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

Optimal. Leaf size=121 \[ -\frac{c (2 b B-3 A c)}{2 b^4 \left (b+c x^2\right )}-\frac{b B-3 A c}{2 b^4 x^2}-\frac{c (b B-A c)}{4 b^3 \left (b+c x^2\right )^2}+\frac{3 c (b B-2 A c) \log \left (b+c x^2\right )}{2 b^5}-\frac{3 c \log (x) (b B-2 A c)}{b^5}-\frac{A}{4 b^3 x^4} \]

[Out]

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

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Rubi [A]  time = 0.1309, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {1584, 446, 77} \[ -\frac{c (2 b B-3 A c)}{2 b^4 \left (b+c x^2\right )}-\frac{b B-3 A c}{2 b^4 x^2}-\frac{c (b B-A c)}{4 b^3 \left (b+c x^2\right )^2}+\frac{3 c (b B-2 A c) \log \left (b+c x^2\right )}{2 b^5}-\frac{3 c \log (x) (b B-2 A c)}{b^5}-\frac{A}{4 b^3 x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(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] && IntegerQ[Simplify[(m + 1)/n]]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

Mathematica [A]  time = 0.0769802, size = 108, normalized size = 0.89 \[ \frac{\frac{b^2 c (A c-b B)}{\left (b+c x^2\right )^2}-\frac{A b^2}{x^4}+\frac{2 b c (3 A c-2 b B)}{b+c x^2}-\frac{2 b (b B-3 A c)}{x^2}+6 c (b B-2 A c) \log \left (b+c x^2\right )+12 c \log (x) (2 A c-b B)}{4 b^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-((A*b^2)/x^4) - (2*b*(b*B - 3*A*c))/x^2 + (b^2*c*(-(b*B) + A*c))/(b + c*x^2)^2 + (2*b*c*(-2*b*B + 3*A*c))/(b
+ c*x^2) + 12*c*(-(b*B) + 2*A*c)*Log[x] + 6*c*(b*B - 2*A*c)*Log[b + c*x^2])/(4*b^5)

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Maple [A]  time = 0.015, size = 150, normalized size = 1.2 \begin{align*} -{\frac{A}{4\,{b}^{3}{x}^{4}}}+{\frac{3\,Ac}{2\,{b}^{4}{x}^{2}}}-{\frac{B}{2\,{b}^{3}{x}^{2}}}+6\,{\frac{A\ln \left ( x \right ){c}^{2}}{{b}^{5}}}-3\,{\frac{Bc\ln \left ( x \right ) }{{b}^{4}}}-3\,{\frac{{c}^{2}\ln \left ( c{x}^{2}+b \right ) A}{{b}^{5}}}+{\frac{3\,c\ln \left ( c{x}^{2}+b \right ) B}{2\,{b}^{4}}}+{\frac{3\,A{c}^{2}}{2\,{b}^{4} \left ( c{x}^{2}+b \right ) }}-{\frac{Bc}{{b}^{3} \left ( c{x}^{2}+b \right ) }}+{\frac{A{c}^{2}}{4\,{b}^{3} \left ( c{x}^{2}+b \right ) ^{2}}}-{\frac{Bc}{4\,{b}^{2} \left ( c{x}^{2}+b \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*A/b^3/x^4+3/2/b^4/x^2*A*c-1/2/b^3/x^2*B+6*c^2/b^5*ln(x)*A-3*c/b^4*ln(x)*B-3/b^5*c^2*ln(c*x^2+b)*A+3/2/b^4
*c*ln(c*x^2+b)*B+3/2/b^4*c^2*A/(c*x^2+b)-1/b^3*c/(c*x^2+b)*B+1/4/b^3*c^2/(c*x^2+b)^2*A-1/4/b^2*c/(c*x^2+b)^2*B

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Maxima [A]  time = 1.1779, size = 185, normalized size = 1.53 \begin{align*} -\frac{6 \,{\left (B b c^{2} - 2 \, A c^{3}\right )} x^{6} + 9 \,{\left (B b^{2} c - 2 \, A b c^{2}\right )} x^{4} + A b^{3} + 2 \,{\left (B b^{3} - 2 \, A b^{2} c\right )} x^{2}}{4 \,{\left (b^{4} c^{2} x^{8} + 2 \, b^{5} c x^{6} + b^{6} x^{4}\right )}} + \frac{3 \,{\left (B b c - 2 \, A c^{2}\right )} \log \left (c x^{2} + b\right )}{2 \, b^{5}} - \frac{3 \,{\left (B b c - 2 \, A c^{2}\right )} \log \left (x^{2}\right )}{2 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(6*(B*b*c^2 - 2*A*c^3)*x^6 + 9*(B*b^2*c - 2*A*b*c^2)*x^4 + A*b^3 + 2*(B*b^3 - 2*A*b^2*c)*x^2)/(b^4*c^2*x^
8 + 2*b^5*c*x^6 + b^6*x^4) + 3/2*(B*b*c - 2*A*c^2)*log(c*x^2 + b)/b^5 - 3/2*(B*b*c - 2*A*c^2)*log(x^2)/b^5

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Fricas [B]  time = 1.10057, size = 474, normalized size = 3.92 \begin{align*} -\frac{6 \,{\left (B b^{2} c^{2} - 2 \, A b c^{3}\right )} x^{6} + A b^{4} + 9 \,{\left (B b^{3} c - 2 \, A b^{2} c^{2}\right )} x^{4} + 2 \,{\left (B b^{4} - 2 \, A b^{3} c\right )} x^{2} - 6 \,{\left ({\left (B b c^{3} - 2 \, A c^{4}\right )} x^{8} + 2 \,{\left (B b^{2} c^{2} - 2 \, A b c^{3}\right )} x^{6} +{\left (B b^{3} c - 2 \, A b^{2} c^{2}\right )} x^{4}\right )} \log \left (c x^{2} + b\right ) + 12 \,{\left ({\left (B b c^{3} - 2 \, A c^{4}\right )} x^{8} + 2 \,{\left (B b^{2} c^{2} - 2 \, A b c^{3}\right )} x^{6} +{\left (B b^{3} c - 2 \, A b^{2} c^{2}\right )} x^{4}\right )} \log \left (x\right )}{4 \,{\left (b^{5} c^{2} x^{8} + 2 \, b^{6} c x^{6} + b^{7} x^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(6*(B*b^2*c^2 - 2*A*b*c^3)*x^6 + A*b^4 + 9*(B*b^3*c - 2*A*b^2*c^2)*x^4 + 2*(B*b^4 - 2*A*b^3*c)*x^2 - 6*((
B*b*c^3 - 2*A*c^4)*x^8 + 2*(B*b^2*c^2 - 2*A*b*c^3)*x^6 + (B*b^3*c - 2*A*b^2*c^2)*x^4)*log(c*x^2 + b) + 12*((B*
b*c^3 - 2*A*c^4)*x^8 + 2*(B*b^2*c^2 - 2*A*b*c^3)*x^6 + (B*b^3*c - 2*A*b^2*c^2)*x^4)*log(x))/(b^5*c^2*x^8 + 2*b
^6*c*x^6 + b^7*x^4)

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Sympy [A]  time = 1.85757, size = 136, normalized size = 1.12 \begin{align*} - \frac{A b^{3} + x^{6} \left (- 12 A c^{3} + 6 B b c^{2}\right ) + x^{4} \left (- 18 A b c^{2} + 9 B b^{2} c\right ) + x^{2} \left (- 4 A b^{2} c + 2 B b^{3}\right )}{4 b^{6} x^{4} + 8 b^{5} c x^{6} + 4 b^{4} c^{2} x^{8}} - \frac{3 c \left (- 2 A c + B b\right ) \log{\left (x \right )}}{b^{5}} + \frac{3 c \left (- 2 A c + B b\right ) \log{\left (\frac{b}{c} + x^{2} \right )}}{2 b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(A*b**3 + x**6*(-12*A*c**3 + 6*B*b*c**2) + x**4*(-18*A*b*c**2 + 9*B*b**2*c) + x**2*(-4*A*b**2*c + 2*B*b**3))/
(4*b**6*x**4 + 8*b**5*c*x**6 + 4*b**4*c**2*x**8) - 3*c*(-2*A*c + B*b)*log(x)/b**5 + 3*c*(-2*A*c + B*b)*log(b/c
+ x**2)/(2*b**5)

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Giac [A]  time = 1.16258, size = 178, normalized size = 1.47 \begin{align*} -\frac{3 \,{\left (B b c - 2 \, A c^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5}} + \frac{3 \,{\left (B b c^{2} - 2 \, A c^{3}\right )} \log \left ({\left | c x^{2} + b \right |}\right )}{2 \, b^{5} c} - \frac{6 \, B b c^{2} x^{6} - 12 \, A c^{3} x^{6} + 9 \, B b^{2} c x^{4} - 18 \, A b c^{2} x^{4} + 2 \, B b^{3} x^{2} - 4 \, A b^{2} c x^{2} + A b^{3}}{4 \,{\left (c x^{4} + b x^{2}\right )}^{2} b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*(B*b*c - 2*A*c^2)*log(abs(x))/b^5 + 3/2*(B*b*c^2 - 2*A*c^3)*log(abs(c*x^2 + b))/(b^5*c) - 1/4*(6*B*b*c^2*x^
6 - 12*A*c^3*x^6 + 9*B*b^2*c*x^4 - 18*A*b*c^2*x^4 + 2*B*b^3*x^2 - 4*A*b^2*c*x^2 + A*b^3)/((c*x^4 + b*x^2)^2*b^
4)