3.217 \(\int \frac{(c+d x^2)^2}{x^4 (a+b x^2)} \, dx\)

Optimal. Leaf size=64 \[ \frac{c (b c-2 a d)}{a^2 x}+\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2} \sqrt{b}}-\frac{c^2}{3 a x^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0546262, antiderivative size = 64, 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} \[ \frac{c (b c-2 a d)}{a^2 x}+\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2} \sqrt{b}}-\frac{c^2}{3 a x^3} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)^2/(x^4*(a + b*x^2)),x]

[Out]

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

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^4 \left (a+b x^2\right )} \, dx &=\int \left (\frac{c^2}{a x^4}+\frac{c (-b c+2 a d)}{a^2 x^2}+\frac{(-b c+a d)^2}{a^2 \left (a+b x^2\right )}\right ) \, dx\\ &=-\frac{c^2}{3 a x^3}+\frac{c (b c-2 a d)}{a^2 x}+\frac{(b c-a d)^2 \int \frac{1}{a+b x^2} \, dx}{a^2}\\ &=-\frac{c^2}{3 a x^3}+\frac{c (b c-2 a d)}{a^2 x}+\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2} \sqrt{b}}\\ \end{align*}

Mathematica [A]  time = 0.0618717, size = 66, normalized size = 1.03 \[ -\frac{c (2 a d-b c)}{a^2 x}+\frac{(a d-b c)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2} \sqrt{b}}-\frac{c^2}{3 a x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2)^2/(x^4*(a + b*x^2)),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 98, normalized size = 1.5 \begin{align*} -{\frac{{c}^{2}}{3\,a{x}^{3}}}-2\,{\frac{cd}{ax}}+{\frac{b{c}^{2}}{{a}^{2}x}}+{{d}^{2}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-2\,{\frac{bcd}{a\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }+{\frac{{b}^{2}{c}^{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)^2/x^4/(b*x^2+a),x)

[Out]

-1/3*c^2/a/x^3-2*c/a/x*d+c^2/a^2/x*b+1/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*d^2-2/a/(a*b)^(1/2)*arctan(b*x/(a*b
)^(1/2))*c*b*d+1/a^2/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*b^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^4/(b*x^2+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.5038, size = 408, normalized size = 6.38 \begin{align*} \left [-\frac{3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{-a b} x^{3} \log \left (\frac{b x^{2} - 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right ) + 2 \, a^{2} b c^{2} - 6 \,{\left (a b^{2} c^{2} - 2 \, a^{2} b c d\right )} x^{2}}{6 \, a^{3} b x^{3}}, \frac{3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{a b} x^{3} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) - a^{2} b c^{2} + 3 \,{\left (a b^{2} c^{2} - 2 \, a^{2} b c d\right )} x^{2}}{3 \, a^{3} b x^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^2/x^4/(b*x^2+a),x, algorithm="fricas")

[Out]

[-1/6*(3*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(-a*b)*x^3*log((b*x^2 - 2*sqrt(-a*b)*x - a)/(b*x^2 + a)) + 2*a^2*
b*c^2 - 6*(a*b^2*c^2 - 2*a^2*b*c*d)*x^2)/(a^3*b*x^3), 1/3*(3*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(a*b)*x^3*arc
tan(sqrt(a*b)*x/a) - a^2*b*c^2 + 3*(a*b^2*c^2 - 2*a^2*b*c*d)*x^2)/(a^3*b*x^3)]

________________________________________________________________________________________

Sympy [B]  time = 0.786519, size = 172, normalized size = 2.69 \begin{align*} - \frac{\sqrt{- \frac{1}{a^{5} b}} \left (a d - b c\right )^{2} \log{\left (- \frac{a^{3} \sqrt{- \frac{1}{a^{5} b}} \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^{5} b}} \left (a d - b c\right )^{2} \log{\left (\frac{a^{3} \sqrt{- \frac{1}{a^{5} b}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} - \frac{a c^{2} + x^{2} \left (6 a c d - 3 b c^{2}\right )}{3 a^{2} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**2+c)**2/x**4/(b*x**2+a),x)

[Out]

-sqrt(-1/(a**5*b))*(a*d - b*c)**2*log(-a**3*sqrt(-1/(a**5*b))*(a*d - b*c)**2/(a**2*d**2 - 2*a*b*c*d + b**2*c**
2) + x)/2 + sqrt(-1/(a**5*b))*(a*d - b*c)**2*log(a**3*sqrt(-1/(a**5*b))*(a*d - b*c)**2/(a**2*d**2 - 2*a*b*c*d
+ b**2*c**2) + x)/2 - (a*c**2 + x**2*(6*a*c*d - 3*b*c**2))/(3*a**2*x**3)

________________________________________________________________________________________

Giac [A]  time = 1.18474, size = 97, normalized size = 1.52 \begin{align*} \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^{2}} + \frac{3 \, b c^{2} x^{2} - 6 \, a c d x^{2} - a c^{2}}{3 \, a^{2} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^2/x^4/(b*x^2+a),x, algorithm="giac")

[Out]

(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^2) + 1/3*(3*b*c^2*x^2 - 6*a*c*d*x^2 - a*c^2
)/(a^2*x^3)