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

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

[Out]

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

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Rubi [A]  time = 0.0988627, 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} \[ -\frac{(b c-a d) (a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{3/2} d^{5/2}}+\frac{x (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}+\frac{b^2 x}{d^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^2/(c + d*x^2)^2,x]

[Out]

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

Mathematica [A]  time = 0.0587676, size = 89, normalized size = 1.09 \[ -\frac{\left (-a^2 d^2-2 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{3/2} d^{5/2}}+\frac{x (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}+\frac{b^2 x}{d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.007, size = 129, normalized size = 1.6 \begin{align*}{\frac{{b}^{2}x}{{d}^{2}}}+{\frac{x{a}^{2}}{2\,c \left ( d{x}^{2}+c \right ) }}-{\frac{abx}{d \left ( d{x}^{2}+c \right ) }}+{\frac{cx{b}^{2}}{2\,{d}^{2} \left ( d{x}^{2}+c \right ) }}+{\frac{{a}^{2}}{2\,c}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{ab}{d}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{3\,{b}^{2}c}{2\,{d}^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.85834, size = 612, normalized size = 7.46 \begin{align*} \left [\frac{4 \, b^{2} c^{2} d^{2} x^{3} +{\left (3 \, b^{2} c^{3} - 2 \, a b c^{2} d - a^{2} c d^{2} +{\left (3 \, b^{2} c^{2} d - 2 \, a b c d^{2} - a^{2} d^{3}\right )} x^{2}\right )} \sqrt{-c d} \log \left (\frac{d x^{2} - 2 \, \sqrt{-c d} x - c}{d x^{2} + c}\right ) + 2 \,{\left (3 \, b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x}{4 \,{\left (c^{2} d^{4} x^{2} + c^{3} d^{3}\right )}}, \frac{2 \, b^{2} c^{2} d^{2} x^{3} -{\left (3 \, b^{2} c^{3} - 2 \, a b c^{2} d - a^{2} c d^{2} +{\left (3 \, b^{2} c^{2} d - 2 \, a b c d^{2} - a^{2} d^{3}\right )} x^{2}\right )} \sqrt{c d} \arctan \left (\frac{\sqrt{c d} x}{c}\right ) +{\left (3 \, b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x}{2 \,{\left (c^{2} d^{4} x^{2} + c^{3} d^{3}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.841904, size = 236, normalized size = 2.88 \begin{align*} \frac{b^{2} x}{d^{2}} + \frac{x \left (a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{2 c^{2} d^{2} + 2 c d^{3} x^{2}} - \frac{\sqrt{- \frac{1}{c^{3} d^{5}}} \left (a d - b c\right ) \left (a d + 3 b c\right ) \log{\left (- \frac{c^{2} d^{2} \sqrt{- \frac{1}{c^{3} d^{5}}} \left (a d - b c\right ) \left (a d + 3 b c\right )}{a^{2} d^{2} + 2 a b c d - 3 b^{2} c^{2}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{c^{3} d^{5}}} \left (a d - b c\right ) \left (a d + 3 b c\right ) \log{\left (\frac{c^{2} d^{2} \sqrt{- \frac{1}{c^{3} d^{5}}} \left (a d - b c\right ) \left (a d + 3 b c\right )}{a^{2} d^{2} + 2 a b c d - 3 b^{2} c^{2}} + x \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.06507, size = 128, normalized size = 1.56 \begin{align*} \frac{b^{2} x}{d^{2}} - \frac{{\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{2 \, \sqrt{c d} c d^{2}} + \frac{b^{2} c^{2} x - 2 \, a b c d x + a^{2} d^{2} x}{2 \,{\left (d x^{2} + c\right )} c d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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