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

Optimal. Leaf size=93 \[ \frac{x \left (\frac{3 a}{d^2}-\frac{5 c}{e^2}\right )}{8 \left (d+e x^2\right )}+\frac{x \left (a+\frac{c d^2}{e^2}\right )}{4 d \left (d+e x^2\right )^2}+\frac{3 \left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{5/2} e^{5/2}} \]

[Out]

((a + (c*d^2)/e^2)*x)/(4*d*(d + e*x^2)^2) + (((3*a)/d^2 - (5*c)/e^2)*x)/(8*(d +
e*x^2)) + (3*(c*d^2 + a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(5/2)*e^(5/2))

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Rubi [A]  time = 0.143192, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{x \left (\frac{3 a}{d^2}-\frac{5 c}{e^2}\right )}{8 \left (d+e x^2\right )}+\frac{x \left (a+\frac{c d^2}{e^2}\right )}{4 d \left (d+e x^2\right )^2}+\frac{3 \left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{5/2} e^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

((a + (c*d^2)/e^2)*x)/(4*d*(d + e*x^2)^2) + (((3*a)/d^2 - (5*c)/e^2)*x)/(8*(d +
e*x^2)) + (3*(c*d^2 + a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(5/2)*e^(5/2))

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Rubi in Sympy [A]  time = 24.203, size = 85, normalized size = 0.91 \[ \frac{x \left (\frac{3 a}{8 d^{2}} - \frac{5 c}{8 e^{2}}\right )}{d + e x^{2}} + \frac{x \left (\frac{a}{4 d} + \frac{c d}{4 e^{2}}\right )}{\left (d + e x^{2}\right )^{2}} + \frac{3 \left (a e^{2} + c d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{8 d^{\frac{5}{2}} e^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(3*a/(8*d**2) - 5*c/(8*e**2))/(d + e*x**2) + x*(a/(4*d) + c*d/(4*e**2))/(d + e
*x**2)**2 + 3*(a*e**2 + c*d**2)*atan(sqrt(e)*x/sqrt(d))/(8*d**(5/2)*e**(5/2))

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Mathematica [A]  time = 0.098541, size = 92, normalized size = 0.99 \[ \frac{a e^2 x \left (5 d+3 e x^2\right )-c d^2 x \left (3 d+5 e x^2\right )}{8 d^2 e^2 \left (d+e x^2\right )^2}+\frac{3 \left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{5/2} e^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(a*e^2*x*(5*d + 3*e*x^2) - c*d^2*x*(3*d + 5*e*x^2))/(8*d^2*e^2*(d + e*x^2)^2) +
(3*(c*d^2 + a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(5/2)*e^(5/2))

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Maple [A]  time = 0.012, size = 99, normalized size = 1.1 \[{\frac{1}{ \left ( e{x}^{2}+d \right ) ^{2}} \left ({\frac{ \left ( 3\,a{e}^{2}-5\,c{d}^{2} \right ){x}^{3}}{8\,{d}^{2}e}}+{\frac{ \left ( 5\,a{e}^{2}-3\,c{d}^{2} \right ) x}{8\,{e}^{2}d}} \right ) }+{\frac{3\,a}{8\,{d}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{3\,c}{8\,{e}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+a)/(e*x^2+d)^3,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.290071, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (c d^{4} + a d^{2} e^{2} +{\left (c d^{2} e^{2} + a e^{4}\right )} x^{4} + 2 \,{\left (c d^{3} e + a d e^{3}\right )} x^{2}\right )} \log \left (\frac{2 \, d e x +{\left (e x^{2} - d\right )} \sqrt{-d e}}{e x^{2} + d}\right ) - 2 \,{\left ({\left (5 \, c d^{2} e - 3 \, a e^{3}\right )} x^{3} +{\left (3 \, c d^{3} - 5 \, a d e^{2}\right )} x\right )} \sqrt{-d e}}{16 \,{\left (d^{2} e^{4} x^{4} + 2 \, d^{3} e^{3} x^{2} + d^{4} e^{2}\right )} \sqrt{-d e}}, \frac{3 \,{\left (c d^{4} + a d^{2} e^{2} +{\left (c d^{2} e^{2} + a e^{4}\right )} x^{4} + 2 \,{\left (c d^{3} e + a d e^{3}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{d e} x}{d}\right ) -{\left ({\left (5 \, c d^{2} e - 3 \, a e^{3}\right )} x^{3} +{\left (3 \, c d^{3} - 5 \, a d e^{2}\right )} x\right )} \sqrt{d e}}{8 \,{\left (d^{2} e^{4} x^{4} + 2 \, d^{3} e^{3} x^{2} + d^{4} e^{2}\right )} \sqrt{d e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(3*(c*d^4 + a*d^2*e^2 + (c*d^2*e^2 + a*e^4)*x^4 + 2*(c*d^3*e + a*d*e^3)*x^
2)*log((2*d*e*x + (e*x^2 - d)*sqrt(-d*e))/(e*x^2 + d)) - 2*((5*c*d^2*e - 3*a*e^3
)*x^3 + (3*c*d^3 - 5*a*d*e^2)*x)*sqrt(-d*e))/((d^2*e^4*x^4 + 2*d^3*e^3*x^2 + d^4
*e^2)*sqrt(-d*e)), 1/8*(3*(c*d^4 + a*d^2*e^2 + (c*d^2*e^2 + a*e^4)*x^4 + 2*(c*d^
3*e + a*d*e^3)*x^2)*arctan(sqrt(d*e)*x/d) - ((5*c*d^2*e - 3*a*e^3)*x^3 + (3*c*d^
3 - 5*a*d*e^2)*x)*sqrt(d*e))/((d^2*e^4*x^4 + 2*d^3*e^3*x^2 + d^4*e^2)*sqrt(d*e))
]

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Sympy [A]  time = 3.11724, size = 219, normalized size = 2.35 \[ - \frac{3 \sqrt{- \frac{1}{d^{5} e^{5}}} \left (a e^{2} + c d^{2}\right ) \log{\left (- \frac{3 d^{3} e^{2} \sqrt{- \frac{1}{d^{5} e^{5}}} \left (a e^{2} + c d^{2}\right )}{3 a e^{2} + 3 c d^{2}} + x \right )}}{16} + \frac{3 \sqrt{- \frac{1}{d^{5} e^{5}}} \left (a e^{2} + c d^{2}\right ) \log{\left (\frac{3 d^{3} e^{2} \sqrt{- \frac{1}{d^{5} e^{5}}} \left (a e^{2} + c d^{2}\right )}{3 a e^{2} + 3 c d^{2}} + x \right )}}{16} + \frac{x^{3} \left (3 a e^{3} - 5 c d^{2} e\right ) + x \left (5 a d e^{2} - 3 c d^{3}\right )}{8 d^{4} e^{2} + 16 d^{3} e^{3} x^{2} + 8 d^{2} e^{4} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*sqrt(-1/(d**5*e**5))*(a*e**2 + c*d**2)*log(-3*d**3*e**2*sqrt(-1/(d**5*e**5))*
(a*e**2 + c*d**2)/(3*a*e**2 + 3*c*d**2) + x)/16 + 3*sqrt(-1/(d**5*e**5))*(a*e**2
+ c*d**2)*log(3*d**3*e**2*sqrt(-1/(d**5*e**5))*(a*e**2 + c*d**2)/(3*a*e**2 + 3*
c*d**2) + x)/16 + (x**3*(3*a*e**3 - 5*c*d**2*e) + x*(5*a*d*e**2 - 3*c*d**3))/(8*
d**4*e**2 + 16*d**3*e**3*x**2 + 8*d**2*e**4*x**4)

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GIAC/XCAS [A]  time = 0.271349, size = 104, normalized size = 1.12 \[ \frac{3 \,{\left (c d^{2} + a e^{2}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{5}{2}\right )}}{8 \, d^{\frac{5}{2}}} - \frac{{\left (5 \, c d^{2} x^{3} e + 3 \, c d^{3} x - 3 \, a x^{3} e^{3} - 5 \, a d x e^{2}\right )} e^{\left (-2\right )}}{8 \,{\left (x^{2} e + d\right )}^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/8*(c*d^2 + a*e^2)*arctan(x*e^(1/2)/sqrt(d))*e^(-5/2)/d^(5/2) - 1/8*(5*c*d^2*x^
3*e + 3*c*d^3*x - 3*a*x^3*e^3 - 5*a*d*x*e^2)*e^(-2)/((x^2*e + d)^2*d^2)