3.115 \(\int \frac{1}{\left (a+b x^4\right )^{3/4} \left (c+d x^4\right )^2} \, dx\)

Optimal. Leaf size=330 \[ -\frac{b^{3/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} (4 b c-a d) F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{4 \sqrt{a} c \left (a+b x^4\right )^{3/4} (b c-a d)^2}-\frac{3 d \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (2 b c-a d) \Pi \left (-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 (b c-a d)^2}-\frac{3 d \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (2 b c-a d) \Pi \left (\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 (b c-a d)^2}-\frac{d x \sqrt [4]{a+b x^4}}{4 c \left (c+d x^4\right ) (b c-a d)} \]

[Out]

-(d*x*(a + b*x^4)^(1/4))/(4*c*(b*c - a*d)*(c + d*x^4)) - (b^(3/2)*(4*b*c - a*d)*
(1 + a/(b*x^4))^(3/4)*x^3*EllipticF[ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(4*Sqrt
[a]*c*(b*c - a*d)^2*(a + b*x^4)^(3/4)) - (3*d*(2*b*c - a*d)*Sqrt[a/(a + b*x^4)]*
Sqrt[a + b*x^4]*EllipticPi[-(Sqrt[b*c - a*d]/(Sqrt[b]*Sqrt[c])), ArcSin[(b^(1/4)
*x)/(a + b*x^4)^(1/4)], -1])/(8*b^(1/4)*c^2*(b*c - a*d)^2) - (3*d*(2*b*c - a*d)*
Sqrt[a/(a + b*x^4)]*Sqrt[a + b*x^4]*EllipticPi[Sqrt[b*c - a*d]/(Sqrt[b]*Sqrt[c])
, ArcSin[(b^(1/4)*x)/(a + b*x^4)^(1/4)], -1])/(8*b^(1/4)*c^2*(b*c - a*d)^2)

_______________________________________________________________________________________

Rubi [A]  time = 0.752525, antiderivative size = 330, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429 \[ -\frac{b^{3/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} (4 b c-a d) F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{4 \sqrt{a} c \left (a+b x^4\right )^{3/4} (b c-a d)^2}-\frac{3 d \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (2 b c-a d) \Pi \left (-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 (b c-a d)^2}-\frac{3 d \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (2 b c-a d) \Pi \left (\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 (b c-a d)^2}-\frac{d x \sqrt [4]{a+b x^4}}{4 c \left (c+d x^4\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x^4)^(3/4)*(c + d*x^4)^2),x]

[Out]

-(d*x*(a + b*x^4)^(1/4))/(4*c*(b*c - a*d)*(c + d*x^4)) - (b^(3/2)*(4*b*c - a*d)*
(1 + a/(b*x^4))^(3/4)*x^3*EllipticF[ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(4*Sqrt
[a]*c*(b*c - a*d)^2*(a + b*x^4)^(3/4)) - (3*d*(2*b*c - a*d)*Sqrt[a/(a + b*x^4)]*
Sqrt[a + b*x^4]*EllipticPi[-(Sqrt[b*c - a*d]/(Sqrt[b]*Sqrt[c])), ArcSin[(b^(1/4)
*x)/(a + b*x^4)^(1/4)], -1])/(8*b^(1/4)*c^2*(b*c - a*d)^2) - (3*d*(2*b*c - a*d)*
Sqrt[a/(a + b*x^4)]*Sqrt[a + b*x^4]*EllipticPi[Sqrt[b*c - a*d]/(Sqrt[b]*Sqrt[c])
, ArcSin[(b^(1/4)*x)/(a + b*x^4)^(1/4)], -1])/(8*b^(1/4)*c^2*(b*c - a*d)^2)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 96.9919, size = 292, normalized size = 0.88 \[ \frac{d x \sqrt [4]{a + b x^{4}}}{4 c \left (c + d x^{4}\right ) \left (a d - b c\right )} + \frac{3 d \sqrt{\frac{a}{a + b x^{4}}} \sqrt{a + b x^{4}} \left (a d - 2 b c\right ) \Pi \left (- \frac{\sqrt{- a d + b c}}{\sqrt{b} \sqrt{c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}\middle | -1\right )}{8 \sqrt [4]{b} c^{2} \left (a d - b c\right )^{2}} + \frac{3 d \sqrt{\frac{a}{a + b x^{4}}} \sqrt{a + b x^{4}} \left (a d - 2 b c\right ) \Pi \left (\frac{\sqrt{- a d + b c}}{\sqrt{b} \sqrt{c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}\middle | -1\right )}{8 \sqrt [4]{b} c^{2} \left (a d - b c\right )^{2}} + \frac{b^{\frac{3}{2}} x^{3} \left (a d - 4 b c\right ) \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{4 \sqrt{a} c \left (a + b x^{4}\right )^{\frac{3}{4}} \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**4+a)**(3/4)/(d*x**4+c)**2,x)

[Out]

d*x*(a + b*x**4)**(1/4)/(4*c*(c + d*x**4)*(a*d - b*c)) + 3*d*sqrt(a/(a + b*x**4)
)*sqrt(a + b*x**4)*(a*d - 2*b*c)*elliptic_pi(-sqrt(-a*d + b*c)/(sqrt(b)*sqrt(c))
, asin(b**(1/4)*x/(a + b*x**4)**(1/4)), -1)/(8*b**(1/4)*c**2*(a*d - b*c)**2) + 3
*d*sqrt(a/(a + b*x**4))*sqrt(a + b*x**4)*(a*d - 2*b*c)*elliptic_pi(sqrt(-a*d + b
*c)/(sqrt(b)*sqrt(c)), asin(b**(1/4)*x/(a + b*x**4)**(1/4)), -1)/(8*b**(1/4)*c**
2*(a*d - b*c)**2) + b**(3/2)*x**3*(a*d - 4*b*c)*(a/(b*x**4) + 1)**(3/4)*elliptic
_f(atan(sqrt(a)/(sqrt(b)*x**2))/2, 2)/(4*sqrt(a)*c*(a + b*x**4)**(3/4)*(a*d - b*
c)**2)

_______________________________________________________________________________________

Mathematica [C]  time = 0.890998, size = 341, normalized size = 1.03 \[ \frac{x \left (\frac{18 a b d x^4 F_1\left (\frac{5}{4};\frac{3}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{x^4 \left (4 a d F_1\left (\frac{9}{4};\frac{3}{4},2;\frac{13}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+3 b c F_1\left (\frac{9}{4};\frac{7}{4},1;\frac{13}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )-9 a c F_1\left (\frac{5}{4};\frac{3}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}+\frac{25 a (3 a d-4 b c) F_1\left (\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{x^4 \left (4 a d F_1\left (\frac{5}{4};\frac{3}{4},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+3 b c F_1\left (\frac{5}{4};\frac{7}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )-5 a c F_1\left (\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}-\frac{5 d \left (a+b x^4\right )}{c}\right )}{20 \left (a+b x^4\right )^{3/4} \left (c+d x^4\right ) (b c-a d)} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/((a + b*x^4)^(3/4)*(c + d*x^4)^2),x]

[Out]

(x*((-5*d*(a + b*x^4))/c + (25*a*(-4*b*c + 3*a*d)*AppellF1[1/4, 3/4, 1, 5/4, -((
b*x^4)/a), -((d*x^4)/c)])/(-5*a*c*AppellF1[1/4, 3/4, 1, 5/4, -((b*x^4)/a), -((d*
x^4)/c)] + x^4*(4*a*d*AppellF1[5/4, 3/4, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + 3
*b*c*AppellF1[5/4, 7/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])) + (18*a*b*d*x^4*Ap
pellF1[5/4, 3/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])/(-9*a*c*AppellF1[5/4, 3/4,
1, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + x^4*(4*a*d*AppellF1[9/4, 3/4, 2, 13/4, -(
(b*x^4)/a), -((d*x^4)/c)] + 3*b*c*AppellF1[9/4, 7/4, 1, 13/4, -((b*x^4)/a), -((d
*x^4)/c)]))))/(20*(b*c - a*d)*(a + b*x^4)^(3/4)*(c + d*x^4))

_______________________________________________________________________________________

Maple [F]  time = 0.059, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( d{x}^{4}+c \right ) ^{2}} \left ( b{x}^{4}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^4+a)^(3/4)/(d*x^4+c)^2,x)

[Out]

int(1/(b*x^4+a)^(3/4)/(d*x^4+c)^2,x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{3}{4}}{\left (d x^{4} + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(3/4)*(d*x^4 + c)^2),x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)^(3/4)*(d*x^4 + c)^2), 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)^(3/4)*(d*x^4 + c)^2),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)**(3/4)/(d*x**4+c)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{3}{4}}{\left (d x^{4} + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(3/4)*(d*x^4 + c)^2),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(3/4)*(d*x^4 + c)^2), x)