3.99 \(\int \frac{1}{\left (a+b x^4\right )^{13/4} \left (c+d x^4\right )} \, dx\)

Optimal. Leaf size=233 \[ \frac{b x (8 b c-17 a d)}{45 a^2 \left (a+b x^4\right )^{5/4} (b c-a d)^2}+\frac{b x \left (113 a^2 d^2-100 a b c d+32 b^2 c^2\right )}{45 a^3 \sqrt [4]{a+b x^4} (b c-a d)^3}-\frac{d^3 \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} (b c-a d)^{13/4}}-\frac{d^3 \tanh ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} (b c-a d)^{13/4}}+\frac{b x}{9 a \left (a+b x^4\right )^{9/4} (b c-a d)} \]

[Out]

(b*x)/(9*a*(b*c - a*d)*(a + b*x^4)^(9/4)) + (b*(8*b*c - 17*a*d)*x)/(45*a^2*(b*c
- a*d)^2*(a + b*x^4)^(5/4)) + (b*(32*b^2*c^2 - 100*a*b*c*d + 113*a^2*d^2)*x)/(45
*a^3*(b*c - a*d)^3*(a + b*x^4)^(1/4)) - (d^3*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)^(13/4)) - (d^3*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)^(13/4))

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Rubi [A]  time = 0.730128, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{b x (8 b c-17 a d)}{45 a^2 \left (a+b x^4\right )^{5/4} (b c-a d)^2}+\frac{b x \left (113 a^2 d^2-100 a b c d+32 b^2 c^2\right )}{45 a^3 \sqrt [4]{a+b x^4} (b c-a d)^3}-\frac{d^3 \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} (b c-a d)^{13/4}}-\frac{d^3 \tanh ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} (b c-a d)^{13/4}}+\frac{b x}{9 a \left (a+b x^4\right )^{9/4} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x^4)^(13/4)*(c + d*x^4)),x]

[Out]

(b*x)/(9*a*(b*c - a*d)*(a + b*x^4)^(9/4)) + (b*(8*b*c - 17*a*d)*x)/(45*a^2*(b*c
- a*d)^2*(a + b*x^4)^(5/4)) + (b*(32*b^2*c^2 - 100*a*b*c*d + 113*a^2*d^2)*x)/(45
*a^3*(b*c - a*d)^3*(a + b*x^4)^(1/4)) - (d^3*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)^(13/4)) - (d^3*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)^(13/4))

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Rubi in Sympy [A]  time = 133.593, size = 212, normalized size = 0.91 \[ - \frac{d^{3} \operatorname{atan}{\left (\frac{x \sqrt [4]{- a d + b c}}{\sqrt [4]{c} \sqrt [4]{a + b x^{4}}} \right )}}{2 c^{\frac{3}{4}} \left (- a d + b c\right )^{\frac{13}{4}}} - \frac{d^{3} \operatorname{atanh}{\left (\frac{x \sqrt [4]{- a d + b c}}{\sqrt [4]{c} \sqrt [4]{a + b x^{4}}} \right )}}{2 c^{\frac{3}{4}} \left (- a d + b c\right )^{\frac{13}{4}}} - \frac{b x}{9 a \left (a + b x^{4}\right )^{\frac{9}{4}} \left (a d - b c\right )} - \frac{b x \left (17 a d - 8 b c\right )}{45 a^{2} \left (a + b x^{4}\right )^{\frac{5}{4}} \left (a d - b c\right )^{2}} - \frac{b x \left (113 a^{2} d^{2} - 100 a b c d + 32 b^{2} c^{2}\right )}{45 a^{3} \sqrt [4]{a + b x^{4}} \left (a d - b c\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**4+a)**(13/4)/(d*x**4+c),x)

[Out]

-d**3*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)**(13/4)) - d**3*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)**(13/4)) - b*x/(9*a*(a + b*x**4)**(9/4)*(a*d - b*
c)) - b*x*(17*a*d - 8*b*c)/(45*a**2*(a + b*x**4)**(5/4)*(a*d - b*c)**2) - b*x*(1
13*a**2*d**2 - 100*a*b*c*d + 32*b**2*c**2)/(45*a**3*(a + b*x**4)**(1/4)*(a*d - b
*c)**3)

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Mathematica [A]  time = 0.683711, size = 231, normalized size = 0.99 \[ \frac{b x \left (\left (a+b x^4\right )^2 \left (113 a^2 d^2-100 a b c d+32 b^2 c^2\right )+5 a^2 (b c-a d)^2+a \left (a+b x^4\right ) (a d-b c) (17 a d-8 b c)\right )}{45 a^3 \left (a+b x^4\right )^{9/4} (b c-a d)^3}-\frac{d^3 \left (-\log \left (\sqrt [4]{c}-\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{a x^4+b}}\right )+\log \left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{a x^4+b}}+\sqrt [4]{c}\right )+2 \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a x^4+b}}\right )\right )}{4 c^{3/4} (b c-a d)^{13/4}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/((a + b*x^4)^(13/4)*(c + d*x^4)),x]

[Out]

(b*x*(5*a^2*(b*c - a*d)^2 + a*(-(b*c) + a*d)*(-8*b*c + 17*a*d)*(a + b*x^4) + (32
*b^2*c^2 - 100*a*b*c*d + 113*a^2*d^2)*(a + b*x^4)^2))/(45*a^3*(b*c - a*d)^3*(a +
b*x^4)^(9/4)) - (d^3*(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)]))/(4*c^(3/4)*(b*c - a*d)^(13/4))

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Maple [F]  time = 0.061, size = 0, normalized size = 0. \[ \int{\frac{1}{d{x}^{4}+c} \left ( b{x}^{4}+a \right ) ^{-{\frac{13}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^4+a)^(13/4)/(d*x^4+c),x)

[Out]

int(1/(b*x^4+a)^(13/4)/(d*x^4+c),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{13}{4}}{\left (d x^{4} + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(13/4)*(d*x^4 + c)),x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)^(13/4)*(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)^(13/4)*(d*x^4 + c)),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)**(13/4)/(d*x**4+c),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{13}{4}}{\left (d x^{4} + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(13/4)*(d*x^4 + c)),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(13/4)*(d*x^4 + c)), x)