3.219 \(\int \frac{1}{(d+e x^2)^2 (-c d^2+b d e+b e^2 x^2+c e^2 x^4)} \, dx\)

Optimal. Leaf size=187 \[ -\frac{\left (3 b^2 e^2-16 b c d e+28 c^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{5/2} \sqrt{e} (2 c d-b e)^3}-\frac{c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{c d-b e}}\right )}{\sqrt{e} \sqrt{c d-b e} (2 c d-b e)^3}-\frac{x (10 c d-3 b e)}{8 d^2 \left (d+e x^2\right ) (2 c d-b e)^2}-\frac{x}{4 d \left (d+e x^2\right )^2 (2 c d-b e)} \]

[Out]

-x/(4*d*(2*c*d - b*e)*(d + e*x^2)^2) - ((10*c*d - 3*b*e)*x)/(8*d^2*(2*c*d - b*e)^2*(d + e*x^2)) - ((28*c^2*d^2
- 16*b*c*d*e + 3*b^2*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(5/2)*Sqrt[e]*(2*c*d - b*e)^3) - (c^(5/2)*ArcTanh
[(Sqrt[c]*Sqrt[e]*x)/Sqrt[c*d - b*e]])/(Sqrt[e]*Sqrt[c*d - b*e]*(2*c*d - b*e)^3)

________________________________________________________________________________________

Rubi [A]  time = 0.277802, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {1149, 414, 527, 522, 205, 208} \[ -\frac{\left (3 b^2 e^2-16 b c d e+28 c^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{5/2} \sqrt{e} (2 c d-b e)^3}-\frac{c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{c d-b e}}\right )}{\sqrt{e} \sqrt{c d-b e} (2 c d-b e)^3}-\frac{x (10 c d-3 b e)}{8 d^2 \left (d+e x^2\right ) (2 c d-b e)^2}-\frac{x}{4 d \left (d+e x^2\right )^2 (2 c d-b e)} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x^2)^2*(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4)),x]

[Out]

-x/(4*d*(2*c*d - b*e)*(d + e*x^2)^2) - ((10*c*d - 3*b*e)*x)/(8*d^2*(2*c*d - b*e)^2*(d + e*x^2)) - ((28*c^2*d^2
- 16*b*c*d*e + 3*b^2*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(5/2)*Sqrt[e]*(2*c*d - b*e)^3) - (c^(5/2)*ArcTanh
[(Sqrt[c]*Sqrt[e]*x)/Sqrt[c*d - b*e]])/(Sqrt[e]*Sqrt[c*d - b*e]*(2*c*d - b*e)^3)

Rule 1149

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[(d + e*x^2)^(p +
q)*(a/d + (c*x^2)/e)^p, x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2
, 0] && IntegerQ[p]

Rule 414

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1)*
(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomial
Q[a, b, c, d, n, p, q, x]

Rule 527

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> -Simp[
((b*e - a*f)*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d
)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)
*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 522

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

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]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{1}{\left (d+e x^2\right )^2 \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx &=\int \frac{1}{\left (d+e x^2\right )^3 \left (\frac{-c d^2+b d e}{d}+c e x^2\right )} \, dx\\ &=-\frac{x}{4 d (2 c d-b e) \left (d+e x^2\right )^2}+\frac{\int \frac{e (7 c d-3 b e)-3 c e^2 x^2}{\left (d+e x^2\right )^2 \left (\frac{-c d^2+b d e}{d}+c e x^2\right )} \, dx}{4 d e (2 c d-b e)}\\ &=-\frac{x}{4 d (2 c d-b e) \left (d+e x^2\right )^2}-\frac{(10 c d-3 b e) x}{8 d^2 (2 c d-b e)^2 \left (d+e x^2\right )}+\frac{\int \frac{e^2 \left (18 c^2 d^2-13 b c d e+3 b^2 e^2\right )-c e^3 (10 c d-3 b e) x^2}{\left (d+e x^2\right ) \left (\frac{-c d^2+b d e}{d}+c e x^2\right )} \, dx}{8 d^2 e^2 (2 c d-b e)^2}\\ &=-\frac{x}{4 d (2 c d-b e) \left (d+e x^2\right )^2}-\frac{(10 c d-3 b e) x}{8 d^2 (2 c d-b e)^2 \left (d+e x^2\right )}+\frac{c^3 \int \frac{1}{\frac{-c d^2+b d e}{d}+c e x^2} \, dx}{(2 c d-b e)^3}-\frac{\left (28 c^2 d^2-16 b c d e+3 b^2 e^2\right ) \int \frac{1}{d+e x^2} \, dx}{8 d^2 (2 c d-b e)^3}\\ &=-\frac{x}{4 d (2 c d-b e) \left (d+e x^2\right )^2}-\frac{(10 c d-3 b e) x}{8 d^2 (2 c d-b e)^2 \left (d+e x^2\right )}-\frac{\left (28 c^2 d^2-16 b c d e+3 b^2 e^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{5/2} \sqrt{e} (2 c d-b e)^3}-\frac{c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{c d-b e}}\right )}{\sqrt{e} \sqrt{c d-b e} (2 c d-b e)^3}\\ \end{align*}

Mathematica [A]  time = 0.424915, size = 177, normalized size = 0.95 \[ \frac{1}{8} \left (-\frac{\left (3 b^2 e^2-16 b c d e+28 c^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d^{5/2} \sqrt{e} (2 c d-b e)^3}-\frac{\frac{8 c^{5/2} \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{b e-c d}}\right )}{\sqrt{e} \sqrt{b e-c d}}+\frac{x (b e-2 c d) \left (2 c d \left (7 d+5 e x^2\right )-b e \left (5 d+3 e x^2\right )\right )}{d^2 \left (d+e x^2\right )^2}}{(b e-2 c d)^3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x^2)^2*(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4)),x]

[Out]

(-(((28*c^2*d^2 - 16*b*c*d*e + 3*b^2*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(d^(5/2)*Sqrt[e]*(2*c*d - b*e)^3)) - ((
(-2*c*d + b*e)*x*(-(b*e*(5*d + 3*e*x^2)) + 2*c*d*(7*d + 5*e*x^2)))/(d^2*(d + e*x^2)^2) + (8*c^(5/2)*ArcTan[(Sq
rt[c]*Sqrt[e]*x)/Sqrt[-(c*d) + b*e]])/(Sqrt[e]*Sqrt[-(c*d) + b*e]))/(-2*c*d + b*e)^3)/8

________________________________________________________________________________________

Maple [A]  time = 0.013, size = 319, normalized size = 1.7 \begin{align*}{\frac{3\,{e}^{3}{x}^{3}{b}^{2}}{8\, \left ( be-2\,cd \right ) ^{3} \left ( e{x}^{2}+d \right ) ^{2}{d}^{2}}}-2\,{\frac{{e}^{2}{x}^{3}bc}{ \left ( be-2\,cd \right ) ^{3} \left ( e{x}^{2}+d \right ) ^{2}d}}+{\frac{5\,e{x}^{3}{c}^{2}}{2\, \left ( be-2\,cd \right ) ^{3} \left ( e{x}^{2}+d \right ) ^{2}}}+{\frac{5\,x{b}^{2}{e}^{2}}{8\, \left ( be-2\,cd \right ) ^{3} \left ( e{x}^{2}+d \right ) ^{2}d}}-3\,{\frac{bxce}{ \left ( be-2\,cd \right ) ^{3} \left ( e{x}^{2}+d \right ) ^{2}}}+{\frac{7\,dx{c}^{2}}{2\, \left ( be-2\,cd \right ) ^{3} \left ( e{x}^{2}+d \right ) ^{2}}}+{\frac{3\,{b}^{2}{e}^{2}}{8\, \left ( be-2\,cd \right ) ^{3}{d}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-2\,{\frac{bce}{ \left ( be-2\,cd \right ) ^{3}d\sqrt{de}}\arctan \left ({\frac{ex}{\sqrt{de}}} \right ) }+{\frac{7\,{c}^{2}}{2\, \left ( be-2\,cd \right ) ^{3}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-{\frac{{c}^{3}}{ \left ( be-2\,cd \right ) ^{3}}\arctan \left ({cex{\frac{1}{\sqrt{ \left ( be-cd \right ) ce}}}} \right ){\frac{1}{\sqrt{ \left ( be-cd \right ) ce}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x^2+d)^2/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x)

[Out]

3/8/(b*e-2*c*d)^3/(e*x^2+d)^2*e^3/d^2*x^3*b^2-2/(b*e-2*c*d)^3/(e*x^2+d)^2*e^2/d*x^3*b*c+5/2/(b*e-2*c*d)^3/(e*x
^2+d)^2*e*x^3*c^2+5/8/(b*e-2*c*d)^3/(e*x^2+d)^2/d*x*b^2*e^2-3/(b*e-2*c*d)^3/(e*x^2+d)^2*x*b*c*e+7/2/(b*e-2*c*d
)^3/(e*x^2+d)^2*d*x*c^2+3/8/(b*e-2*c*d)^3/d^2/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*b^2*e^2-2/(b*e-2*c*d)^3/d/(d
*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*b*c*e+7/2/(b*e-2*c*d)^3/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*c^2-c^3/(b*e-2*c
*d)^3/((b*e-c*d)*c*e)^(1/2)*arctan(c*e*x/((b*e-c*d)*c*e)^(1/2))

________________________________________________________________________________________

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(1/(e*x^2+d)^2/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 7.54463, size = 3611, normalized size = 19.31 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x^2+d)^2/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x, algorithm="fricas")

[Out]

[-1/16*(2*(20*c^2*d^3*e^2 - 16*b*c*d^2*e^3 + 3*b^2*d*e^4)*x^3 + 8*(c^2*d^3*e^3*x^4 + 2*c^2*d^4*e^2*x^2 + c^2*d
^5*e)*sqrt(c/(c*d*e - b*e^2))*log((c*e*x^2 + 2*(c*d*e - b*e^2)*x*sqrt(c/(c*d*e - b*e^2)) + c*d - b*e)/(c*e*x^2
- c*d + b*e)) - (28*c^2*d^4 - 16*b*c*d^3*e + 3*b^2*d^2*e^2 + (28*c^2*d^2*e^2 - 16*b*c*d*e^3 + 3*b^2*e^4)*x^4
+ 2*(28*c^2*d^3*e - 16*b*c*d^2*e^2 + 3*b^2*d*e^3)*x^2)*sqrt(-d*e)*log((e*x^2 - 2*sqrt(-d*e)*x - d)/(e*x^2 + d)
) + 2*(28*c^2*d^4*e - 24*b*c*d^3*e^2 + 5*b^2*d^2*e^3)*x)/(8*c^3*d^8*e - 12*b*c^2*d^7*e^2 + 6*b^2*c*d^6*e^3 - b
^3*d^5*e^4 + (8*c^3*d^6*e^3 - 12*b*c^2*d^5*e^4 + 6*b^2*c*d^4*e^5 - b^3*d^3*e^6)*x^4 + 2*(8*c^3*d^7*e^2 - 12*b*
c^2*d^6*e^3 + 6*b^2*c*d^5*e^4 - b^3*d^4*e^5)*x^2), -1/8*((20*c^2*d^3*e^2 - 16*b*c*d^2*e^3 + 3*b^2*d*e^4)*x^3 +
(28*c^2*d^4 - 16*b*c*d^3*e + 3*b^2*d^2*e^2 + (28*c^2*d^2*e^2 - 16*b*c*d*e^3 + 3*b^2*e^4)*x^4 + 2*(28*c^2*d^3*
e - 16*b*c*d^2*e^2 + 3*b^2*d*e^3)*x^2)*sqrt(d*e)*arctan(sqrt(d*e)*x/d) + 4*(c^2*d^3*e^3*x^4 + 2*c^2*d^4*e^2*x^
2 + c^2*d^5*e)*sqrt(c/(c*d*e - b*e^2))*log((c*e*x^2 + 2*(c*d*e - b*e^2)*x*sqrt(c/(c*d*e - b*e^2)) + c*d - b*e)
/(c*e*x^2 - c*d + b*e)) + (28*c^2*d^4*e - 24*b*c*d^3*e^2 + 5*b^2*d^2*e^3)*x)/(8*c^3*d^8*e - 12*b*c^2*d^7*e^2 +
6*b^2*c*d^6*e^3 - b^3*d^5*e^4 + (8*c^3*d^6*e^3 - 12*b*c^2*d^5*e^4 + 6*b^2*c*d^4*e^5 - b^3*d^3*e^6)*x^4 + 2*(8
*c^3*d^7*e^2 - 12*b*c^2*d^6*e^3 + 6*b^2*c*d^5*e^4 - b^3*d^4*e^5)*x^2), -1/16*(2*(20*c^2*d^3*e^2 - 16*b*c*d^2*e
^3 + 3*b^2*d*e^4)*x^3 - 16*(c^2*d^3*e^3*x^4 + 2*c^2*d^4*e^2*x^2 + c^2*d^5*e)*sqrt(-c/(c*d*e - b*e^2))*arctan(e
*x*sqrt(-c/(c*d*e - b*e^2))) - (28*c^2*d^4 - 16*b*c*d^3*e + 3*b^2*d^2*e^2 + (28*c^2*d^2*e^2 - 16*b*c*d*e^3 + 3
*b^2*e^4)*x^4 + 2*(28*c^2*d^3*e - 16*b*c*d^2*e^2 + 3*b^2*d*e^3)*x^2)*sqrt(-d*e)*log((e*x^2 - 2*sqrt(-d*e)*x -
d)/(e*x^2 + d)) + 2*(28*c^2*d^4*e - 24*b*c*d^3*e^2 + 5*b^2*d^2*e^3)*x)/(8*c^3*d^8*e - 12*b*c^2*d^7*e^2 + 6*b^2
*c*d^6*e^3 - b^3*d^5*e^4 + (8*c^3*d^6*e^3 - 12*b*c^2*d^5*e^4 + 6*b^2*c*d^4*e^5 - b^3*d^3*e^6)*x^4 + 2*(8*c^3*d
^7*e^2 - 12*b*c^2*d^6*e^3 + 6*b^2*c*d^5*e^4 - b^3*d^4*e^5)*x^2), -1/8*((20*c^2*d^3*e^2 - 16*b*c*d^2*e^3 + 3*b^
2*d*e^4)*x^3 - 8*(c^2*d^3*e^3*x^4 + 2*c^2*d^4*e^2*x^2 + c^2*d^5*e)*sqrt(-c/(c*d*e - b*e^2))*arctan(e*x*sqrt(-c
/(c*d*e - b*e^2))) + (28*c^2*d^4 - 16*b*c*d^3*e + 3*b^2*d^2*e^2 + (28*c^2*d^2*e^2 - 16*b*c*d*e^3 + 3*b^2*e^4)*
x^4 + 2*(28*c^2*d^3*e - 16*b*c*d^2*e^2 + 3*b^2*d*e^3)*x^2)*sqrt(d*e)*arctan(sqrt(d*e)*x/d) + (28*c^2*d^4*e - 2
4*b*c*d^3*e^2 + 5*b^2*d^2*e^3)*x)/(8*c^3*d^8*e - 12*b*c^2*d^7*e^2 + 6*b^2*c*d^6*e^3 - b^3*d^5*e^4 + (8*c^3*d^6
*e^3 - 12*b*c^2*d^5*e^4 + 6*b^2*c*d^4*e^5 - b^3*d^3*e^6)*x^4 + 2*(8*c^3*d^7*e^2 - 12*b*c^2*d^6*e^3 + 6*b^2*c*d
^5*e^4 - b^3*d^4*e^5)*x^2)]

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x**2+d)**2/(c*e**2*x**4+b*e**2*x**2+b*d*e-c*d**2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x^2+d)^2/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x, algorithm="giac")

[Out]

Timed out