3.171 \(\int \frac {x^9}{(a+b x^2)^3} \, dx\)

Optimal. Leaf size=74 \[ -\frac {a^4}{4 b^5 \left (a+b x^2\right )^2}+\frac {2 a^3}{b^5 \left (a+b x^2\right )}+\frac {3 a^2 \log \left (a+b x^2\right )}{b^5}-\frac {3 a x^2}{2 b^4}+\frac {x^4}{4 b^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 74, 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} \[ -\frac {a^4}{4 b^5 \left (a+b x^2\right )^2}+\frac {2 a^3}{b^5 \left (a+b x^2\right )}+\frac {3 a^2 \log \left (a+b x^2\right )}{b^5}-\frac {3 a x^2}{2 b^4}+\frac {x^4}{4 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^9/(a + b*x^2)^3,x]

[Out]

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

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^9}{\left (a+b x^2\right )^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^4}{(a+b x)^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {3 a}{b^4}+\frac {x}{b^3}+\frac {a^4}{b^4 (a+b x)^3}-\frac {4 a^3}{b^4 (a+b x)^2}+\frac {6 a^2}{b^4 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {3 a x^2}{2 b^4}+\frac {x^4}{4 b^3}-\frac {a^4}{4 b^5 \left (a+b x^2\right )^2}+\frac {2 a^3}{b^5 \left (a+b x^2\right )}+\frac {3 a^2 \log \left (a+b x^2\right )}{b^5}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 63, normalized size = 0.85 \[ \frac {-\frac {a^4}{\left (a+b x^2\right )^2}+\frac {8 a^3}{a+b x^2}+12 a^2 \log \left (a+b x^2\right )-6 a b x^2+b^2 x^4}{4 b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^9/(a + b*x^2)^3,x]

[Out]

(-6*a*b*x^2 + b^2*x^4 - a^4/(a + b*x^2)^2 + (8*a^3)/(a + b*x^2) + 12*a^2*Log[a + b*x^2])/(4*b^5)

________________________________________________________________________________________

fricas [A]  time = 0.79, size = 103, normalized size = 1.39 \[ \frac {b^{4} x^{8} - 4 \, a b^{3} x^{6} - 11 \, a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + 7 \, a^{4} + 12 \, {\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )} \log \left (b x^{2} + a\right )}{4 \, {\left (b^{7} x^{4} + 2 \, a b^{6} x^{2} + a^{2} b^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(b*x^2+a)^3,x, algorithm="fricas")

[Out]

1/4*(b^4*x^8 - 4*a*b^3*x^6 - 11*a^2*b^2*x^4 + 2*a^3*b*x^2 + 7*a^4 + 12*(a^2*b^2*x^4 + 2*a^3*b*x^2 + a^4)*log(b
*x^2 + a))/(b^7*x^4 + 2*a*b^6*x^2 + a^2*b^5)

________________________________________________________________________________________

giac [A]  time = 0.61, size = 80, normalized size = 1.08 \[ \frac {3 \, a^{2} \log \left ({\left | b x^{2} + a \right |}\right )}{b^{5}} + \frac {b^{3} x^{4} - 6 \, a b^{2} x^{2}}{4 \, b^{6}} - \frac {18 \, a^{2} b^{2} x^{4} + 28 \, a^{3} b x^{2} + 11 \, a^{4}}{4 \, {\left (b x^{2} + a\right )}^{2} b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(b*x^2+a)^3,x, algorithm="giac")

[Out]

3*a^2*log(abs(b*x^2 + a))/b^5 + 1/4*(b^3*x^4 - 6*a*b^2*x^2)/b^6 - 1/4*(18*a^2*b^2*x^4 + 28*a^3*b*x^2 + 11*a^4)
/((b*x^2 + a)^2*b^5)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 69, normalized size = 0.93 \[ \frac {x^{4}}{4 b^{3}}-\frac {a^{4}}{4 \left (b \,x^{2}+a \right )^{2} b^{5}}-\frac {3 a \,x^{2}}{2 b^{4}}+\frac {2 a^{3}}{\left (b \,x^{2}+a \right ) b^{5}}+\frac {3 a^{2} \ln \left (b \,x^{2}+a \right )}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9/(b*x^2+a)^3,x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.34, size = 77, normalized size = 1.04 \[ \frac {8 \, a^{3} b x^{2} + 7 \, a^{4}}{4 \, {\left (b^{7} x^{4} + 2 \, a b^{6} x^{2} + a^{2} b^{5}\right )}} + \frac {3 \, a^{2} \log \left (b x^{2} + a\right )}{b^{5}} + \frac {b x^{4} - 6 \, a x^{2}}{4 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(b*x^2+a)^3,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.08, size = 78, normalized size = 1.05 \[ \frac {\frac {7\,a^4}{4\,b}+2\,a^3\,x^2}{a^2\,b^4+2\,a\,b^5\,x^2+b^6\,x^4}+\frac {x^4}{4\,b^3}-\frac {3\,a\,x^2}{2\,b^4}+\frac {3\,a^2\,\ln \left (b\,x^2+a\right )}{b^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9/(a + b*x^2)^3,x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.42, size = 78, normalized size = 1.05 \[ \frac {3 a^{2} \log {\left (a + b x^{2} \right )}}{b^{5}} - \frac {3 a x^{2}}{2 b^{4}} + \frac {7 a^{4} + 8 a^{3} b x^{2}}{4 a^{2} b^{5} + 8 a b^{6} x^{2} + 4 b^{7} x^{4}} + \frac {x^{4}}{4 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**9/(b*x**2+a)**3,x)

[Out]

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

________________________________________________________________________________________