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

Optimal. Leaf size=211 \[ \frac{\left (77 \sqrt{b} c-15 \sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{15/4} b^{3/4}}+\frac{\left (15 \sqrt{a} e+77 \sqrt{b} c\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{15/4} b^{3/4}}+\frac{5 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} \sqrt{b}}+\frac{x \left (77 c+60 d x+45 e x^2\right )}{384 a^3 \left (a-b x^4\right )}+\frac{x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a-b x^4\right )^2}+\frac{x \left (c+d x+e x^2\right )}{12 a \left (a-b x^4\right )^3} \]

[Out]

(x*(c + d*x + e*x^2))/(12*a*(a - b*x^4)^3) + (x*(11*c + 10*d*x + 9*e*x^2))/(96*a
^2*(a - b*x^4)^2) + (x*(77*c + 60*d*x + 45*e*x^2))/(384*a^3*(a - b*x^4)) + ((77*
Sqrt[b]*c - 15*Sqrt[a]*e)*ArcTan[(b^(1/4)*x)/a^(1/4)])/(256*a^(15/4)*b^(3/4)) +
((77*Sqrt[b]*c + 15*Sqrt[a]*e)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(256*a^(15/4)*b^(3/
4)) + (5*d*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a]])/(32*a^(7/2)*Sqrt[b])

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Rubi [A]  time = 0.441644, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{\left (77 \sqrt{b} c-15 \sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{15/4} b^{3/4}}+\frac{\left (15 \sqrt{a} e+77 \sqrt{b} c\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{15/4} b^{3/4}}+\frac{5 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} \sqrt{b}}+\frac{x \left (77 c+60 d x+45 e x^2\right )}{384 a^3 \left (a-b x^4\right )}+\frac{x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a-b x^4\right )^2}+\frac{x \left (c+d x+e x^2\right )}{12 a \left (a-b x^4\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

(x*(c + d*x + e*x^2))/(12*a*(a - b*x^4)^3) + (x*(11*c + 10*d*x + 9*e*x^2))/(96*a
^2*(a - b*x^4)^2) + (x*(77*c + 60*d*x + 45*e*x^2))/(384*a^3*(a - b*x^4)) + ((77*
Sqrt[b]*c - 15*Sqrt[a]*e)*ArcTan[(b^(1/4)*x)/a^(1/4)])/(256*a^(15/4)*b^(3/4)) +
((77*Sqrt[b]*c + 15*Sqrt[a]*e)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(256*a^(15/4)*b^(3/
4)) + (5*d*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a]])/(32*a^(7/2)*Sqrt[b])

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Rubi in Sympy [A]  time = 69.4565, size = 197, normalized size = 0.93 \[ \frac{x \left (c + d x + e x^{2}\right )}{12 a \left (a - b x^{4}\right )^{3}} + \frac{x \left (11 c + 10 d x + 9 e x^{2}\right )}{96 a^{2} \left (a - b x^{4}\right )^{2}} + \frac{x \left (77 c + 60 d x + 45 e x^{2}\right )}{384 a^{3} \left (a - b x^{4}\right )} + \frac{5 d \operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{32 a^{\frac{7}{2}} \sqrt{b}} - \frac{\left (15 \sqrt{a} e - 77 \sqrt{b} c\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{256 a^{\frac{15}{4}} b^{\frac{3}{4}}} + \frac{\left (15 \sqrt{a} e + 77 \sqrt{b} c\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{256 a^{\frac{15}{4}} b^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(c + d*x + e*x**2)/(12*a*(a - b*x**4)**3) + x*(11*c + 10*d*x + 9*e*x**2)/(96*a
**2*(a - b*x**4)**2) + x*(77*c + 60*d*x + 45*e*x**2)/(384*a**3*(a - b*x**4)) + 5
*d*atanh(sqrt(b)*x**2/sqrt(a))/(32*a**(7/2)*sqrt(b)) - (15*sqrt(a)*e - 77*sqrt(b
)*c)*atan(b**(1/4)*x/a**(1/4))/(256*a**(15/4)*b**(3/4)) + (15*sqrt(a)*e + 77*sqr
t(b)*c)*atanh(b**(1/4)*x/a**(1/4))/(256*a**(15/4)*b**(3/4))

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Mathematica [A]  time = 0.532324, size = 276, normalized size = 1.31 \[ \frac{-\frac{3 \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (15 a^{3/4} e+77 \sqrt [4]{a} \sqrt{b} c+40 \sqrt{a} \sqrt [4]{b} d\right )}{b^{3/4}}+\frac{3 \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (15 a^{3/4} e+77 \sqrt [4]{a} \sqrt{b} c-40 \sqrt{a} \sqrt [4]{b} d\right )}{b^{3/4}}+\frac{128 a^3 x (c+x (d+e x))}{\left (a-b x^4\right )^3}+\frac{16 a^2 x (11 c+x (10 d+9 e x))}{\left (a-b x^4\right )^2}+\frac{6 \sqrt [4]{a} \left (77 \sqrt{b} c-15 \sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{b^{3/4}}+\frac{4 a x (77 c+15 x (4 d+3 e x))}{a-b x^4}+\frac{120 \sqrt{a} d \log \left (\sqrt{a}+\sqrt{b} x^2\right )}{\sqrt{b}}}{1536 a^4} \]

Antiderivative was successfully verified.

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

[Out]

((128*a^3*x*(c + x*(d + e*x)))/(a - b*x^4)^3 + (4*a*x*(77*c + 15*x*(4*d + 3*e*x)
))/(a - b*x^4) + (16*a^2*x*(11*c + x*(10*d + 9*e*x)))/(a - b*x^4)^2 + (6*a^(1/4)
*(77*Sqrt[b]*c - 15*Sqrt[a]*e)*ArcTan[(b^(1/4)*x)/a^(1/4)])/b^(3/4) - (3*(77*a^(
1/4)*Sqrt[b]*c + 40*Sqrt[a]*b^(1/4)*d + 15*a^(3/4)*e)*Log[a^(1/4) - b^(1/4)*x])/
b^(3/4) + (3*(77*a^(1/4)*Sqrt[b]*c - 40*Sqrt[a]*b^(1/4)*d + 15*a^(3/4)*e)*Log[a^
(1/4) + b^(1/4)*x])/b^(3/4) + (120*Sqrt[a]*d*Log[Sqrt[a] + Sqrt[b]*x^2])/Sqrt[b]
)/(1536*a^4)

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Maple [A]  time = 0.019, size = 281, normalized size = 1.3 \[{\frac{1}{ \left ( b{x}^{4}-a \right ) ^{3}} \left ( -{\frac{15\,{b}^{2}e{x}^{11}}{128\,{a}^{3}}}-{\frac{5\,{b}^{2}d{x}^{10}}{32\,{a}^{3}}}-{\frac{77\,{b}^{2}c{x}^{9}}{384\,{a}^{3}}}+{\frac{21\,be{x}^{7}}{64\,{a}^{2}}}+{\frac{5\,bd{x}^{6}}{12\,{a}^{2}}}+{\frac{33\,bc{x}^{5}}{64\,{a}^{2}}}-{\frac{113\,e{x}^{3}}{384\,a}}-{\frac{11\,d{x}^{2}}{32\,a}}-{\frac{51\,cx}{128\,a}} \right ) }+{\frac{77\,c}{512\,{a}^{4}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{77\,c}{256\,{a}^{4}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }-{\frac{5\,d}{64}\ln \left ({1 \left ( -{a}^{4}+{x}^{2}\sqrt{b{a}^{7}} \right ) \left ( -{a}^{4}-{x}^{2}\sqrt{b{a}^{7}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{b{a}^{7}}}}}-{\frac{15\,e}{256\,{a}^{3}b}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{15\,e}{512\,{a}^{3}b}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-15/128*e/a^3*b^2*x^11-5/32*d/a^3*b^2*x^10-77/384*c/a^3*b^2*x^9+21/64/a^2*b*e*x
^7+5/12/a^2*d*b*x^6+33/64/a^2*c*b*x^5-113/384/a*e*x^3-11/32*d/a*x^2-51/128/a*c*x
)/(b*x^4-a)^3+77/512*c*(a/b)^(1/4)/a^4*ln((x+(a/b)^(1/4))/(x-(a/b)^(1/4)))+77/25
6*c*(a/b)^(1/4)/a^4*arctan(x/(a/b)^(1/4))-5/64*d/(b*a^7)^(1/2)*ln((-a^4+x^2*(b*a
^7)^(1/2))/(-a^4-x^2*(b*a^7)^(1/2)))-15/256*e/a^3/b/(a/b)^(1/4)*arctan(x/(a/b)^(
1/4))+15/512*e/a^3/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)^4,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)^4,x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 34.2366, size = 612, normalized size = 2.9 \[ \operatorname{RootSum}{\left (68719476736 t^{4} a^{15} b^{3} + t^{2} \left (- 1211105280 a^{8} b^{2} c e - 838860800 a^{8} b^{2} d^{2}\right ) + t \left (18432000 a^{5} b d e^{2} + 485703680 a^{4} b^{2} c^{2} d\right ) - 50625 a^{2} e^{4} + 2668050 a b c^{2} e^{2} - 7392000 a b c d^{2} e + 2560000 a b d^{4} - 35153041 b^{2} c^{4}, \left ( t \mapsto t \log{\left (x + \frac{452984832000 t^{3} a^{13} b^{2} e^{3} + 11936653639680 t^{3} a^{12} b^{3} c^{2} e - 33071248179200 t^{3} a^{12} b^{3} c d^{2} + 544997376000 t^{2} a^{9} b^{2} c d e^{2} - 503316480000 t^{2} a^{9} b^{2} d^{3} e - 4787095470080 t^{2} a^{8} b^{3} c^{3} d - 5987520000 t a^{6} b c e^{4} - 8294400000 t a^{6} b d^{2} e^{3} - 210370406400 t a^{5} b^{2} c^{3} e^{2} + 655699968000 t a^{5} b^{2} c^{2} d^{2} e + 201850880000 t a^{5} b^{2} c d^{4} - 1385873488384 t a^{4} b^{3} c^{5} + 91125000 a^{3} d e^{5} - 5544000000 a^{2} b c d^{3} e^{2} + 3072000000 a^{2} b d^{5} e + 105459123000 a b^{2} c^{4} d e - 146090560000 a b^{2} c^{3} d^{3}}{11390625 a^{3} e^{6} + 300155625 a^{2} b c^{2} e^{4} - 3326400000 a^{2} b c d^{2} e^{3} + 2304000000 a^{2} b d^{4} e^{2} - 7909434225 a b^{2} c^{4} e^{2} + 87654336000 a b^{2} c^{3} d^{2} e - 60712960000 a b^{2} c^{2} d^{4} - 208422380089 b^{3} c^{6}} \right )} \right )\right )} - \frac{153 a^{2} c x + 132 a^{2} d x^{2} + 113 a^{2} e x^{3} - 198 a b c x^{5} - 160 a b d x^{6} - 126 a b e x^{7} + 77 b^{2} c x^{9} + 60 b^{2} d x^{10} + 45 b^{2} e x^{11}}{- 384 a^{6} + 1152 a^{5} b x^{4} - 1152 a^{4} b^{2} x^{8} + 384 a^{3} b^{3} x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(68719476736*_t**4*a**15*b**3 + _t**2*(-1211105280*a**8*b**2*c*e - 838860
800*a**8*b**2*d**2) + _t*(18432000*a**5*b*d*e**2 + 485703680*a**4*b**2*c**2*d) -
50625*a**2*e**4 + 2668050*a*b*c**2*e**2 - 7392000*a*b*c*d**2*e + 2560000*a*b*d*
*4 - 35153041*b**2*c**4, Lambda(_t, _t*log(x + (452984832000*_t**3*a**13*b**2*e*
*3 + 11936653639680*_t**3*a**12*b**3*c**2*e - 33071248179200*_t**3*a**12*b**3*c*
d**2 + 544997376000*_t**2*a**9*b**2*c*d*e**2 - 503316480000*_t**2*a**9*b**2*d**3
*e - 4787095470080*_t**2*a**8*b**3*c**3*d - 5987520000*_t*a**6*b*c*e**4 - 829440
0000*_t*a**6*b*d**2*e**3 - 210370406400*_t*a**5*b**2*c**3*e**2 + 655699968000*_t
*a**5*b**2*c**2*d**2*e + 201850880000*_t*a**5*b**2*c*d**4 - 1385873488384*_t*a**
4*b**3*c**5 + 91125000*a**3*d*e**5 - 5544000000*a**2*b*c*d**3*e**2 + 3072000000*
a**2*b*d**5*e + 105459123000*a*b**2*c**4*d*e - 146090560000*a*b**2*c**3*d**3)/(1
1390625*a**3*e**6 + 300155625*a**2*b*c**2*e**4 - 3326400000*a**2*b*c*d**2*e**3 +
2304000000*a**2*b*d**4*e**2 - 7909434225*a*b**2*c**4*e**2 + 87654336000*a*b**2*
c**3*d**2*e - 60712960000*a*b**2*c**2*d**4 - 208422380089*b**3*c**6)))) - (153*a
**2*c*x + 132*a**2*d*x**2 + 113*a**2*e*x**3 - 198*a*b*c*x**5 - 160*a*b*d*x**6 -
126*a*b*e*x**7 + 77*b**2*c*x**9 + 60*b**2*d*x**10 + 45*b**2*e*x**11)/(-384*a**6
+ 1152*a**5*b*x**4 - 1152*a**4*b**2*x**8 + 384*a**3*b**3*x**12)

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GIAC/XCAS [A]  time = 0.222059, size = 531, normalized size = 2.52 \[ \frac{\sqrt{2}{\left (40 \, \sqrt{2} \sqrt{-a b} b^{2} d + 77 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c + 15 \, \left (-a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{512 \, a^{4} b^{3}} + \frac{\sqrt{2}{\left (40 \, \sqrt{2} \sqrt{-a b} b^{2} d + 77 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c + 15 \, \left (-a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{512 \, a^{4} b^{3}} + \frac{\sqrt{2}{\left (77 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - 15 \, \left (-a b^{3}\right )^{\frac{3}{4}} e\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{1024 \, a^{4} b^{3}} - \frac{\sqrt{2}{\left (77 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - 15 \, \left (-a b^{3}\right )^{\frac{3}{4}} e\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{1024 \, a^{4} b^{3}} - \frac{45 \, b^{2} x^{11} e + 60 \, b^{2} d x^{10} + 77 \, b^{2} c x^{9} - 126 \, a b x^{7} e - 160 \, a b d x^{6} - 198 \, a b c x^{5} + 113 \, a^{2} x^{3} e + 132 \, a^{2} d x^{2} + 153 \, a^{2} c x}{384 \,{\left (b x^{4} - a\right )}^{3} a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/512*sqrt(2)*(40*sqrt(2)*sqrt(-a*b)*b^2*d + 77*(-a*b^3)^(1/4)*b^2*c + 15*(-a*b^
3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/(a^4*b
^3) + 1/512*sqrt(2)*(40*sqrt(2)*sqrt(-a*b)*b^2*d + 77*(-a*b^3)^(1/4)*b^2*c + 15*
(-a*b^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/
(a^4*b^3) + 1/1024*sqrt(2)*(77*(-a*b^3)^(1/4)*b^2*c - 15*(-a*b^3)^(3/4)*e)*ln(x^
2 + sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/(a^4*b^3) - 1/1024*sqrt(2)*(77*(-a*b^3)
^(1/4)*b^2*c - 15*(-a*b^3)^(3/4)*e)*ln(x^2 - sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b)
)/(a^4*b^3) - 1/384*(45*b^2*x^11*e + 60*b^2*d*x^10 + 77*b^2*c*x^9 - 126*a*b*x^7*
e - 160*a*b*d*x^6 - 198*a*b*c*x^5 + 113*a^2*x^3*e + 132*a^2*d*x^2 + 153*a^2*c*x)
/((b*x^4 - a)^3*a^3)