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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0498151, 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} \[ -\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} b^{3/2}}-\frac{c^2}{a x}+\frac{d^2 x}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0509508, size = 55, normalized size = 1. \[ -\frac{(a d-b c)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} b^{3/2}}-\frac{c^2}{a x}+\frac{d^2 x}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 85, normalized size = 1.6 \begin{align*}{\frac{{d}^{2}x}{b}}-{\frac{{c}^{2}}{ax}}-{\frac{a{d}^{2}}{b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+2\,{\frac{cd}{\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }-{\frac{b{c}^{2}}{a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.46248, size = 344, normalized size = 6.25 \begin{align*} \left [\frac{2 \, a^{2} b d^{2} x^{2} - 2 \, a b^{2} c^{2} -{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{-a b} x \log \left (\frac{b x^{2} + 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right )}{2 \, a^{2} b^{2} x}, \frac{a^{2} b d^{2} x^{2} - a b^{2} c^{2} -{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{a b} x \arctan \left (\frac{\sqrt{a b} x}{a}\right )}{a^{2} b^{2} x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [B]  time = 0.691259, size = 165, normalized size = 3. \begin{align*} \frac{\sqrt{- \frac{1}{a^{3} b^{3}}} \left (a d - b c\right )^{2} \log{\left (- \frac{a^{2} b \sqrt{- \frac{1}{a^{3} b^{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}{a^{3} b^{3}}} \left (a d - b c\right )^{2} \log{\left (\frac{a^{2} b \sqrt{- \frac{1}{a^{3} b^{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{d^{2} x}{b} - \frac{c^{2}}{a x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.1572, size = 85, normalized size = 1.55 \begin{align*} \frac{d^{2} x}{b} - \frac{c^{2}}{a x} - \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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