3.109 \(\int \frac{1}{\sqrt [4]{a+b x^4} \left (c+d x^4\right )^2} \, dx\)

Optimal. Leaf size=162 \[ \frac{(4 b c-3 a d) \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{5/4}}+\frac{(4 b c-3 a d) \tanh ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{5/4}}-\frac{d x \left (a+b x^4\right )^{3/4}}{4 c \left (c+d x^4\right ) (b c-a d)} \]

[Out]

-(d*x*(a + b*x^4)^(3/4))/(4*c*(b*c - a*d)*(c + d*x^4)) + ((4*b*c - 3*a*d)*ArcTan
[((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))])/(8*c^(7/4)*(b*c - a*d)^(5/4
)) + ((4*b*c - 3*a*d)*ArcTanh[((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))]
)/(8*c^(7/4)*(b*c - a*d)^(5/4))

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Rubi [A]  time = 0.300709, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{(4 b c-3 a d) \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{5/4}}+\frac{(4 b c-3 a d) \tanh ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{5/4}}-\frac{d x \left (a+b x^4\right )^{3/4}}{4 c \left (c+d x^4\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x^4)^(1/4)*(c + d*x^4)^2),x]

[Out]

-(d*x*(a + b*x^4)^(3/4))/(4*c*(b*c - a*d)*(c + d*x^4)) + ((4*b*c - 3*a*d)*ArcTan
[((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))])/(8*c^(7/4)*(b*c - a*d)^(5/4
)) + ((4*b*c - 3*a*d)*ArcTanh[((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))]
)/(8*c^(7/4)*(b*c - a*d)^(5/4))

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Rubi in Sympy [A]  time = 36.6484, size = 141, normalized size = 0.87 \[ \frac{d x \left (a + b x^{4}\right )^{\frac{3}{4}}}{4 c \left (c + d x^{4}\right ) \left (a d - b c\right )} - \frac{\left (3 a d - 4 b c\right ) \operatorname{atan}{\left (\frac{x \sqrt [4]{- a d + b c}}{\sqrt [4]{c} \sqrt [4]{a + b x^{4}}} \right )}}{8 c^{\frac{7}{4}} \left (- a d + b c\right )^{\frac{5}{4}}} - \frac{\left (3 a d - 4 b c\right ) \operatorname{atanh}{\left (\frac{x \sqrt [4]{- a d + b c}}{\sqrt [4]{c} \sqrt [4]{a + b x^{4}}} \right )}}{8 c^{\frac{7}{4}} \left (- a d + b c\right )^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**4+a)**(1/4)/(d*x**4+c)**2,x)

[Out]

d*x*(a + b*x**4)**(3/4)/(4*c*(c + d*x**4)*(a*d - b*c)) - (3*a*d - 4*b*c)*atan(x*
(-a*d + b*c)**(1/4)/(c**(1/4)*(a + b*x**4)**(1/4)))/(8*c**(7/4)*(-a*d + b*c)**(5
/4)) - (3*a*d - 4*b*c)*atanh(x*(-a*d + b*c)**(1/4)/(c**(1/4)*(a + b*x**4)**(1/4)
))/(8*c**(7/4)*(-a*d + b*c)**(5/4))

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Mathematica [A]  time = 0.285242, size = 171, normalized size = 1.06 \[ \frac{(4 b c-3 a d) \left (-\log \left (\sqrt [4]{c}-\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{a x^4+b}}\right )+\log \left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{a x^4+b}}+\sqrt [4]{c}\right )+2 \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a x^4+b}}\right )\right )}{16 c^{7/4} (b c-a d)^{5/4}}-\frac{d x \left (a+b x^4\right )^{3/4}}{4 c \left (c+d x^4\right ) (b c-a d)} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/((a + b*x^4)^(1/4)*(c + d*x^4)^2),x]

[Out]

-(d*x*(a + b*x^4)^(3/4))/(4*c*(b*c - a*d)*(c + d*x^4)) + ((4*b*c - 3*a*d)*(2*Arc
Tan[((b*c - a*d)^(1/4)*x)/(c^(1/4)*(b + a*x^4)^(1/4))] - Log[c^(1/4) - ((b*c - a
*d)^(1/4)*x)/(b + a*x^4)^(1/4)] + Log[c^(1/4) + ((b*c - a*d)^(1/4)*x)/(b + a*x^4
)^(1/4)]))/(16*c^(7/4)*(b*c - a*d)^(5/4))

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( d{x}^{4}+c \right ) ^{2}}{\frac{1}{\sqrt [4]{b{x}^{4}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^4+a)^(1/4)/(d*x^4+c)^2,x)

[Out]

int(1/(b*x^4+a)^(1/4)/(d*x^4+c)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{1}{4}}{\left (d x^{4} + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(1/4)*(d*x^4 + c)^2),x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)^(1/4)*(d*x^4 + c)^2), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(1/4)*(d*x^4 + c)^2),x, algorithm="fricas")

[Out]

Timed out

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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(1/(b*x**4+a)**(1/4)/(d*x**4+c)**2,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{1}{4}}{\left (d x^{4} + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(1/4)*(d*x^4 + c)^2),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(1/4)*(d*x^4 + c)^2), x)