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

Optimal. Leaf size=155 \[ \frac{\left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{16 a^{7/2} c^{5/2}}-\frac{(d+e x) \left (a e^2+5 c d^2\right ) (a e-c d x)}{16 a^3 c^2 \left (a+c x^2\right )}-\frac{(d+e x)^3 (a e-5 c d x)}{24 a^2 c \left (a+c x^2\right )^2}+\frac{x (d+e x)^4}{6 a \left (a+c x^2\right )^3} \]

[Out]

(x*(d + e*x)^4)/(6*a*(a + c*x^2)^3) - ((a*e - 5*c*d*x)*(d + e*x)^3)/(24*a^2*c*(a
+ c*x^2)^2) - ((5*c*d^2 + a*e^2)*(a*e - c*d*x)*(d + e*x))/(16*a^3*c^2*(a + c*x^
2)) + ((c*d^2 + a*e^2)*(5*c*d^2 + a*e^2)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(16*a^(7/2
)*c^(5/2))

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Rubi [A]  time = 0.25645, antiderivative size = 155, 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 \[ \frac{\left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{16 a^{7/2} c^{5/2}}-\frac{(d+e x) \left (a e^2+5 c d^2\right ) (a e-c d x)}{16 a^3 c^2 \left (a+c x^2\right )}-\frac{(d+e x)^3 (a e-5 c d x)}{24 a^2 c \left (a+c x^2\right )^2}+\frac{x (d+e x)^4}{6 a \left (a+c x^2\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

(x*(d + e*x)^4)/(6*a*(a + c*x^2)^3) - ((a*e - 5*c*d*x)*(d + e*x)^3)/(24*a^2*c*(a
+ c*x^2)^2) - ((5*c*d^2 + a*e^2)*(a*e - c*d*x)*(d + e*x))/(16*a^3*c^2*(a + c*x^
2)) + ((c*d^2 + a*e^2)*(5*c*d^2 + a*e^2)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(16*a^(7/2
)*c^(5/2))

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Rubi in Sympy [A]  time = 27.7009, size = 139, normalized size = 0.9 \[ \frac{x \left (d + e x\right )^{4}}{6 a \left (a + c x^{2}\right )^{3}} - \frac{\left (d + e x\right )^{3} \left (a e - 5 c d x\right )}{24 a^{2} c \left (a + c x^{2}\right )^{2}} - \frac{\left (d + e x\right ) \left (a e - c d x\right ) \left (a e^{2} + 5 c d^{2}\right )}{16 a^{3} c^{2} \left (a + c x^{2}\right )} + \frac{\left (a e^{2} + c d^{2}\right ) \left (a e^{2} + 5 c d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{16 a^{\frac{7}{2}} c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(d + e*x)**4/(6*a*(a + c*x**2)**3) - (d + e*x)**3*(a*e - 5*c*d*x)/(24*a**2*c*(
a + c*x**2)**2) - (d + e*x)*(a*e - c*d*x)*(a*e**2 + 5*c*d**2)/(16*a**3*c**2*(a +
c*x**2)) + (a*e**2 + c*d**2)*(a*e**2 + 5*c*d**2)*atan(sqrt(c)*x/sqrt(a))/(16*a*
*(7/2)*c**(5/2))

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Mathematica [A]  time = 0.316039, size = 197, normalized size = 1.27 \[ \frac{\left (a^2 e^4+6 a c d^2 e^2+5 c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{16 a^{7/2} c^{5/2}}+\frac{-a^4 e^3 (16 d+3 e x)-2 a^3 c e \left (16 d^3+9 d^2 e x+24 d e^2 x^2+4 e^3 x^3\right )+3 a^2 c^2 x \left (11 d^4+16 d^2 e^2 x^2+e^4 x^4\right )+2 a c^3 d^2 x^3 \left (20 d^2+9 e^2 x^2\right )+15 c^4 d^4 x^5}{48 a^3 c^2 \left (a+c x^2\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

(15*c^4*d^4*x^5 - a^4*e^3*(16*d + 3*e*x) + 2*a*c^3*d^2*x^3*(20*d^2 + 9*e^2*x^2)
- 2*a^3*c*e*(16*d^3 + 9*d^2*e*x + 24*d*e^2*x^2 + 4*e^3*x^3) + 3*a^2*c^2*x*(11*d^
4 + 16*d^2*e^2*x^2 + e^4*x^4))/(48*a^3*c^2*(a + c*x^2)^3) + ((5*c^2*d^4 + 6*a*c*
d^2*e^2 + a^2*e^4)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(16*a^(7/2)*c^(5/2))

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Maple [A]  time = 0.013, size = 225, normalized size = 1.5 \[{\frac{1}{ \left ( c{x}^{2}+a \right ) ^{3}} \left ({\frac{ \left ({a}^{2}{e}^{4}+6\,ac{d}^{2}{e}^{2}+5\,{c}^{2}{d}^{4} \right ){x}^{5}}{16\,{a}^{3}}}-{\frac{ \left ({a}^{2}{e}^{4}-6\,ac{d}^{2}{e}^{2}-5\,{c}^{2}{d}^{4} \right ){x}^{3}}{6\,{a}^{2}c}}-{\frac{d{e}^{3}{x}^{2}}{c}}-{\frac{ \left ({a}^{2}{e}^{4}+6\,ac{d}^{2}{e}^{2}-11\,{c}^{2}{d}^{4} \right ) x}{16\,a{c}^{2}}}-{\frac{de \left ( a{e}^{2}+2\,c{d}^{2} \right ) }{3\,{c}^{2}}} \right ) }+{\frac{{e}^{4}}{16\,a{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,{d}^{2}{e}^{2}}{8\,{a}^{2}c}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{5\,{d}^{4}}{16\,{a}^{3}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^4/(c*x^2+a)^4,x)

[Out]

(1/16*(a^2*e^4+6*a*c*d^2*e^2+5*c^2*d^4)/a^3*x^5-1/6*(a^2*e^4-6*a*c*d^2*e^2-5*c^2
*d^4)/a^2/c*x^3-d*e^3*x^2/c-1/16*(a^2*e^4+6*a*c*d^2*e^2-11*c^2*d^4)/a/c^2*x-1/3*
d*e*(a*e^2+2*c*d^2)/c^2)/(c*x^2+a)^3+1/16/a/c^2/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/
2))*e^4+3/8/a^2/c/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*d^2*e^2+5/16/a^3/(a*c)^(1/
2)*arctan(c*x/(a*c)^(1/2))*d^4

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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((e*x + d)^4/(c*x^2 + a)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.239503, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (5 \, a^{3} c^{2} d^{4} + 6 \, a^{4} c d^{2} e^{2} + a^{5} e^{4} +{\left (5 \, c^{5} d^{4} + 6 \, a c^{4} d^{2} e^{2} + a^{2} c^{3} e^{4}\right )} x^{6} + 3 \,{\left (5 \, a c^{4} d^{4} + 6 \, a^{2} c^{3} d^{2} e^{2} + a^{3} c^{2} e^{4}\right )} x^{4} + 3 \,{\left (5 \, a^{2} c^{3} d^{4} + 6 \, a^{3} c^{2} d^{2} e^{2} + a^{4} c e^{4}\right )} x^{2}\right )} \log \left (\frac{2 \, a c x +{\left (c x^{2} - a\right )} \sqrt{-a c}}{c x^{2} + a}\right ) - 2 \,{\left (48 \, a^{3} c d e^{3} x^{2} + 32 \, a^{3} c d^{3} e + 16 \, a^{4} d e^{3} - 3 \,{\left (5 \, c^{4} d^{4} + 6 \, a c^{3} d^{2} e^{2} + a^{2} c^{2} e^{4}\right )} x^{5} - 8 \,{\left (5 \, a c^{3} d^{4} + 6 \, a^{2} c^{2} d^{2} e^{2} - a^{3} c e^{4}\right )} x^{3} - 3 \,{\left (11 \, a^{2} c^{2} d^{4} - 6 \, a^{3} c d^{2} e^{2} - a^{4} e^{4}\right )} x\right )} \sqrt{-a c}}{96 \,{\left (a^{3} c^{5} x^{6} + 3 \, a^{4} c^{4} x^{4} + 3 \, a^{5} c^{3} x^{2} + a^{6} c^{2}\right )} \sqrt{-a c}}, \frac{3 \,{\left (5 \, a^{3} c^{2} d^{4} + 6 \, a^{4} c d^{2} e^{2} + a^{5} e^{4} +{\left (5 \, c^{5} d^{4} + 6 \, a c^{4} d^{2} e^{2} + a^{2} c^{3} e^{4}\right )} x^{6} + 3 \,{\left (5 \, a c^{4} d^{4} + 6 \, a^{2} c^{3} d^{2} e^{2} + a^{3} c^{2} e^{4}\right )} x^{4} + 3 \,{\left (5 \, a^{2} c^{3} d^{4} + 6 \, a^{3} c^{2} d^{2} e^{2} + a^{4} c e^{4}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) -{\left (48 \, a^{3} c d e^{3} x^{2} + 32 \, a^{3} c d^{3} e + 16 \, a^{4} d e^{3} - 3 \,{\left (5 \, c^{4} d^{4} + 6 \, a c^{3} d^{2} e^{2} + a^{2} c^{2} e^{4}\right )} x^{5} - 8 \,{\left (5 \, a c^{3} d^{4} + 6 \, a^{2} c^{2} d^{2} e^{2} - a^{3} c e^{4}\right )} x^{3} - 3 \,{\left (11 \, a^{2} c^{2} d^{4} - 6 \, a^{3} c d^{2} e^{2} - a^{4} e^{4}\right )} x\right )} \sqrt{a c}}{48 \,{\left (a^{3} c^{5} x^{6} + 3 \, a^{4} c^{4} x^{4} + 3 \, a^{5} c^{3} x^{2} + a^{6} c^{2}\right )} \sqrt{a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/96*(3*(5*a^3*c^2*d^4 + 6*a^4*c*d^2*e^2 + a^5*e^4 + (5*c^5*d^4 + 6*a*c^4*d^2*e
^2 + a^2*c^3*e^4)*x^6 + 3*(5*a*c^4*d^4 + 6*a^2*c^3*d^2*e^2 + a^3*c^2*e^4)*x^4 +
3*(5*a^2*c^3*d^4 + 6*a^3*c^2*d^2*e^2 + a^4*c*e^4)*x^2)*log((2*a*c*x + (c*x^2 - a
)*sqrt(-a*c))/(c*x^2 + a)) - 2*(48*a^3*c*d*e^3*x^2 + 32*a^3*c*d^3*e + 16*a^4*d*e
^3 - 3*(5*c^4*d^4 + 6*a*c^3*d^2*e^2 + a^2*c^2*e^4)*x^5 - 8*(5*a*c^3*d^4 + 6*a^2*
c^2*d^2*e^2 - a^3*c*e^4)*x^3 - 3*(11*a^2*c^2*d^4 - 6*a^3*c*d^2*e^2 - a^4*e^4)*x)
*sqrt(-a*c))/((a^3*c^5*x^6 + 3*a^4*c^4*x^4 + 3*a^5*c^3*x^2 + a^6*c^2)*sqrt(-a*c)
), 1/48*(3*(5*a^3*c^2*d^4 + 6*a^4*c*d^2*e^2 + a^5*e^4 + (5*c^5*d^4 + 6*a*c^4*d^2
*e^2 + a^2*c^3*e^4)*x^6 + 3*(5*a*c^4*d^4 + 6*a^2*c^3*d^2*e^2 + a^3*c^2*e^4)*x^4
+ 3*(5*a^2*c^3*d^4 + 6*a^3*c^2*d^2*e^2 + a^4*c*e^4)*x^2)*arctan(sqrt(a*c)*x/a) -
(48*a^3*c*d*e^3*x^2 + 32*a^3*c*d^3*e + 16*a^4*d*e^3 - 3*(5*c^4*d^4 + 6*a*c^3*d^
2*e^2 + a^2*c^2*e^4)*x^5 - 8*(5*a*c^3*d^4 + 6*a^2*c^2*d^2*e^2 - a^3*c*e^4)*x^3 -
3*(11*a^2*c^2*d^4 - 6*a^3*c*d^2*e^2 - a^4*e^4)*x)*sqrt(a*c))/((a^3*c^5*x^6 + 3*
a^4*c^4*x^4 + 3*a^5*c^3*x^2 + a^6*c^2)*sqrt(a*c))]

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Sympy [A]  time = 12.9792, size = 413, normalized size = 2.66 \[ - \frac{\sqrt{- \frac{1}{a^{7} c^{5}}} \left (a e^{2} + c d^{2}\right ) \left (a e^{2} + 5 c d^{2}\right ) \log{\left (- \frac{a^{4} c^{2} \sqrt{- \frac{1}{a^{7} c^{5}}} \left (a e^{2} + c d^{2}\right ) \left (a e^{2} + 5 c d^{2}\right )}{a^{2} e^{4} + 6 a c d^{2} e^{2} + 5 c^{2} d^{4}} + x \right )}}{32} + \frac{\sqrt{- \frac{1}{a^{7} c^{5}}} \left (a e^{2} + c d^{2}\right ) \left (a e^{2} + 5 c d^{2}\right ) \log{\left (\frac{a^{4} c^{2} \sqrt{- \frac{1}{a^{7} c^{5}}} \left (a e^{2} + c d^{2}\right ) \left (a e^{2} + 5 c d^{2}\right )}{a^{2} e^{4} + 6 a c d^{2} e^{2} + 5 c^{2} d^{4}} + x \right )}}{32} + \frac{- 16 a^{4} d e^{3} - 32 a^{3} c d^{3} e - 48 a^{3} c d e^{3} x^{2} + x^{5} \left (3 a^{2} c^{2} e^{4} + 18 a c^{3} d^{2} e^{2} + 15 c^{4} d^{4}\right ) + x^{3} \left (- 8 a^{3} c e^{4} + 48 a^{2} c^{2} d^{2} e^{2} + 40 a c^{3} d^{4}\right ) + x \left (- 3 a^{4} e^{4} - 18 a^{3} c d^{2} e^{2} + 33 a^{2} c^{2} d^{4}\right )}{48 a^{6} c^{2} + 144 a^{5} c^{3} x^{2} + 144 a^{4} c^{4} x^{4} + 48 a^{3} c^{5} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-1/(a**7*c**5))*(a*e**2 + c*d**2)*(a*e**2 + 5*c*d**2)*log(-a**4*c**2*sqrt(
-1/(a**7*c**5))*(a*e**2 + c*d**2)*(a*e**2 + 5*c*d**2)/(a**2*e**4 + 6*a*c*d**2*e*
*2 + 5*c**2*d**4) + x)/32 + sqrt(-1/(a**7*c**5))*(a*e**2 + c*d**2)*(a*e**2 + 5*c
*d**2)*log(a**4*c**2*sqrt(-1/(a**7*c**5))*(a*e**2 + c*d**2)*(a*e**2 + 5*c*d**2)/
(a**2*e**4 + 6*a*c*d**2*e**2 + 5*c**2*d**4) + x)/32 + (-16*a**4*d*e**3 - 32*a**3
*c*d**3*e - 48*a**3*c*d*e**3*x**2 + x**5*(3*a**2*c**2*e**4 + 18*a*c**3*d**2*e**2
+ 15*c**4*d**4) + x**3*(-8*a**3*c*e**4 + 48*a**2*c**2*d**2*e**2 + 40*a*c**3*d**
4) + x*(-3*a**4*e**4 - 18*a**3*c*d**2*e**2 + 33*a**2*c**2*d**4))/(48*a**6*c**2 +
144*a**5*c**3*x**2 + 144*a**4*c**4*x**4 + 48*a**3*c**5*x**6)

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GIAC/XCAS [A]  time = 0.212405, size = 277, normalized size = 1.79 \[ \frac{{\left (5 \, c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{16 \, \sqrt{a c} a^{3} c^{2}} + \frac{15 \, c^{4} d^{4} x^{5} + 18 \, a c^{3} d^{2} x^{5} e^{2} + 40 \, a c^{3} d^{4} x^{3} + 3 \, a^{2} c^{2} x^{5} e^{4} + 48 \, a^{2} c^{2} d^{2} x^{3} e^{2} + 33 \, a^{2} c^{2} d^{4} x - 8 \, a^{3} c x^{3} e^{4} - 48 \, a^{3} c d x^{2} e^{3} - 18 \, a^{3} c d^{2} x e^{2} - 32 \, a^{3} c d^{3} e - 3 \, a^{4} x e^{4} - 16 \, a^{4} d e^{3}}{48 \,{\left (c x^{2} + a\right )}^{3} a^{3} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*(5*c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4)*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*a^3*
c^2) + 1/48*(15*c^4*d^4*x^5 + 18*a*c^3*d^2*x^5*e^2 + 40*a*c^3*d^4*x^3 + 3*a^2*c^
2*x^5*e^4 + 48*a^2*c^2*d^2*x^3*e^2 + 33*a^2*c^2*d^4*x - 8*a^3*c*x^3*e^4 - 48*a^3
*c*d*x^2*e^3 - 18*a^3*c*d^2*x*e^2 - 32*a^3*c*d^3*e - 3*a^4*x*e^4 - 16*a^4*d*e^3)
/((c*x^2 + a)^3*a^3*c^2)