3.287 \(\int \frac{a+b x^2+c x^4}{x^8 (d+e x^2)^2} \, dx\)

Optimal. Leaf size=167 \[ \frac{e^2 x \left (a e^2-b d e+c d^2\right )}{2 d^5 \left (d+e x^2\right )}+\frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (5 c d^2-e (7 b d-9 a e)\right )}{2 d^{11/2}}-\frac{c d^2-e (2 b d-3 a e)}{3 d^4 x^3}+\frac{e \left (2 c d^2-e (3 b d-4 a e)\right )}{d^5 x}-\frac{b d-2 a e}{5 d^3 x^5}-\frac{a}{7 d^2 x^7} \]

[Out]

-a/(7*d^2*x^7) - (b*d - 2*a*e)/(5*d^3*x^5) - (c*d^2 - e*(2*b*d - 3*a*e))/(3*d^4*x^3) + (e*(2*c*d^2 - e*(3*b*d
- 4*a*e)))/(d^5*x) + (e^2*(c*d^2 - b*d*e + a*e^2)*x)/(2*d^5*(d + e*x^2)) + (e^(3/2)*(5*c*d^2 - e*(7*b*d - 9*a*
e))*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(11/2))

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Rubi [A]  time = 0.33021, antiderivative size = 167, 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} \[ \frac{e^2 x \left (a e^2-b d e+c d^2\right )}{2 d^5 \left (d+e x^2\right )}+\frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (5 c d^2-e (7 b d-9 a e)\right )}{2 d^{11/2}}-\frac{c d^2-e (2 b d-3 a e)}{3 d^4 x^3}+\frac{e \left (2 c d^2-e (3 b d-4 a e)\right )}{d^5 x}-\frac{b d-2 a e}{5 d^3 x^5}-\frac{a}{7 d^2 x^7} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)/(x^8*(d + e*x^2)^2),x]

[Out]

-a/(7*d^2*x^7) - (b*d - 2*a*e)/(5*d^3*x^5) - (c*d^2 - e*(2*b*d - 3*a*e))/(3*d^4*x^3) + (e*(2*c*d^2 - e*(3*b*d
- 4*a*e)))/(d^5*x) + (e^2*(c*d^2 - b*d*e + a*e^2)*x)/(2*d^5*(d + e*x^2)) + (e^(3/2)*(5*c*d^2 - e*(7*b*d - 9*a*
e))*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(11/2))

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^8 \left (d+e x^2\right )^2} \, dx &=\frac{e^2 \left (c d^2-b d e+a e^2\right ) x}{2 d^5 \left (d+e x^2\right )}+\frac{\int \frac{2 a d^4 e^2+2 d^3 e^2 (b d-a e) x^2+2 d^2 e^2 \left (c d^2-b d e+a e^2\right ) x^4-2 d e^3 \left (c d^2-b d e+a e^2\right ) x^6+e^4 \left (c d^2-b d e+a e^2\right ) x^8}{x^8 \left (d+e x^2\right )} \, dx}{2 d^5 e^2}\\ &=\frac{e^2 \left (c d^2-b d e+a e^2\right ) x}{2 d^5 \left (d+e x^2\right )}+\frac{\int \left (\frac{2 a d^3 e^2}{x^8}+\frac{2 d^2 e^2 (b d-2 a e)}{x^6}+\frac{2 d e^2 \left (c d^2-e (2 b d-3 a e)\right )}{x^4}+\frac{2 e^3 \left (-2 c d^2+e (3 b d-4 a e)\right )}{x^2}+\frac{e^4 \left (5 c d^2-e (7 b d-9 a e)\right )}{d+e x^2}\right ) \, dx}{2 d^5 e^2}\\ &=-\frac{a}{7 d^2 x^7}-\frac{b d-2 a e}{5 d^3 x^5}-\frac{c d^2-e (2 b d-3 a e)}{3 d^4 x^3}+\frac{e \left (2 c d^2-e (3 b d-4 a e)\right )}{d^5 x}+\frac{e^2 \left (c d^2-b d e+a e^2\right ) x}{2 d^5 \left (d+e x^2\right )}+\frac{\left (e^2 \left (5 c d^2-e (7 b d-9 a e)\right )\right ) \int \frac{1}{d+e x^2} \, dx}{2 d^5}\\ &=-\frac{a}{7 d^2 x^7}-\frac{b d-2 a e}{5 d^3 x^5}-\frac{c d^2-e (2 b d-3 a e)}{3 d^4 x^3}+\frac{e \left (2 c d^2-e (3 b d-4 a e)\right )}{d^5 x}+\frac{e^2 \left (c d^2-b d e+a e^2\right ) x}{2 d^5 \left (d+e x^2\right )}+\frac{e^{3/2} \left (5 c d^2-e (7 b d-9 a e)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{11/2}}\\ \end{align*}

Mathematica [A]  time = 0.0950441, size = 166, normalized size = 0.99 \[ \frac{e^2 x \left (a e^2-b d e+c d^2\right )}{2 d^5 \left (d+e x^2\right )}+\frac{-3 a e^2+2 b d e-c d^2}{3 d^4 x^3}+\frac{e \left (4 a e^2-3 b d e+2 c d^2\right )}{d^5 x}+\frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (9 a e^2-7 b d e+5 c d^2\right )}{2 d^{11/2}}+\frac{2 a e-b d}{5 d^3 x^5}-\frac{a}{7 d^2 x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)/(x^8*(d + e*x^2)^2),x]

[Out]

-a/(7*d^2*x^7) + (-(b*d) + 2*a*e)/(5*d^3*x^5) + (-(c*d^2) + 2*b*d*e - 3*a*e^2)/(3*d^4*x^3) + (e*(2*c*d^2 - 3*b
*d*e + 4*a*e^2))/(d^5*x) + (e^2*(c*d^2 - b*d*e + a*e^2)*x)/(2*d^5*(d + e*x^2)) + (e^(3/2)*(5*c*d^2 - 7*b*d*e +
9*a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(11/2))

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Maple [A]  time = 0.016, size = 221, normalized size = 1.3 \begin{align*} -{\frac{a}{7\,{d}^{2}{x}^{7}}}+{\frac{2\,ae}{5\,{d}^{3}{x}^{5}}}-{\frac{b}{5\,{d}^{2}{x}^{5}}}-{\frac{a{e}^{2}}{{d}^{4}{x}^{3}}}+{\frac{2\,be}{3\,{d}^{3}{x}^{3}}}-{\frac{c}{3\,{d}^{2}{x}^{3}}}+4\,{\frac{{e}^{3}a}{{d}^{5}x}}-3\,{\frac{{e}^{2}b}{{d}^{4}x}}+2\,{\frac{ce}{{d}^{3}x}}+{\frac{{e}^{4}xa}{2\,{d}^{5} \left ( e{x}^{2}+d \right ) }}-{\frac{{e}^{3}xb}{2\,{d}^{4} \left ( e{x}^{2}+d \right ) }}+{\frac{{e}^{2}xc}{2\,{d}^{3} \left ( e{x}^{2}+d \right ) }}+{\frac{9\,{e}^{4}a}{2\,{d}^{5}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-{\frac{7\,{e}^{3}b}{2\,{d}^{4}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{5\,{e}^{2}c}{2\,{d}^{3}}\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^8/(e*x^2+d)^2,x)

[Out]

-1/7*a/d^2/x^7+2/5/d^3/x^5*a*e-1/5/d^2/x^5*b-1/d^4/x^3*a*e^2+2/3/d^3/x^3*e*b-1/3/d^2/x^3*c+4*e^3/d^5/x*a-3*e^2
/d^4/x*b+2*e/d^3/x*c+1/2*e^4/d^5*x/(e*x^2+d)*a-1/2*e^3/d^4*x/(e*x^2+d)*b+1/2*e^2/d^3*x/(e*x^2+d)*c+9/2*e^4/d^5
/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))*a-7/2*e^3/d^4/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))*b+5/2*e^2/d^3/(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^8/(e*x^2+d)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.81143, size = 938, normalized size = 5.62 \begin{align*} \left [\frac{210 \,{\left (5 \, c d^{2} e^{2} - 7 \, b d e^{3} + 9 \, a e^{4}\right )} x^{8} + 140 \,{\left (5 \, c d^{3} e - 7 \, b d^{2} e^{2} + 9 \, a d e^{3}\right )} x^{6} - 60 \, a d^{4} - 28 \,{\left (5 \, c d^{4} - 7 \, b d^{3} e + 9 \, a d^{2} e^{2}\right )} x^{4} - 12 \,{\left (7 \, b d^{4} - 9 \, a d^{3} e\right )} x^{2} + 105 \,{\left ({\left (5 \, c d^{2} e^{2} - 7 \, b d e^{3} + 9 \, a e^{4}\right )} x^{9} +{\left (5 \, c d^{3} e - 7 \, b d^{2} e^{2} + 9 \, a d e^{3}\right )} x^{7}\right )} \sqrt{-\frac{e}{d}} \log \left (\frac{e x^{2} + 2 \, d x \sqrt{-\frac{e}{d}} - d}{e x^{2} + d}\right )}{420 \,{\left (d^{5} e x^{9} + d^{6} x^{7}\right )}}, \frac{105 \,{\left (5 \, c d^{2} e^{2} - 7 \, b d e^{3} + 9 \, a e^{4}\right )} x^{8} + 70 \,{\left (5 \, c d^{3} e - 7 \, b d^{2} e^{2} + 9 \, a d e^{3}\right )} x^{6} - 30 \, a d^{4} - 14 \,{\left (5 \, c d^{4} - 7 \, b d^{3} e + 9 \, a d^{2} e^{2}\right )} x^{4} - 6 \,{\left (7 \, b d^{4} - 9 \, a d^{3} e\right )} x^{2} + 105 \,{\left ({\left (5 \, c d^{2} e^{2} - 7 \, b d e^{3} + 9 \, a e^{4}\right )} x^{9} +{\left (5 \, c d^{3} e - 7 \, b d^{2} e^{2} + 9 \, a d e^{3}\right )} x^{7}\right )} \sqrt{\frac{e}{d}} \arctan \left (x \sqrt{\frac{e}{d}}\right )}{210 \,{\left (d^{5} e x^{9} + d^{6} x^{7}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^8/(e*x^2+d)^2,x, algorithm="fricas")

[Out]

[1/420*(210*(5*c*d^2*e^2 - 7*b*d*e^3 + 9*a*e^4)*x^8 + 140*(5*c*d^3*e - 7*b*d^2*e^2 + 9*a*d*e^3)*x^6 - 60*a*d^4
- 28*(5*c*d^4 - 7*b*d^3*e + 9*a*d^2*e^2)*x^4 - 12*(7*b*d^4 - 9*a*d^3*e)*x^2 + 105*((5*c*d^2*e^2 - 7*b*d*e^3 +
9*a*e^4)*x^9 + (5*c*d^3*e - 7*b*d^2*e^2 + 9*a*d*e^3)*x^7)*sqrt(-e/d)*log((e*x^2 + 2*d*x*sqrt(-e/d) - d)/(e*x^
2 + d)))/(d^5*e*x^9 + d^6*x^7), 1/210*(105*(5*c*d^2*e^2 - 7*b*d*e^3 + 9*a*e^4)*x^8 + 70*(5*c*d^3*e - 7*b*d^2*e
^2 + 9*a*d*e^3)*x^6 - 30*a*d^4 - 14*(5*c*d^4 - 7*b*d^3*e + 9*a*d^2*e^2)*x^4 - 6*(7*b*d^4 - 9*a*d^3*e)*x^2 + 10
5*((5*c*d^2*e^2 - 7*b*d*e^3 + 9*a*e^4)*x^9 + (5*c*d^3*e - 7*b*d^2*e^2 + 9*a*d*e^3)*x^7)*sqrt(e/d)*arctan(x*sqr
t(e/d)))/(d^5*e*x^9 + d^6*x^7)]

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Sympy [B]  time = 3.36566, size = 328, normalized size = 1.96 \begin{align*} - \frac{\sqrt{- \frac{e^{3}}{d^{11}}} \left (9 a e^{2} - 7 b d e + 5 c d^{2}\right ) \log{\left (- \frac{d^{6} \sqrt{- \frac{e^{3}}{d^{11}}} \left (9 a e^{2} - 7 b d e + 5 c d^{2}\right )}{9 a e^{4} - 7 b d e^{3} + 5 c d^{2} e^{2}} + x \right )}}{4} + \frac{\sqrt{- \frac{e^{3}}{d^{11}}} \left (9 a e^{2} - 7 b d e + 5 c d^{2}\right ) \log{\left (\frac{d^{6} \sqrt{- \frac{e^{3}}{d^{11}}} \left (9 a e^{2} - 7 b d e + 5 c d^{2}\right )}{9 a e^{4} - 7 b d e^{3} + 5 c d^{2} e^{2}} + x \right )}}{4} + \frac{- 30 a d^{4} + x^{8} \left (945 a e^{4} - 735 b d e^{3} + 525 c d^{2} e^{2}\right ) + x^{6} \left (630 a d e^{3} - 490 b d^{2} e^{2} + 350 c d^{3} e\right ) + x^{4} \left (- 126 a d^{2} e^{2} + 98 b d^{3} e - 70 c d^{4}\right ) + x^{2} \left (54 a d^{3} e - 42 b d^{4}\right )}{210 d^{6} x^{7} + 210 d^{5} e x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2+a)/x**8/(e*x**2+d)**2,x)

[Out]

-sqrt(-e**3/d**11)*(9*a*e**2 - 7*b*d*e + 5*c*d**2)*log(-d**6*sqrt(-e**3/d**11)*(9*a*e**2 - 7*b*d*e + 5*c*d**2)
/(9*a*e**4 - 7*b*d*e**3 + 5*c*d**2*e**2) + x)/4 + sqrt(-e**3/d**11)*(9*a*e**2 - 7*b*d*e + 5*c*d**2)*log(d**6*s
qrt(-e**3/d**11)*(9*a*e**2 - 7*b*d*e + 5*c*d**2)/(9*a*e**4 - 7*b*d*e**3 + 5*c*d**2*e**2) + x)/4 + (-30*a*d**4
+ x**8*(945*a*e**4 - 735*b*d*e**3 + 525*c*d**2*e**2) + x**6*(630*a*d*e**3 - 490*b*d**2*e**2 + 350*c*d**3*e) +
x**4*(-126*a*d**2*e**2 + 98*b*d**3*e - 70*c*d**4) + x**2*(54*a*d**3*e - 42*b*d**4))/(210*d**6*x**7 + 210*d**5*
e*x**9)

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Giac [A]  time = 1.09087, size = 221, normalized size = 1.32 \begin{align*} \frac{{\left (5 \, c d^{2} e^{2} - 7 \, b d e^{3} + 9 \, a e^{4}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{1}{2}\right )}}{2 \, d^{\frac{11}{2}}} + \frac{c d^{2} x e^{2} - b d x e^{3} + a x e^{4}}{2 \,{\left (x^{2} e + d\right )} d^{5}} + \frac{210 \, c d^{2} x^{6} e - 315 \, b d x^{6} e^{2} - 35 \, c d^{3} x^{4} + 420 \, a x^{6} e^{3} + 70 \, b d^{2} x^{4} e - 105 \, a d x^{4} e^{2} - 21 \, b d^{3} x^{2} + 42 \, a d^{2} x^{2} e - 15 \, a d^{3}}{105 \, d^{5} x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^8/(e*x^2+d)^2,x, algorithm="giac")

[Out]

1/2*(5*c*d^2*e^2 - 7*b*d*e^3 + 9*a*e^4)*arctan(x*e^(1/2)/sqrt(d))*e^(-1/2)/d^(11/2) + 1/2*(c*d^2*x*e^2 - b*d*x
*e^3 + a*x*e^4)/((x^2*e + d)*d^5) + 1/105*(210*c*d^2*x^6*e - 315*b*d*x^6*e^2 - 35*c*d^3*x^4 + 420*a*x^6*e^3 +
70*b*d^2*x^4*e - 105*a*d*x^4*e^2 - 21*b*d^3*x^2 + 42*a*d^2*x^2*e - 15*a*d^3)/(d^5*x^7)