3.80 \(\int \frac{x^7 (A+B x^2)}{(b x^2+c x^4)^3} \, dx\)

Optimal. Leaf size=32 \[ \frac{\left (A+B x^2\right )^2}{4 \left (b+c x^2\right )^2 (b B-A c)} \]

[Out]

(A + B*x^2)^2/(4*(b*B - A*c)*(b + c*x^2)^2)

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Rubi [A]  time = 0.0307202, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1584, 444, 37} \[ \frac{\left (A+B x^2\right )^2}{4 \left (b+c x^2\right )^2 (b B-A c)} \]

Antiderivative was successfully verified.

[In]

Int[(x^7*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]

[Out]

(A + B*x^2)^2/(4*(b*B - A*c)*(b + c*x^2)^2)

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{x^7 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac{x \left (A+B x^2\right )}{\left (b+c x^2\right )^3} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{A+B x}{(b+c x)^3} \, dx,x,x^2\right )\\ &=\frac{\left (A+B x^2\right )^2}{4 (b B-A c) \left (b+c x^2\right )^2}\\ \end{align*}

Mathematica [A]  time = 0.0130782, size = 30, normalized size = 0.94 \[ -\frac{c \left (A+2 B x^2\right )+b B}{4 c^2 \left (b+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^7*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]

[Out]

-(b*B + c*(A + 2*B*x^2))/(4*c^2*(b + c*x^2)^2)

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Maple [A]  time = 0.007, size = 39, normalized size = 1.2 \begin{align*} -{\frac{B}{2\,{c}^{2} \left ( c{x}^{2}+b \right ) }}-{\frac{Ac-Bb}{4\,{c}^{2} \left ( c{x}^{2}+b \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(B*x^2+A)/(c*x^4+b*x^2)^3,x)

[Out]

-1/2*B/c^2/(c*x^2+b)-1/4*(A*c-B*b)/c^2/(c*x^2+b)^2

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Maxima [A]  time = 1.14028, size = 57, normalized size = 1.78 \begin{align*} -\frac{2 \, B c x^{2} + B b + A c}{4 \,{\left (c^{4} x^{4} + 2 \, b c^{3} x^{2} + b^{2} c^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(B*x^2+A)/(c*x^4+b*x^2)^3,x, algorithm="maxima")

[Out]

-1/4*(2*B*c*x^2 + B*b + A*c)/(c^4*x^4 + 2*b*c^3*x^2 + b^2*c^2)

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Fricas [A]  time = 1.01252, size = 86, normalized size = 2.69 \begin{align*} -\frac{2 \, B c x^{2} + B b + A c}{4 \,{\left (c^{4} x^{4} + 2 \, b c^{3} x^{2} + b^{2} c^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(B*x^2+A)/(c*x^4+b*x^2)^3,x, algorithm="fricas")

[Out]

-1/4*(2*B*c*x^2 + B*b + A*c)/(c^4*x^4 + 2*b*c^3*x^2 + b^2*c^2)

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Sympy [A]  time = 0.672229, size = 42, normalized size = 1.31 \begin{align*} - \frac{A c + B b + 2 B c x^{2}}{4 b^{2} c^{2} + 8 b c^{3} x^{2} + 4 c^{4} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7*(B*x**2+A)/(c*x**4+b*x**2)**3,x)

[Out]

-(A*c + B*b + 2*B*c*x**2)/(4*b**2*c**2 + 8*b*c**3*x**2 + 4*c**4*x**4)

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Giac [A]  time = 1.14302, size = 38, normalized size = 1.19 \begin{align*} -\frac{2 \, B c x^{2} + B b + A c}{4 \,{\left (c x^{2} + b\right )}^{2} c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(B*x^2+A)/(c*x^4+b*x^2)^3,x, algorithm="giac")

[Out]

-1/4*(2*B*c*x^2 + B*b + A*c)/((c*x^2 + b)^2*c^2)