3.286 \(\int \frac{a+b x^2+c x^4}{x^6 (d+e x^2)^2} \, dx\)

Optimal. Leaf size=136 \[ -\frac{e x \left (a e^2-b d e+c d^2\right )}{2 d^4 \left (d+e x^2\right )}-\frac{c d^2-e (2 b d-3 a e)}{d^4 x}-\frac{\sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (3 c d^2-e (5 b d-7 a e)\right )}{2 d^{9/2}}-\frac{b d-2 a e}{3 d^3 x^3}-\frac{a}{5 d^2 x^5} \]

[Out]

-a/(5*d^2*x^5) - (b*d - 2*a*e)/(3*d^3*x^3) - (c*d^2 - e*(2*b*d - 3*a*e))/(d^4*x) - (e*(c*d^2 - b*d*e + a*e^2)*
x)/(2*d^4*(d + e*x^2)) - (Sqrt[e]*(3*c*d^2 - e*(5*b*d - 7*a*e))*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(9/2))

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Rubi [A]  time = 0.252272, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {1259, 1802, 205} \[ -\frac{e x \left (a e^2-b d e+c d^2\right )}{2 d^4 \left (d+e x^2\right )}-\frac{c d^2-e (2 b d-3 a e)}{d^4 x}-\frac{\sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (3 c d^2-e (5 b d-7 a e)\right )}{2 d^{9/2}}-\frac{b d-2 a e}{3 d^3 x^3}-\frac{a}{5 d^2 x^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)/(x^6*(d + e*x^2)^2),x]

[Out]

-a/(5*d^2*x^5) - (b*d - 2*a*e)/(3*d^3*x^3) - (c*d^2 - e*(2*b*d - 3*a*e))/(d^4*x) - (e*(c*d^2 - b*d*e + a*e^2)*
x)/(2*d^4*(d + e*x^2)) - (Sqrt[e]*(3*c*d^2 - e*(5*b*d - 7*a*e))*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(9/2))

Rule 1259

Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[((-d)^(
m/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*x*(d + e*x^2)^(q + 1))/(2*e^(2*p + m/2)*(q + 1)), x] + Dist[(-d)^(m/2 - 1)/
(2*e^(2*p)*(q + 1)), Int[x^m*(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1*(2*(-d)^(-(m/2) + 1)*e^(2*p)*(q + 1)*
(a + b*x^2 + c*x^4)^p - ((c*d^2 - b*d*e + a*e^2)^p/(e^(m/2)*x^m))*(d + e*(2*q + 3)*x^2)))/(d + e*x^2)], x], x]
, x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] && ILtQ[m/2, 0]

Rule 1802

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x
^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

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{a+b x^2+c x^4}{x^6 \left (d+e x^2\right )^2} \, dx &=-\frac{e \left (c d^2-b d e+a e^2\right ) x}{2 d^4 \left (d+e x^2\right )}-\frac{\int \frac{-2 a d^3 e^2-2 d^2 e^2 (b d-a e) x^2-2 d e^2 \left (c d^2-b d e+a e^2\right ) x^4+e^3 \left (c d^2-b d e+a e^2\right ) x^6}{x^6 \left (d+e x^2\right )} \, dx}{2 d^4 e^2}\\ &=-\frac{e \left (c d^2-b d e+a e^2\right ) x}{2 d^4 \left (d+e x^2\right )}-\frac{\int \left (-\frac{2 a d^2 e^2}{x^6}-\frac{2 d e^2 (b d-2 a e)}{x^4}+\frac{2 e^2 \left (-c d^2+e (2 b d-3 a e)\right )}{x^2}+\frac{e^3 \left (3 c d^2-e (5 b d-7 a e)\right )}{d+e x^2}\right ) \, dx}{2 d^4 e^2}\\ &=-\frac{a}{5 d^2 x^5}-\frac{b d-2 a e}{3 d^3 x^3}-\frac{c d^2-e (2 b d-3 a e)}{d^4 x}-\frac{e \left (c d^2-b d e+a e^2\right ) x}{2 d^4 \left (d+e x^2\right )}-\frac{\left (e \left (3 c d^2-e (5 b d-7 a e)\right )\right ) \int \frac{1}{d+e x^2} \, dx}{2 d^4}\\ &=-\frac{a}{5 d^2 x^5}-\frac{b d-2 a e}{3 d^3 x^3}-\frac{c d^2-e (2 b d-3 a e)}{d^4 x}-\frac{e \left (c d^2-b d e+a e^2\right ) x}{2 d^4 \left (d+e x^2\right )}-\frac{\sqrt{e} \left (3 c d^2-e (5 b d-7 a e)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.0901613, size = 135, normalized size = 0.99 \[ -\frac{e x \left (a e^2-b d e+c d^2\right )}{2 d^4 \left (d+e x^2\right )}+\frac{-3 a e^2+2 b d e-c d^2}{d^4 x}-\frac{\sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (7 a e^2-5 b d e+3 c d^2\right )}{2 d^{9/2}}+\frac{2 a e-b d}{3 d^3 x^3}-\frac{a}{5 d^2 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)/(x^6*(d + e*x^2)^2),x]

[Out]

-a/(5*d^2*x^5) + (-(b*d) + 2*a*e)/(3*d^3*x^3) + (-(c*d^2) + 2*b*d*e - 3*a*e^2)/(d^4*x) - (e*(c*d^2 - b*d*e + a
*e^2)*x)/(2*d^4*(d + e*x^2)) - (Sqrt[e]*(3*c*d^2 - 5*b*d*e + 7*a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(9/2))

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Maple [A]  time = 0.013, size = 183, normalized size = 1.4 \begin{align*} -{\frac{a}{5\,{d}^{2}{x}^{5}}}+{\frac{2\,ae}{3\,{d}^{3}{x}^{3}}}-{\frac{b}{3\,{d}^{2}{x}^{3}}}-3\,{\frac{a{e}^{2}}{{d}^{4}x}}+2\,{\frac{be}{{d}^{3}x}}-{\frac{c}{{d}^{2}x}}-{\frac{{e}^{3}xa}{2\,{d}^{4} \left ( e{x}^{2}+d \right ) }}+{\frac{{e}^{2}xb}{2\,{d}^{3} \left ( e{x}^{2}+d \right ) }}-{\frac{exc}{2\,{d}^{2} \left ( e{x}^{2}+d \right ) }}-{\frac{7\,{e}^{3}a}{2\,{d}^{4}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{5\,{e}^{2}b}{2\,{d}^{3}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-{\frac{3\,ec}{2\,{d}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2+a)/x^6/(e*x^2+d)^2,x)

[Out]

-1/5*a/d^2/x^5+2/3/d^3/x^3*a*e-1/3/d^2/x^3*b-3/d^4/x*a*e^2+2/d^3/x*e*b-1/d^2/x*c-1/2*e^3/d^4*x/(e*x^2+d)*a+1/2
*e^2/d^3*x/(e*x^2+d)*b-1/2*e/d^2*x/(e*x^2+d)*c-7/2*e^3/d^4/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))*a+5/2*e^2/d^3/(
d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))*b-3/2*e/d^2/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))*c

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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((c*x^4+b*x^2+a)/x^6/(e*x^2+d)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.90378, size = 775, normalized size = 5.7 \begin{align*} \left [-\frac{30 \,{\left (3 \, c d^{2} e - 5 \, b d e^{2} + 7 \, a e^{3}\right )} x^{6} + 20 \,{\left (3 \, c d^{3} - 5 \, b d^{2} e + 7 \, a d e^{2}\right )} x^{4} + 12 \, a d^{3} + 4 \,{\left (5 \, b d^{3} - 7 \, a d^{2} e\right )} x^{2} - 15 \,{\left ({\left (3 \, c d^{2} e - 5 \, b d e^{2} + 7 \, a e^{3}\right )} x^{7} +{\left (3 \, c d^{3} - 5 \, b d^{2} e + 7 \, a d e^{2}\right )} x^{5}\right )} \sqrt{-\frac{e}{d}} \log \left (\frac{e x^{2} - 2 \, d x \sqrt{-\frac{e}{d}} - d}{e x^{2} + d}\right )}{60 \,{\left (d^{4} e x^{7} + d^{5} x^{5}\right )}}, -\frac{15 \,{\left (3 \, c d^{2} e - 5 \, b d e^{2} + 7 \, a e^{3}\right )} x^{6} + 10 \,{\left (3 \, c d^{3} - 5 \, b d^{2} e + 7 \, a d e^{2}\right )} x^{4} + 6 \, a d^{3} + 2 \,{\left (5 \, b d^{3} - 7 \, a d^{2} e\right )} x^{2} + 15 \,{\left ({\left (3 \, c d^{2} e - 5 \, b d e^{2} + 7 \, a e^{3}\right )} x^{7} +{\left (3 \, c d^{3} - 5 \, b d^{2} e + 7 \, a d e^{2}\right )} x^{5}\right )} \sqrt{\frac{e}{d}} \arctan \left (x \sqrt{\frac{e}{d}}\right )}{30 \,{\left (d^{4} e x^{7} + d^{5} x^{5}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^6/(e*x^2+d)^2,x, algorithm="fricas")

[Out]

[-1/60*(30*(3*c*d^2*e - 5*b*d*e^2 + 7*a*e^3)*x^6 + 20*(3*c*d^3 - 5*b*d^2*e + 7*a*d*e^2)*x^4 + 12*a*d^3 + 4*(5*
b*d^3 - 7*a*d^2*e)*x^2 - 15*((3*c*d^2*e - 5*b*d*e^2 + 7*a*e^3)*x^7 + (3*c*d^3 - 5*b*d^2*e + 7*a*d*e^2)*x^5)*sq
rt(-e/d)*log((e*x^2 - 2*d*x*sqrt(-e/d) - d)/(e*x^2 + d)))/(d^4*e*x^7 + d^5*x^5), -1/30*(15*(3*c*d^2*e - 5*b*d*
e^2 + 7*a*e^3)*x^6 + 10*(3*c*d^3 - 5*b*d^2*e + 7*a*d*e^2)*x^4 + 6*a*d^3 + 2*(5*b*d^3 - 7*a*d^2*e)*x^2 + 15*((3
*c*d^2*e - 5*b*d*e^2 + 7*a*e^3)*x^7 + (3*c*d^3 - 5*b*d^2*e + 7*a*d*e^2)*x^5)*sqrt(e/d)*arctan(x*sqrt(e/d)))/(d
^4*e*x^7 + d^5*x^5)]

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Sympy [B]  time = 2.49606, size = 284, normalized size = 2.09 \begin{align*} \frac{\sqrt{- \frac{e}{d^{9}}} \left (7 a e^{2} - 5 b d e + 3 c d^{2}\right ) \log{\left (- \frac{d^{5} \sqrt{- \frac{e}{d^{9}}} \left (7 a e^{2} - 5 b d e + 3 c d^{2}\right )}{7 a e^{3} - 5 b d e^{2} + 3 c d^{2} e} + x \right )}}{4} - \frac{\sqrt{- \frac{e}{d^{9}}} \left (7 a e^{2} - 5 b d e + 3 c d^{2}\right ) \log{\left (\frac{d^{5} \sqrt{- \frac{e}{d^{9}}} \left (7 a e^{2} - 5 b d e + 3 c d^{2}\right )}{7 a e^{3} - 5 b d e^{2} + 3 c d^{2} e} + x \right )}}{4} - \frac{6 a d^{3} + x^{6} \left (105 a e^{3} - 75 b d e^{2} + 45 c d^{2} e\right ) + x^{4} \left (70 a d e^{2} - 50 b d^{2} e + 30 c d^{3}\right ) + x^{2} \left (- 14 a d^{2} e + 10 b d^{3}\right )}{30 d^{5} x^{5} + 30 d^{4} e x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2+a)/x**6/(e*x**2+d)**2,x)

[Out]

sqrt(-e/d**9)*(7*a*e**2 - 5*b*d*e + 3*c*d**2)*log(-d**5*sqrt(-e/d**9)*(7*a*e**2 - 5*b*d*e + 3*c*d**2)/(7*a*e**
3 - 5*b*d*e**2 + 3*c*d**2*e) + x)/4 - sqrt(-e/d**9)*(7*a*e**2 - 5*b*d*e + 3*c*d**2)*log(d**5*sqrt(-e/d**9)*(7*
a*e**2 - 5*b*d*e + 3*c*d**2)/(7*a*e**3 - 5*b*d*e**2 + 3*c*d**2*e) + x)/4 - (6*a*d**3 + x**6*(105*a*e**3 - 75*b
*d*e**2 + 45*c*d**2*e) + x**4*(70*a*d*e**2 - 50*b*d**2*e + 30*c*d**3) + x**2*(-14*a*d**2*e + 10*b*d**3))/(30*d
**5*x**5 + 30*d**4*e*x**7)

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Giac [A]  time = 1.10171, size = 177, normalized size = 1.3 \begin{align*} -\frac{{\left (3 \, c d^{2} e - 5 \, b d e^{2} + 7 \, a e^{3}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{1}{2}\right )}}{2 \, d^{\frac{9}{2}}} - \frac{c d^{2} x e - b d x e^{2} + a x e^{3}}{2 \,{\left (x^{2} e + d\right )} d^{4}} - \frac{15 \, c d^{2} x^{4} - 30 \, b d x^{4} e + 45 \, a x^{4} e^{2} + 5 \, b d^{2} x^{2} - 10 \, a d x^{2} e + 3 \, a d^{2}}{15 \, d^{4} x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^6/(e*x^2+d)^2,x, algorithm="giac")

[Out]

-1/2*(3*c*d^2*e - 5*b*d*e^2 + 7*a*e^3)*arctan(x*e^(1/2)/sqrt(d))*e^(-1/2)/d^(9/2) - 1/2*(c*d^2*x*e - b*d*x*e^2
+ a*x*e^3)/((x^2*e + d)*d^4) - 1/15*(15*c*d^2*x^4 - 30*b*d*x^4*e + 45*a*x^4*e^2 + 5*b*d^2*x^2 - 10*a*d*x^2*e
+ 3*a*d^2)/(d^4*x^5)