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

Optimal. Leaf size=322 \[ -\frac{\left (3 \sqrt{c} d^2-\sqrt{a} e^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{7/4} c^{3/4}}+\frac{\left (3 \sqrt{c} d^2-\sqrt{a} e^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{7/4} c^{3/4}}-\frac{\left (\sqrt{a} e^2+3 \sqrt{c} d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} c^{3/4}}+\frac{\left (\sqrt{a} e^2+3 \sqrt{c} d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} c^{3/4}}+\frac{d e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{c}}+\frac{x (d+e x)^2}{4 a \left (a+c x^4\right )} \]

[Out]

(x*(d + e*x)^2)/(4*a*(a + c*x^4)) + (d*e*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(2*a^(3/
2)*Sqrt[c]) - ((3*Sqrt[c]*d^2 + Sqrt[a]*e^2)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1
/4)])/(8*Sqrt[2]*a^(7/4)*c^(3/4)) + ((3*Sqrt[c]*d^2 + Sqrt[a]*e^2)*ArcTan[1 + (S
qrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*c^(3/4)) - ((3*Sqrt[c]*d^2 - Sqrt
[a]*e^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(
7/4)*c^(3/4)) + ((3*Sqrt[c]*d^2 - Sqrt[a]*e^2)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(
1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(7/4)*c^(3/4))

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Rubi [A]  time = 0.579997, antiderivative size = 322, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.588 \[ -\frac{\left (3 \sqrt{c} d^2-\sqrt{a} e^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{7/4} c^{3/4}}+\frac{\left (3 \sqrt{c} d^2-\sqrt{a} e^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{7/4} c^{3/4}}-\frac{\left (\sqrt{a} e^2+3 \sqrt{c} d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} c^{3/4}}+\frac{\left (\sqrt{a} e^2+3 \sqrt{c} d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} c^{3/4}}+\frac{d e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{c}}+\frac{x (d+e x)^2}{4 a \left (a+c x^4\right )} \]

Antiderivative was successfully verified.

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

[Out]

(x*(d + e*x)^2)/(4*a*(a + c*x^4)) + (d*e*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(2*a^(3/
2)*Sqrt[c]) - ((3*Sqrt[c]*d^2 + Sqrt[a]*e^2)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1
/4)])/(8*Sqrt[2]*a^(7/4)*c^(3/4)) + ((3*Sqrt[c]*d^2 + Sqrt[a]*e^2)*ArcTan[1 + (S
qrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*c^(3/4)) - ((3*Sqrt[c]*d^2 - Sqrt
[a]*e^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(
7/4)*c^(3/4)) + ((3*Sqrt[c]*d^2 - Sqrt[a]*e^2)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(
1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(7/4)*c^(3/4))

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Rubi in Sympy [A]  time = 98.9791, size = 299, normalized size = 0.93 \[ \frac{x \left (d + e x\right )^{2}}{4 a \left (a + c x^{4}\right )} + \frac{d e \operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}} \sqrt{c}} + \frac{\sqrt{2} \left (\sqrt{a} e^{2} - 3 \sqrt{c} d^{2}\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} x + \sqrt{a} \sqrt{c} + c x^{2} \right )}}{32 a^{\frac{7}{4}} c^{\frac{3}{4}}} - \frac{\sqrt{2} \left (\sqrt{a} e^{2} - 3 \sqrt{c} d^{2}\right ) \log{\left (\sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} x + \sqrt{a} \sqrt{c} + c x^{2} \right )}}{32 a^{\frac{7}{4}} c^{\frac{3}{4}}} - \frac{\sqrt{2} \left (\sqrt{a} e^{2} + 3 \sqrt{c} d^{2}\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{16 a^{\frac{7}{4}} c^{\frac{3}{4}}} + \frac{\sqrt{2} \left (\sqrt{a} e^{2} + 3 \sqrt{c} d^{2}\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{16 a^{\frac{7}{4}} c^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(d + e*x)**2/(4*a*(a + c*x**4)) + d*e*atan(sqrt(c)*x**2/sqrt(a))/(2*a**(3/2)*s
qrt(c)) + sqrt(2)*(sqrt(a)*e**2 - 3*sqrt(c)*d**2)*log(-sqrt(2)*a**(1/4)*c**(3/4)
*x + sqrt(a)*sqrt(c) + c*x**2)/(32*a**(7/4)*c**(3/4)) - sqrt(2)*(sqrt(a)*e**2 -
3*sqrt(c)*d**2)*log(sqrt(2)*a**(1/4)*c**(3/4)*x + sqrt(a)*sqrt(c) + c*x**2)/(32*
a**(7/4)*c**(3/4)) - sqrt(2)*(sqrt(a)*e**2 + 3*sqrt(c)*d**2)*atan(1 - sqrt(2)*c*
*(1/4)*x/a**(1/4))/(16*a**(7/4)*c**(3/4)) + sqrt(2)*(sqrt(a)*e**2 + 3*sqrt(c)*d*
*2)*atan(1 + sqrt(2)*c**(1/4)*x/a**(1/4))/(16*a**(7/4)*c**(3/4))

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Mathematica [A]  time = 0.653972, size = 321, normalized size = 1. \[ \frac{\frac{\sqrt{2} \left (a^{3/4} e^2-3 \sqrt [4]{a} \sqrt{c} d^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{c^{3/4}}+\frac{\sqrt{2} \left (3 \sqrt [4]{a} \sqrt{c} d^2-a^{3/4} e^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{c^{3/4}}-\frac{2 \sqrt [4]{a} \left (8 \sqrt [4]{a} \sqrt [4]{c} d e+\sqrt{2} \sqrt{a} e^2+3 \sqrt{2} \sqrt{c} d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{c^{3/4}}+\frac{2 \sqrt [4]{a} \left (-8 \sqrt [4]{a} \sqrt [4]{c} d e+\sqrt{2} \sqrt{a} e^2+3 \sqrt{2} \sqrt{c} d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{c^{3/4}}+\frac{8 a x (d+e x)^2}{a+c x^4}}{32 a^2} \]

Antiderivative was successfully verified.

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

[Out]

((8*a*x*(d + e*x)^2)/(a + c*x^4) - (2*a^(1/4)*(3*Sqrt[2]*Sqrt[c]*d^2 + 8*a^(1/4)
*c^(1/4)*d*e + Sqrt[2]*Sqrt[a]*e^2)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/c^(
3/4) + (2*a^(1/4)*(3*Sqrt[2]*Sqrt[c]*d^2 - 8*a^(1/4)*c^(1/4)*d*e + Sqrt[2]*Sqrt[
a]*e^2)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/c^(3/4) + (Sqrt[2]*(-3*a^(1/4)*
Sqrt[c]*d^2 + a^(3/4)*e^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2
])/c^(3/4) + (Sqrt[2]*(3*a^(1/4)*Sqrt[c]*d^2 - a^(3/4)*e^2)*Log[Sqrt[a] + Sqrt[2
]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/c^(3/4))/(32*a^2)

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Maple [A]  time = 0.007, size = 362, normalized size = 1.1 \[{\frac{{d}^{2}x}{4\,a \left ( c{x}^{4}+a \right ) }}+{\frac{3\,{d}^{2}\sqrt{2}}{32\,{a}^{2}}\sqrt [4]{{\frac{a}{c}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }+{\frac{3\,{d}^{2}\sqrt{2}}{16\,{a}^{2}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }+{\frac{3\,{d}^{2}\sqrt{2}}{16\,{a}^{2}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }+{\frac{de{x}^{2}}{2\,a \left ( c{x}^{4}+a \right ) }}+{\frac{de}{2\,a}\arctan \left ({x}^{2}\sqrt{{\frac{c}{a}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{{e}^{2}{x}^{3}}{4\,a \left ( c{x}^{4}+a \right ) }}+{\frac{{e}^{2}\sqrt{2}}{32\,ac}\ln \left ({1 \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{{e}^{2}\sqrt{2}}{16\,ac}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{{e}^{2}\sqrt{2}}{16\,ac}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^2/(c*x^4+a)^2,x)

[Out]

1/4*d^2*x/a/(c*x^4+a)+3/32*d^2/a^2*(1/c*a)^(1/4)*2^(1/2)*ln((x^2+(1/c*a)^(1/4)*x
*2^(1/2)+(1/c*a)^(1/2))/(x^2-(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2)))+3/16*d^2/a^
2*(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x+1)+3/16*d^2/a^2*(1/c*a)^(
1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x-1)+1/2*d*e*x^2/a/(c*x^4+a)+1/2*d*e/a
/(a*c)^(1/2)*arctan(x^2*(c/a)^(1/2))+1/4*e^2*x^3/a/(c*x^4+a)+1/32*e^2/a/c/(1/c*a
)^(1/4)*2^(1/2)*ln((x^2-(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2))/(x^2+(1/c*a)^(1/4
)*x*2^(1/2)+(1/c*a)^(1/2)))+1/16*e^2/a/c/(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1
/c*a)^(1/4)*x+1)+1/16*e^2/a/c/(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)
*x-1)

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

[Out]

Exception raised: ValueError

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 13.061, size = 318, normalized size = 0.99 \[ \operatorname{RootSum}{\left (65536 t^{4} a^{7} c^{3} + 11264 t^{2} a^{4} c^{2} d^{2} e^{2} + t \left (256 a^{3} c d e^{5} - 2304 a^{2} c^{2} d^{5} e\right ) + a^{2} e^{8} + 82 a c d^{4} e^{4} + 81 c^{2} d^{8}, \left ( t \mapsto t \log{\left (x + \frac{4096 t^{3} a^{7} c^{2} e^{6} + 356352 t^{3} a^{6} c^{3} d^{4} e^{2} - 23552 t^{2} a^{5} c^{2} d^{3} e^{5} + 27648 t^{2} a^{4} c^{3} d^{7} e + 912 t a^{4} c d^{2} e^{8} + 43584 t a^{3} c^{2} d^{6} e^{4} + 3888 t a^{2} c^{3} d^{10} + 12 a^{3} d e^{11} - 1088 a^{2} c d^{5} e^{7} - 7020 a c^{2} d^{9} e^{3}}{a^{3} e^{12} - 649 a^{2} c d^{4} e^{8} - 5841 a c^{2} d^{8} e^{4} + 729 c^{3} d^{12}} \right )} \right )\right )} + \frac{d^{2} x + 2 d e x^{2} + e^{2} x^{3}}{4 a^{2} + 4 a c x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(65536*_t**4*a**7*c**3 + 11264*_t**2*a**4*c**2*d**2*e**2 + _t*(256*a**3*c
*d*e**5 - 2304*a**2*c**2*d**5*e) + a**2*e**8 + 82*a*c*d**4*e**4 + 81*c**2*d**8,
Lambda(_t, _t*log(x + (4096*_t**3*a**7*c**2*e**6 + 356352*_t**3*a**6*c**3*d**4*e
**2 - 23552*_t**2*a**5*c**2*d**3*e**5 + 27648*_t**2*a**4*c**3*d**7*e + 912*_t*a*
*4*c*d**2*e**8 + 43584*_t*a**3*c**2*d**6*e**4 + 3888*_t*a**2*c**3*d**10 + 12*a**
3*d*e**11 - 1088*a**2*c*d**5*e**7 - 7020*a*c**2*d**9*e**3)/(a**3*e**12 - 649*a**
2*c*d**4*e**8 - 5841*a*c**2*d**8*e**4 + 729*c**3*d**12)))) + (d**2*x + 2*d*e*x**
2 + e**2*x**3)/(4*a**2 + 4*a*c*x**4)

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GIAC/XCAS [A]  time = 0.270812, size = 436, normalized size = 1.35 \[ \frac{x^{3} e^{2} + 2 \, d x^{2} e + d^{2} x}{4 \,{\left (c x^{4} + a\right )} a} + \frac{\sqrt{2}{\left (4 \, \sqrt{2} \sqrt{a c} c^{2} d e + 3 \, \left (a c^{3}\right )^{\frac{1}{4}} c^{2} d^{2} + \left (a c^{3}\right )^{\frac{3}{4}} e^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{16 \, a^{2} c^{3}} + \frac{\sqrt{2}{\left (4 \, \sqrt{2} \sqrt{a c} c^{2} d e + 3 \, \left (a c^{3}\right )^{\frac{1}{4}} c^{2} d^{2} + \left (a c^{3}\right )^{\frac{3}{4}} e^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{16 \, a^{2} c^{3}} + \frac{\sqrt{2}{\left (3 \, \left (a c^{3}\right )^{\frac{1}{4}} c^{2} d^{2} - \left (a c^{3}\right )^{\frac{3}{4}} e^{2}\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{32 \, a^{2} c^{3}} - \frac{\sqrt{2}{\left (3 \, \left (a c^{3}\right )^{\frac{1}{4}} c^{2} d^{2} - \left (a c^{3}\right )^{\frac{3}{4}} e^{2}\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{32 \, a^{2} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(x^3*e^2 + 2*d*x^2*e + d^2*x)/((c*x^4 + a)*a) + 1/16*sqrt(2)*(4*sqrt(2)*sqrt
(a*c)*c^2*d*e + 3*(a*c^3)^(1/4)*c^2*d^2 + (a*c^3)^(3/4)*e^2)*arctan(1/2*sqrt(2)*
(2*x + sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a^2*c^3) + 1/16*sqrt(2)*(4*sqrt(2)*sqr
t(a*c)*c^2*d*e + 3*(a*c^3)^(1/4)*c^2*d^2 + (a*c^3)^(3/4)*e^2)*arctan(1/2*sqrt(2)
*(2*x - sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a^2*c^3) + 1/32*sqrt(2)*(3*(a*c^3)^(1
/4)*c^2*d^2 - (a*c^3)^(3/4)*e^2)*ln(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^
2*c^3) - 1/32*sqrt(2)*(3*(a*c^3)^(1/4)*c^2*d^2 - (a*c^3)^(3/4)*e^2)*ln(x^2 - sqr
t(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^2*c^3)