3.117 \(\int \frac{c+d x}{\left (a-b x^4\right )^2} \, dx\)

Optimal. Leaf size=110 \[ \frac{3 c \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{7/4} \sqrt [4]{b}}+\frac{3 c \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{7/4} \sqrt [4]{b}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b}}+\frac{x (c+d x)}{4 a \left (a-b x^4\right )} \]

[Out]

(x*(c + d*x))/(4*a*(a - b*x^4)) + (3*c*ArcTan[(b^(1/4)*x)/a^(1/4)])/(8*a^(7/4)*b
^(1/4)) + (3*c*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(8*a^(7/4)*b^(1/4)) + (d*ArcTanh[(S
qrt[b]*x^2)/Sqrt[a]])/(4*a^(3/2)*Sqrt[b])

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Rubi [A]  time = 0.18865, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ \frac{3 c \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{7/4} \sqrt [4]{b}}+\frac{3 c \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{7/4} \sqrt [4]{b}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b}}+\frac{x (c+d x)}{4 a \left (a-b x^4\right )} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)/(a - b*x^4)^2,x]

[Out]

(x*(c + d*x))/(4*a*(a - b*x^4)) + (3*c*ArcTan[(b^(1/4)*x)/a^(1/4)])/(8*a^(7/4)*b
^(1/4)) + (3*c*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(8*a^(7/4)*b^(1/4)) + (d*ArcTanh[(S
qrt[b]*x^2)/Sqrt[a]])/(4*a^(3/2)*Sqrt[b])

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Rubi in Sympy [A]  time = 32.9588, size = 102, normalized size = 0.93 \[ \frac{x \left (c + d x\right )}{4 a \left (a - b x^{4}\right )} + \frac{d \operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{4 a^{\frac{3}{2}} \sqrt{b}} + \frac{3 c \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{8 a^{\frac{7}{4}} \sqrt [4]{b}} + \frac{3 c \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{8 a^{\frac{7}{4}} \sqrt [4]{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(c + d*x)/(4*a*(a - b*x**4)) + d*atanh(sqrt(b)*x**2/sqrt(a))/(4*a**(3/2)*sqrt(
b)) + 3*c*atan(b**(1/4)*x/a**(1/4))/(8*a**(7/4)*b**(1/4)) + 3*c*atanh(b**(1/4)*x
/a**(1/4))/(8*a**(7/4)*b**(1/4))

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Mathematica [A]  time = 0.343325, size = 168, normalized size = 1.53 \[ \frac{\frac{4 a x (c+d x)}{a-b x^4}-\frac{\left (3 \sqrt [4]{a} \sqrt [4]{b} c+2 \sqrt{a} d\right ) \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right )}{\sqrt{b}}+\frac{\left (3 \sqrt [4]{a} \sqrt [4]{b} c-2 \sqrt{a} d\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )}{\sqrt{b}}+\frac{6 \sqrt [4]{a} c \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt [4]{b}}+\frac{2 \sqrt{a} d \log \left (\sqrt{a}+\sqrt{b} x^2\right )}{\sqrt{b}}}{16 a^2} \]

Antiderivative was successfully verified.

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

[Out]

((4*a*x*(c + d*x))/(a - b*x^4) + (6*a^(1/4)*c*ArcTan[(b^(1/4)*x)/a^(1/4)])/b^(1/
4) - ((3*a^(1/4)*b^(1/4)*c + 2*Sqrt[a]*d)*Log[a^(1/4) - b^(1/4)*x])/Sqrt[b] + ((
3*a^(1/4)*b^(1/4)*c - 2*Sqrt[a]*d)*Log[a^(1/4) + b^(1/4)*x])/Sqrt[b] + (2*Sqrt[a
]*d*Log[Sqrt[a] + Sqrt[b]*x^2])/Sqrt[b])/(16*a^2)

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Maple [A]  time = 0.007, size = 142, normalized size = 1.3 \[ -{\frac{cx}{4\,a \left ( b{x}^{4}-a \right ) }}+{\frac{3\,c}{16\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{3\,c}{8\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }-{\frac{d{x}^{2}}{4\,a \left ( b{x}^{4}-a \right ) }}-{\frac{d}{8\,a}\ln \left ({1 \left ( -a+{x}^{2}\sqrt{ab} \right ) \left ( -a-{x}^{2}\sqrt{ab} \right ) ^{-1}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)/(-b*x^4+a)^2,x)

[Out]

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

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 2.41347, size = 155, normalized size = 1.41 \[ \operatorname{RootSum}{\left (65536 t^{4} a^{7} b^{2} - 2048 t^{2} a^{4} b d^{2} + 1152 t a^{2} b c^{2} d + 16 a d^{4} - 81 b c^{4}, \left ( t \mapsto t \log{\left (x + \frac{32768 t^{3} a^{6} b d^{2} + 4608 t^{2} a^{4} b c^{2} d - 512 t a^{3} d^{4} + 1296 t a^{2} b c^{4} + 360 a c^{2} d^{3}}{192 a c d^{4} + 243 b c^{5}} \right )} \right )\right )} - \frac{c x + d x^{2}}{- 4 a^{2} + 4 a b x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(65536*_t**4*a**7*b**2 - 2048*_t**2*a**4*b*d**2 + 1152*_t*a**2*b*c**2*d +
16*a*d**4 - 81*b*c**4, Lambda(_t, _t*log(x + (32768*_t**3*a**6*b*d**2 + 4608*_t
**2*a**4*b*c**2*d - 512*_t*a**3*d**4 + 1296*_t*a**2*b*c**4 + 360*a*c**2*d**3)/(1
92*a*c*d**4 + 243*b*c**5)))) - (c*x + d*x**2)/(-4*a**2 + 4*a*b*x**4)

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GIAC/XCAS [A]  time = 0.221915, size = 343, normalized size = 3.12 \[ \frac{3 \, \sqrt{2} \left (-a b^{3}\right )^{\frac{1}{4}} c{\rm ln}\left (x^{2} + \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{32 \, a^{2} b} - \frac{3 \, \sqrt{2} \left (-a b^{3}\right )^{\frac{1}{4}} c{\rm ln}\left (x^{2} - \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{32 \, a^{2} b} - \frac{d x^{2} + c x}{4 \,{\left (b x^{4} - a\right )} a} - \frac{\sqrt{2}{\left (2 \, \sqrt{2} \sqrt{-a b} b d - 3 \, \left (-a b^{3}\right )^{\frac{1}{4}} b c\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{16 \, a^{2} b^{2}} - \frac{\sqrt{2}{\left (2 \, \sqrt{2} \sqrt{-a b} b d - 3 \, \left (-a b^{3}\right )^{\frac{1}{4}} b c\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{16 \, a^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)/(b*x^4 - a)^2,x, algorithm="giac")

[Out]

3/32*sqrt(2)*(-a*b^3)^(1/4)*c*ln(x^2 + sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/(a^2
*b) - 3/32*sqrt(2)*(-a*b^3)^(1/4)*c*ln(x^2 - sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b)
)/(a^2*b) - 1/4*(d*x^2 + c*x)/((b*x^4 - a)*a) - 1/16*sqrt(2)*(2*sqrt(2)*sqrt(-a*
b)*b*d - 3*(-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^2*b^2) - 1/16*sqrt(2)*(2*sqrt(2)*sqrt(-a*b)*b*d - 3*(-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^2*b^2)