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

Optimal. Leaf size=53 \[ \frac {b c \log \left (a+b x^2\right )}{a^3}-\frac {2 b c \log (x)}{a^3}-\frac {b c}{2 a^2 \left (a+b x^2\right )}-\frac {c}{2 a^2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {21, 266, 44} \[ -\frac {b c}{2 a^2 \left (a+b x^2\right )}+\frac {b c \log \left (a+b x^2\right )}{a^3}-\frac {2 b c \log (x)}{a^3}-\frac {c}{2 a^2 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-c/(2*a^2*x^2) - (b*c)/(2*a^2*(a + b*x^2)) - (2*b*c*Log[x])/a^3 + (b*c*Log[a + b*x^2])/a^3

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 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])

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]]

Rubi steps

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

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Mathematica [A]  time = 0.04, size = 42, normalized size = 0.79 \[ -\frac {c \left (a \left (\frac {b}{a+b x^2}+\frac {1}{x^2}\right )-2 b \log \left (a+b x^2\right )+4 b \log (x)\right )}{2 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(c*(a*(x^(-2) + b/(a + b*x^2)) + 4*b*Log[x] - 2*b*Log[a + b*x^2]))/a^3

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fricas [A]  time = 0.46, size = 80, normalized size = 1.51 \[ -\frac {2 \, a b c x^{2} + a^{2} c - 2 \, {\left (b^{2} c x^{4} + a b c x^{2}\right )} \log \left (b x^{2} + a\right ) + 4 \, {\left (b^{2} c x^{4} + a b c x^{2}\right )} \log \relax (x)}{2 \, {\left (a^{3} b x^{4} + a^{4} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*a*b*c*x^2 + a^2*c - 2*(b^2*c*x^4 + a*b*c*x^2)*log(b*x^2 + a) + 4*(b^2*c*x^4 + a*b*c*x^2)*log(x))/(a^3*
b*x^4 + a^4*x^2)

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giac [A]  time = 0.30, size = 56, normalized size = 1.06 \[ -\frac {b c \log \left (x^{2}\right )}{a^{3}} + \frac {b c \log \left ({\left | b x^{2} + a \right |}\right )}{a^{3}} - \frac {2 \, b c x^{2} + a c}{2 \, {\left (b x^{4} + a x^{2}\right )} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b*c*log(x^2)/a^3 + b*c*log(abs(b*x^2 + a))/a^3 - 1/2*(2*b*c*x^2 + a*c)/((b*x^4 + a*x^2)*a^2)

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maple [A]  time = 0.02, size = 50, normalized size = 0.94 \[ -\frac {b c}{2 \left (b \,x^{2}+a \right ) a^{2}}-\frac {2 b c \ln \relax (x )}{a^{3}}+\frac {b c \ln \left (b \,x^{2}+a \right )}{a^{3}}-\frac {c}{2 a^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.05, size = 57, normalized size = 1.08 \[ -\frac {2 \, b c x^{2} + a c}{2 \, {\left (a^{2} b x^{4} + a^{3} x^{2}\right )}} + \frac {b c \log \left (b x^{2} + a\right )}{a^{3}} - \frac {b c \log \left (x^{2}\right )}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*b*c*x^2 + a*c)/(a^2*b*x^4 + a^3*x^2) + b*c*log(b*x^2 + a)/a^3 - b*c*log(x^2)/a^3

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mupad [B]  time = 0.10, size = 55, normalized size = 1.04 \[ \frac {b\,c\,\ln \left (b\,x^2+a\right )}{a^3}-\frac {\frac {c}{2\,a}+\frac {b\,c\,x^2}{a^2}}{b\,x^4+a\,x^2}-\frac {2\,b\,c\,\ln \relax (x)}{a^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*c*log(a + b*x^2))/a^3 - (c/(2*a) + (b*c*x^2)/a^2)/(a*x^2 + b*x^4) - (2*b*c*log(x))/a^3

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sympy [A]  time = 0.38, size = 53, normalized size = 1.00 \[ c \left (\frac {- a - 2 b x^{2}}{2 a^{3} x^{2} + 2 a^{2} b x^{4}} - \frac {2 b \log {\relax (x )}}{a^{3}} + \frac {b \log {\left (\frac {a}{b} + x^{2} \right )}}{a^{3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*((-a - 2*b*x**2)/(2*a**3*x**2 + 2*a**2*b*x**4) - 2*b*log(x)/a**3 + b*log(a/b + x**2)/a**3)

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