3.204 \(\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5}{\left (a-b x^4\right )^4} \, dx\)

Optimal. Leaf size=293 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-15 \sqrt{a} \sqrt{b} e-7 a g+77 b c\right )}{256 a^{15/4} b^{5/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (15 \sqrt{a} \sqrt{b} e-7 a g+77 b c\right )}{256 a^{15/4} b^{5/4}}+\frac{(5 b d-a h) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} b^{3/2}}+\frac{x \left (7 (11 b c-a g)+12 x (5 b d-a h)+45 b e x^2\right )}{384 a^3 b \left (a-b x^4\right )}+\frac{x \left (2 x (5 b d-a h)-a g+11 b c+9 b e x^2\right )+8 a f}{96 a^2 b \left (a-b x^4\right )^2}+\frac{x \left (x (a h+b d)+a g+b c+b e x^2+b f x^3\right )}{12 a b \left (a-b x^4\right )^3} \]

[Out]

(x*(b*c + a*g + (b*d + a*h)*x + b*e*x^2 + b*f*x^3))/(12*a*b*(a - b*x^4)^3) + (x*
(7*(11*b*c - a*g) + 12*(5*b*d - a*h)*x + 45*b*e*x^2))/(384*a^3*b*(a - b*x^4)) +
(8*a*f + x*(11*b*c - a*g + 2*(5*b*d - a*h)*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*b*d - a*h)*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a
]])/(32*a^(7/2)*b^(3/2))

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Rubi [A]  time = 0.900172, antiderivative size = 293, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-15 \sqrt{a} \sqrt{b} e-7 a g+77 b c\right )}{256 a^{15/4} b^{5/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (15 \sqrt{a} \sqrt{b} e-7 a g+77 b c\right )}{256 a^{15/4} b^{5/4}}+\frac{(5 b d-a h) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} b^{3/2}}+\frac{x \left (7 (11 b c-a g)+12 x (5 b d-a h)+45 b e x^2\right )}{384 a^3 b \left (a-b x^4\right )}+\frac{x \left (2 x (5 b d-a h)-a g+11 b c+9 b e x^2\right )+8 a f}{96 a^2 b \left (a-b x^4\right )^2}+\frac{x \left (x (a h+b d)+a g+b c+b e x^2+b f x^3\right )}{12 a b \left (a-b x^4\right )^3} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(a - b*x^4)^4,x]

[Out]

(x*(b*c + a*g + (b*d + a*h)*x + b*e*x^2 + b*f*x^3))/(12*a*b*(a - b*x^4)^3) + (x*
(7*(11*b*c - a*g) + 12*(5*b*d - a*h)*x + 45*b*e*x^2))/(384*a^3*b*(a - b*x^4)) +
(8*a*f + x*(11*b*c - a*g + 2*(5*b*d - a*h)*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*b*d - a*h)*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a
]])/(32*a^(7/2)*b^(3/2))

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Rubi in Sympy [A]  time = 145.162, size = 274, normalized size = 0.94 \[ \frac{x \left (a g + b c + b e x^{2} + b f x^{3} + x \left (a h + b d\right )\right )}{12 a b \left (a - b x^{4}\right )^{3}} + \frac{8 a f - x \left (a g - 11 b c - 9 b e x^{2} + 2 x \left (a h - 5 b d\right )\right )}{96 a^{2} b \left (a - b x^{4}\right )^{2}} - \frac{x \left (7 a g - 77 b c - 45 b e x^{2} + 12 x \left (a h - 5 b d\right )\right )}{384 a^{3} b \left (a - b x^{4}\right )} - \frac{\left (a h - 5 b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{32 a^{\frac{7}{2}} b^{\frac{3}{2}}} - \frac{\left (- 15 \sqrt{a} \sqrt{b} e + 7 a g - 77 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{256 a^{\frac{15}{4}} b^{\frac{5}{4}}} - \frac{\left (15 \sqrt{a} \sqrt{b} e + 7 a g - 77 b c\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{256 a^{\frac{15}{4}} b^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((h*x**5+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*e*x**2 + b*f*x**3 + x*(a*h + b*d))/(12*a*b*(a - b*x**4)**3) + (
8*a*f - x*(a*g - 11*b*c - 9*b*e*x**2 + 2*x*(a*h - 5*b*d)))/(96*a**2*b*(a - b*x**
4)**2) - x*(7*a*g - 77*b*c - 45*b*e*x**2 + 12*x*(a*h - 5*b*d))/(384*a**3*b*(a -
b*x**4)) - (a*h - 5*b*d)*atanh(sqrt(b)*x**2/sqrt(a))/(32*a**(7/2)*b**(3/2)) - (-
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 = 0.764228, size = 360, normalized size = 1.23 \[ \frac{-3 \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (-8 a^{5/4} h+15 \sqrt{a} b^{3/4} e+40 \sqrt [4]{a} b d-7 a \sqrt [4]{b} g+77 b^{5/4} c\right )+3 \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (8 a^{5/4} h+15 \sqrt{a} b^{3/4} e-40 \sqrt [4]{a} b d-7 a \sqrt [4]{b} g+77 b^{5/4} c\right )+\frac{128 a^{11/4} \sqrt{b} (a (f+x (g+h x))+b x (c+x (d+e x)))}{\left (a-b x^4\right )^3}+\frac{16 a^{7/4} \sqrt{b} x (-a (g+2 h x)+11 b c+b x (10 d+9 e x))}{\left (a-b x^4\right )^2}+\frac{4 a^{3/4} \sqrt{b} x \left (-7 a g-12 a h x+77 b c+60 b d x+45 b e x^2\right )}{a-b x^4}+6 \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-15 \sqrt{a} \sqrt{b} e-7 a g+77 b c\right )-24 \sqrt [4]{a} (a h-5 b d) \log \left (\sqrt{a}+\sqrt{b} x^2\right )}{1536 a^{15/4} b^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(a - b*x^4)^4,x]

[Out]

((4*a^(3/4)*Sqrt[b]*x*(77*b*c - 7*a*g + 60*b*d*x - 12*a*h*x + 45*b*e*x^2))/(a -
b*x^4) + (16*a^(7/4)*Sqrt[b]*x*(11*b*c + b*x*(10*d + 9*e*x) - a*(g + 2*h*x)))/(a
- b*x^4)^2 + (128*a^(11/4)*Sqrt[b]*(b*x*(c + x*(d + e*x)) + a*(f + x*(g + h*x))
))/(a - b*x^4)^3 + 6*b^(1/4)*(77*b*c - 15*Sqrt[a]*Sqrt[b]*e - 7*a*g)*ArcTan[(b^(
1/4)*x)/a^(1/4)] - 3*(77*b^(5/4)*c + 40*a^(1/4)*b*d + 15*Sqrt[a]*b^(3/4)*e - 7*a
*b^(1/4)*g - 8*a^(5/4)*h)*Log[a^(1/4) - b^(1/4)*x] + 3*(77*b^(5/4)*c - 40*a^(1/4
)*b*d + 15*Sqrt[a]*b^(3/4)*e - 7*a*b^(1/4)*g + 8*a^(5/4)*h)*Log[a^(1/4) + b^(1/4
)*x] - 24*a^(1/4)*(-5*b*d + a*h)*Log[Sqrt[a] + Sqrt[b]*x^2])/(1536*a^(15/4)*b^(3
/2))

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Maple [A]  time = 0.021, size = 463, normalized size = 1.6 \[{\frac{1}{ \left ( b{x}^{4}-a \right ) ^{3}} \left ( -{\frac{15\,{b}^{2}e{x}^{11}}{128\,{a}^{3}}}+{\frac{ \left ( ah-5\,bd \right ) b{x}^{10}}{32\,{a}^{3}}}+{\frac{ \left ( 7\,ag-77\,bc \right ) b{x}^{9}}{384\,{a}^{3}}}+{\frac{21\,be{x}^{7}}{64\,{a}^{2}}}-{\frac{ \left ( ah-5\,bd \right ){x}^{6}}{12\,{a}^{2}}}-{\frac{ \left ( 3\,ag-33\,bc \right ){x}^{5}}{64\,{a}^{2}}}-{\frac{113\,e{x}^{3}}{384\,a}}-{\frac{ \left ( ah+11\,bd \right ){x}^{2}}{32\,ab}}-{\frac{ \left ( 7\,ag+51\,bc \right ) x}{128\,ab}}-{\frac{f}{12\,b}} \right ) }-{\frac{7\,g}{256\,{a}^{3}b}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }+{\frac{77\,c}{256\,{a}^{4}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }-{\frac{7\,g}{512\,{a}^{3}b}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{77\,c}{512\,{a}^{4}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{ah}{64}\ln \left ({1 \left ( -{a}^{4}b+{x}^{2}\sqrt{{a}^{7}{b}^{3}} \right ) \left ( -{a}^{4}b-{x}^{2}\sqrt{{a}^{7}{b}^{3}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{a}^{7}{b}^{3}}}}}-{\frac{5\,bd}{64}\ln \left ({1 \left ( -{a}^{4}b+{x}^{2}\sqrt{{a}^{7}{b}^{3}} \right ) \left ( -{a}^{4}b-{x}^{2}\sqrt{{a}^{7}{b}^{3}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{a}^{7}{b}^{3}}}}}-{\frac{15\,e}{256\,{a}^{3}b}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{15\,e}{512\,{a}^{3}b}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((h*x^5+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+1/32*(a*h-5*b*d)/a^3*b*x^10+7/384*(a*g-11*b*c)/a^3*b*x^9
+21/64/a^2*b*e*x^7-1/12/a^2*(a*h-5*b*d)*x^6-3/64/a^2*(a*g-11*b*c)*x^5-113/384/a*
e*x^3-1/32*(a*h+11*b*d)/a/b*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*arctan(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)))+1/64/(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)))*a*h-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((h*x^5 + 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((h*x^5 + 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((h*x**5+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 = 0.22365, size = 738, normalized size = 2.52 \[ -\frac{\sqrt{2}{\left (40 \, \sqrt{2} \sqrt{-a b} b^{2} d - 8 \, \sqrt{2} \sqrt{-a b} a b h - 77 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c + 7 \, \left (-a b^{3}\right )^{\frac{1}{4}} a b g - 15 \, \left (-a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{512 \, a^{4} b^{3}} - \frac{\sqrt{2}{\left (40 \, \sqrt{2} \sqrt{-a b} b^{2} d - 8 \, \sqrt{2} \sqrt{-a b} a b h - 77 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c + 7 \, \left (-a b^{3}\right )^{\frac{1}{4}} a b g - 15 \, \left (-a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{512 \, a^{4} b^{3}} + \frac{\sqrt{2}{\left (77 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - 7 \, \left (-a b^{3}\right )^{\frac{1}{4}} a b g - 15 \, \left (-a b^{3}\right )^{\frac{3}{4}} e\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{1024 \, a^{4} b^{3}} - \frac{\sqrt{2}{\left (77 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - 7 \, \left (-a b^{3}\right )^{\frac{1}{4}} a b g - 15 \, \left (-a b^{3}\right )^{\frac{3}{4}} e\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{1024 \, a^{4} b^{3}} - \frac{45 \, b^{3} x^{11} e + 60 \, b^{3} d x^{10} - 12 \, a b^{2} h x^{10} + 77 \, b^{3} c x^{9} - 7 \, a b^{2} g x^{9} - 126 \, a b^{2} x^{7} e - 160 \, a b^{2} d x^{6} + 32 \, a^{2} b h x^{6} - 198 \, a b^{2} c x^{5} + 18 \, a^{2} b g x^{5} + 113 \, a^{2} b x^{3} e + 132 \, a^{2} b d x^{2} + 12 \, a^{3} h x^{2} + 153 \, a^{2} b c x + 21 \, a^{3} g x + 32 \, a^{3} f}{384 \,{\left (b x^{4} - a\right )}^{3} a^{3} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((h*x^5 + 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 - 8*sqrt(2)*sqrt(-a*b)*a*b*h - 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*sq
rt(2)*(2*x + sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/(a^4*b^3) - 1/512*sqrt(2)*(40*s
qrt(2)*sqrt(-a*b)*b^2*d - 8*sqrt(2)*sqrt(-a*b)*a*b*h - 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) + 1/1024*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) - 1/1024*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 - 12*a*b^2*h*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 + 32*a^2*b*h*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 + 12*a^3*h*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)