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

Optimal. Leaf size=82 \[ \frac{(b c-a d) (3 a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} b^{5/2}}+\frac{x (b c-a d)^2}{2 a b^2 \left (a+b x^2\right )}+\frac{d^2 x}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.103651, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {390, 385, 205} \[ \frac{(b c-a d) (3 a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} b^{5/2}}+\frac{x (b c-a d)^2}{2 a b^2 \left (a+b x^2\right )}+\frac{d^2 x}{b^2} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)^2/(a + b*x^2)^2,x]

[Out]

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

Rule 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

Rule 385

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

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

Mathematica [A]  time = 0.0630599, size = 88, normalized size = 1.07 \[ \frac{\left (-3 a^2 d^2+2 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} b^{5/2}}+\frac{x (b c-a d)^2}{2 a b^2 \left (a+b x^2\right )}+\frac{d^2 x}{b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.008, size = 129, normalized size = 1.6 \begin{align*}{\frac{{d}^{2}x}{{b}^{2}}}+{\frac{ax{d}^{2}}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{cxd}{b \left ( b{x}^{2}+a \right ) }}+{\frac{x{c}^{2}}{2\,a \left ( b{x}^{2}+a \right ) }}-{\frac{3\,a{d}^{2}}{2\,{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{cd}{b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{{c}^{2}}{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/(b*x^2+a)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.79625, size = 612, normalized size = 7.46 \begin{align*} \left [\frac{4 \, a^{2} b^{2} d^{2} x^{3} +{\left (a b^{2} c^{2} + 2 \, a^{2} b c d - 3 \, a^{3} d^{2} +{\left (b^{3} c^{2} + 2 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x^{2}\right )} \sqrt{-a b} \log \left (\frac{b x^{2} + 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right ) + 2 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2}\right )} x}{4 \,{\left (a^{2} b^{4} x^{2} + a^{3} b^{3}\right )}}, \frac{2 \, a^{2} b^{2} d^{2} x^{3} +{\left (a b^{2} c^{2} + 2 \, a^{2} b c d - 3 \, a^{3} d^{2} +{\left (b^{3} c^{2} + 2 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x^{2}\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2}\right )} x}{2 \,{\left (a^{2} b^{4} x^{2} + a^{3} b^{3}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.861887, size = 236, normalized size = 2.88 \begin{align*} \frac{x \left (a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{2 a^{2} b^{2} + 2 a b^{3} x^{2}} + \frac{\sqrt{- \frac{1}{a^{3} b^{5}}} \left (a d - b c\right ) \left (3 a d + b c\right ) \log{\left (- \frac{a^{2} b^{2} \sqrt{- \frac{1}{a^{3} b^{5}}} \left (a d - b c\right ) \left (3 a d + b c\right )}{3 a^{2} d^{2} - 2 a b c d - b^{2} c^{2}} + x \right )}}{4} - \frac{\sqrt{- \frac{1}{a^{3} b^{5}}} \left (a d - b c\right ) \left (3 a d + b c\right ) \log{\left (\frac{a^{2} b^{2} \sqrt{- \frac{1}{a^{3} b^{5}}} \left (a d - b c\right ) \left (3 a d + b c\right )}{3 a^{2} d^{2} - 2 a b c d - b^{2} c^{2}} + x \right )}}{4} + \frac{d^{2} x}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15406, size = 127, normalized size = 1.55 \begin{align*} \frac{d^{2} x}{b^{2}} + \frac{{\left (b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a b^{2}} + \frac{b^{2} c^{2} x - 2 \, a b c d x + a^{2} d^{2} x}{2 \,{\left (b x^{2} + a\right )} a b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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