\(\int \frac {a+b x^2}{(a-b x^2)^2} \, dx\) [115]

Optimal result

Integrand size = 18, antiderivative size = 12 \[ \int \frac {a+b x^2}{\left (a-b x^2\right )^2} \, dx=\frac {x}{a-b x^2} \]

[Out]

x/(-b*x^2+a)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {391} \[ \int \frac {a+b x^2}{\left (a-b x^2\right )^2} \, dx=\frac {x}{a-b x^2} \]

[In]

Int[(a + b*x^2)/(a - b*x^2)^2,x]

[Out]

x/(a - b*x^2)

Rule 391

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[c*x*((a + b*x^n)^(p + 1)/a), x]
/; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[a*d - b*c*(n*(p + 1) + 1), 0]

Rubi steps \begin{align*} \text {integral}& = \frac {x}{a-b x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {a+b x^2}{\left (a-b x^2\right )^2} \, dx=-\frac {x}{-a+b x^2} \]

[In]

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

[Out]

-(x/(-a + b*x^2))

Maple [A] (verified)

Time = 2.51 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.08

method	result	size
gosper	\(\frac {x}{-b \,x^{2}+a}\)	\(13\)
default	\(\frac {x}{-b \,x^{2}+a}\)	\(13\)
norman	\(\frac {x}{-b \,x^{2}+a}\)	\(13\)
risch	\(\frac {x}{-b \,x^{2}+a}\)	\(13\)
parallelrisch	\(-\frac {x}{b \,x^{2}-a}\)	\(15\)

[In]

int((b*x^2+a)/(-b*x^2+a)^2,x,method=_RETURNVERBOSE)

[Out]

x/(-b*x^2+a)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {a+b x^2}{\left (a-b x^2\right )^2} \, dx=-\frac {x}{b x^{2} - a} \]

[In]

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

[Out]

-x/(b*x^2 - a)

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int \frac {a+b x^2}{\left (a-b x^2\right )^2} \, dx=- \frac {x}{- a + b x^{2}} \]

[In]

integrate((b*x**2+a)/(-b*x**2+a)**2,x)

[Out]

-x/(-a + b*x**2)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {a+b x^2}{\left (a-b x^2\right )^2} \, dx=-\frac {x}{b x^{2} - a} \]

[In]

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

[Out]

-x/(b*x^2 - a)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {a+b x^2}{\left (a-b x^2\right )^2} \, dx=-\frac {x}{b x^{2} - a} \]

[In]

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

[Out]

-x/(b*x^2 - a)

Mupad [B] (verification not implemented)

Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {a+b x^2}{\left (a-b x^2\right )^2} \, dx=\frac {x}{a-b\,x^2} \]

[In]

int((a + b*x^2)/(a - b*x^2)^2,x)

[Out]

x/(a - b*x^2)