3.120 \(\int \frac{c+d x}{\left (a+b x^4\right )^3} \, dx\)

Optimal. Leaf size=266 \[ -\frac{21 c \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{11/4} \sqrt [4]{b}}+\frac{21 c \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{11/4} \sqrt [4]{b}}-\frac{21 c \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{11/4} \sqrt [4]{b}}+\frac{21 c \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} a^{11/4} \sqrt [4]{b}}+\frac{3 d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{b}}+\frac{x (7 c+6 d x)}{32 a^2 \left (a+b x^4\right )}+\frac{x (c+d x)}{8 a \left (a+b x^4\right )^2} \]

[Out]

(x*(c + d*x))/(8*a*(a + b*x^4)^2) + (x*(7*c + 6*d*x))/(32*a^2*(a + b*x^4)) + (3*
d*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(16*a^(5/2)*Sqrt[b]) - (21*c*ArcTan[1 - (Sqrt[2
]*b^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(11/4)*b^(1/4)) + (21*c*ArcTan[1 + (Sqrt[2]
*b^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(11/4)*b^(1/4)) - (21*c*Log[Sqrt[a] - Sqrt[2
]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(128*Sqrt[2]*a^(11/4)*b^(1/4)) + (21*c*Log[S
qrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(128*Sqrt[2]*a^(11/4)*b^(1/4)
)

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Rubi [A]  time = 0.506791, antiderivative size = 266, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667 \[ -\frac{21 c \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{11/4} \sqrt [4]{b}}+\frac{21 c \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{11/4} \sqrt [4]{b}}-\frac{21 c \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{11/4} \sqrt [4]{b}}+\frac{21 c \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} a^{11/4} \sqrt [4]{b}}+\frac{3 d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{b}}+\frac{x (7 c+6 d x)}{32 a^2 \left (a+b x^4\right )}+\frac{x (c+d x)}{8 a \left (a+b x^4\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)/(a + b*x^4)^3,x]

[Out]

(x*(c + d*x))/(8*a*(a + b*x^4)^2) + (x*(7*c + 6*d*x))/(32*a^2*(a + b*x^4)) + (3*
d*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(16*a^(5/2)*Sqrt[b]) - (21*c*ArcTan[1 - (Sqrt[2
]*b^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(11/4)*b^(1/4)) + (21*c*ArcTan[1 + (Sqrt[2]
*b^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(11/4)*b^(1/4)) - (21*c*Log[Sqrt[a] - Sqrt[2
]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(128*Sqrt[2]*a^(11/4)*b^(1/4)) + (21*c*Log[S
qrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(128*Sqrt[2]*a^(11/4)*b^(1/4)
)

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Rubi in Sympy [A]  time = 86.4273, size = 257, normalized size = 0.97 \[ \frac{x \left (c + d x\right )}{8 a \left (a + b x^{4}\right )^{2}} + \frac{x \left (7 c + 6 d x\right )}{32 a^{2} \left (a + b x^{4}\right )} + \frac{3 d \operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{16 a^{\frac{5}{2}} \sqrt{b}} - \frac{21 \sqrt{2} c \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x + \sqrt{a} + \sqrt{b} x^{2} \right )}}{256 a^{\frac{11}{4}} \sqrt [4]{b}} + \frac{21 \sqrt{2} c \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x + \sqrt{a} + \sqrt{b} x^{2} \right )}}{256 a^{\frac{11}{4}} \sqrt [4]{b}} - \frac{21 \sqrt{2} c \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{128 a^{\frac{11}{4}} \sqrt [4]{b}} + \frac{21 \sqrt{2} c \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{128 a^{\frac{11}{4}} \sqrt [4]{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)/(b*x**4+a)**3,x)

[Out]

x*(c + d*x)/(8*a*(a + b*x**4)**2) + x*(7*c + 6*d*x)/(32*a**2*(a + b*x**4)) + 3*d
*atan(sqrt(b)*x**2/sqrt(a))/(16*a**(5/2)*sqrt(b)) - 21*sqrt(2)*c*log(-sqrt(2)*a*
*(1/4)*b**(1/4)*x + sqrt(a) + sqrt(b)*x**2)/(256*a**(11/4)*b**(1/4)) + 21*sqrt(2
)*c*log(sqrt(2)*a**(1/4)*b**(1/4)*x + sqrt(a) + sqrt(b)*x**2)/(256*a**(11/4)*b**
(1/4)) - 21*sqrt(2)*c*atan(1 - sqrt(2)*b**(1/4)*x/a**(1/4))/(128*a**(11/4)*b**(1
/4)) + 21*sqrt(2)*c*atan(1 + sqrt(2)*b**(1/4)*x/a**(1/4))/(128*a**(11/4)*b**(1/4
))

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Mathematica [A]  time = 0.410365, size = 249, normalized size = 0.94 \[ \frac{\frac{32 a^{7/4} x (c+d x)}{\left (a+b x^4\right )^2}+\frac{8 a^{3/4} x (7 c+6 d x)}{a+b x^4}-\frac{6 \left (8 \sqrt [4]{a} d+7 \sqrt{2} \sqrt [4]{b} c\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt{b}}+\frac{6 \left (7 \sqrt{2} \sqrt [4]{b} c-8 \sqrt [4]{a} d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt{b}}-\frac{21 \sqrt{2} c \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{\sqrt [4]{b}}+\frac{21 \sqrt{2} c \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{\sqrt [4]{b}}}{256 a^{11/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)/(a + b*x^4)^3,x]

[Out]

((32*a^(7/4)*x*(c + d*x))/(a + b*x^4)^2 + (8*a^(3/4)*x*(7*c + 6*d*x))/(a + b*x^4
) - (6*(7*Sqrt[2]*b^(1/4)*c + 8*a^(1/4)*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4
)])/Sqrt[b] + (6*(7*Sqrt[2]*b^(1/4)*c - 8*a^(1/4)*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)
*x)/a^(1/4)])/Sqrt[b] - (21*Sqrt[2]*c*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x +
Sqrt[b]*x^2])/b^(1/4) + (21*Sqrt[2]*c*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x +
Sqrt[b]*x^2])/b^(1/4))/(256*a^(11/4))

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Maple [A]  time = 0.008, size = 222, normalized size = 0.8 \[{\frac{cx}{8\,a \left ( b{x}^{4}+a \right ) ^{2}}}+{\frac{7\,cx}{32\,{a}^{2} \left ( b{x}^{4}+a \right ) }}+{\frac{21\,c\sqrt{2}}{256\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{21\,c\sqrt{2}}{128\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{21\,c\sqrt{2}}{128\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{d{x}^{2}}{8\,a \left ( b{x}^{4}+a \right ) ^{2}}}+{\frac{3\,d{x}^{2}}{16\,{a}^{2} \left ( b{x}^{4}+a \right ) }}+{\frac{3\,d}{16\,{a}^{2}}\arctan \left ({x}^{2}\sqrt{{\frac{b}{a}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)/(b*x^4+a)^3,x)

[Out]

1/8*c*x/a/(b*x^4+a)^2+7/32*c/a^2*x/(b*x^4+a)+21/256*c/a^3*(a/b)^(1/4)*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))
)+21/128*c/a^3*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x+1)+21/128*c/a^3*
(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x-1)+1/8*d*x^2/a/(b*x^4+a)^2+3/16
*d/a^2*x^2/(b*x^4+a)+3/16*d/a^2/(a*b)^(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)^3,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)^3,x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 4.26566, size = 192, normalized size = 0.72 \[ \operatorname{RootSum}{\left (268435456 t^{4} a^{11} b^{2} + 4718592 t^{2} a^{6} b d^{2} - 2709504 t a^{3} b c^{2} d + 20736 a d^{4} + 194481 b c^{4}, \left ( t \mapsto t \log{\left (x + \frac{- 67108864 t^{3} a^{9} b d^{2} - 9633792 t^{2} a^{6} b c^{2} d - 589824 t a^{4} d^{4} - 2765952 t a^{3} b c^{4} + 423360 a c^{2} d^{3}}{193536 a c d^{4} - 453789 b c^{5}} \right )} \right )\right )} + \frac{11 a c x + 10 a d x^{2} + 7 b c x^{5} + 6 b d x^{6}}{32 a^{4} + 64 a^{3} b x^{4} + 32 a^{2} b^{2} x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)/(b*x**4+a)**3,x)

[Out]

RootSum(268435456*_t**4*a**11*b**2 + 4718592*_t**2*a**6*b*d**2 - 2709504*_t*a**3
*b*c**2*d + 20736*a*d**4 + 194481*b*c**4, Lambda(_t, _t*log(x + (-67108864*_t**3
*a**9*b*d**2 - 9633792*_t**2*a**6*b*c**2*d - 589824*_t*a**4*d**4 - 2765952*_t*a*
*3*b*c**4 + 423360*a*c**2*d**3)/(193536*a*c*d**4 - 453789*b*c**5)))) + (11*a*c*x
+ 10*a*d*x**2 + 7*b*c*x**5 + 6*b*d*x**6)/(32*a**4 + 64*a**3*b*x**4 + 32*a**2*b*
*2*x**8)

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GIAC/XCAS [A]  time = 0.221637, size = 346, normalized size = 1.3 \[ \frac{21 \, \sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}} c{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{256 \, a^{3} b} - \frac{21 \, \sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}} c{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{256 \, a^{3} b} + \frac{3 \, \sqrt{2}{\left (4 \, \sqrt{2} \sqrt{a b} b d + 7 \, \left (a b^{3}\right )^{\frac{1}{4}} b c\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{128 \, a^{3} b^{2}} + \frac{3 \, \sqrt{2}{\left (4 \, \sqrt{2} \sqrt{a b} b d + 7 \, \left (a b^{3}\right )^{\frac{1}{4}} b c\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{128 \, a^{3} b^{2}} + \frac{6 \, b d x^{6} + 7 \, b c x^{5} + 10 \, a d x^{2} + 11 \, a c x}{32 \,{\left (b x^{4} + a\right )}^{2} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)/(b*x^4 + a)^3,x, algorithm="giac")

[Out]

21/256*sqrt(2)*(a*b^3)^(1/4)*c*ln(x^2 + sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^3*
b) - 21/256*sqrt(2)*(a*b^3)^(1/4)*c*ln(x^2 - sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/
(a^3*b) + 3/128*sqrt(2)*(4*sqrt(2)*sqrt(a*b)*b*d + 7*(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^3*b^2) + 3/128*sqrt(2)*(4
*sqrt(2)*sqrt(a*b)*b*d + 7*(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^3*b^2) + 1/32*(6*b*d*x^6 + 7*b*c*x^5 + 10*a*d*x^2 +
11*a*c*x)/((b*x^4 + a)^2*a^2)