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

Optimal. Leaf size=55 \[ -\frac{a^2}{c x}-\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{3/2} d^{3/2}}+\frac{b^2 x}{d} \]

[Out]

-(a^2/(c*x)) + (b^2*x)/d - ((b*c - a*d)^2*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(3/2)*d^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.0488449, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {461, 205} \[ -\frac{a^2}{c x}-\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{3/2} d^{3/2}}+\frac{b^2 x}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^2/(c*x)) + (b^2*x)/d - ((b*c - a*d)^2*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(3/2)*d^(3/2))

Rule 461

Int[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegr
and[((e*x)^m*(a + b*x^n)^p)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n
, 0] && IGtQ[p, 0] && (IntegerQ[m] || IGtQ[2*(m + 1), 0] ||  !RationalQ[m])

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2\right )^2}{x^2 \left (c+d x^2\right )} \, dx &=\int \left (\frac{b^2}{d}+\frac{a^2}{c x^2}-\frac{(b c-a d)^2}{c d \left (c+d x^2\right )}\right ) \, dx\\ &=-\frac{a^2}{c x}+\frac{b^2 x}{d}-\frac{(b c-a d)^2 \int \frac{1}{c+d x^2} \, dx}{c d}\\ &=-\frac{a^2}{c x}+\frac{b^2 x}{d}-\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{3/2} d^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0448656, size = 55, normalized size = 1. \[ -\frac{a^2}{c x}-\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{3/2} d^{3/2}}+\frac{b^2 x}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^2/(c*x)) + (b^2*x)/d - ((b*c - a*d)^2*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(3/2)*d^(3/2))

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 85, normalized size = 1.6 \begin{align*}{\frac{{b}^{2}x}{d}}-{\frac{{a}^{2}d}{c}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+2\,{\frac{ab}{\sqrt{cd}}\arctan \left ({\frac{dx}{\sqrt{cd}}} \right ) }-{\frac{{b}^{2}c}{d}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{{a}^{2}}{cx}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.32094, size = 344, normalized size = 6.25 \begin{align*} \left [\frac{2 \, b^{2} c^{2} d x^{2} - 2 \, a^{2} c d^{2} -{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{-c d} x \log \left (\frac{d x^{2} + 2 \, \sqrt{-c d} x - c}{d x^{2} + c}\right )}{2 \, c^{2} d^{2} x}, \frac{b^{2} c^{2} d x^{2} - a^{2} c d^{2} -{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{c d} x \arctan \left (\frac{\sqrt{c d} x}{c}\right )}{c^{2} d^{2} x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(2*b^2*c^2*d*x^2 - 2*a^2*c*d^2 - (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(-c*d)*x*log((d*x^2 + 2*sqrt(-c*d)*x
- c)/(d*x^2 + c)))/(c^2*d^2*x), (b^2*c^2*d*x^2 - a^2*c*d^2 - (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(c*d)*x*arct
an(sqrt(c*d)*x/c))/(c^2*d^2*x)]

________________________________________________________________________________________

Sympy [B]  time = 0.669569, size = 165, normalized size = 3. \begin{align*} - \frac{a^{2}}{c x} + \frac{b^{2} x}{d} + \frac{\sqrt{- \frac{1}{c^{3} d^{3}}} \left (a d - b c\right )^{2} \log{\left (- \frac{c^{2} d \sqrt{- \frac{1}{c^{3} d^{3}}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} - \frac{\sqrt{- \frac{1}{c^{3} d^{3}}} \left (a d - b c\right )^{2} \log{\left (\frac{c^{2} d \sqrt{- \frac{1}{c^{3} d^{3}}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**2/(c*x) + b**2*x/d + sqrt(-1/(c**3*d**3))*(a*d - b*c)**2*log(-c**2*d*sqrt(-1/(c**3*d**3))*(a*d - b*c)**2/(
a**2*d**2 - 2*a*b*c*d + b**2*c**2) + x)/2 - sqrt(-1/(c**3*d**3))*(a*d - b*c)**2*log(c**2*d*sqrt(-1/(c**3*d**3)
)*(a*d - b*c)**2/(a**2*d**2 - 2*a*b*c*d + b**2*c**2) + x)/2

________________________________________________________________________________________

Giac [A]  time = 1.1792, size = 85, normalized size = 1.55 \begin{align*} \frac{b^{2} x}{d} - \frac{a^{2}}{c x} - \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{\sqrt{c d} c d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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