\(\int \frac {(d x)^m}{(a^2+2 a b x^2+b^2 x^4)^3} \, dx\) [790]

Optimal result

Integrand size = 26, antiderivative size = 44 \[ \int \frac {(d x)^m}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {(d x)^{1+m} \operatorname {Hypergeometric2F1}\left (6,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )}{a^6 d (1+m)} \]

[Out]

(d*x)^(1+m)*hypergeom([6, 1/2+1/2*m],[3/2+1/2*m],-b*x^2/a)/a^6/d/(1+m)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {28, 371} \[ \int \frac {(d x)^m}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {(d x)^{m+1} \operatorname {Hypergeometric2F1}\left (6,\frac {m+1}{2},\frac {m+3}{2},-\frac {b x^2}{a}\right )}{a^6 d (m+1)} \]

[In]

Int[(d*x)^m/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

((d*x)^(1 + m)*Hypergeometric2F1[6, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(a^6*d*(1 + m))

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps \begin{align*} \text {integral}& = b^6 \int \frac {(d x)^m}{\left (a b+b^2 x^2\right )^6} \, dx \\ & = \frac {(d x)^{1+m} \, _2F_1\left (6,\frac {1+m}{2};\frac {3+m}{2};-\frac {b x^2}{a}\right )}{a^6 d (1+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.95 \[ \int \frac {(d x)^m}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {x (d x)^m \operatorname {Hypergeometric2F1}\left (6,\frac {1+m}{2},1+\frac {1+m}{2},-\frac {b x^2}{a}\right )}{a^6 (1+m)} \]

[In]

Integrate[(d*x)^m/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(x*(d*x)^m*Hypergeometric2F1[6, (1 + m)/2, 1 + (1 + m)/2, -((b*x^2)/a)])/(a^6*(1 + m))

Maple [F]

\[\int \frac {\left (d x \right )^{m}}{\left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{3}}d x\]

[In]

int((d*x)^m/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

int((d*x)^m/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

Fricas [F]

\[ \int \frac {(d x)^m}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\int { \frac {\left (d x\right )^{m}}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{3}} \,d x } \]

[In]

integrate((d*x)^m/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")

[Out]

integral((d*x)^m/(b^6*x^12 + 6*a*b^5*x^10 + 15*a^2*b^4*x^8 + 20*a^3*b^3*x^6 + 15*a^4*b^2*x^4 + 6*a^5*b*x^2 + a
^6), x)

Sympy [F]

\[ \int \frac {(d x)^m}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\int \frac {\left (d x\right )^{m}}{\left (a + b x^{2}\right )^{6}}\, dx \]

[In]

integrate((d*x)**m/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

Integral((d*x)**m/(a + b*x**2)**6, x)

Maxima [F]

\[ \int \frac {(d x)^m}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\int { \frac {\left (d x\right )^{m}}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{3}} \,d x } \]

[In]

integrate((d*x)^m/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")

[Out]

integrate((d*x)^m/(b^2*x^4 + 2*a*b*x^2 + a^2)^3, x)

Giac [F]

\[ \int \frac {(d x)^m}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\int { \frac {\left (d x\right )^{m}}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{3}} \,d x } \]

[In]

integrate((d*x)^m/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")

[Out]

integrate((d*x)^m/(b^2*x^4 + 2*a*b*x^2 + a^2)^3, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(d x)^m}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\int \frac {{\left (d\,x\right )}^m}{{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^3} \,d x \]

[In]

int((d*x)^m/(a^2 + b^2*x^4 + 2*a*b*x^2)^3,x)

[Out]

int((d*x)^m/(a^2 + b^2*x^4 + 2*a*b*x^2)^3, x)