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

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

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {446, 77} \begin {gather*} -\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 )} \end {gather*}

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 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]))))

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]]

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 = 0.03, size = 46, normalized size = 0.90 \begin {gather*} \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} \end {gather*}

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)

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {c+d x^2}{x \left (a+b x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.84, size = 71, normalized size = 1.39 \begin {gather*} \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 \relax (x)}{2 \, {\left (a^{2} b^{2} x^{2} + a^{3} b\right )}} \end {gather*}

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)

________________________________________________________________________________________

giac [A]  time = 0.32, size = 63, normalized size = 1.24 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 53, normalized size = 1.04 \begin {gather*} \frac {c}{2 \left (b \,x^{2}+a \right ) a}+\frac {c \ln \relax (x )}{a^{2}}-\frac {c \ln \left (b \,x^{2}+a \right )}{2 a^{2}}-\frac {d}{2 \left (b \,x^{2}+a \right ) b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.07, size = 51, normalized size = 1.00 \begin {gather*} \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 {gather*}

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

________________________________________________________________________________________

mupad [B]  time = 0.06, size = 47, normalized size = 0.92 \begin {gather*} \frac {c\,\ln \relax (x)}{a^2}-\frac {c\,\ln \left (b\,x^2+a\right )}{2\,a^2}-\frac {a\,d-b\,c}{2\,a\,b\,\left (b\,x^2+a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^2)/(x*(a + b*x^2)^2),x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.41, size = 46, normalized size = 0.90 \begin {gather*} \frac {- a d + b c}{2 a^{2} b + 2 a b^{2} x^{2}} + \frac {c \log {\relax (x )}}{a^{2}} - \frac {c \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{2}} \end {gather*}

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)

________________________________________________________________________________________