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

Optimal. Leaf size=38 \[ \frac{b c \log \left (a+b x^2\right )}{2 a^2}-\frac{b c \log (x)}{a^2}-\frac{c}{2 a x^2} \]

[Out]

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

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Rubi [A]  time = 0.0237782, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {21, 266, 44} \[ \frac{b c \log \left (a+b x^2\right )}{2 a^2}-\frac{b c \log (x)}{a^2}-\frac{c}{2 a x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0068979, size = 37, normalized size = 0.97 \[ c \left (\frac{b \log \left (a+b x^2\right )}{2 a^2}-\frac{b \log (x)}{a^2}-\frac{1}{2 a x^2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

c*(-1/(2*a*x^2) - (b*Log[x])/a^2 + (b*Log[a + b*x^2])/(2*a^2))

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Maple [A]  time = 0.007, size = 35, normalized size = 0.9 \begin{align*} -{\frac{c}{2\,a{x}^{2}}}-{\frac{bc\ln \left ( x \right ) }{{a}^{2}}}+{\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((b*c*x^2+a*c)/x^3/(b*x^2+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.03755, size = 49, normalized size = 1.29 \begin{align*} \frac{b c \log \left (b x^{2} + a\right )}{2 \, a^{2}} - \frac{b c \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((b*c*x^2+a*c)/x^3/(b*x^2+a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.25337, size = 88, normalized size = 2.32 \begin{align*} \frac{b c x^{2} \log \left (b x^{2} + a\right ) - 2 \, b c 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((b*c*x^2+a*c)/x^3/(b*x^2+a)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.408551, size = 32, normalized size = 0.84 \begin{align*} c \left (- \frac{1}{2 a x^{2}} - \frac{b \log{\left (x \right )}}{a^{2}} + \frac{b \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*c*x**2+a*c)/x**3/(b*x**2+a)**2,x)

[Out]

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

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Giac [A]  time = 1.18102, size = 63, normalized size = 1.66 \begin{align*} -\frac{b c \log \left (x^{2}\right )}{2 \, a^{2}} + \frac{b c \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{2}} + \frac{b c x^{2} - a c}{2 \, a^{2} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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