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

Optimal. Leaf size=308 \[ -\frac{\left (3 \sqrt{b} c-\sqrt{a} e\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{3/4}}+\frac{\left (3 \sqrt{b} c-\sqrt{a} e\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{3/4}}-\frac{\left (\sqrt{a} e+3 \sqrt{b} c\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} b^{3/4}}+\frac{\left (\sqrt{a} e+3 \sqrt{b} c\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} b^{3/4}}+\frac{d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b}}+\frac{x \left (c+d x+e x^2\right )}{4 a \left (a+b x^4\right )} \]

[Out]

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

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Rubi [A]  time = 0.567999, antiderivative size = 308, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ -\frac{\left (3 \sqrt{b} c-\sqrt{a} e\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{3/4}}+\frac{\left (3 \sqrt{b} c-\sqrt{a} e\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{3/4}}-\frac{\left (\sqrt{a} e+3 \sqrt{b} c\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} b^{3/4}}+\frac{\left (\sqrt{a} e+3 \sqrt{b} c\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} b^{3/4}}+\frac{d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b}}+\frac{x \left (c+d x+e x^2\right )}{4 a \left (a+b x^4\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 88.0503, size = 287, normalized size = 0.93 \[ \frac{x \left (c + d x + e x^{2}\right )}{4 a \left (a + b x^{4}\right )} + \frac{d \operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{4 a^{\frac{3}{2}} \sqrt{b}} + \frac{\sqrt{2} \left (\sqrt{a} e - 3 \sqrt{b} c\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} b^{\frac{3}{4}} x + \sqrt{a} \sqrt{b} + b x^{2} \right )}}{32 a^{\frac{7}{4}} b^{\frac{3}{4}}} - \frac{\sqrt{2} \left (\sqrt{a} e - 3 \sqrt{b} c\right ) \log{\left (\sqrt{2} \sqrt [4]{a} b^{\frac{3}{4}} x + \sqrt{a} \sqrt{b} + b x^{2} \right )}}{32 a^{\frac{7}{4}} b^{\frac{3}{4}}} - \frac{\sqrt{2} \left (\sqrt{a} e + 3 \sqrt{b} c\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{16 a^{\frac{7}{4}} b^{\frac{3}{4}}} + \frac{\sqrt{2} \left (\sqrt{a} e + 3 \sqrt{b} c\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{16 a^{\frac{7}{4}} b^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.81362, size = 305, normalized size = 0.99 \[ \frac{\frac{\sqrt{2} \left (a^{3/4} e-3 \sqrt [4]{a} \sqrt{b} c\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{b^{3/4}}+\frac{\sqrt{2} \left (3 \sqrt [4]{a} \sqrt{b} c-a^{3/4} e\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{b^{3/4}}-\frac{2 \sqrt [4]{a} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (4 \sqrt [4]{a} \sqrt [4]{b} d+\sqrt{2} \sqrt{a} e+3 \sqrt{2} \sqrt{b} c\right )}{b^{3/4}}+\frac{2 \sqrt [4]{a} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (-4 \sqrt [4]{a} \sqrt [4]{b} d+\sqrt{2} \sqrt{a} e+3 \sqrt{2} \sqrt{b} c\right )}{b^{3/4}}+\frac{8 a x (c+x (d+e x))}{a+b x^4}}{32 a^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 344, normalized size = 1.1 \[{\frac{cx}{4\,a \left ( b{x}^{4}+a \right ) }}+{\frac{3\,c\sqrt{2}}{32\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{3\,c\sqrt{2}}{16\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{3\,c\sqrt{2}}{16\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{d{x}^{2}}{4\,a \left ( b{x}^{4}+a \right ) }}+{\frac{d}{4\,a}\arctan \left ({x}^{2}\sqrt{{\frac{b}{a}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{e{x}^{3}}{4\,a \left ( b{x}^{4}+a \right ) }}+{\frac{e\sqrt{2}}{32\,ab}\ln \left ({1 \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{e\sqrt{2}}{16\,ab}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{e\sqrt{2}}{16\,ab}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*c*x/a/(b*x^4+a)+3/32*c/a^2*(a/b)^(1/4)*2^(1/2)*ln((x^2+(a/b)^(1/4)*x*2^(1/2)
+(a/b)^(1/2))/(x^2-(a/b)^(1/4)*x*2^(1/2)+(a/b)^(1/2)))+3/16*c/a^2*(a/b)^(1/4)*2^
(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x+1)+3/16*c/a^2*(a/b)^(1/4)*2^(1/2)*arctan(2^(1
/2)/(a/b)^(1/4)*x-1)+1/4*d*x^2/a/(b*x^4+a)+1/4*d/a/(a*b)^(1/2)*arctan(x^2*(b/a)^
(1/2))+1/4*e*x^3/a/(b*x^4+a)+1/32*e/a/b/(a/b)^(1/4)*2^(1/2)*ln((x^2-(a/b)^(1/4)*
x*2^(1/2)+(a/b)^(1/2))/(x^2+(a/b)^(1/4)*x*2^(1/2)+(a/b)^(1/2)))+1/16*e/a/b/(a/b)
^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x+1)+1/16*e/a/b/(a/b)^(1/4)*2^(1/2)*ar
ctan(2^(1/2)/(a/b)^(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^2 + d*x + c)/(b*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^2 + d*x + c)/(b*x^4 + a)^2,x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 13.705, size = 505, normalized size = 1.64 \[ \operatorname{RootSum}{\left (65536 t^{4} a^{7} b^{3} + t^{2} \left (3072 a^{4} b^{2} c e + 2048 a^{4} b^{2} d^{2}\right ) + t \left (128 a^{3} b d e^{2} - 1152 a^{2} b^{2} c^{2} d\right ) + a^{2} e^{4} + 18 a b c^{2} e^{2} - 48 a b c d^{2} e + 16 a b d^{4} + 81 b^{2} c^{4}, \left ( t \mapsto t \log{\left (x + \frac{4096 t^{3} a^{7} b^{2} e^{3} - 36864 t^{3} a^{6} b^{3} c^{2} e + 98304 t^{3} a^{6} b^{3} c d^{2} + 4608 t^{2} a^{5} b^{2} c d e^{2} - 4096 t^{2} a^{5} b^{2} d^{3} e + 13824 t^{2} a^{4} b^{3} c^{3} d + 144 t a^{4} b c e^{4} + 192 t a^{4} b d^{2} e^{3} - 1728 t a^{3} b^{2} c^{3} e^{2} + 5184 t a^{3} b^{2} c^{2} d^{2} e + 1536 t a^{3} b^{2} c d^{4} + 3888 t a^{2} b^{3} c^{5} + 6 a^{3} d e^{5} + 120 a^{2} b c d^{3} e^{2} - 64 a^{2} b d^{5} e + 810 a b^{2} c^{4} d e - 1080 a b^{2} c^{3} d^{3}}{a^{3} e^{6} - 9 a^{2} b c^{2} e^{4} + 96 a^{2} b c d^{2} e^{3} - 64 a^{2} b d^{4} e^{2} - 81 a b^{2} c^{4} e^{2} + 864 a b^{2} c^{3} d^{2} e - 576 a b^{2} c^{2} d^{4} + 729 b^{3} c^{6}} \right )} \right )\right )} + \frac{c x + d x^{2} + e x^{3}}{4 a^{2} + 4 a b x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(65536*_t**4*a**7*b**3 + _t**2*(3072*a**4*b**2*c*e + 2048*a**4*b**2*d**2)
+ _t*(128*a**3*b*d*e**2 - 1152*a**2*b**2*c**2*d) + a**2*e**4 + 18*a*b*c**2*e**2
- 48*a*b*c*d**2*e + 16*a*b*d**4 + 81*b**2*c**4, Lambda(_t, _t*log(x + (4096*_t*
*3*a**7*b**2*e**3 - 36864*_t**3*a**6*b**3*c**2*e + 98304*_t**3*a**6*b**3*c*d**2
+ 4608*_t**2*a**5*b**2*c*d*e**2 - 4096*_t**2*a**5*b**2*d**3*e + 13824*_t**2*a**4
*b**3*c**3*d + 144*_t*a**4*b*c*e**4 + 192*_t*a**4*b*d**2*e**3 - 1728*_t*a**3*b**
2*c**3*e**2 + 5184*_t*a**3*b**2*c**2*d**2*e + 1536*_t*a**3*b**2*c*d**4 + 3888*_t
*a**2*b**3*c**5 + 6*a**3*d*e**5 + 120*a**2*b*c*d**3*e**2 - 64*a**2*b*d**5*e + 81
0*a*b**2*c**4*d*e - 1080*a*b**2*c**3*d**3)/(a**3*e**6 - 9*a**2*b*c**2*e**4 + 96*
a**2*b*c*d**2*e**3 - 64*a**2*b*d**4*e**2 - 81*a*b**2*c**4*e**2 + 864*a*b**2*c**3
*d**2*e - 576*a*b**2*c**2*d**4 + 729*b**3*c**6)))) + (c*x + d*x**2 + e*x**3)/(4*
a**2 + 4*a*b*x**4)

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GIAC/XCAS [A]  time = 0.220738, size = 413, normalized size = 1.34 \[ \frac{x^{3} e + d x^{2} + c x}{4 \,{\left (b x^{4} + a\right )} a} + \frac{\sqrt{2}{\left (2 \, \sqrt{2} \sqrt{a b} b^{2} d + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c + \left (a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{16 \, a^{2} b^{3}} + \frac{\sqrt{2}{\left (2 \, \sqrt{2} \sqrt{a b} b^{2} d + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c + \left (a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{16 \, a^{2} b^{3}} + \frac{\sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c - \left (a b^{3}\right )^{\frac{3}{4}} e\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{32 \, a^{2} b^{3}} - \frac{\sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c - \left (a b^{3}\right )^{\frac{3}{4}} e\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{32 \, a^{2} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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