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

Optimal. Leaf size=81 \[ -\frac{x (-2 a d f-3 b c f+3 b d e)}{3 f^2}+\frac{(b e-a f) (d e-c f) \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e} f^{5/2}}+\frac{d x \left (a+b x^2\right )}{3 f} \]

[Out]

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

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Rubi [A]  time = 0.0798565, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {528, 388, 205} \[ -\frac{x (-2 a d f-3 b c f+3 b d e)}{3 f^2}+\frac{(b e-a f) (d e-c f) \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e} f^{5/2}}+\frac{d x \left (a+b x^2\right )}{3 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 528

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[
(f*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(b*(n*(p + q + 1) + 1)), x] + Dist[1/(b*(n*(p + q + 1) + 1)), Int[(a +
b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f + b*e*n*(p + q + 1)) + (d*(b*e - a*f) + f*n*q*(b*c - a*d) + b*
d*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && GtQ[q, 0] && NeQ[n*(p + q + 1) + 1
, 0]

Rule 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 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 ) \left (c+d x^2\right )}{e+f x^2} \, dx &=\frac{d x \left (a+b x^2\right )}{3 f}+\frac{\int \frac{-a (d e-3 c f)-(3 b d e-3 b c f-2 a d f) x^2}{e+f x^2} \, dx}{3 f}\\ &=-\frac{(3 b d e-3 b c f-2 a d f) x}{3 f^2}+\frac{d x \left (a+b x^2\right )}{3 f}+\frac{((b e-a f) (d e-c f)) \int \frac{1}{e+f x^2} \, dx}{f^2}\\ &=-\frac{(3 b d e-3 b c f-2 a d f) x}{3 f^2}+\frac{d x \left (a+b x^2\right )}{3 f}+\frac{(b e-a f) (d e-c f) \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e} f^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.0534778, size = 72, normalized size = 0.89 \[ \frac{x (a d f+b c f-b d e)}{f^2}+\frac{(b e-a f) (d e-c f) \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e} f^{5/2}}+\frac{b d x^3}{3 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.006, size = 119, normalized size = 1.5 \begin{align*}{\frac{{x}^{3}bd}{3\,f}}+{\frac{adx}{f}}+{\frac{bcx}{f}}-{\frac{bdex}{{f}^{2}}}+{ac\arctan \left ({fx{\frac{1}{\sqrt{ef}}}} \right ){\frac{1}{\sqrt{ef}}}}-{\frac{ade}{f}\arctan \left ({fx{\frac{1}{\sqrt{ef}}}} \right ){\frac{1}{\sqrt{ef}}}}-{\frac{bce}{f}\arctan \left ({fx{\frac{1}{\sqrt{ef}}}} \right ){\frac{1}{\sqrt{ef}}}}+{\frac{bd{e}^{2}}{{f}^{2}}\arctan \left ({fx{\frac{1}{\sqrt{ef}}}} \right ){\frac{1}{\sqrt{ef}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.48632, size = 423, normalized size = 5.22 \begin{align*} \left [\frac{2 \, b d e f^{2} x^{3} - 3 \,{\left (b d e^{2} + a c f^{2} -{\left (b c + a d\right )} e f\right )} \sqrt{-e f} \log \left (\frac{f x^{2} - 2 \, \sqrt{-e f} x - e}{f x^{2} + e}\right ) - 6 \,{\left (b d e^{2} f -{\left (b c + a d\right )} e f^{2}\right )} x}{6 \, e f^{3}}, \frac{b d e f^{2} x^{3} + 3 \,{\left (b d e^{2} + a c f^{2} -{\left (b c + a d\right )} e f\right )} \sqrt{e f} \arctan \left (\frac{\sqrt{e f} x}{e}\right ) - 3 \,{\left (b d e^{2} f -{\left (b c + a d\right )} e f^{2}\right )} x}{3 \, e f^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.836548, size = 206, normalized size = 2.54 \begin{align*} \frac{b d x^{3}}{3 f} - \frac{\sqrt{- \frac{1}{e f^{5}}} \left (a f - b e\right ) \left (c f - d e\right ) \log{\left (- \frac{e f^{2} \sqrt{- \frac{1}{e f^{5}}} \left (a f - b e\right ) \left (c f - d e\right )}{a c f^{2} - a d e f - b c e f + b d e^{2}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{e f^{5}}} \left (a f - b e\right ) \left (c f - d e\right ) \log{\left (\frac{e f^{2} \sqrt{- \frac{1}{e f^{5}}} \left (a f - b e\right ) \left (c f - d e\right )}{a c f^{2} - a d e f - b c e f + b d e^{2}} + x \right )}}{2} + \frac{x \left (a d f + b c f - b d e\right )}{f^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.19611, size = 108, normalized size = 1.33 \begin{align*} \frac{{\left (a c f^{2} - b c f e - a d f e + b d e^{2}\right )} \arctan \left (\sqrt{f} x e^{\left (-\frac{1}{2}\right )}\right ) e^{\left (-\frac{1}{2}\right )}}{f^{\frac{5}{2}}} + \frac{b d f^{2} x^{3} + 3 \, b c f^{2} x + 3 \, a d f^{2} x - 3 \, b d f x e}{3 \, f^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*c*f^2 - b*c*f*e - a*d*f*e + b*d*e^2)*arctan(sqrt(f)*x*e^(-1/2))*e^(-1/2)/f^(5/2) + 1/3*(b*d*f^2*x^3 + 3*b*c
*f^2*x + 3*a*d*f^2*x - 3*b*d*f*x*e)/f^3