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

Optimal. Leaf size=146 \[ \frac{\left (3 \sqrt{b} c-\sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{7/4} b^{3/4}}+\frac{\left (\sqrt{a} e+3 \sqrt{b} c\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{7/4} b^{3/4}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b}}+\frac{x \left (c+d x+e x^2\right )}{4 a \left (a-b x^4\right )} \]

[Out]

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

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Rubi [A]  time = 0.270704, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{\left (3 \sqrt{b} c-\sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{7/4} b^{3/4}}+\frac{\left (\sqrt{a} e+3 \sqrt{b} c\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{7/4} b^{3/4}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b}}+\frac{x \left (c+d x+e x^2\right )}{4 a \left (a-b x^4\right )} \]

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

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Rubi in Sympy [A]  time = 41.703, size = 131, normalized size = 0.9 \[ \frac{x \left (c + d x + e x^{2}\right )}{4 a \left (a - b x^{4}\right )} + \frac{d \operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{4 a^{\frac{3}{2}} \sqrt{b}} - \frac{\left (\sqrt{a} e - 3 \sqrt{b} c\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{8 a^{\frac{7}{4}} b^{\frac{3}{4}}} + \frac{\left (\sqrt{a} e + 3 \sqrt{b} c\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{8 a^{\frac{7}{4}} b^{\frac{3}{4}}} \]

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*atanh(sqrt(b)*x**2/sqrt(a))/(4*a**(3
/2)*sqrt(b)) - (sqrt(a)*e - 3*sqrt(b)*c)*atan(b**(1/4)*x/a**(1/4))/(8*a**(7/4)*b
**(3/4)) + (sqrt(a)*e + 3*sqrt(b)*c)*atanh(b**(1/4)*x/a**(1/4))/(8*a**(7/4)*b**(
3/4))

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Mathematica [A]  time = 0.542465, size = 211, normalized size = 1.45 \[ \frac{-\frac{\log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (a^{3/4} e+3 \sqrt [4]{a} \sqrt{b} c+2 \sqrt{a} \sqrt [4]{b} d\right )}{b^{3/4}}+\frac{\log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (a^{3/4} e+3 \sqrt [4]{a} \sqrt{b} c-2 \sqrt{a} \sqrt [4]{b} d\right )}{b^{3/4}}-\frac{2 \sqrt [4]{a} \left (\sqrt{a} e-3 \sqrt{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{b^{3/4}}+\frac{4 a x (c+x (d+e x))}{a-b x^4}+\frac{2 \sqrt{a} d \log \left (\sqrt{a}+\sqrt{b} x^2\right )}{\sqrt{b}}}{16 a^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.007, size = 228, normalized size = 1.6 \[ -{\frac{cx}{4\,a \left ( b{x}^{4}-a \right ) }}+{\frac{3\,c}{16\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{3\,c}{8\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }-{\frac{d{x}^{2}}{4\,a \left ( b{x}^{4}-a \right ) }}-{\frac{d}{8\,a}\ln \left ({1 \left ( -a+{x}^{2}\sqrt{ab} \right ) \left ( -a-{x}^{2}\sqrt{ab} \right ) ^{-1}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{e{x}^{3}}{4\,a \left ( b{x}^{4}-a \right ) }}-{\frac{e}{8\,ab}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{e}{16\,ab}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \]

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)+3/16*c/a^2*(a/b)^(1/4)*ln((x+(a/b)^(1/4))/(x-(a/b)^(1/4)))+
3/8*c/a^2*(a/b)^(1/4)*arctan(x/(a/b)^(1/4))-1/4*d*x^2/a/(b*x^4-a)-1/8*d/a/(a*b)^
(1/2)*ln((-a+x^2*(a*b)^(1/2))/(-a-x^2*(a*b)^(1/2)))-1/4*e*x^3/a/(b*x^4-a)-1/8*e/
a/b/(a/b)^(1/4)*arctan(x/(a/b)^(1/4))+1/16*e/a/b/(a/b)^(1/4)*ln((x+(a/b)^(1/4))/
(x-(a/b)^(1/4)))

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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^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 = 13.6669, size = 507, normalized size = 3.47 \[ \operatorname{RootSum}{\left (65536 t^{4} a^{7} b^{3} + t^{2} \left (- 3072 a^{4} b^{2} c e - 2048 a^{4} b^{2} d^{2}\right ) + t \left (128 a^{3} b d e^{2} + 1152 a^{2} b^{2} c^{2} d\right ) - a^{2} e^{4} + 18 a b c^{2} e^{2} - 48 a b c d^{2} e + 16 a b d^{4} - 81 b^{2} c^{4}, \left ( t \mapsto t \log{\left (x + \frac{4096 t^{3} a^{7} b^{2} e^{3} + 36864 t^{3} a^{6} b^{3} c^{2} e - 98304 t^{3} a^{6} b^{3} c d^{2} + 4608 t^{2} a^{5} b^{2} c d e^{2} - 4096 t^{2} a^{5} b^{2} d^{3} e - 13824 t^{2} a^{4} b^{3} c^{3} d - 144 t a^{4} b c e^{4} - 192 t a^{4} b d^{2} e^{3} - 1728 t a^{3} b^{2} c^{3} e^{2} + 5184 t a^{3} b^{2} c^{2} d^{2} e + 1536 t a^{3} b^{2} c d^{4} - 3888 t a^{2} b^{3} c^{5} + 6 a^{3} d e^{5} - 120 a^{2} b c d^{3} e^{2} + 64 a^{2} b d^{5} e + 810 a b^{2} c^{4} d e - 1080 a b^{2} c^{3} d^{3}}{a^{3} e^{6} + 9 a^{2} b c^{2} e^{4} - 96 a^{2} b c d^{2} e^{3} + 64 a^{2} b d^{4} e^{2} - 81 a b^{2} c^{4} e^{2} + 864 a b^{2} c^{3} d^{2} e - 576 a b^{2} c^{2} d^{4} - 729 b^{3} c^{6}} \right )} \right )\right )} - \frac{c x + d x^{2} + e x^{3}}{- 4 a^{2} + 4 a b x^{4}} \]

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 + 8
10*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 = 0.220682, size = 440, normalized size = 3.01 \[ -\frac{x^{3} e + d x^{2} + c x}{4 \,{\left (b x^{4} - a\right )} a} + \frac{\sqrt{2}{\left (2 \, \sqrt{2} \sqrt{-a b} b^{2} d + 3 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c + \left (-a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{16 \, a^{2} b^{3}} + \frac{\sqrt{2}{\left (2 \, \sqrt{2} \sqrt{-a b} b^{2} d + 3 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c + \left (-a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{16 \, a^{2} b^{3}} + \frac{\sqrt{2}{\left (3 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - \left (-a b^{3}\right )^{\frac{3}{4}} e\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{32 \, a^{2} b^{3}} - \frac{\sqrt{2}{\left (3 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - \left (-a b^{3}\right )^{\frac{3}{4}} e\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{32 \, a^{2} b^{3}} \]

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 + sqr
t(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 - sqr
t(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)