3.174 \(\int \frac{c+d x+e x^2+f x^3+g x^4}{\left (a-b x^4\right )^4} \, dx\)

Optimal. Leaf size=266 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-15 \sqrt{a} \sqrt{b} e-7 a g+77 b c\right )}{256 a^{15/4} b^{5/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (15 \sqrt{a} \sqrt{b} e-7 a g+77 b c\right )}{256 a^{15/4} b^{5/4}}+\frac{5 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} \sqrt{b}}+\frac{x \left (7 (11 b c-a g)+60 b d x+45 b e x^2\right )}{384 a^3 b \left (a-b x^4\right )}+\frac{x \left (-a g+11 b c+10 b d x+9 b e x^2\right )+8 a f}{96 a^2 b \left (a-b x^4\right )^2}+\frac{x \left (a g+b c+b d x+b e x^2+b f x^3\right )}{12 a b \left (a-b x^4\right )^3} \]

[Out]

(x*(b*c + a*g + b*d*x + b*e*x^2 + b*f*x^3))/(12*a*b*(a - b*x^4)^3) + (x*(7*(11*b
*c - a*g) + 60*b*d*x + 45*b*e*x^2))/(384*a^3*b*(a - b*x^4)) + (8*a*f + x*(11*b*c
- a*g + 10*b*d*x + 9*b*e*x^2))/(96*a^2*b*(a - b*x^4)^2) + ((77*b*c - 15*Sqrt[a]
*Sqrt[b]*e - 7*a*g)*ArcTan[(b^(1/4)*x)/a^(1/4)])/(256*a^(15/4)*b^(5/4)) + ((77*b
*c + 15*Sqrt[a]*Sqrt[b]*e - 7*a*g)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(256*a^(15/4)*b
^(5/4)) + (5*d*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a]])/(32*a^(7/2)*Sqrt[b])

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Rubi [A]  time = 0.649152, antiderivative size = 266, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.258 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-15 \sqrt{a} \sqrt{b} e-7 a g+77 b c\right )}{256 a^{15/4} b^{5/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (15 \sqrt{a} \sqrt{b} e-7 a g+77 b c\right )}{256 a^{15/4} b^{5/4}}+\frac{5 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} \sqrt{b}}+\frac{x \left (7 (11 b c-a g)+60 b d x+45 b e x^2\right )}{384 a^3 b \left (a-b x^4\right )}+\frac{x \left (-a g+11 b c+10 b d x+9 b e x^2\right )+8 a f}{96 a^2 b \left (a-b x^4\right )^2}+\frac{x \left (a g+b c+b d x+b e x^2+b f x^3\right )}{12 a b \left (a-b x^4\right )^3} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x + e*x^2 + f*x^3 + g*x^4)/(a - b*x^4)^4,x]

[Out]

(x*(b*c + a*g + b*d*x + b*e*x^2 + b*f*x^3))/(12*a*b*(a - b*x^4)^3) + (x*(7*(11*b
*c - a*g) + 60*b*d*x + 45*b*e*x^2))/(384*a^3*b*(a - b*x^4)) + (8*a*f + x*(11*b*c
- a*g + 10*b*d*x + 9*b*e*x^2))/(96*a^2*b*(a - b*x^4)^2) + ((77*b*c - 15*Sqrt[a]
*Sqrt[b]*e - 7*a*g)*ArcTan[(b^(1/4)*x)/a^(1/4)])/(256*a^(15/4)*b^(5/4)) + ((77*b
*c + 15*Sqrt[a]*Sqrt[b]*e - 7*a*g)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(256*a^(15/4)*b
^(5/4)) + (5*d*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a]])/(32*a^(7/2)*Sqrt[b])

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Rubi in Sympy [A]  time = 119.523, size = 255, normalized size = 0.96 \[ \frac{x \left (a g + b c + b d x + b e x^{2} + b f x^{3}\right )}{12 a b \left (a - b x^{4}\right )^{3}} + \frac{8 a f - x \left (a g - 11 b c - 10 b d x - 9 b e x^{2}\right )}{96 a^{2} b \left (a - b x^{4}\right )^{2}} - \frac{x \left (7 a g - 77 b c - 60 b d x - 45 b e x^{2}\right )}{384 a^{3} b \left (a - b x^{4}\right )} + \frac{5 d \operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{32 a^{\frac{7}{2}} \sqrt{b}} - \frac{\left (- 15 \sqrt{a} \sqrt{b} e + 7 a g - 77 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{256 a^{\frac{15}{4}} b^{\frac{5}{4}}} - \frac{\left (15 \sqrt{a} \sqrt{b} e + 7 a g - 77 b c\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{256 a^{\frac{15}{4}} b^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x**4+f*x**3+e*x**2+d*x+c)/(-b*x**4+a)**4,x)

[Out]

x*(a*g + b*c + b*d*x + b*e*x**2 + b*f*x**3)/(12*a*b*(a - b*x**4)**3) + (8*a*f -
x*(a*g - 11*b*c - 10*b*d*x - 9*b*e*x**2))/(96*a**2*b*(a - b*x**4)**2) - x*(7*a*g
- 77*b*c - 60*b*d*x - 45*b*e*x**2)/(384*a**3*b*(a - b*x**4)) + 5*d*atanh(sqrt(b
)*x**2/sqrt(a))/(32*a**(7/2)*sqrt(b)) - (-15*sqrt(a)*sqrt(b)*e + 7*a*g - 77*b*c)
*atanh(b**(1/4)*x/a**(1/4))/(256*a**(15/4)*b**(5/4)) - (15*sqrt(a)*sqrt(b)*e + 7
*a*g - 77*b*c)*atan(b**(1/4)*x/a**(1/4))/(256*a**(15/4)*b**(5/4))

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Mathematica [A]  time = 0.628692, size = 313, normalized size = 1.18 \[ \frac{\frac{128 a^{11/4} \sqrt [4]{b} (a (f+g x)+b x (c+x (d+e x)))}{\left (a-b x^4\right )^3}+\frac{16 a^{7/4} \sqrt [4]{b} x (-a g+11 b c+b x (10 d+9 e x))}{\left (a-b x^4\right )^2}+\frac{4 a^{3/4} \sqrt [4]{b} x (-7 a g+77 b c+15 b x (4 d+3 e x))}{a-b x^4}-3 \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (40 \sqrt [4]{a} b^{3/4} d+15 \sqrt{a} \sqrt{b} e-7 a g+77 b c\right )+3 \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (-40 \sqrt [4]{a} b^{3/4} d+15 \sqrt{a} \sqrt{b} e-7 a g+77 b c\right )+120 \sqrt [4]{a} b^{3/4} d \log \left (\sqrt{a}+\sqrt{b} x^2\right )+6 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-15 \sqrt{a} \sqrt{b} e-7 a g+77 b c\right )}{1536 a^{15/4} b^{5/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4)/(a - b*x^4)^4,x]

[Out]

((4*a^(3/4)*b^(1/4)*x*(77*b*c - 7*a*g + 15*b*x*(4*d + 3*e*x)))/(a - b*x^4) + (16
*a^(7/4)*b^(1/4)*x*(11*b*c - a*g + b*x*(10*d + 9*e*x)))/(a - b*x^4)^2 + (128*a^(
11/4)*b^(1/4)*(a*(f + g*x) + b*x*(c + x*(d + e*x))))/(a - b*x^4)^3 + 6*(77*b*c -
15*Sqrt[a]*Sqrt[b]*e - 7*a*g)*ArcTan[(b^(1/4)*x)/a^(1/4)] - 3*(77*b*c + 40*a^(1
/4)*b^(3/4)*d + 15*Sqrt[a]*Sqrt[b]*e - 7*a*g)*Log[a^(1/4) - b^(1/4)*x] + 3*(77*b
*c - 40*a^(1/4)*b^(3/4)*d + 15*Sqrt[a]*Sqrt[b]*e - 7*a*g)*Log[a^(1/4) + b^(1/4)*
x] + 120*a^(1/4)*b^(3/4)*d*Log[Sqrt[a] + Sqrt[b]*x^2])/(1536*a^(15/4)*b^(5/4))

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Maple [A]  time = 0.021, size = 384, normalized size = 1.4 \[{\frac{1}{ \left ( b{x}^{4}-a \right ) ^{3}} \left ( -{\frac{15\,{b}^{2}e{x}^{11}}{128\,{a}^{3}}}-{\frac{5\,{b}^{2}d{x}^{10}}{32\,{a}^{3}}}+{\frac{ \left ( 7\,ag-77\,bc \right ) b{x}^{9}}{384\,{a}^{3}}}+{\frac{21\,be{x}^{7}}{64\,{a}^{2}}}+{\frac{5\,bd{x}^{6}}{12\,{a}^{2}}}-{\frac{ \left ( 3\,ag-33\,bc \right ){x}^{5}}{64\,{a}^{2}}}-{\frac{113\,e{x}^{3}}{384\,a}}-{\frac{11\,d{x}^{2}}{32\,a}}-{\frac{ \left ( 7\,ag+51\,bc \right ) x}{128\,ab}}-{\frac{f}{12\,b}} \right ) }-{\frac{7\,g}{256\,{a}^{3}b}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }+{\frac{77\,c}{256\,{a}^{4}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }-{\frac{7\,g}{512\,{a}^{3}b}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{77\,c}{512\,{a}^{4}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }-{\frac{5\,bd}{64}\ln \left ({1 \left ( -{a}^{4}b+{x}^{2}\sqrt{{a}^{7}{b}^{3}} \right ) \left ( -{a}^{4}b-{x}^{2}\sqrt{{a}^{7}{b}^{3}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{a}^{7}{b}^{3}}}}}-{\frac{15\,e}{256\,{a}^{3}b}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{15\,e}{512\,{a}^{3}b}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((g*x^4+f*x^3+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+7/384*(a*g-11*b*c)/a^3*b*x^9+21/64/a
^2*b*e*x^7+5/12/a^2*d*b*x^6-3/64/a^2*(a*g-11*b*c)*x^5-113/384/a*e*x^3-11/32*d/a*
x^2-1/128*(7*a*g+51*b*c)/a/b*x-1/12*f/b)/(b*x^4-a)^3-7/256*(a/b)^(1/4)/a^3/b*arc
tan(x/(a/b)^(1/4))*g+77/256*c*(a/b)^(1/4)/a^4*arctan(x/(a/b)^(1/4))-7/512*(a/b)^
(1/4)/a^3/b*ln((x+(a/b)^(1/4))/(x-(a/b)^(1/4)))*g+77/512*c*(a/b)^(1/4)/a^4*ln((x
+(a/b)^(1/4))/(x-(a/b)^(1/4)))-5/64*b*d/(a^7*b^3)^(1/2)*ln((-a^4*b+x^2*(a^7*b^3)
^(1/2))/(-a^4*b-x^2*(a^7*b^3)^(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((g*x^4 + f*x^3 + 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((g*x^4 + f*x^3 + e*x^2 + d*x + c)/(b*x^4 - a)^4,x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x**4+f*x**3+e*x**2+d*x+c)/(-b*x**4+a)**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.221355, size = 662, normalized size = 2.49 \[ \frac{\sqrt{2}{\left (40 \, \sqrt{2} \sqrt{-a b} b^{2} d + 77 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - 7 \, \left (-a b^{3}\right )^{\frac{1}{4}} a b g + 15 \, \left (-a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{512 \, a^{4} b^{3}} + \frac{\sqrt{2}{\left (40 \, \sqrt{2} \sqrt{-a b} b^{2} d + 77 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - 7 \, \left (-a b^{3}\right )^{\frac{1}{4}} a b g + 15 \, \left (-a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{512 \, a^{4} b^{3}} + \frac{\sqrt{2}{\left (77 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - 7 \, \left (-a b^{3}\right )^{\frac{1}{4}} a b g - 15 \, \left (-a b^{3}\right )^{\frac{3}{4}} e\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{1024 \, a^{4} b^{3}} - \frac{\sqrt{2}{\left (77 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - 7 \, \left (-a b^{3}\right )^{\frac{1}{4}} a b g - 15 \, \left (-a b^{3}\right )^{\frac{3}{4}} e\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{1024 \, a^{4} b^{3}} - \frac{45 \, b^{3} x^{11} e + 60 \, b^{3} d x^{10} + 77 \, b^{3} c x^{9} - 7 \, a b^{2} g x^{9} - 126 \, a b^{2} x^{7} e - 160 \, a b^{2} d x^{6} - 198 \, a b^{2} c x^{5} + 18 \, a^{2} b g x^{5} + 113 \, a^{2} b x^{3} e + 132 \, a^{2} b d x^{2} + 153 \, a^{2} b c x + 21 \, a^{3} g x + 32 \, a^{3} f}{384 \,{\left (b x^{4} - a\right )}^{3} a^{3} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x^4 + f*x^3 + 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 - 7*(-a*b^3
)^(1/4)*a*b*g + 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 - 7*(-a*b^3)^(1/4)*a*b*g + 15*(-a*b^3)^(3/4)*e)*arctan(1/2*s
qrt(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 - 7*(-a*b^3)^(1/4)*a*b*g - 15*(-a*b^3)^(3/4)*e)*ln(x^2 + s
qrt(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 - 7*(-a*b^3)^(1/4)*a*b*g - 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^3*x^11*e + 60*b^3*d*x^10 + 77*b^3*
c*x^9 - 7*a*b^2*g*x^9 - 126*a*b^2*x^7*e - 160*a*b^2*d*x^6 - 198*a*b^2*c*x^5 + 18
*a^2*b*g*x^5 + 113*a^2*b*x^3*e + 132*a^2*b*d*x^2 + 153*a^2*b*c*x + 21*a^3*g*x +
32*a^3*f)/((b*x^4 - a)^3*a^3*b)