3.285 \(\int \frac{a+b x^2+c x^4}{x^4 \left (d+e x^2\right )^3} \, dx\)

Optimal. Leaf size=142 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (35 a e^2-15 b d e+3 c d^2\right )}{8 d^{9/2} \sqrt{e}}+\frac{x \left (3 c d^2-e (7 b d-11 a e)\right )}{8 d^4 \left (d+e x^2\right )}+\frac{x \left (a e^2-b d e+c d^2\right )}{4 d^3 \left (d+e x^2\right )^2}-\frac{b d-3 a e}{d^4 x}-\frac{a}{3 d^3 x^3} \]

[Out]

-a/(3*d^3*x^3) - (b*d - 3*a*e)/(d^4*x) + ((c*d^2 - b*d*e + a*e^2)*x)/(4*d^3*(d +
e*x^2)^2) + ((3*c*d^2 - e*(7*b*d - 11*a*e))*x)/(8*d^4*(d + e*x^2)) + ((3*c*d^2
- 15*b*d*e + 35*a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(9/2)*Sqrt[e])

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Rubi [A]  time = 0.423639, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (35 a e^2-15 b d e+3 c d^2\right )}{8 d^{9/2} \sqrt{e}}+\frac{x \left (3 c d^2-e (7 b d-11 a e)\right )}{8 d^4 \left (d+e x^2\right )}+\frac{x \left (a e^2-b d e+c d^2\right )}{4 d^3 \left (d+e x^2\right )^2}-\frac{b d-3 a e}{d^4 x}-\frac{a}{3 d^3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2 + c*x^4)/(x^4*(d + e*x^2)^3),x]

[Out]

-a/(3*d^3*x^3) - (b*d - 3*a*e)/(d^4*x) + ((c*d^2 - b*d*e + a*e^2)*x)/(4*d^3*(d +
e*x^2)^2) + ((3*c*d^2 - e*(7*b*d - 11*a*e))*x)/(8*d^4*(d + e*x^2)) + ((3*c*d^2
- 15*b*d*e + 35*a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(9/2)*Sqrt[e])

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Rubi in Sympy [A]  time = 62.8802, size = 133, normalized size = 0.94 \[ - \frac{a}{3 d^{3} x^{3}} + \frac{x \left (a e^{2} - b d e + c d^{2}\right )}{4 d^{3} \left (d + e x^{2}\right )^{2}} + \frac{x \left (11 a e^{2} - 7 b d e + 3 c d^{2}\right )}{8 d^{4} \left (d + e x^{2}\right )} + \frac{3 a e - b d}{d^{4} x} + \frac{\left (35 a e^{2} - 15 b d e + 3 c d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{8 d^{\frac{9}{2}} \sqrt{e}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2+a)/x**4/(e*x**2+d)**3,x)

[Out]

-a/(3*d**3*x**3) + x*(a*e**2 - b*d*e + c*d**2)/(4*d**3*(d + e*x**2)**2) + x*(11*
a*e**2 - 7*b*d*e + 3*c*d**2)/(8*d**4*(d + e*x**2)) + (3*a*e - b*d)/(d**4*x) + (3
5*a*e**2 - 15*b*d*e + 3*c*d**2)*atan(sqrt(e)*x/sqrt(d))/(8*d**(9/2)*sqrt(e))

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Mathematica [A]  time = 0.145062, size = 141, normalized size = 0.99 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (35 a e^2-15 b d e+3 c d^2\right )}{8 d^{9/2} \sqrt{e}}+\frac{x \left (11 a e^2-7 b d e+3 c d^2\right )}{8 d^4 \left (d+e x^2\right )}+\frac{x \left (a e^2-b d e+c d^2\right )}{4 d^3 \left (d+e x^2\right )^2}+\frac{3 a e-b d}{d^4 x}-\frac{a}{3 d^3 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2 + c*x^4)/(x^4*(d + e*x^2)^3),x]

[Out]

-a/(3*d^3*x^3) + (-(b*d) + 3*a*e)/(d^4*x) + ((c*d^2 - b*d*e + a*e^2)*x)/(4*d^3*(
d + e*x^2)^2) + ((3*c*d^2 - 7*b*d*e + 11*a*e^2)*x)/(8*d^4*(d + e*x^2)) + ((3*c*d
^2 - 15*b*d*e + 35*a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(9/2)*Sqrt[e])

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Maple [A]  time = 0.022, size = 207, normalized size = 1.5 \[ -{\frac{a}{3\,{d}^{3}{x}^{3}}}+3\,{\frac{ae}{{d}^{4}x}}-{\frac{b}{{d}^{3}x}}+{\frac{11\,{x}^{3}a{e}^{3}}{8\,{d}^{4} \left ( e{x}^{2}+d \right ) ^{2}}}-{\frac{7\,b{x}^{3}{e}^{2}}{8\,{d}^{3} \left ( e{x}^{2}+d \right ) ^{2}}}+{\frac{3\,c{x}^{3}e}{8\,{d}^{2} \left ( e{x}^{2}+d \right ) ^{2}}}+{\frac{13\,a{e}^{2}x}{8\,{d}^{3} \left ( e{x}^{2}+d \right ) ^{2}}}-{\frac{9\,bex}{8\,{d}^{2} \left ( e{x}^{2}+d \right ) ^{2}}}+{\frac{5\,cx}{8\,d \left ( e{x}^{2}+d \right ) ^{2}}}+{\frac{35\,a{e}^{2}}{8\,{d}^{4}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-{\frac{15\,be}{8\,{d}^{3}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{3\,c}{8\,{d}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2+a)/x^4/(e*x^2+d)^3,x)

[Out]

-1/3*a/d^3/x^3+3/d^4/x*a*e-1/d^3/x*b+11/8/d^4/(e*x^2+d)^2*x^3*a*e^3-7/8/d^3/(e*x
^2+d)^2*x^3*b*e^2+3/8/d^2/(e*x^2+d)^2*x^3*c*e+13/8/d^3/(e*x^2+d)^2*e^2*a*x-9/8/d
^2/(e*x^2+d)^2*b*e*x+5/8/d/(e*x^2+d)^2*c*x+35/8/d^4/(d*e)^(1/2)*arctan(x*e/(d*e)
^(1/2))*a*e^2-15/8/d^3/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*b*e+3/8/d^2/(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)^3*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.277585, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left ({\left (3 \, c d^{2} e^{2} - 15 \, b d e^{3} + 35 \, a e^{4}\right )} x^{7} + 2 \,{\left (3 \, c d^{3} e - 15 \, b d^{2} e^{2} + 35 \, a d e^{3}\right )} x^{5} +{\left (3 \, c d^{4} - 15 \, b d^{3} e + 35 \, a d^{2} e^{2}\right )} x^{3}\right )} \log \left (\frac{2 \, d e x +{\left (e x^{2} - d\right )} \sqrt{-d e}}{e x^{2} + d}\right ) + 2 \,{\left (3 \,{\left (3 \, c d^{2} e - 15 \, b d e^{2} + 35 \, a e^{3}\right )} x^{6} + 5 \,{\left (3 \, c d^{3} - 15 \, b d^{2} e + 35 \, a d e^{2}\right )} x^{4} - 8 \, a d^{3} - 8 \,{\left (3 \, b d^{3} - 7 \, a d^{2} e\right )} x^{2}\right )} \sqrt{-d e}}{48 \,{\left (d^{4} e^{2} x^{7} + 2 \, d^{5} e x^{5} + d^{6} x^{3}\right )} \sqrt{-d e}}, \frac{3 \,{\left ({\left (3 \, c d^{2} e^{2} - 15 \, b d e^{3} + 35 \, a e^{4}\right )} x^{7} + 2 \,{\left (3 \, c d^{3} e - 15 \, b d^{2} e^{2} + 35 \, a d e^{3}\right )} x^{5} +{\left (3 \, c d^{4} - 15 \, b d^{3} e + 35 \, a d^{2} e^{2}\right )} x^{3}\right )} \arctan \left (\frac{\sqrt{d e} x}{d}\right ) +{\left (3 \,{\left (3 \, c d^{2} e - 15 \, b d e^{2} + 35 \, a e^{3}\right )} x^{6} + 5 \,{\left (3 \, c d^{3} - 15 \, b d^{2} e + 35 \, a d e^{2}\right )} x^{4} - 8 \, a d^{3} - 8 \,{\left (3 \, b d^{3} - 7 \, a d^{2} e\right )} x^{2}\right )} \sqrt{d e}}{24 \,{\left (d^{4} e^{2} x^{7} + 2 \, d^{5} e x^{5} + d^{6} x^{3}\right )} \sqrt{d e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)/((e*x^2 + d)^3*x^4),x, algorithm="fricas")

[Out]

[1/48*(3*((3*c*d^2*e^2 - 15*b*d*e^3 + 35*a*e^4)*x^7 + 2*(3*c*d^3*e - 15*b*d^2*e^
2 + 35*a*d*e^3)*x^5 + (3*c*d^4 - 15*b*d^3*e + 35*a*d^2*e^2)*x^3)*log((2*d*e*x +
(e*x^2 - d)*sqrt(-d*e))/(e*x^2 + d)) + 2*(3*(3*c*d^2*e - 15*b*d*e^2 + 35*a*e^3)*
x^6 + 5*(3*c*d^3 - 15*b*d^2*e + 35*a*d*e^2)*x^4 - 8*a*d^3 - 8*(3*b*d^3 - 7*a*d^2
*e)*x^2)*sqrt(-d*e))/((d^4*e^2*x^7 + 2*d^5*e*x^5 + d^6*x^3)*sqrt(-d*e)), 1/24*(3
*((3*c*d^2*e^2 - 15*b*d*e^3 + 35*a*e^4)*x^7 + 2*(3*c*d^3*e - 15*b*d^2*e^2 + 35*a
*d*e^3)*x^5 + (3*c*d^4 - 15*b*d^3*e + 35*a*d^2*e^2)*x^3)*arctan(sqrt(d*e)*x/d) +
(3*(3*c*d^2*e - 15*b*d*e^2 + 35*a*e^3)*x^6 + 5*(3*c*d^3 - 15*b*d^2*e + 35*a*d*e
^2)*x^4 - 8*a*d^3 - 8*(3*b*d^3 - 7*a*d^2*e)*x^2)*sqrt(d*e))/((d^4*e^2*x^7 + 2*d^
5*e*x^5 + d^6*x^3)*sqrt(d*e))]

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Sympy [A]  time = 12.595, size = 214, normalized size = 1.51 \[ - \frac{\sqrt{- \frac{1}{d^{9} e}} \left (35 a e^{2} - 15 b d e + 3 c d^{2}\right ) \log{\left (- d^{5} \sqrt{- \frac{1}{d^{9} e}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{d^{9} e}} \left (35 a e^{2} - 15 b d e + 3 c d^{2}\right ) \log{\left (d^{5} \sqrt{- \frac{1}{d^{9} e}} + x \right )}}{16} + \frac{- 8 a d^{3} + x^{6} \left (105 a e^{3} - 45 b d e^{2} + 9 c d^{2} e\right ) + x^{4} \left (175 a d e^{2} - 75 b d^{2} e + 15 c d^{3}\right ) + x^{2} \left (56 a d^{2} e - 24 b d^{3}\right )}{24 d^{6} x^{3} + 48 d^{5} e x^{5} + 24 d^{4} e^{2} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+b*x**2+a)/x**4/(e*x**2+d)**3,x)

[Out]

-sqrt(-1/(d**9*e))*(35*a*e**2 - 15*b*d*e + 3*c*d**2)*log(-d**5*sqrt(-1/(d**9*e))
+ x)/16 + sqrt(-1/(d**9*e))*(35*a*e**2 - 15*b*d*e + 3*c*d**2)*log(d**5*sqrt(-1/
(d**9*e)) + x)/16 + (-8*a*d**3 + x**6*(105*a*e**3 - 45*b*d*e**2 + 9*c*d**2*e) +
x**4*(175*a*d*e**2 - 75*b*d**2*e + 15*c*d**3) + x**2*(56*a*d**2*e - 24*b*d**3))/
(24*d**6*x**3 + 48*d**5*e*x**5 + 24*d**4*e**2*x**7)

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GIAC/XCAS [A]  time = 0.272068, size = 173, normalized size = 1.22 \[ \frac{{\left (3 \, c d^{2} - 15 \, b d e + 35 \, a e^{2}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{1}{2}\right )}}{8 \, d^{\frac{9}{2}}} + \frac{3 \, c d^{2} x^{3} e - 7 \, b d x^{3} e^{2} + 5 \, c d^{3} x + 11 \, a x^{3} e^{3} - 9 \, b d^{2} x e + 13 \, a d x e^{2}}{8 \,{\left (x^{2} e + d\right )}^{2} d^{4}} - \frac{3 \, b d x^{2} - 9 \, a x^{2} e + a d}{3 \, d^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)/((e*x^2 + d)^3*x^4),x, algorithm="giac")

[Out]

1/8*(3*c*d^2 - 15*b*d*e + 35*a*e^2)*arctan(x*e^(1/2)/sqrt(d))*e^(-1/2)/d^(9/2) +
1/8*(3*c*d^2*x^3*e - 7*b*d*x^3*e^2 + 5*c*d^3*x + 11*a*x^3*e^3 - 9*b*d^2*x*e + 1
3*a*d*x*e^2)/((x^2*e + d)^2*d^4) - 1/3*(3*b*d*x^2 - 9*a*x^2*e + a*d)/(d^4*x^3)