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

Optimal. Leaf size=134 \[ -\frac{d \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} (b c-a d)^{5/4}}-\frac{d \tanh ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} (b c-a d)^{5/4}}+\frac{b x}{a \sqrt [4]{a+b x^4} (b c-a d)} \]

[Out]

(b*x)/(a*(b*c - a*d)*(a + b*x^4)^(1/4)) - (d*ArcTan[((b*c - a*d)^(1/4)*x)/(c^(1/
4)*(a + b*x^4)^(1/4))])/(2*c^(3/4)*(b*c - a*d)^(5/4)) - (d*ArcTanh[((b*c - a*d)^
(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))])/(2*c^(3/4)*(b*c - a*d)^(5/4))

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Rubi [A]  time = 0.235548, antiderivative size = 134, 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 \[ -\frac{d \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} (b c-a d)^{5/4}}-\frac{d \tanh ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} (b c-a d)^{5/4}}+\frac{b x}{a \sqrt [4]{a+b x^4} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

(b*x)/(a*(b*c - a*d)*(a + b*x^4)^(1/4)) - (d*ArcTan[((b*c - a*d)^(1/4)*x)/(c^(1/
4)*(a + b*x^4)^(1/4))])/(2*c^(3/4)*(b*c - a*d)^(5/4)) - (d*ArcTanh[((b*c - a*d)^
(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))])/(2*c^(3/4)*(b*c - a*d)^(5/4))

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Rubi in Sympy [A]  time = 34.0446, size = 117, normalized size = 0.87 \[ - \frac{d \operatorname{atan}{\left (\frac{x \sqrt [4]{- a d + b c}}{\sqrt [4]{c} \sqrt [4]{a + b x^{4}}} \right )}}{2 c^{\frac{3}{4}} \left (- a d + b c\right )^{\frac{5}{4}}} - \frac{d \operatorname{atanh}{\left (\frac{x \sqrt [4]{- a d + b c}}{\sqrt [4]{c} \sqrt [4]{a + b x^{4}}} \right )}}{2 c^{\frac{3}{4}} \left (- a d + b c\right )^{\frac{5}{4}}} - \frac{b x}{a \sqrt [4]{a + b x^{4}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-d*atan(x*(-a*d + b*c)**(1/4)/(c**(1/4)*(a + b*x**4)**(1/4)))/(2*c**(3/4)*(-a*d
+ b*c)**(5/4)) - d*atanh(x*(-a*d + b*c)**(1/4)/(c**(1/4)*(a + b*x**4)**(1/4)))/(
2*c**(3/4)*(-a*d + b*c)**(5/4)) - b*x/(a*(a + b*x**4)**(1/4)*(a*d - b*c))

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Mathematica [A]  time = 0.367768, size = 153, normalized size = 1.14 \[ \frac{\frac{d \left (-\log \left (\sqrt [4]{c}-\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{a x^4+b}}\right )+\log \left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{a x^4+b}}+\sqrt [4]{c}\right )+2 \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a x^4+b}}\right )\right )}{c^{3/4} \sqrt [4]{b c-a d}}-\frac{4 b x}{a \sqrt [4]{a+b x^4}}}{4 (a d-b c)} \]

Warning: Unable to verify antiderivative.

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

[Out]

((-4*b*x)/(a*(a + b*x^4)^(1/4)) + (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)]))/(c^(3/4)*(b*c - a*d)^(1/4)
))/(4*(-(b*c) + a*d))

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Maple [F]  time = 0.055, size = 0, normalized size = 0. \[ \int{\frac{1}{d{x}^{4}+c} \left ( b{x}^{4}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((b*x^4 + a)^(5/4)*(d*x^4 + c)), 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)),x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/((a + b*x**4)**(5/4)*(c + d*x**4)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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