*b
**11*x**18)

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Giac [A]  time = 1.69398, size = 30, normalized size = 0.88 \begin{align*} -\frac{9 \, b x^{2} + a}{144 \,{\left (b x^{2} + a\right )}^{9} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^2+a)^10,x, algorithm="giac")

[Out]

-1/144*(9*b*x^2 + a)/((b*x^2 + a)^9*b^2)