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

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

[Out]

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

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Rubi [A]  time = 0.0409947, 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} \[ \frac{d x (2 b c-a d)}{b^2}+\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{5/2}}+\frac{d^2 x^3}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)^2/(a + b*x^2),x]

[Out]

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

Mathematica [A]  time = 0.0491206, size = 59, normalized size = 0.94 \[ \frac{d x \left (-3 a d+6 b c+b d x^2\right )}{3 b^2}+\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 95, normalized size = 1.5 \begin{align*}{\frac{{d}^{2}{x}^{3}}{3\,b}}-{\frac{a{d}^{2}x}{{b}^{2}}}+2\,{\frac{dxc}{b}}+{\frac{{a}^{2}{d}^{2}}{{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-2\,{\frac{acd}{b\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }+{{c}^{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/(b*x^2+a),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.08632, size = 390, normalized size = 6.19 \begin{align*} \left [\frac{2 \, a b^{2} d^{2} x^{3} - 3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{-a b} \log \left (\frac{b x^{2} - 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right ) + 6 \,{\left (2 \, a b^{2} c d - a^{2} b d^{2}\right )} x}{6 \, a b^{3}}, \frac{a b^{2} d^{2} x^{3} + 3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) + 3 \,{\left (2 \, a b^{2} c d - a^{2} b d^{2}\right )} x}{3 \, a b^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.537803, size = 172, normalized size = 2.73 \begin{align*} - \frac{\sqrt{- \frac{1}{a b^{5}}} \left (a d - b c\right )^{2} \log{\left (- \frac{a b^{2} \sqrt{- \frac{1}{a b^{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}{a b^{5}}} \left (a d - b c\right )^{2} \log{\left (\frac{a b^{2} \sqrt{- \frac{1}{a b^{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{d^{2} x^{3}}{3 b} - \frac{x \left (a d^{2} - 2 b c d\right )}{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),x)

[Out]

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

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Giac [A]  time = 1.12288, 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{b x}{\sqrt{a b}}\right )}{\sqrt{a b} b^{2}} + \frac{b^{2} d^{2} x^{3} + 6 \, b^{2} c d x - 3 \, a b d^{2} x}{3 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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