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

Optimal. Leaf size=291 \[ -\frac{77 c \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{512 \sqrt{2} a^{15/4} \sqrt [4]{b}}+\frac{77 c \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{512 \sqrt{2} a^{15/4} \sqrt [4]{b}}-\frac{77 c \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt{2} a^{15/4} \sqrt [4]{b}}+\frac{77 c \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{256 \sqrt{2} a^{15/4} \sqrt [4]{b}}+\frac{5 d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} \sqrt{b}}+\frac{x (77 c+60 d x)}{384 a^3 \left (a+b x^4\right )}+\frac{x (11 c+10 d x)}{96 a^2 \left (a+b x^4\right )^2}+\frac{x (c+d x)}{12 a \left (a+b x^4\right )^3} \]

[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)) + (5*d*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]]
)/(32*a^(7/2)*Sqrt[b]) - (77*c*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(256*Sqr
t[2]*a^(15/4)*b^(1/4)) + (77*c*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(256*Sqr
t[2]*a^(15/4)*b^(1/4)) - (77*c*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]
*x^2])/(512*Sqrt[2]*a^(15/4)*b^(1/4)) + (77*c*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1
/4)*x + Sqrt[b]*x^2])/(512*Sqrt[2]*a^(15/4)*b^(1/4))

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Rubi [A]  time = 0.580991, antiderivative size = 291, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 10, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667 \[ -\frac{77 c \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{512 \sqrt{2} a^{15/4} \sqrt [4]{b}}+\frac{77 c \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{512 \sqrt{2} a^{15/4} \sqrt [4]{b}}-\frac{77 c \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt{2} a^{15/4} \sqrt [4]{b}}+\frac{77 c \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{256 \sqrt{2} a^{15/4} \sqrt [4]{b}}+\frac{5 d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} \sqrt{b}}+\frac{x (77 c+60 d x)}{384 a^3 \left (a+b x^4\right )}+\frac{x (11 c+10 d x)}{96 a^2 \left (a+b x^4\right )^2}+\frac{x (c+d x)}{12 a \left (a+b x^4\right )^3} \]

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)) + (5*d*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]]
)/(32*a^(7/2)*Sqrt[b]) - (77*c*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(256*Sqr
t[2]*a^(15/4)*b^(1/4)) + (77*c*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(256*Sqr
t[2]*a^(15/4)*b^(1/4)) - (77*c*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]
*x^2])/(512*Sqrt[2]*a^(15/4)*b^(1/4)) + (77*c*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1
/4)*x + Sqrt[b]*x^2])/(512*Sqrt[2]*a^(15/4)*b^(1/4))

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Rubi in Sympy [A]  time = 98.4271, size = 280, normalized size = 0.96 \[ \frac{x \left (c + d x\right )}{12 a \left (a + b x^{4}\right )^{3}} + \frac{x \left (11 c + 10 d x\right )}{96 a^{2} \left (a + b x^{4}\right )^{2}} + \frac{x \left (77 c + 60 d x\right )}{384 a^{3} \left (a + b x^{4}\right )} + \frac{5 d \operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{32 a^{\frac{7}{2}} \sqrt{b}} - \frac{77 \sqrt{2} c \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x + \sqrt{a} + \sqrt{b} x^{2} \right )}}{1024 a^{\frac{15}{4}} \sqrt [4]{b}} + \frac{77 \sqrt{2} c \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x + \sqrt{a} + \sqrt{b} x^{2} \right )}}{1024 a^{\frac{15}{4}} \sqrt [4]{b}} - \frac{77 \sqrt{2} c \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{512 a^{\frac{15}{4}} \sqrt [4]{b}} + \frac{77 \sqrt{2} c \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{512 a^{\frac{15}{4}} \sqrt [4]{b}} \]

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*atan(sqrt(b)*x**2/sqrt(a))/(3
2*a**(7/2)*sqrt(b)) - 77*sqrt(2)*c*log(-sqrt(2)*a**(1/4)*b**(1/4)*x + sqrt(a) +
sqrt(b)*x**2)/(1024*a**(15/4)*b**(1/4)) + 77*sqrt(2)*c*log(sqrt(2)*a**(1/4)*b**(
1/4)*x + sqrt(a) + sqrt(b)*x**2)/(1024*a**(15/4)*b**(1/4)) - 77*sqrt(2)*c*atan(1
- sqrt(2)*b**(1/4)*x/a**(1/4))/(512*a**(15/4)*b**(1/4)) + 77*sqrt(2)*c*atan(1 +
sqrt(2)*b**(1/4)*x/a**(1/4))/(512*a**(15/4)*b**(1/4))

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Mathematica [A]  time = 0.510721, size = 274, normalized size = 0.94 \[ \frac{\frac{256 a^{11/4} x (c+d x)}{\left (a+b x^4\right )^3}+\frac{32 a^{7/4} x (11 c+10 d x)}{\left (a+b x^4\right )^2}+\frac{8 a^{3/4} x (77 c+60 d x)}{a+b x^4}-\frac{6 \left (80 \sqrt [4]{a} d+77 \sqrt{2} \sqrt [4]{b} c\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt{b}}+\frac{6 \left (77 \sqrt{2} \sqrt [4]{b} c-80 \sqrt [4]{a} d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt{b}}-\frac{231 \sqrt{2} c \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{\sqrt [4]{b}}+\frac{231 \sqrt{2} c \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{\sqrt [4]{b}}}{3072 a^{15/4}} \]

Antiderivative was successfully verified.

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

[Out]

((256*a^(11/4)*x*(c + d*x))/(a + b*x^4)^3 + (32*a^(7/4)*x*(11*c + 10*d*x))/(a +
b*x^4)^2 + (8*a^(3/4)*x*(77*c + 60*d*x))/(a + b*x^4) - (6*(77*Sqrt[2]*b^(1/4)*c
+ 80*a^(1/4)*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/Sqrt[b] + (6*(77*Sqrt[2
]*b^(1/4)*c - 80*a^(1/4)*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/Sqrt[b] - (
231*Sqrt[2]*c*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/b^(1/4) +
(231*Sqrt[2]*c*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/b^(1/4))/
(3072*a^(15/4))

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Maple [A]  time = 0.024, size = 224, normalized size = 0.8 \[{\frac{1}{ \left ( b{x}^{4}+a \right ) ^{3}} \left ({\frac{5\,{b}^{2}d{x}^{10}}{32\,{a}^{3}}}+{\frac{77\,{b}^{2}c{x}^{9}}{384\,{a}^{3}}}+{\frac{5\,bd{x}^{6}}{12\,{a}^{2}}}+{\frac{33\,bc{x}^{5}}{64\,{a}^{2}}}+{\frac{11\,d{x}^{2}}{32\,a}}+{\frac{51\,cx}{128\,a}} \right ) }+{\frac{77\,c\sqrt{2}}{1024\,{a}^{4}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{77\,c\sqrt{2}}{512\,{a}^{4}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{77\,c\sqrt{2}}{512\,{a}^{4}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{5\,d}{32}\arctan \left ({x}^{2}\sqrt{{\frac{b}{a}}} \right ){\frac{1}{\sqrt{b{a}^{7}}}}} \]

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+77/1024*c*(a/b)^(1/4)/a^4*2^(1/2)*ln((x^2+(
a/b)^(1/4)*x*2^(1/2)+(a/b)^(1/2))/(x^2-(a/b)^(1/4)*x*2^(1/2)+(a/b)^(1/2)))+77/51
2*c*(a/b)^(1/4)/a^4*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x+1)+77/512*c*(a/b)^(1/4)
/a^4*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x-1)+5/32*d/(b*a^7)^(1/2)*arctan(x^2*(b/
a)^(1/2))

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

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 16.1953, size = 231, normalized size = 0.79 \[ \operatorname{RootSum}{\left (68719476736 t^{4} a^{15} b^{2} + 838860800 t^{2} a^{8} b d^{2} - 485703680 t a^{4} b c^{2} d + 2560000 a d^{4} + 35153041 b c^{4}, \left ( t \mapsto t \log{\left (x + \frac{- 429496729600 t^{3} a^{12} b d^{2} - 62170071040 t^{2} a^{8} b c^{2} d - 2621440000 t a^{5} d^{4} - 17998356992 t a^{4} b c^{4} + 1897280000 a c^{2} d^{3}}{788480000 a c d^{4} - 2706784157 b c^{5}} \right )} \right )\right )} + \frac{153 a^{2} c x + 132 a^{2} d x^{2} + 198 a b c x^{5} + 160 a b d x^{6} + 77 b^{2} c x^{9} + 60 b^{2} d x^{10}}{384 a^{6} + 1152 a^{5} b x^{4} + 1152 a^{4} b^{2} x^{8} + 384 a^{3} b^{3} x^{12}} \]

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 + (-4294
96729600*_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 +
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)

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GIAC/XCAS [A]  time = 0.219318, size = 378, normalized size = 1.3 \[ \frac{77 \, \sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}} c{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{1024 \, a^{4} b} - \frac{77 \, \sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}} c{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{1024 \, a^{4} b} + \frac{\sqrt{2}{\left (40 \, \sqrt{2} \sqrt{a b} b d + 77 \, \left (a b^{3}\right )^{\frac{1}{4}} b c\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{512 \, a^{4} b^{2}} + \frac{\sqrt{2}{\left (40 \, \sqrt{2} \sqrt{a b} b d + 77 \, \left (a b^{3}\right )^{\frac{1}{4}} b c\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{512 \, a^{4} b^{2}} + \frac{60 \, b^{2} d x^{10} + 77 \, b^{2} c x^{9} + 160 \, a b d x^{6} + 198 \, a b c x^{5} + 132 \, a^{2} d x^{2} + 153 \, a^{2} c x}{384 \,{\left (b x^{4} + a\right )}^{3} a^{3}} \]

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)*arct
an(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 - sq
rt(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)