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

Optimal. Leaf size=162 \[ \frac{77 c \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{15/4} \sqrt [4]{b}}+\frac{77 c \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{15/4} \sqrt [4]{b}}+\frac{5 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} \sqrt{b}}+\frac{x (77 c+60 d x)}{384 a^3 \left (a-b x^4\right )}+\frac{x (11 c+10 d x)}{96 a^2 \left (a-b x^4\right )^2}+\frac{x (c+d x)}{12 a \left (a-b x^4\right )^3} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.292439, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ \frac{77 c \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{15/4} \sqrt [4]{b}}+\frac{77 c \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{15/4} \sqrt [4]{b}}+\frac{5 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} \sqrt{b}}+\frac{x (77 c+60 d x)}{384 a^3 \left (a-b x^4\right )}+\frac{x (11 c+10 d x)}{96 a^2 \left (a-b x^4\right )^2}+\frac{x (c+d x)}{12 a \left (a-b x^4\right )^3} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)/(a - b*x^4)^4,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 52.3235, size = 151, normalized size = 0.93 \[ \frac{x \left (c + d x\right )}{12 a \left (a - b x^{4}\right )^{3}} + \frac{x \left (11 c + 10 d x\right )}{96 a^{2} \left (a - b x^{4}\right )^{2}} + \frac{x \left (77 c + 60 d x\right )}{384 a^{3} \left (a - b x^{4}\right )} + \frac{5 d \operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{32 a^{\frac{7}{2}} \sqrt{b}} + \frac{77 c \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{256 a^{\frac{15}{4}} \sqrt [4]{b}} + \frac{77 c \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{256 a^{\frac{15}{4}} \sqrt [4]{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.431859, size = 217, normalized size = 1.34 \[ \frac{\frac{128 a^3 x (c+d x)}{\left (a-b x^4\right )^3}+\frac{16 a^2 x (11 c+10 d x)}{\left (a-b x^4\right )^2}+\frac{4 a x (77 c+60 d x)}{a-b x^4}-\frac{3 \left (77 \sqrt [4]{a} \sqrt [4]{b} c+40 \sqrt{a} d\right ) \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right )}{\sqrt{b}}+\frac{3 \left (77 \sqrt [4]{a} \sqrt [4]{b} c-40 \sqrt{a} d\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )}{\sqrt{b}}+\frac{462 \sqrt [4]{a} c \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt [4]{b}}+\frac{120 \sqrt{a} d \log \left (\sqrt{a}+\sqrt{b} x^2\right )}{\sqrt{b}}}{1536 a^4} \]

Antiderivative was successfully verified.

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

[Out]

((128*a^3*x*(c + d*x))/(a - b*x^4)^3 + (16*a^2*x*(11*c + 10*d*x))/(a - b*x^4)^2
+ (4*a*x*(77*c + 60*d*x))/(a - b*x^4) + (462*a^(1/4)*c*ArcTan[(b^(1/4)*x)/a^(1/4
)])/b^(1/4) - (3*(77*a^(1/4)*b^(1/4)*c + 40*Sqrt[a]*d)*Log[a^(1/4) - b^(1/4)*x])
/Sqrt[b] + (3*(77*a^(1/4)*b^(1/4)*c - 40*Sqrt[a]*d)*Log[a^(1/4) + b^(1/4)*x])/Sq
rt[b] + (120*Sqrt[a]*d*Log[Sqrt[a] + Sqrt[b]*x^2])/Sqrt[b])/(1536*a^4)

_______________________________________________________________________________________

Maple [A]  time = 0.022, size = 184, normalized size = 1.1 \[{\frac{1}{ \left ( b{x}^{4}-a \right ) ^{3}} \left ( -{\frac{5\,{b}^{2}d{x}^{10}}{32\,{a}^{3}}}-{\frac{77\,{b}^{2}c{x}^{9}}{384\,{a}^{3}}}+{\frac{5\,bd{x}^{6}}{12\,{a}^{2}}}+{\frac{33\,bc{x}^{5}}{64\,{a}^{2}}}-{\frac{11\,d{x}^{2}}{32\,a}}-{\frac{51\,cx}{128\,a}} \right ) }+{\frac{77\,c}{512\,{a}^{4}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{77\,c}{256\,{a}^{4}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }-{\frac{5\,d}{64}\ln \left ({1 \left ( -{a}^{4}+{x}^{2}\sqrt{b{a}^{7}} \right ) \left ( -{a}^{4}-{x}^{2}\sqrt{b{a}^{7}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{b{a}^{7}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)/(-b*x^4+a)^4,x)

[Out]

(-5/32*d/a^3*b^2*x^10-77/384*c/a^3*b^2*x^9+5/12/a^2*d*b*x^6+33/64/a^2*c*b*x^5-11
/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)))+77/256*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)))

_______________________________________________________________________________________

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((d*x + c)/(b*x^4 - a)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

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((d*x + c)/(b*x^4 - a)^4,x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

_______________________________________________________________________________________

Sympy [A]  time = 16.2026, size = 231, normalized size = 1.43 \[ \operatorname{RootSum}{\left (68719476736 t^{4} a^{15} b^{2} - 838860800 t^{2} a^{8} b d^{2} + 485703680 t a^{4} b c^{2} d + 2560000 a d^{4} - 35153041 b c^{4}, \left ( t \mapsto t \log{\left (x + \frac{429496729600 t^{3} a^{12} b d^{2} + 62170071040 t^{2} a^{8} b c^{2} d - 2621440000 t a^{5} d^{4} + 17998356992 t a^{4} b c^{4} + 1897280000 a c^{2} d^{3}}{788480000 a c d^{4} + 2706784157 b c^{5}} \right )} \right )\right )} - \frac{153 a^{2} c x + 132 a^{2} d x^{2} - 198 a b c x^{5} - 160 a b d x^{6} + 77 b^{2} c x^{9} + 60 b^{2} d x^{10}}{- 384 a^{6} + 1152 a^{5} b x^{4} - 1152 a^{4} b^{2} x^{8} + 384 a^{3} b^{3} x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(68719476736*_t**4*a**15*b**2 - 838860800*_t**2*a**8*b*d**2 + 485703680*_
t*a**4*b*c**2*d + 2560000*a*d**4 - 35153041*b*c**4, Lambda(_t, _t*log(x + (42949
6729600*_t**3*a**12*b*d**2 + 62170071040*_t**2*a**8*b*c**2*d - 2621440000*_t*a**
5*d**4 + 17998356992*_t*a**4*b*c**4 + 1897280000*a*c**2*d**3)/(788480000*a*c*d**
4 + 2706784157*b*c**5)))) - (153*a**2*c*x + 132*a**2*d*x**2 - 198*a*b*c*x**5 - 1
60*a*b*d*x**6 + 77*b**2*c*x**9 + 60*b**2*d*x**10)/(-384*a**6 + 1152*a**5*b*x**4
- 1152*a**4*b**2*x**8 + 384*a**3*b**3*x**12)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.223605, size = 400, normalized size = 2.47 \[ \frac{77 \, \sqrt{2} \left (-a b^{3}\right )^{\frac{1}{4}} c{\rm ln}\left (x^{2} + \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{1024 \, a^{4} b} - \frac{77 \, \sqrt{2} \left (-a b^{3}\right )^{\frac{1}{4}} c{\rm ln}\left (x^{2} - \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{1024 \, a^{4} b} - \frac{\sqrt{2}{\left (40 \, \sqrt{2} \sqrt{-a b} b d - 77 \, \left (-a b^{3}\right )^{\frac{1}{4}} b c\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{512 \, a^{4} b^{2}} - \frac{\sqrt{2}{\left (40 \, \sqrt{2} \sqrt{-a b} b d - 77 \, \left (-a b^{3}\right )^{\frac{1}{4}} b c\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{512 \, a^{4} b^{2}} - \frac{60 \, b^{2} d x^{10} + 77 \, b^{2} c x^{9} - 160 \, a b d x^{6} - 198 \, a b c x^{5} + 132 \, a^{2} d x^{2} + 153 \, a^{2} c x}{384 \,{\left (b x^{4} - a\right )}^{3} a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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