3.98 \(\int \frac{1}{\left (a+b x^4\right )^{9/4} \left (c+d x^4\right )} \, dx\)

Optimal. Leaf size=180 \[ \frac{b x (4 b c-9 a d)}{5 a^2 \sqrt [4]{a+b x^4} (b c-a d)^2}+\frac{d^2 \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} (b c-a d)^{9/4}}+\frac{d^2 \tanh ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} (b c-a d)^{9/4}}+\frac{b x}{5 a \left (a+b x^4\right )^{5/4} (b c-a d)} \]

[Out]

(b*x)/(5*a*(b*c - a*d)*(a + b*x^4)^(5/4)) + (b*(4*b*c - 9*a*d)*x)/(5*a^2*(b*c -
a*d)^2*(a + b*x^4)^(1/4)) + (d^2*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)^(9/4)) + (d^2*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)^(9/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.489959, antiderivative size = 180, 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 \[ \frac{b x (4 b c-9 a d)}{5 a^2 \sqrt [4]{a+b x^4} (b c-a d)^2}+\frac{d^2 \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} (b c-a d)^{9/4}}+\frac{d^2 \tanh ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} (b c-a d)^{9/4}}+\frac{b x}{5 a \left (a+b x^4\right )^{5/4} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x^4)^(9/4)*(c + d*x^4)),x]

[Out]

(b*x)/(5*a*(b*c - a*d)*(a + b*x^4)^(5/4)) + (b*(4*b*c - 9*a*d)*x)/(5*a^2*(b*c -
a*d)^2*(a + b*x^4)^(1/4)) + (d^2*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)^(9/4)) + (d^2*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)^(9/4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 80.6157, size = 158, normalized size = 0.88 \[ \frac{d^{2} \operatorname{atan}{\left (\frac{x \sqrt [4]{- a d + b c}}{\sqrt [4]{c} \sqrt [4]{a + b x^{4}}} \right )}}{2 c^{\frac{3}{4}} \left (- a d + b c\right )^{\frac{9}{4}}} + \frac{d^{2} \operatorname{atanh}{\left (\frac{x \sqrt [4]{- a d + b c}}{\sqrt [4]{c} \sqrt [4]{a + b x^{4}}} \right )}}{2 c^{\frac{3}{4}} \left (- a d + b c\right )^{\frac{9}{4}}} - \frac{b x}{5 a \left (a + b x^{4}\right )^{\frac{5}{4}} \left (a d - b c\right )} - \frac{b x \left (9 a d - 4 b c\right )}{5 a^{2} \sqrt [4]{a + b x^{4}} \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**4+a)**(9/4)/(d*x**4+c),x)

[Out]

d**2*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)**(9/4)) + d**2*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)**(9/4)) - b*x/(5*a*(a + b*x**4)**(5/4)*(a*d - b*c))
- b*x*(9*a*d - 4*b*c)/(5*a**2*(a + b*x**4)**(1/4)*(a*d - b*c)**2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.563055, size = 184, normalized size = 1.02 \[ \frac{b x \left (\left (a+b x^4\right ) (4 b c-9 a d)+a (b c-a d)\right )}{5 a^2 \left (a+b x^4\right )^{5/4} (b c-a d)^2}+\frac{d^2 \left (-\log \left (\sqrt [4]{c}-\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{a x^4+b}}\right )+\log \left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{a x^4+b}}+\sqrt [4]{c}\right )+2 \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a x^4+b}}\right )\right )}{4 c^{3/4} (b c-a d)^{9/4}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/((a + b*x^4)^(9/4)*(c + d*x^4)),x]

[Out]

(b*x*(a*(b*c - a*d) + (4*b*c - 9*a*d)*(a + b*x^4)))/(5*a^2*(b*c - a*d)^2*(a + b*
x^4)^(5/4)) + (d^2*(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)^(9/4))

_______________________________________________________________________________________

Maple [F]  time = 0.057, size = 0, normalized size = 0. \[ \int{\frac{1}{d{x}^{4}+c} \left ( b{x}^{4}+a \right ) ^{-{\frac{9}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^4+a)^(9/4)/(d*x^4+c),x)

[Out]

int(1/(b*x^4+a)^(9/4)/(d*x^4+c),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{9}{4}}{\left (d x^{4} + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(9/4)*(d*x^4 + c)),x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)^(9/4)*(d*x^4 + c)), x)

_______________________________________________________________________________________

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)^(9/4)*(d*x^4 + c)),x, algorithm="fricas")

[Out]

Timed out

_______________________________________________________________________________________

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)**(9/4)/(d*x**4+c),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{9}{4}}{\left (d x^{4} + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(9/4)*(d*x^4 + c)),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(9/4)*(d*x^4 + c)), x)