3.267 \(\int \frac{c+d x^2}{x (a+b x^2)^2} \, dx\)

Optimal. Leaf size=51 \[ -\frac{c \log \left (a+b x^2\right )}{2 a^2}+\frac{c \log (x)}{a^2}+\frac{b c-a d}{2 a b \left (a+b x^2\right )} \]

[Out]

(b*c - a*d)/(2*a*b*(a + b*x^2)) + (c*Log[x])/a^2 - (c*Log[a + b*x^2])/(2*a^2)

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Rubi [A]  time = 0.0441073, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {446, 77} \[ -\frac{c \log \left (a+b x^2\right )}{2 a^2}+\frac{c \log (x)}{a^2}+\frac{b c-a d}{2 a b \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)/(x*(a + b*x^2)^2),x]

[Out]

(b*c - a*d)/(2*a*b*(a + b*x^2)) + (c*Log[x])/a^2 - (c*Log[a + b*x^2])/(2*a^2)

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

Mathematica [A]  time = 0.0284382, size = 46, normalized size = 0.9 \[ \frac{\frac{a (b c-a d)}{b \left (a+b x^2\right )}-c \log \left (a+b x^2\right )+2 c \log (x)}{2 a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2)/(x*(a + b*x^2)^2),x]

[Out]

((a*(b*c - a*d))/(b*(a + b*x^2)) + 2*c*Log[x] - c*Log[a + b*x^2])/(2*a^2)

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Maple [A]  time = 0.012, size = 53, normalized size = 1. \begin{align*}{\frac{c\ln \left ( x \right ) }{{a}^{2}}}-{\frac{c\ln \left ( b{x}^{2}+a \right ) }{2\,{a}^{2}}}-{\frac{d}{2\,b \left ( b{x}^{2}+a \right ) }}+{\frac{c}{2\,a \left ( b{x}^{2}+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^2+c)/x/(b*x^2+a)^2,x)

[Out]

c*ln(x)/a^2-1/2*c*ln(b*x^2+a)/a^2-1/2/b/(b*x^2+a)*d+1/2*c/a/(b*x^2+a)

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Maxima [A]  time = 1.11155, size = 69, normalized size = 1.35 \begin{align*} \frac{b c - a d}{2 \,{\left (a b^{2} x^{2} + a^{2} b\right )}} - \frac{c \log \left (b x^{2} + a\right )}{2 \, a^{2}} + \frac{c \log \left (x^{2}\right )}{2 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b*c - a*d)/(a*b^2*x^2 + a^2*b) - 1/2*c*log(b*x^2 + a)/a^2 + 1/2*c*log(x^2)/a^2

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Fricas [A]  time = 1.49893, size = 150, normalized size = 2.94 \begin{align*} \frac{a b c - a^{2} d -{\left (b^{2} c x^{2} + a b c\right )} \log \left (b x^{2} + a\right ) + 2 \,{\left (b^{2} c x^{2} + a b c\right )} \log \left (x\right )}{2 \,{\left (a^{2} b^{2} x^{2} + a^{3} b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(a*b*c - a^2*d - (b^2*c*x^2 + a*b*c)*log(b*x^2 + a) + 2*(b^2*c*x^2 + a*b*c)*log(x))/(a^2*b^2*x^2 + a^3*b)

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Sympy [A]  time = 0.550187, size = 46, normalized size = 0.9 \begin{align*} - \frac{a d - b c}{2 a^{2} b + 2 a b^{2} x^{2}} + \frac{c \log{\left (x \right )}}{a^{2}} - \frac{c \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**2+c)/x/(b*x**2+a)**2,x)

[Out]

-(a*d - b*c)/(2*a**2*b + 2*a*b**2*x**2) + c*log(x)/a**2 - c*log(a/b + x**2)/(2*a**2)

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Giac [A]  time = 1.12414, size = 85, normalized size = 1.67 \begin{align*} \frac{c \log \left (x^{2}\right )}{2 \, a^{2}} - \frac{c \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{2}} + \frac{b^{2} c x^{2} + 2 \, a b c - a^{2} d}{2 \,{\left (b x^{2} + a\right )} a^{2} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*c*log(x^2)/a^2 - 1/2*c*log(abs(b*x^2 + a))/a^2 + 1/2*(b^2*c*x^2 + 2*a*b*c - a^2*d)/((b*x^2 + a)*a^2*b)