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

Optimal. Leaf size=126 \[ \frac{x \left (\frac{5 a}{d^2}-\frac{7 c}{e^2}\right )}{24 \left (d+e x^2\right )^2}+\frac{x \left (a+\frac{c d^2}{e^2}\right )}{6 d \left (d+e x^2\right )^3}+\frac{\left (5 a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{16 d^{7/2} e^{5/2}}+\frac{x \left (5 a e^2+c d^2\right )}{16 d^3 e^2 \left (d+e x^2\right )} \]

[Out]

((a + (c*d^2)/e^2)*x)/(6*d*(d + e*x^2)^3) + (((5*a)/d^2 - (7*c)/e^2)*x)/(24*(d +
e*x^2)^2) + ((c*d^2 + 5*a*e^2)*x)/(16*d^3*e^2*(d + e*x^2)) + ((c*d^2 + 5*a*e^2)
*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(16*d^(7/2)*e^(5/2))

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Rubi [A]  time = 0.222384, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{x \left (\frac{5 a}{d^2}-\frac{7 c}{e^2}\right )}{24 \left (d+e x^2\right )^2}+\frac{x \left (a+\frac{c d^2}{e^2}\right )}{6 d \left (d+e x^2\right )^3}+\frac{\left (5 a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{16 d^{7/2} e^{5/2}}+\frac{x \left (5 a e^2+c d^2\right )}{16 d^3 e^2 \left (d+e x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

((a + (c*d^2)/e^2)*x)/(6*d*(d + e*x^2)^3) + (((5*a)/d^2 - (7*c)/e^2)*x)/(24*(d +
e*x^2)^2) + ((c*d^2 + 5*a*e^2)*x)/(16*d^3*e^2*(d + e*x^2)) + ((c*d^2 + 5*a*e^2)
*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(16*d^(7/2)*e^(5/2))

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Rubi in Sympy [A]  time = 28.812, size = 112, normalized size = 0.89 \[ \frac{x \left (\frac{5 a}{16 d^{3}} + \frac{c}{16 d e^{2}}\right )}{d + e x^{2}} + \frac{x \left (\frac{5 a}{24 d^{2}} - \frac{7 c}{24 e^{2}}\right )}{\left (d + e x^{2}\right )^{2}} + \frac{x \left (\frac{a}{6 d} + \frac{c d}{6 e^{2}}\right )}{\left (d + e x^{2}\right )^{3}} + \frac{\left (5 a e^{2} + c d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{16 d^{\frac{7}{2}} e^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(5*a/(16*d**3) + c/(16*d*e**2))/(d + e*x**2) + x*(5*a/(24*d**2) - 7*c/(24*e**2
))/(d + e*x**2)**2 + x*(a/(6*d) + c*d/(6*e**2))/(d + e*x**2)**3 + (5*a*e**2 + c*
d**2)*atan(sqrt(e)*x/sqrt(d))/(16*d**(7/2)*e**(5/2))

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Mathematica [A]  time = 0.139761, size = 113, normalized size = 0.9 \[ \frac{\left (5 a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{16 d^{7/2} e^{5/2}}+\frac{x \left (a e^2 \left (33 d^2+40 d e x^2+15 e^2 x^4\right )+c d^2 \left (-3 d^2-8 d e x^2+3 e^2 x^4\right )\right )}{48 d^3 e^2 \left (d+e x^2\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

(x*(c*d^2*(-3*d^2 - 8*d*e*x^2 + 3*e^2*x^4) + a*e^2*(33*d^2 + 40*d*e*x^2 + 15*e^2
*x^4)))/(48*d^3*e^2*(d + e*x^2)^3) + ((c*d^2 + 5*a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[
d]])/(16*d^(7/2)*e^(5/2))

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Maple [A]  time = 0.013, size = 122, normalized size = 1. \[{\frac{1}{ \left ( e{x}^{2}+d \right ) ^{3}} \left ({\frac{ \left ( 5\,a{e}^{2}+c{d}^{2} \right ){x}^{5}}{16\,{d}^{3}}}+{\frac{ \left ( 5\,a{e}^{2}-c{d}^{2} \right ){x}^{3}}{6\,{d}^{2}e}}+{\frac{ \left ( 11\,a{e}^{2}-c{d}^{2} \right ) x}{16\,{e}^{2}d}} \right ) }+{\frac{5\,a}{16\,{d}^{3}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{c}{16\,{e}^{2}d}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+a)/(e*x^2+d)^4,x)

[Out]

(1/16*(5*a*e^2+c*d^2)/d^3*x^5+1/6*(5*a*e^2-c*d^2)/d^2/e*x^3+1/16*(11*a*e^2-c*d^2
)/e^2/d*x)/(e*x^2+d)^3+5/16/d^3/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*a+1/16/d/e^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 + a)/(e*x^2 + d)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.287885, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left ({\left (c d^{2} e^{3} + 5 \, a e^{5}\right )} x^{6} + c d^{5} + 5 \, a d^{3} e^{2} + 3 \,{\left (c d^{3} e^{2} + 5 \, a d e^{4}\right )} x^{4} + 3 \,{\left (c d^{4} e + 5 \, a d^{2} e^{3}\right )} x^{2}\right )} \log \left (\frac{2 \, d e x +{\left (e x^{2} - d\right )} \sqrt{-d e}}{e x^{2} + d}\right ) + 2 \,{\left (3 \,{\left (c d^{2} e^{2} + 5 \, a e^{4}\right )} x^{5} - 8 \,{\left (c d^{3} e - 5 \, a d e^{3}\right )} x^{3} - 3 \,{\left (c d^{4} - 11 \, a d^{2} e^{2}\right )} x\right )} \sqrt{-d e}}{96 \,{\left (d^{3} e^{5} x^{6} + 3 \, d^{4} e^{4} x^{4} + 3 \, d^{5} e^{3} x^{2} + d^{6} e^{2}\right )} \sqrt{-d e}}, \frac{3 \,{\left ({\left (c d^{2} e^{3} + 5 \, a e^{5}\right )} x^{6} + c d^{5} + 5 \, a d^{3} e^{2} + 3 \,{\left (c d^{3} e^{2} + 5 \, a d e^{4}\right )} x^{4} + 3 \,{\left (c d^{4} e + 5 \, a d^{2} e^{3}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{d e} x}{d}\right ) +{\left (3 \,{\left (c d^{2} e^{2} + 5 \, a e^{4}\right )} x^{5} - 8 \,{\left (c d^{3} e - 5 \, a d e^{3}\right )} x^{3} - 3 \,{\left (c d^{4} - 11 \, a d^{2} e^{2}\right )} x\right )} \sqrt{d e}}{48 \,{\left (d^{3} e^{5} x^{6} + 3 \, d^{4} e^{4} x^{4} + 3 \, d^{5} e^{3} x^{2} + d^{6} e^{2}\right )} \sqrt{d e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/96*(3*((c*d^2*e^3 + 5*a*e^5)*x^6 + c*d^5 + 5*a*d^3*e^2 + 3*(c*d^3*e^2 + 5*a*d
*e^4)*x^4 + 3*(c*d^4*e + 5*a*d^2*e^3)*x^2)*log((2*d*e*x + (e*x^2 - d)*sqrt(-d*e)
)/(e*x^2 + d)) + 2*(3*(c*d^2*e^2 + 5*a*e^4)*x^5 - 8*(c*d^3*e - 5*a*d*e^3)*x^3 -
3*(c*d^4 - 11*a*d^2*e^2)*x)*sqrt(-d*e))/((d^3*e^5*x^6 + 3*d^4*e^4*x^4 + 3*d^5*e^
3*x^2 + d^6*e^2)*sqrt(-d*e)), 1/48*(3*((c*d^2*e^3 + 5*a*e^5)*x^6 + c*d^5 + 5*a*d
^3*e^2 + 3*(c*d^3*e^2 + 5*a*d*e^4)*x^4 + 3*(c*d^4*e + 5*a*d^2*e^3)*x^2)*arctan(s
qrt(d*e)*x/d) + (3*(c*d^2*e^2 + 5*a*e^4)*x^5 - 8*(c*d^3*e - 5*a*d*e^3)*x^3 - 3*(
c*d^4 - 11*a*d^2*e^2)*x)*sqrt(d*e))/((d^3*e^5*x^6 + 3*d^4*e^4*x^4 + 3*d^5*e^3*x^
2 + d^6*e^2)*sqrt(d*e))]

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Sympy [A]  time = 4.03331, size = 204, normalized size = 1.62 \[ - \frac{\sqrt{- \frac{1}{d^{7} e^{5}}} \left (5 a e^{2} + c d^{2}\right ) \log{\left (- d^{4} e^{2} \sqrt{- \frac{1}{d^{7} e^{5}}} + x \right )}}{32} + \frac{\sqrt{- \frac{1}{d^{7} e^{5}}} \left (5 a e^{2} + c d^{2}\right ) \log{\left (d^{4} e^{2} \sqrt{- \frac{1}{d^{7} e^{5}}} + x \right )}}{32} + \frac{x^{5} \left (15 a e^{4} + 3 c d^{2} e^{2}\right ) + x^{3} \left (40 a d e^{3} - 8 c d^{3} e\right ) + x \left (33 a d^{2} e^{2} - 3 c d^{4}\right )}{48 d^{6} e^{2} + 144 d^{5} e^{3} x^{2} + 144 d^{4} e^{4} x^{4} + 48 d^{3} e^{5} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-1/(d**7*e**5))*(5*a*e**2 + c*d**2)*log(-d**4*e**2*sqrt(-1/(d**7*e**5)) +
x)/32 + sqrt(-1/(d**7*e**5))*(5*a*e**2 + c*d**2)*log(d**4*e**2*sqrt(-1/(d**7*e**
5)) + x)/32 + (x**5*(15*a*e**4 + 3*c*d**2*e**2) + x**3*(40*a*d*e**3 - 8*c*d**3*e
) + x*(33*a*d**2*e**2 - 3*c*d**4))/(48*d**6*e**2 + 144*d**5*e**3*x**2 + 144*d**4
*e**4*x**4 + 48*d**3*e**5*x**6)

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GIAC/XCAS [A]  time = 0.27065, size = 135, normalized size = 1.07 \[ \frac{{\left (c d^{2} + 5 \, a e^{2}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{5}{2}\right )}}{16 \, d^{\frac{7}{2}}} + \frac{{\left (3 \, c d^{2} x^{5} e^{2} - 8 \, c d^{3} x^{3} e + 15 \, a x^{5} e^{4} - 3 \, c d^{4} x + 40 \, a d x^{3} e^{3} + 33 \, a d^{2} x e^{2}\right )} e^{\left (-2\right )}}{48 \,{\left (x^{2} e + d\right )}^{3} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*(c*d^2 + 5*a*e^2)*arctan(x*e^(1/2)/sqrt(d))*e^(-5/2)/d^(7/2) + 1/48*(3*c*d^
2*x^5*e^2 - 8*c*d^3*x^3*e + 15*a*x^5*e^4 - 3*c*d^4*x + 40*a*d*x^3*e^3 + 33*a*d^2
*x*e^2)*e^(-2)/((x^2*e + d)^3*d^3)