3.110 \(\int \frac{1}{\left (a+b x^4\right )^{5/4} \left (c+d x^4\right )^2} \, dx\)

Optimal. Leaf size=205 \[ -\frac{d (8 b c-3 a d) \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{9/4}}-\frac{d (8 b c-3 a d) \tanh ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{9/4}}+\frac{b x (a d+4 b c)}{4 a c \sqrt [4]{a+b x^4} (b c-a d)^2}-\frac{d x}{4 c \sqrt [4]{a+b x^4} \left (c+d x^4\right ) (b c-a d)} \]

[Out]

(b*(4*b*c + a*d)*x)/(4*a*c*(b*c - a*d)^2*(a + b*x^4)^(1/4)) - (d*x)/(4*c*(b*c -
a*d)*(a + b*x^4)^(1/4)*(c + d*x^4)) - (d*(8*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)^(9/4)) - (d*(8*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)^(9/4))

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Rubi [A]  time = 0.495085, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{d (8 b c-3 a d) \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{9/4}}-\frac{d (8 b c-3 a d) \tanh ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{9/4}}+\frac{b x (a d+4 b c)}{4 a c \sqrt [4]{a+b x^4} (b c-a d)^2}-\frac{d x}{4 c \sqrt [4]{a+b x^4} \left (c+d x^4\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x^4)^(5/4)*(c + d*x^4)^2),x]

[Out]

(b*(4*b*c + a*d)*x)/(4*a*c*(b*c - a*d)^2*(a + b*x^4)^(1/4)) - (d*x)/(4*c*(b*c -
a*d)*(a + b*x^4)^(1/4)*(c + d*x^4)) - (d*(8*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)^(9/4)) - (d*(8*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)^(9/4))

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Rubi in Sympy [A]  time = 77.0004, size = 180, normalized size = 0.88 \[ \frac{d x}{4 c \sqrt [4]{a + b x^{4}} \left (c + d x^{4}\right ) \left (a d - b c\right )} + \frac{d \left (3 a d - 8 b c\right ) \operatorname{atan}{\left (\frac{x \sqrt [4]{- a d + b c}}{\sqrt [4]{c} \sqrt [4]{a + b x^{4}}} \right )}}{8 c^{\frac{7}{4}} \left (- a d + b c\right )^{\frac{9}{4}}} + \frac{d \left (3 a d - 8 b c\right ) \operatorname{atanh}{\left (\frac{x \sqrt [4]{- a d + b c}}{\sqrt [4]{c} \sqrt [4]{a + b x^{4}}} \right )}}{8 c^{\frac{7}{4}} \left (- a d + b c\right )^{\frac{9}{4}}} + \frac{b x \left (a d + 4 b c\right )}{4 a c \sqrt [4]{a + b x^{4}} \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**4+a)**(5/4)/(d*x**4+c)**2,x)

[Out]

d*x/(4*c*(a + b*x**4)**(1/4)*(c + d*x**4)*(a*d - b*c)) + d*(3*a*d - 8*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)**
(9/4)) + d*(3*a*d - 8*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)**(9/4)) + b*x*(a*d + 4*b*c)/(4*a*c*(a + b*x**4)*
*(1/4)*(a*d - b*c)**2)

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Mathematica [A]  time = 0.449289, size = 204, normalized size = 1. \[ \frac{x \left (a^2 d^2+a b d^2 x^4+4 b^2 c \left (c+d x^4\right )\right )}{4 a c \sqrt [4]{a+b x^4} \left (c+d x^4\right ) (b c-a d)^2}+\frac{d (3 a d-8 b c) \left (-\log \left (\sqrt [4]{c}-\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{a x^4+b}}\right )+\log \left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{a x^4+b}}+\sqrt [4]{c}\right )+2 \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a x^4+b}}\right )\right )}{16 c^{7/4} (b c-a d)^{9/4}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/((a + b*x^4)^(5/4)*(c + d*x^4)^2),x]

[Out]

(x*(a^2*d^2 + a*b*d^2*x^4 + 4*b^2*c*(c + d*x^4)))/(4*a*c*(b*c - a*d)^2*(a + b*x^
4)^(1/4)*(c + d*x^4)) + (d*(-8*b*c + 3*a*d)*(2*ArcTan[((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)^(9/4))

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Maple [F]  time = 0.059, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( d{x}^{4}+c \right ) ^{2}} \left ( b{x}^{4}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^4+a)^(5/4)/(d*x^4+c)^2,x)

[Out]

int(1/(b*x^4+a)^(5/4)/(d*x^4+c)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{5}{4}}{\left (d x^{4} + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(5/4)*(d*x^4 + c)^2),x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)^(5/4)*(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)^(5/4)*(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)**(5/4)/(d*x**4+c)**2,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{5}{4}}{\left (d x^{4} + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(5/4)*(d*x^4 + c)^2),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(5/4)*(d*x^4 + c)^2), x)