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

Optimal. Leaf size=71 \[ -\frac{x (b c-a d)}{2 a^2 \left (a+b x^2\right )}-\frac{(3 b c-a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2} \sqrt{b}}-\frac{c}{a^2 x} \]

[Out]

-(c/(a^2*x)) - ((b*c - a*d)*x)/(2*a^2*(a + b*x^2)) - ((3*b*c - a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(5/2)*Sq
rt[b])

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Rubi [A]  time = 0.0529184, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {456, 453, 205} \[ -\frac{x (b c-a d)}{2 a^2 \left (a+b x^2\right )}-\frac{(3 b c-a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2} \sqrt{b}}-\frac{c}{a^2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c/(a^2*x)) - ((b*c - a*d)*x)/(2*a^2*(a + b*x^2)) - ((3*b*c - a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(5/2)*Sq
rt[b])

Rule 456

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[((-a)^(m/2 - 1)*(b*c - a*d)*
x*(a + b*x^2)^(p + 1))/(2*b^(m/2 + 1)*(p + 1)), x] + Dist[1/(2*b^(m/2 + 1)*(p + 1)), Int[x^m*(a + b*x^2)^(p +
1)*ExpandToSum[2*b*(p + 1)*Together[(b^(m/2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d)*x^(-m + 2))/(a + b*x^2)]
- ((-a)^(m/2 - 1)*(b*c - a*d))/x^m, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &
& ILtQ[m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])

Rule 453

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

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{c+d x^2}{x^2 \left (a+b x^2\right )^2} \, dx &=-\frac{(b c-a d) x}{2 a^2 \left (a+b x^2\right )}-\frac{1}{2} \int \frac{-\frac{2 c}{a}+\frac{(b c-a d) x^2}{a^2}}{x^2 \left (a+b x^2\right )} \, dx\\ &=-\frac{c}{a^2 x}-\frac{(b c-a d) x}{2 a^2 \left (a+b x^2\right )}-\frac{(3 b c-a d) \int \frac{1}{a+b x^2} \, dx}{2 a^2}\\ &=-\frac{c}{a^2 x}-\frac{(b c-a d) x}{2 a^2 \left (a+b x^2\right )}-\frac{(3 b c-a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2} \sqrt{b}}\\ \end{align*}

Mathematica [A]  time = 0.03296, size = 70, normalized size = 0.99 \[ \frac{x (a d-b c)}{2 a^2 \left (a+b x^2\right )}+\frac{(a d-3 b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2} \sqrt{b}}-\frac{c}{a^2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c/(a^2*x)) + ((-(b*c) + a*d)*x)/(2*a^2*(a + b*x^2)) + ((-3*b*c + a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(5/2
)*Sqrt[b])

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Maple [A]  time = 0.01, size = 85, normalized size = 1.2 \begin{align*} -{\frac{c}{{a}^{2}x}}+{\frac{dx}{2\,a \left ( b{x}^{2}+a \right ) }}-{\frac{bcx}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{d}{2\,a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{3\,bc}{2\,{a}^{2}}\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)/x^2/(b*x^2+a)^2,x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.56468, size = 447, normalized size = 6.3 \begin{align*} \left [-\frac{4 \, a^{2} b c + 2 \,{\left (3 \, a b^{2} c - a^{2} b d\right )} x^{2} -{\left ({\left (3 \, b^{2} c - a b d\right )} x^{3} +{\left (3 \, a b c - a^{2} d\right )} x\right )} \sqrt{-a b} \log \left (\frac{b x^{2} - 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right )}{4 \,{\left (a^{3} b^{2} x^{3} + a^{4} b x\right )}}, -\frac{2 \, a^{2} b c +{\left (3 \, a b^{2} c - a^{2} b d\right )} x^{2} +{\left ({\left (3 \, b^{2} c - a b d\right )} x^{3} +{\left (3 \, a b c - a^{2} d\right )} x\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{a}\right )}{2 \,{\left (a^{3} b^{2} x^{3} + a^{4} b x\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*(4*a^2*b*c + 2*(3*a*b^2*c - a^2*b*d)*x^2 - ((3*b^2*c - a*b*d)*x^3 + (3*a*b*c - a^2*d)*x)*sqrt(-a*b)*log(
(b*x^2 - 2*sqrt(-a*b)*x - a)/(b*x^2 + a)))/(a^3*b^2*x^3 + a^4*b*x), -1/2*(2*a^2*b*c + (3*a*b^2*c - a^2*b*d)*x^
2 + ((3*b^2*c - a*b*d)*x^3 + (3*a*b*c - a^2*d)*x)*sqrt(a*b)*arctan(sqrt(a*b)*x/a))/(a^3*b^2*x^3 + a^4*b*x)]

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Sympy [A]  time = 0.626656, size = 114, normalized size = 1.61 \begin{align*} - \frac{\sqrt{- \frac{1}{a^{5} b}} \left (a d - 3 b c\right ) \log{\left (- a^{3} \sqrt{- \frac{1}{a^{5} b}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{a^{5} b}} \left (a d - 3 b c\right ) \log{\left (a^{3} \sqrt{- \frac{1}{a^{5} b}} + x \right )}}{4} + \frac{- 2 a c + x^{2} \left (a d - 3 b c\right )}{2 a^{3} x + 2 a^{2} b x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-1/(a**5*b))*(a*d - 3*b*c)*log(-a**3*sqrt(-1/(a**5*b)) + x)/4 + sqrt(-1/(a**5*b))*(a*d - 3*b*c)*log(a**3
*sqrt(-1/(a**5*b)) + x)/4 + (-2*a*c + x**2*(a*d - 3*b*c))/(2*a**3*x + 2*a**2*b*x**3)

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Giac [A]  time = 1.15805, size = 86, normalized size = 1.21 \begin{align*} -\frac{{\left (3 \, b c - a d\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{2}} - \frac{3 \, b c x^{2} - a d x^{2} + 2 \, a c}{2 \,{\left (b x^{3} + a x\right )} a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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