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

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

[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 = 0.584933, 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 \[ \frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (5 c d^2-e (7 b d-9 a e)\right )}{2 d^{11/2}}+\frac{e^2 x \left (a e^2-b d e+c d^2\right )}{2 d^5 \left (d+e x^2\right )}+\frac{e \left (2 c d^2-e (3 b d-4 a e)\right )}{d^5 x}-\frac{c d^2-e (2 b d-3 a e)}{3 d^4 x^3}-\frac{b d-2 a e}{5 d^3 x^5}-\frac{a}{7 d^2 x^7} \]

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))

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Rubi in Sympy [A]  time = 88.3198, size = 158, normalized size = 0.95 \[ - \frac{a}{7 d^{2} x^{7}} + \frac{2 a e - b d}{5 d^{3} x^{5}} - \frac{3 a e^{2} - 2 b d e + c d^{2}}{3 d^{4} x^{3}} + \frac{e^{2} x \left (a e^{2} - b d e + c d^{2}\right )}{2 d^{5} \left (d + e x^{2}\right )} + \frac{e \left (4 a e^{2} - 3 b d e + 2 c d^{2}\right )}{d^{5} x} + \frac{e^{\frac{3}{2}} \left (9 a e^{2} - 7 b d e + 5 c d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{2 d^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a/(7*d**2*x**7) + (2*a*e - b*d)/(5*d**3*x**5) - (3*a*e**2 - 2*b*d*e + c*d**2)/(
3*d**4*x**3) + e**2*x*(a*e**2 - b*d*e + c*d**2)/(2*d**5*(d + e*x**2)) + e*(4*a*e
**2 - 3*b*d*e + 2*c*d**2)/(d**5*x) + e**(3/2)*(9*a*e**2 - 7*b*d*e + 5*c*d**2)*at
an(sqrt(e)*x/sqrt(d))/(2*d**(11/2))

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Mathematica [A]  time = 0.166284, size = 166, normalized size = 0.99 \[ \frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (9 a e^2-7 b d e+5 c d^2\right )}{2 d^{11/2}}+\frac{e^2 x \left (a e^2-b d e+c d^2\right )}{2 d^5 \left (d+e x^2\right )}+\frac{e \left (4 a e^2-3 b d e+2 c d^2\right )}{d^5 x}+\frac{-3 a e^2+2 b d e-c d^2}{3 d^4 x^3}+\frac{2 a e-b d}{5 d^3 x^5}-\frac{a}{7 d^2 x^7} \]

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[(Sq
rt[e]*x)/Sqrt[d]])/(2*d^(11/2))

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Maple [A]  time = 0.023, size = 221, normalized size = 1.3 \[ -{\frac{a}{7\,{d}^{2}{x}^{7}}}+{\frac{2\,ae}{5\,{d}^{3}{x}^{5}}}-{\frac{b}{5\,{d}^{2}{x}^{5}}}-{\frac{a{e}^{2}}{{d}^{4}{x}^{3}}}+{\frac{2\,be}{3\,{d}^{3}{x}^{3}}}-{\frac{c}{3\,{d}^{2}{x}^{3}}}+4\,{\frac{{e}^{3}a}{{d}^{5}x}}-3\,{\frac{b{e}^{2}}{{d}^{4}x}}+2\,{\frac{ce}{{d}^{3}x}}+{\frac{{e}^{4}xa}{2\,{d}^{5} \left ( e{x}^{2}+d \right ) }}-{\frac{{e}^{3}xb}{2\,{d}^{4} \left ( e{x}^{2}+d \right ) }}+{\frac{{e}^{2}xc}{2\,{d}^{3} \left ( e{x}^{2}+d \right ) }}+{\frac{9\,a{e}^{4}}{2\,{d}^{5}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-{\frac{7\,b{e}^{3}}{2\,{d}^{4}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{5\,{e}^{2}c}{2\,{d}^{3}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}} \]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.266977, size = 1, normalized size = 0.01 \[ \left [\frac{210 \,{\left (5 \, c d^{2} e^{2} - 7 \, b d e^{3} + 9 \, a e^{4}\right )} x^{8} + 140 \,{\left (5 \, c d^{3} e - 7 \, b d^{2} e^{2} + 9 \, a d e^{3}\right )} x^{6} - 60 \, a d^{4} - 28 \,{\left (5 \, c d^{4} - 7 \, b d^{3} e + 9 \, a d^{2} e^{2}\right )} x^{4} - 12 \,{\left (7 \, b d^{4} - 9 \, a d^{3} e\right )} x^{2} + 105 \,{\left ({\left (5 \, c d^{2} e^{2} - 7 \, b d e^{3} + 9 \, a e^{4}\right )} x^{9} +{\left (5 \, c d^{3} e - 7 \, b d^{2} e^{2} + 9 \, a d e^{3}\right )} x^{7}\right )} \sqrt{-\frac{e}{d}} \log \left (\frac{e x^{2} + 2 \, d x \sqrt{-\frac{e}{d}} - d}{e x^{2} + d}\right )}{420 \,{\left (d^{5} e x^{9} + d^{6} x^{7}\right )}}, \frac{105 \,{\left (5 \, c d^{2} e^{2} - 7 \, b d e^{3} + 9 \, a e^{4}\right )} x^{8} + 70 \,{\left (5 \, c d^{3} e - 7 \, b d^{2} e^{2} + 9 \, a d e^{3}\right )} x^{6} - 30 \, a d^{4} - 14 \,{\left (5 \, c d^{4} - 7 \, b d^{3} e + 9 \, a d^{2} e^{2}\right )} x^{4} - 6 \,{\left (7 \, b d^{4} - 9 \, a d^{3} e\right )} x^{2} + 105 \,{\left ({\left (5 \, c d^{2} e^{2} - 7 \, b d e^{3} + 9 \, a e^{4}\right )} x^{9} +{\left (5 \, c d^{3} e - 7 \, b d^{2} e^{2} + 9 \, a d e^{3}\right )} x^{7}\right )} \sqrt{\frac{e}{d}} \arctan \left (\frac{e x}{d \sqrt{\frac{e}{d}}}\right )}{210 \,{\left (d^{5} e x^{9} + d^{6} x^{7}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)/((e*x^2 + d)^2*x^8),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 + 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)*sq
rt(e/d)*arctan(e*x/(d*sqrt(e/d))))/(d^5*e*x^9 + d^6*x^7)]

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Sympy [A]  time = 13.0207, size = 328, normalized size = 1.96 \[ - \frac{\sqrt{- \frac{e^{3}}{d^{11}}} \left (9 a e^{2} - 7 b d e + 5 c d^{2}\right ) \log{\left (- \frac{d^{6} \sqrt{- \frac{e^{3}}{d^{11}}} \left (9 a e^{2} - 7 b d e + 5 c d^{2}\right )}{9 a e^{4} - 7 b d e^{3} + 5 c d^{2} e^{2}} + x \right )}}{4} + \frac{\sqrt{- \frac{e^{3}}{d^{11}}} \left (9 a e^{2} - 7 b d e + 5 c d^{2}\right ) \log{\left (\frac{d^{6} \sqrt{- \frac{e^{3}}{d^{11}}} \left (9 a e^{2} - 7 b d e + 5 c d^{2}\right )}{9 a e^{4} - 7 b d e^{3} + 5 c d^{2} e^{2}} + x \right )}}{4} + \frac{- 30 a d^{4} + x^{8} \left (945 a e^{4} - 735 b d e^{3} + 525 c d^{2} e^{2}\right ) + x^{6} \left (630 a d e^{3} - 490 b d^{2} e^{2} + 350 c d^{3} e\right ) + x^{4} \left (- 126 a d^{2} e^{2} + 98 b d^{3} e - 70 c d^{4}\right ) + x^{2} \left (54 a d^{3} e - 42 b d^{4}\right )}{210 d^{6} x^{7} + 210 d^{5} e x^{9}} \]

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*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 + (-
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 - 7
0*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/XCAS [A]  time = 0.269673, size = 221, normalized size = 1.32 \[ \frac{{\left (5 \, c d^{2} e^{2} - 7 \, b d e^{3} + 9 \, a e^{4}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{1}{2}\right )}}{2 \, d^{\frac{11}{2}}} + \frac{c d^{2} x e^{2} - b d x e^{3} + a x e^{4}}{2 \,{\left (x^{2} e + d\right )} d^{5}} + \frac{210 \, c d^{2} x^{6} e - 315 \, b d x^{6} e^{2} - 35 \, c d^{3} x^{4} + 420 \, a x^{6} e^{3} + 70 \, b d^{2} x^{4} e - 105 \, a d x^{4} e^{2} - 21 \, b d^{3} x^{2} + 42 \, a d^{2} x^{2} e - 15 \, a d^{3}}{105 \, d^{5} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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