3.201 \(\int \frac{x^7}{(a+b x^2)^{10}} \, dx\)

Optimal. Leaf size=72 \[ \frac{a^3}{18 b^4 \left (a+b x^2\right )^9}-\frac{3 a^2}{16 b^4 \left (a+b x^2\right )^8}+\frac{3 a}{14 b^4 \left (a+b x^2\right )^7}-\frac{1}{12 b^4 \left (a+b x^2\right )^6} \]

[Out]

a^3/(18*b^4*(a + b*x^2)^9) - (3*a^2)/(16*b^4*(a + b*x^2)^8) + (3*a)/(14*b^4*(a + b*x^2)^7) - 1/(12*b^4*(a + b*
x^2)^6)

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Rubi [A]  time = 0.0530002, antiderivative size = 72, 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} \[ \frac{a^3}{18 b^4 \left (a+b x^2\right )^9}-\frac{3 a^2}{16 b^4 \left (a+b x^2\right )^8}+\frac{3 a}{14 b^4 \left (a+b x^2\right )^7}-\frac{1}{12 b^4 \left (a+b x^2\right )^6} \]

Antiderivative was successfully verified.

[In]

Int[x^7/(a + b*x^2)^10,x]

[Out]

a^3/(18*b^4*(a + b*x^2)^9) - (3*a^2)/(16*b^4*(a + b*x^2)^8) + (3*a)/(14*b^4*(a + b*x^2)^7) - 1/(12*b^4*(a + b*
x^2)^6)

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^7}{\left (a+b x^2\right )^{10}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^3}{(a+b x)^{10}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a^3}{b^3 (a+b x)^{10}}+\frac{3 a^2}{b^3 (a+b x)^9}-\frac{3 a}{b^3 (a+b x)^8}+\frac{1}{b^3 (a+b x)^7}\right ) \, dx,x,x^2\right )\\ &=\frac{a^3}{18 b^4 \left (a+b x^2\right )^9}-\frac{3 a^2}{16 b^4 \left (a+b x^2\right )^8}+\frac{3 a}{14 b^4 \left (a+b x^2\right )^7}-\frac{1}{12 b^4 \left (a+b x^2\right )^6}\\ \end{align*}

Mathematica [A]  time = 0.0105129, size = 46, normalized size = 0.64 \[ -\frac{9 a^2 b x^2+a^3+36 a b^2 x^4+84 b^3 x^6}{1008 b^4 \left (a+b x^2\right )^9} \]

Antiderivative was successfully verified.

[In]

Integrate[x^7/(a + b*x^2)^10,x]

[Out]

-(a^3 + 9*a^2*b*x^2 + 36*a*b^2*x^4 + 84*b^3*x^6)/(1008*b^4*(a + b*x^2)^9)

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Maple [A]  time = 0.008, size = 65, normalized size = 0.9 \begin{align*}{\frac{{a}^{3}}{18\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{3\,{a}^{2}}{16\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{8}}}+{\frac{3\,a}{14\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{7}}}-{\frac{1}{12\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(b*x^2+a)^10,x)

[Out]

1/18*a^3/b^4/(b*x^2+a)^9-3/16*a^2/b^4/(b*x^2+a)^8+3/14*a/b^4/(b*x^2+a)^7-1/12/b^4/(b*x^2+a)^6

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Maxima [B]  time = 2.81793, size = 182, normalized size = 2.53 \begin{align*} -\frac{84 \, b^{3} x^{6} + 36 \, a b^{2} x^{4} + 9 \, a^{2} b x^{2} + a^{3}}{1008 \,{\left (b^{13} x^{18} + 9 \, a b^{12} x^{16} + 36 \, a^{2} b^{11} x^{14} + 84 \, a^{3} b^{10} x^{12} + 126 \, a^{4} b^{9} x^{10} + 126 \, a^{5} b^{8} x^{8} + 84 \, a^{6} b^{7} x^{6} + 36 \, a^{7} b^{6} x^{4} + 9 \, a^{8} b^{5} x^{2} + a^{9} b^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(b*x^2+a)^10,x, algorithm="maxima")

[Out]

-1/1008*(84*b^3*x^6 + 36*a*b^2*x^4 + 9*a^2*b*x^2 + a^3)/(b^13*x^18 + 9*a*b^12*x^16 + 36*a^2*b^11*x^14 + 84*a^3
*b^10*x^12 + 126*a^4*b^9*x^10 + 126*a^5*b^8*x^8 + 84*a^6*b^7*x^6 + 36*a^7*b^6*x^4 + 9*a^8*b^5*x^2 + a^9*b^4)

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Fricas [B]  time = 1.19305, size = 297, normalized size = 4.12 \begin{align*} -\frac{84 \, b^{3} x^{6} + 36 \, a b^{2} x^{4} + 9 \, a^{2} b x^{2} + a^{3}}{1008 \,{\left (b^{13} x^{18} + 9 \, a b^{12} x^{16} + 36 \, a^{2} b^{11} x^{14} + 84 \, a^{3} b^{10} x^{12} + 126 \, a^{4} b^{9} x^{10} + 126 \, a^{5} b^{8} x^{8} + 84 \, a^{6} b^{7} x^{6} + 36 \, a^{7} b^{6} x^{4} + 9 \, a^{8} b^{5} x^{2} + a^{9} b^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(b*x^2+a)^10,x, algorithm="fricas")

[Out]

-1/1008*(84*b^3*x^6 + 36*a*b^2*x^4 + 9*a^2*b*x^2 + a^3)/(b^13*x^18 + 9*a*b^12*x^16 + 36*a^2*b^11*x^14 + 84*a^3
*b^10*x^12 + 126*a^4*b^9*x^10 + 126*a^5*b^8*x^8 + 84*a^6*b^7*x^6 + 36*a^7*b^6*x^4 + 9*a^8*b^5*x^2 + a^9*b^4)

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Sympy [B]  time = 7.12123, size = 143, normalized size = 1.99 \begin{align*} - \frac{a^{3} + 9 a^{2} b x^{2} + 36 a b^{2} x^{4} + 84 b^{3} x^{6}}{1008 a^{9} b^{4} + 9072 a^{8} b^{5} x^{2} + 36288 a^{7} b^{6} x^{4} + 84672 a^{6} b^{7} x^{6} + 127008 a^{5} b^{8} x^{8} + 127008 a^{4} b^{9} x^{10} + 84672 a^{3} b^{10} x^{12} + 36288 a^{2} b^{11} x^{14} + 9072 a b^{12} x^{16} + 1008 b^{13} x^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7/(b*x**2+a)**10,x)

[Out]

-(a**3 + 9*a**2*b*x**2 + 36*a*b**2*x**4 + 84*b**3*x**6)/(1008*a**9*b**4 + 9072*a**8*b**5*x**2 + 36288*a**7*b**
6*x**4 + 84672*a**6*b**7*x**6 + 127008*a**5*b**8*x**8 + 127008*a**4*b**9*x**10 + 84672*a**3*b**10*x**12 + 3628
8*a**2*b**11*x**14 + 9072*a*b**12*x**16 + 1008*b**13*x**18)

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Giac [A]  time = 2.53198, size = 59, normalized size = 0.82 \begin{align*} -\frac{84 \, b^{3} x^{6} + 36 \, a b^{2} x^{4} + 9 \, a^{2} b x^{2} + a^{3}}{1008 \,{\left (b x^{2} + a\right )}^{9} b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(b*x^2+a)^10,x, algorithm="giac")

[Out]

-1/1008*(84*b^3*x^6 + 36*a*b^2*x^4 + 9*a^2*b*x^2 + a^3)/((b*x^2 + a)^9*b^4)