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

Optimal. Leaf size=63 \[ -\frac{b x (b c-2 a d)}{d^2}+\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} d^{5/2}}+\frac{b^2 x^3}{3 d} \]

[Out]

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

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Rubi [A]  time = 0.0428152, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {390, 205} \[ -\frac{b x (b c-2 a d)}{d^2}+\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} d^{5/2}}+\frac{b^2 x^3}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^2/(c + d*x^2),x]

[Out]

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

Mathematica [A]  time = 0.0482854, size = 59, normalized size = 0.94 \[ \frac{b x \left (6 a d-3 b c+b d x^2\right )}{3 d^2}+\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} d^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*x*(-3*b*c + 6*a*d + b*d*x^2))/(3*d^2) + ((b*c - a*d)^2*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*d^(5/2))

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Maple [A]  time = 0.003, size = 95, normalized size = 1.5 \begin{align*}{\frac{{b}^{2}{x}^{3}}{3\,d}}+2\,{\frac{abx}{d}}-{\frac{{b}^{2}xc}{{d}^{2}}}+{{a}^{2}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-2\,{\frac{abc}{d\sqrt{cd}}\arctan \left ({\frac{dx}{\sqrt{cd}}} \right ) }+{\frac{{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),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.84056, size = 390, normalized size = 6.19 \begin{align*} \left [\frac{2 \, b^{2} c d^{2} x^{3} - 3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{-c d} \log \left (\frac{d x^{2} - 2 \, \sqrt{-c d} x - c}{d x^{2} + c}\right ) - 6 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2}\right )} x}{6 \, c d^{3}}, \frac{b^{2} c d^{2} x^{3} + 3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{c d} \arctan \left (\frac{\sqrt{c d} x}{c}\right ) - 3 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2}\right )} x}{3 \, c d^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.525615, size = 172, normalized size = 2.73 \begin{align*} \frac{b^{2} x^{3}}{3 d} - \frac{\sqrt{- \frac{1}{c d^{5}}} \left (a d - b c\right )^{2} \log{\left (- \frac{c d^{2} \sqrt{- \frac{1}{c d^{5}}} \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}{c d^{5}}} \left (a d - b c\right )^{2} \log{\left (\frac{c d^{2} \sqrt{- \frac{1}{c d^{5}}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} + \frac{x \left (2 a b d - b^{2} c\right )}{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),x)

[Out]

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

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Giac [A]  time = 1.09229, size = 97, normalized size = 1.54 \begin{align*} \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{\sqrt{c d} d^{2}} + \frac{b^{2} d^{2} x^{3} - 3 \, b^{2} c d x + 6 \, a b d^{2} x}{3 \, d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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