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

Optimal. Leaf size=34 \[ \frac{(b B-A c) \log \left (b+c x^2\right )}{2 b c}+\frac{A \log (x)}{b} \]

[Out]

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

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Rubi [A]  time = 0.0407605, antiderivative size = 34, 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, 72} \[ \frac{(b B-A c) \log \left (b+c x^2\right )}{2 b c}+\frac{A \log (x)}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

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

Mathematica [A]  time = 0.0118481, size = 34, normalized size = 1. \[ \frac{(b B-A c) \log \left (b+c x^2\right )}{2 b c}+\frac{A \log (x)}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.004, size = 37, normalized size = 1.1 \begin{align*}{\frac{A\ln \left ( x \right ) }{b}}-{\frac{\ln \left ( c{x}^{2}+b \right ) A}{2\,b}}+{\frac{\ln \left ( c{x}^{2}+b \right ) B}{2\,c}} \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),x)

[Out]

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

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Maxima [A]  time = 1.2096, size = 47, normalized size = 1.38 \begin{align*} \frac{A \log \left (x^{2}\right )}{2 \, b} + \frac{{\left (B b - A c\right )} \log \left (c x^{2} + b\right )}{2 \, b c} \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),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.772343, size = 74, normalized size = 2.18 \begin{align*} \frac{2 \, A c \log \left (x\right ) +{\left (B b - A c\right )} \log \left (c x^{2} + b\right )}{2 \, b c} \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),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.65411, size = 26, normalized size = 0.76 \begin{align*} \frac{A \log{\left (x \right )}}{b} + \frac{\left (- A c + B b\right ) \log{\left (\frac{b}{c} + x^{2} \right )}}{2 b c} \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),x)

[Out]

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

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Giac [A]  time = 1.25781, size = 46, normalized size = 1.35 \begin{align*} \frac{A \log \left ({\left | x \right |}\right )}{b} + \frac{{\left (B b - A c\right )} \log \left ({\left | c x^{2} + b \right |}\right )}{2 \, b c} \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),x, algorithm="giac")

[Out]

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