3.206 \(\int \frac{c+d x^2}{x^3 (a+b x^2)} \, dx\)

Optimal. Leaf size=50 \[ \frac{(b c-a d) \log \left (a+b x^2\right )}{2 a^2}-\frac{\log (x) (b c-a d)}{a^2}-\frac{c}{2 a x^2} \]

[Out]

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

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Rubi [A]  time = 0.045265, antiderivative size = 50, 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} \[ \frac{(b c-a d) \log \left (a+b x^2\right )}{2 a^2}-\frac{\log (x) (b c-a d)}{a^2}-\frac{c}{2 a x^2} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)/(x^3*(a + b*x^2)),x]

[Out]

-c/(2*a*x^2) - ((b*c - a*d)*Log[x])/a^2 + ((b*c - a*d)*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^3 \left (a+b x^2\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{c+d x}{x^2 (a+b x)} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{c}{a x^2}+\frac{-b c+a d}{a^2 x}-\frac{b (-b c+a d)}{a^2 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{c}{2 a x^2}-\frac{(b c-a d) \log (x)}{a^2}+\frac{(b c-a d) \log \left (a+b x^2\right )}{2 a^2}\\ \end{align*}

Mathematica [A]  time = 0.0201228, size = 49, normalized size = 0.98 \[ \frac{(b c-a d) \log \left (a+b x^2\right )}{2 a^2}+\frac{\log (x) (a d-b c)}{a^2}-\frac{c}{2 a x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.005, size = 56, normalized size = 1.1 \begin{align*} -{\frac{c}{2\,a{x}^{2}}}+{\frac{\ln \left ( x \right ) d}{a}}-{\frac{bc\ln \left ( x \right ) }{{a}^{2}}}-{\frac{\ln \left ( b{x}^{2}+a \right ) d}{2\,a}}+{\frac{bc\ln \left ( b{x}^{2}+a \right ) }{2\,{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.976455, size = 65, normalized size = 1.3 \begin{align*} \frac{{\left (b c - a d\right )} \log \left (b x^{2} + a\right )}{2 \, a^{2}} - \frac{{\left (b c - a d\right )} \log \left (x^{2}\right )}{2 \, a^{2}} - \frac{c}{2 \, a x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.4719, size = 109, normalized size = 2.18 \begin{align*} \frac{{\left (b c - a d\right )} x^{2} \log \left (b x^{2} + a\right ) - 2 \,{\left (b c - a d\right )} x^{2} \log \left (x\right ) - a c}{2 \, a^{2} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.782153, size = 41, normalized size = 0.82 \begin{align*} - \frac{c}{2 a x^{2}} + \frac{\left (a d - b c\right ) \log{\left (x \right )}}{a^{2}} - \frac{\left (a d - b c\right ) \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**3/(b*x**2+a),x)

[Out]

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

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Giac [A]  time = 1.14478, size = 97, normalized size = 1.94 \begin{align*} -\frac{{\left (b c - a d\right )} \log \left (x^{2}\right )}{2 \, a^{2}} + \frac{{\left (b^{2} c - a b d\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{2} b} + \frac{b c x^{2} - a d x^{2} - a c}{2 \, a^{2} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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