3.144 \(\int \frac{\left (a+c x^4\right )^2}{\left (d+e x^2\right )^5} \, dx\)

Optimal. Leaf size=226 \[ -\frac{x \left (-35 a^2 e^4-6 a c d^2 e^2+93 c^2 d^4\right )}{128 d^4 e^4 \left (d+e x^2\right )}+\frac{\left (35 a^2 e^4+6 a c d^2 e^2+35 c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{128 d^{9/2} e^{9/2}}+\frac{x \left (35 a^2 e^4+6 a c d^2 e^2+163 c^2 d^4\right )}{192 d^3 e^4 \left (d+e x^2\right )^2}+\frac{x \left (a e^2+c d^2\right )^2}{8 d e^4 \left (d+e x^2\right )^4}-\frac{x \left (25 c d^2-7 a e^2\right ) \left (a e^2+c d^2\right )}{48 d^2 e^4 \left (d+e x^2\right )^3} \]

[Out]

((c*d^2 + a*e^2)^2*x)/(8*d*e^4*(d + e*x^2)^4) - ((25*c*d^2 - 7*a*e^2)*(c*d^2 + a
*e^2)*x)/(48*d^2*e^4*(d + e*x^2)^3) + ((163*c^2*d^4 + 6*a*c*d^2*e^2 + 35*a^2*e^4
)*x)/(192*d^3*e^4*(d + e*x^2)^2) - ((93*c^2*d^4 - 6*a*c*d^2*e^2 - 35*a^2*e^4)*x)
/(128*d^4*e^4*(d + e*x^2)) + ((35*c^2*d^4 + 6*a*c*d^2*e^2 + 35*a^2*e^4)*ArcTan[(
Sqrt[e]*x)/Sqrt[d]])/(128*d^(9/2)*e^(9/2))

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Rubi [A]  time = 0.571897, antiderivative size = 226, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{x \left (-35 a^2 e^4-6 a c d^2 e^2+93 c^2 d^4\right )}{128 d^4 e^4 \left (d+e x^2\right )}+\frac{\left (35 a^2 e^4+6 a c d^2 e^2+35 c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{128 d^{9/2} e^{9/2}}+\frac{x \left (35 a^2 e^4+6 a c d^2 e^2+163 c^2 d^4\right )}{192 d^3 e^4 \left (d+e x^2\right )^2}+\frac{x \left (a e^2+c d^2\right )^2}{8 d e^4 \left (d+e x^2\right )^4}-\frac{x \left (25 c d^2-7 a e^2\right ) \left (a e^2+c d^2\right )}{48 d^2 e^4 \left (d+e x^2\right )^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^4)^2/(d + e*x^2)^5,x]

[Out]

((c*d^2 + a*e^2)^2*x)/(8*d*e^4*(d + e*x^2)^4) - ((25*c*d^2 - 7*a*e^2)*(c*d^2 + a
*e^2)*x)/(48*d^2*e^4*(d + e*x^2)^3) + ((163*c^2*d^4 + 6*a*c*d^2*e^2 + 35*a^2*e^4
)*x)/(192*d^3*e^4*(d + e*x^2)^2) - ((93*c^2*d^4 - 6*a*c*d^2*e^2 - 35*a^2*e^4)*x)
/(128*d^4*e^4*(d + e*x^2)) + ((35*c^2*d^4 + 6*a*c*d^2*e^2 + 35*a^2*e^4)*ArcTan[(
Sqrt[e]*x)/Sqrt[d]])/(128*d^(9/2)*e^(9/2))

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Rubi in Sympy [A]  time = 92.5557, size = 255, normalized size = 1.13 \[ - \frac{c^{2} x^{7}}{e \left (d + e x^{2}\right )^{4}} + \frac{x \left (a^{2} e^{4} + 2 a c d^{2} e^{2} - 7 c^{2} d^{4}\right )}{8 d e^{4} \left (d + e x^{2}\right )^{4}} + \frac{x \left (7 a^{2} e^{4} - 18 a c d^{2} e^{2} + 119 c^{2} d^{4}\right )}{48 d^{2} e^{4} \left (d + e x^{2}\right )^{3}} + \frac{x \left (35 a^{2} e^{4} + 6 a c d^{2} e^{2} - 413 c^{2} d^{4}\right )}{192 d^{3} e^{4} \left (d + e x^{2}\right )^{2}} + \frac{x \left (35 a^{2} e^{4} + 6 a c d^{2} e^{2} + 35 c^{2} d^{4}\right )}{128 d^{4} e^{4} \left (d + e x^{2}\right )} + \frac{\left (35 a^{2} e^{4} + 6 a c d^{2} e^{2} + 35 c^{2} d^{4}\right ) \operatorname{atan}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{128 d^{\frac{9}{2}} e^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+a)**2/(e*x**2+d)**5,x)

[Out]

-c**2*x**7/(e*(d + e*x**2)**4) + x*(a**2*e**4 + 2*a*c*d**2*e**2 - 7*c**2*d**4)/(
8*d*e**4*(d + e*x**2)**4) + x*(7*a**2*e**4 - 18*a*c*d**2*e**2 + 119*c**2*d**4)/(
48*d**2*e**4*(d + e*x**2)**3) + x*(35*a**2*e**4 + 6*a*c*d**2*e**2 - 413*c**2*d**
4)/(192*d**3*e**4*(d + e*x**2)**2) + x*(35*a**2*e**4 + 6*a*c*d**2*e**2 + 35*c**2
*d**4)/(128*d**4*e**4*(d + e*x**2)) + (35*a**2*e**4 + 6*a*c*d**2*e**2 + 35*c**2*
d**4)*atan(sqrt(e)*x/sqrt(d))/(128*d**(9/2)*e**(9/2))

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Mathematica [A]  time = 0.330338, size = 200, normalized size = 0.88 \[ \frac{3 \left (35 a^2 e^4+6 a c d^2 e^2+35 c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )+\frac{\sqrt{d} \sqrt{e} x \left (a^2 e^4 \left (279 d^3+511 d^2 e x^2+385 d e^2 x^4+105 e^3 x^6\right )-6 a c d^2 e^2 \left (3 d^3+11 d^2 e x^2-11 d e^2 x^4-3 e^3 x^6\right )-c^2 d^4 \left (105 d^3+385 d^2 e x^2+511 d e^2 x^4+279 e^3 x^6\right )\right )}{\left (d+e x^2\right )^4}}{384 d^{9/2} e^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^4)^2/(d + e*x^2)^5,x]

[Out]

((Sqrt[d]*Sqrt[e]*x*(-6*a*c*d^2*e^2*(3*d^3 + 11*d^2*e*x^2 - 11*d*e^2*x^4 - 3*e^3
*x^6) + a^2*e^4*(279*d^3 + 511*d^2*e*x^2 + 385*d*e^2*x^4 + 105*e^3*x^6) - c^2*d^
4*(105*d^3 + 385*d^2*e*x^2 + 511*d*e^2*x^4 + 279*e^3*x^6)))/(d + e*x^2)^4 + 3*(3
5*c^2*d^4 + 6*a*c*d^2*e^2 + 35*a^2*e^4)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(384*d^(9/2
)*e^(9/2))

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Maple [A]  time = 0.016, size = 231, normalized size = 1. \[{\frac{1}{ \left ( e{x}^{2}+d \right ) ^{4}} \left ({\frac{ \left ( 35\,{a}^{2}{e}^{4}+6\,ac{d}^{2}{e}^{2}-93\,{c}^{2}{d}^{4} \right ){x}^{7}}{128\,{d}^{4}e}}+{\frac{ \left ( 385\,{a}^{2}{e}^{4}+66\,ac{d}^{2}{e}^{2}-511\,{c}^{2}{d}^{4} \right ){x}^{5}}{384\,{d}^{3}{e}^{2}}}+{\frac{ \left ( 511\,{a}^{2}{e}^{4}-66\,ac{d}^{2}{e}^{2}-385\,{c}^{2}{d}^{4} \right ){x}^{3}}{384\,{d}^{2}{e}^{3}}}+{\frac{ \left ( 93\,{a}^{2}{e}^{4}-6\,ac{d}^{2}{e}^{2}-35\,{c}^{2}{d}^{4} \right ) x}{128\,d{e}^{4}}} \right ) }+{\frac{35\,{a}^{2}}{128\,{d}^{4}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{3\,ac}{64\,{d}^{2}{e}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{35\,{c}^{2}}{128\,{e}^{4}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+a)^2/(e*x^2+d)^5,x)

[Out]

(1/128*(35*a^2*e^4+6*a*c*d^2*e^2-93*c^2*d^4)/d^4/e*x^7+1/384*(385*a^2*e^4+66*a*c
*d^2*e^2-511*c^2*d^4)/d^3/e^2*x^5+1/384*(511*a^2*e^4-66*a*c*d^2*e^2-385*c^2*d^4)
/d^2/e^3*x^3+1/128*(93*a^2*e^4-6*a*c*d^2*e^2-35*c^2*d^4)/d/e^4*x)/(e*x^2+d)^4+35
/128/d^4/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*a^2+3/64/d^2/e^2/(d*e)^(1/2)*arctan
(x*e/(d*e)^(1/2))*a*c+35/128/e^4/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*c^2

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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)^2/(e*x^2 + d)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.281606, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + a)^2/(e*x^2 + d)^5,x, algorithm="fricas")

[Out]

[1/768*(3*(35*c^2*d^8 + 6*a*c*d^6*e^2 + 35*a^2*d^4*e^4 + (35*c^2*d^4*e^4 + 6*a*c
*d^2*e^6 + 35*a^2*e^8)*x^8 + 4*(35*c^2*d^5*e^3 + 6*a*c*d^3*e^5 + 35*a^2*d*e^7)*x
^6 + 6*(35*c^2*d^6*e^2 + 6*a*c*d^4*e^4 + 35*a^2*d^2*e^6)*x^4 + 4*(35*c^2*d^7*e +
6*a*c*d^5*e^3 + 35*a^2*d^3*e^5)*x^2)*log((2*d*e*x + (e*x^2 - d)*sqrt(-d*e))/(e*
x^2 + d)) - 2*(3*(93*c^2*d^4*e^3 - 6*a*c*d^2*e^5 - 35*a^2*e^7)*x^7 + (511*c^2*d^
5*e^2 - 66*a*c*d^3*e^4 - 385*a^2*d*e^6)*x^5 + (385*c^2*d^6*e + 66*a*c*d^4*e^3 -
511*a^2*d^2*e^5)*x^3 + 3*(35*c^2*d^7 + 6*a*c*d^5*e^2 - 93*a^2*d^3*e^4)*x)*sqrt(-
d*e))/((d^4*e^8*x^8 + 4*d^5*e^7*x^6 + 6*d^6*e^6*x^4 + 4*d^7*e^5*x^2 + d^8*e^4)*s
qrt(-d*e)), 1/384*(3*(35*c^2*d^8 + 6*a*c*d^6*e^2 + 35*a^2*d^4*e^4 + (35*c^2*d^4*
e^4 + 6*a*c*d^2*e^6 + 35*a^2*e^8)*x^8 + 4*(35*c^2*d^5*e^3 + 6*a*c*d^3*e^5 + 35*a
^2*d*e^7)*x^6 + 6*(35*c^2*d^6*e^2 + 6*a*c*d^4*e^4 + 35*a^2*d^2*e^6)*x^4 + 4*(35*
c^2*d^7*e + 6*a*c*d^5*e^3 + 35*a^2*d^3*e^5)*x^2)*arctan(sqrt(d*e)*x/d) - (3*(93*
c^2*d^4*e^3 - 6*a*c*d^2*e^5 - 35*a^2*e^7)*x^7 + (511*c^2*d^5*e^2 - 66*a*c*d^3*e^
4 - 385*a^2*d*e^6)*x^5 + (385*c^2*d^6*e + 66*a*c*d^4*e^3 - 511*a^2*d^2*e^5)*x^3
+ 3*(35*c^2*d^7 + 6*a*c*d^5*e^2 - 93*a^2*d^3*e^4)*x)*sqrt(d*e))/((d^4*e^8*x^8 +
4*d^5*e^7*x^6 + 6*d^6*e^6*x^4 + 4*d^7*e^5*x^2 + d^8*e^4)*sqrt(d*e))]

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Sympy [A]  time = 17.5187, size = 335, normalized size = 1.48 \[ - \frac{\sqrt{- \frac{1}{d^{9} e^{9}}} \left (35 a^{2} e^{4} + 6 a c d^{2} e^{2} + 35 c^{2} d^{4}\right ) \log{\left (- d^{5} e^{4} \sqrt{- \frac{1}{d^{9} e^{9}}} + x \right )}}{256} + \frac{\sqrt{- \frac{1}{d^{9} e^{9}}} \left (35 a^{2} e^{4} + 6 a c d^{2} e^{2} + 35 c^{2} d^{4}\right ) \log{\left (d^{5} e^{4} \sqrt{- \frac{1}{d^{9} e^{9}}} + x \right )}}{256} + \frac{x^{7} \left (105 a^{2} e^{7} + 18 a c d^{2} e^{5} - 279 c^{2} d^{4} e^{3}\right ) + x^{5} \left (385 a^{2} d e^{6} + 66 a c d^{3} e^{4} - 511 c^{2} d^{5} e^{2}\right ) + x^{3} \left (511 a^{2} d^{2} e^{5} - 66 a c d^{4} e^{3} - 385 c^{2} d^{6} e\right ) + x \left (279 a^{2} d^{3} e^{4} - 18 a c d^{5} e^{2} - 105 c^{2} d^{7}\right )}{384 d^{8} e^{4} + 1536 d^{7} e^{5} x^{2} + 2304 d^{6} e^{6} x^{4} + 1536 d^{5} e^{7} x^{6} + 384 d^{4} e^{8} x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+a)**2/(e*x**2+d)**5,x)

[Out]

-sqrt(-1/(d**9*e**9))*(35*a**2*e**4 + 6*a*c*d**2*e**2 + 35*c**2*d**4)*log(-d**5*
e**4*sqrt(-1/(d**9*e**9)) + x)/256 + sqrt(-1/(d**9*e**9))*(35*a**2*e**4 + 6*a*c*
d**2*e**2 + 35*c**2*d**4)*log(d**5*e**4*sqrt(-1/(d**9*e**9)) + x)/256 + (x**7*(1
05*a**2*e**7 + 18*a*c*d**2*e**5 - 279*c**2*d**4*e**3) + x**5*(385*a**2*d*e**6 +
66*a*c*d**3*e**4 - 511*c**2*d**5*e**2) + x**3*(511*a**2*d**2*e**5 - 66*a*c*d**4*
e**3 - 385*c**2*d**6*e) + x*(279*a**2*d**3*e**4 - 18*a*c*d**5*e**2 - 105*c**2*d*
*7))/(384*d**8*e**4 + 1536*d**7*e**5*x**2 + 2304*d**6*e**6*x**4 + 1536*d**5*e**7
*x**6 + 384*d**4*e**8*x**8)

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GIAC/XCAS [A]  time = 0.275597, size = 267, normalized size = 1.18 \[ \frac{{\left (35 \, c^{2} d^{4} + 6 \, a c d^{2} e^{2} + 35 \, a^{2} e^{4}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{9}{2}\right )}}{128 \, d^{\frac{9}{2}}} - \frac{{\left (279 \, c^{2} d^{4} x^{7} e^{3} + 511 \, c^{2} d^{5} x^{5} e^{2} - 18 \, a c d^{2} x^{7} e^{5} + 385 \, c^{2} d^{6} x^{3} e - 66 \, a c d^{3} x^{5} e^{4} + 105 \, c^{2} d^{7} x - 105 \, a^{2} x^{7} e^{7} + 66 \, a c d^{4} x^{3} e^{3} - 385 \, a^{2} d x^{5} e^{6} + 18 \, a c d^{5} x e^{2} - 511 \, a^{2} d^{2} x^{3} e^{5} - 279 \, a^{2} d^{3} x e^{4}\right )} e^{\left (-4\right )}}{384 \,{\left (x^{2} e + d\right )}^{4} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + a)^2/(e*x^2 + d)^5,x, algorithm="giac")

[Out]

1/128*(35*c^2*d^4 + 6*a*c*d^2*e^2 + 35*a^2*e^4)*arctan(x*e^(1/2)/sqrt(d))*e^(-9/
2)/d^(9/2) - 1/384*(279*c^2*d^4*x^7*e^3 + 511*c^2*d^5*x^5*e^2 - 18*a*c*d^2*x^7*e
^5 + 385*c^2*d^6*x^3*e - 66*a*c*d^3*x^5*e^4 + 105*c^2*d^7*x - 105*a^2*x^7*e^7 +
66*a*c*d^4*x^3*e^3 - 385*a^2*d*x^5*e^6 + 18*a*c*d^5*x*e^2 - 511*a^2*d^2*x^3*e^5
- 279*a^2*d^3*x*e^4)*e^(-4)/((x^2*e + d)^4*d^4)