3.104 \(\int \frac{1}{\left (a+b x^4\right )^{7/4} \left (c+d x^4\right )} \, dx\)

Optimal. Leaf size=304 \[ -\frac{b^{3/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} (2 b c-5 a d) F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{3 a^{3/2} \left (a+b x^4\right )^{3/4} (b c-a d)^2}+\frac{d^2 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c (b c-a d)^2}+\frac{d^2 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c (b c-a d)^2}+\frac{b x}{3 a \left (a+b x^4\right )^{3/4} (b c-a d)} \]

[Out]

(b*x)/(3*a*(b*c - a*d)*(a + b*x^4)^(3/4)) - (b^(3/2)*(2*b*c - 5*a*d)*(1 + a/(b*x
^4))^(3/4)*x^3*EllipticF[ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(3*a^(3/2)*(b*c -
a*d)^2*(a + b*x^4)^(3/4)) + (d^2*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]
)/(2*b^(1/4)*c*(b*c - a*d)^2) + (d^2*Sqrt[a/(a + b*x^4)]*Sqrt[a + b*x^4]*Ellipti
cPi[Sqrt[b*c - a*d]/(Sqrt[b]*Sqrt[c]), ArcSin[(b^(1/4)*x)/(a + b*x^4)^(1/4)], -1
])/(2*b^(1/4)*c*(b*c - a*d)^2)

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Rubi [A]  time = 0.69783, antiderivative size = 304, 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 \[ -\frac{b^{3/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} (2 b c-5 a d) F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{3 a^{3/2} \left (a+b x^4\right )^{3/4} (b c-a d)^2}+\frac{d^2 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c (b c-a d)^2}+\frac{d^2 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c (b c-a d)^2}+\frac{b x}{3 a \left (a+b x^4\right )^{3/4} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x^4)^(7/4)*(c + d*x^4)),x]

[Out]

(b*x)/(3*a*(b*c - a*d)*(a + b*x^4)^(3/4)) - (b^(3/2)*(2*b*c - 5*a*d)*(1 + a/(b*x
^4))^(3/4)*x^3*EllipticF[ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(3*a^(3/2)*(b*c -
a*d)^2*(a + b*x^4)^(3/4)) + (d^2*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]
)/(2*b^(1/4)*c*(b*c - a*d)^2) + (d^2*Sqrt[a/(a + b*x^4)]*Sqrt[a + b*x^4]*Ellipti
cPi[Sqrt[b*c - a*d]/(Sqrt[b]*Sqrt[c]), ArcSin[(b^(1/4)*x)/(a + b*x^4)^(1/4)], -1
])/(2*b^(1/4)*c*(b*c - a*d)^2)

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Rubi in Sympy [A]  time = 99.6203, size = 265, normalized size = 0.87 \[ \frac{d^{2} \sqrt{\frac{a}{a + b x^{4}}} \sqrt{a + b x^{4}} \Pi \left (- \frac{\sqrt{- a d + b c}}{\sqrt{b} \sqrt{c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}\middle | -1\right )}{2 \sqrt [4]{b} c \left (a d - b c\right )^{2}} + \frac{d^{2} \sqrt{\frac{a}{a + b x^{4}}} \sqrt{a + b x^{4}} \Pi \left (\frac{\sqrt{- a d + b c}}{\sqrt{b} \sqrt{c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}\middle | -1\right )}{2 \sqrt [4]{b} c \left (a d - b c\right )^{2}} - \frac{b x}{3 a \left (a + b x^{4}\right )^{\frac{3}{4}} \left (a d - b c\right )} + \frac{b^{\frac{3}{2}} x^{3} \left (5 a d - 2 b c\right ) \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{3 a^{\frac{3}{2}} \left (a + b x^{4}\right )^{\frac{3}{4}} \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**4+a)**(7/4)/(d*x**4+c),x)

[Out]

d**2*sqrt(a/(a + b*x**4))*sqrt(a + b*x**4)*elliptic_pi(-sqrt(-a*d + b*c)/(sqrt(b
)*sqrt(c)), asin(b**(1/4)*x/(a + b*x**4)**(1/4)), -1)/(2*b**(1/4)*c*(a*d - b*c)*
*2) + d**2*sqrt(a/(a + b*x**4))*sqrt(a + b*x**4)*elliptic_pi(sqrt(-a*d + b*c)/(s
qrt(b)*sqrt(c)), asin(b**(1/4)*x/(a + b*x**4)**(1/4)), -1)/(2*b**(1/4)*c*(a*d -
b*c)**2) - b*x/(3*a*(a + b*x**4)**(3/4)*(a*d - b*c)) + b**(3/2)*x**3*(5*a*d - 2*
b*c)*(a/(b*x**4) + 1)**(3/4)*elliptic_f(atan(sqrt(a)/(sqrt(b)*x**2))/2, 2)/(3*a*
*(3/2)*(a + b*x**4)**(3/4)*(a*d - b*c)**2)

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Mathematica [C]  time = 0.536398, size = 343, normalized size = 1.13 \[ \frac{x \left (\frac{18 b c d x^4 F_1\left (\frac{5}{4};\frac{3}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{\left (c+d x^4\right ) \left (x^4 \left (4 a d F_1\left (\frac{9}{4};\frac{3}{4},2;\frac{13}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+3 b c F_1\left (\frac{9}{4};\frac{7}{4},1;\frac{13}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )-9 a c F_1\left (\frac{5}{4};\frac{3}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )}+\frac{25 c (2 b c-3 a d) F_1\left (\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{\left (c+d x^4\right ) \left (x^4 \left (4 a d F_1\left (\frac{5}{4};\frac{3}{4},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+3 b c F_1\left (\frac{5}{4};\frac{7}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )-5 a c F_1\left (\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )}-\frac{5 b}{a}\right )}{15 \left (a+b x^4\right )^{3/4} (a d-b c)} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/((a + b*x^4)^(7/4)*(c + d*x^4)),x]

[Out]

(x*((-5*b)/a + (25*c*(2*b*c - 3*a*d)*AppellF1[1/4, 3/4, 1, 5/4, -((b*x^4)/a), -(
(d*x^4)/c)])/((c + d*x^4)*(-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*b*c*d*x^4*A
ppellF1[5/4, 3/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])/((c + d*x^4)*(-9*a*c*Appe
llF1[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)])))))/(15*(-(b*c) + a*d)*(a + b*x^4)^(3/4))

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Maple [F]  time = 0.054, size = 0, normalized size = 0. \[ \int{\frac{1}{d{x}^{4}+c} \left ( b{x}^{4}+a \right ) ^{-{\frac{7}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^4+a)^(7/4)/(d*x^4+c),x)

[Out]

int(1/(b*x^4+a)^(7/4)/(d*x^4+c),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{7}{4}}{\left (d x^{4} + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(7/4)*(d*x^4 + c)),x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)^(7/4)*(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)^(7/4)*(d*x^4 + c)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x^{4}\right )^{\frac{7}{4}} \left (c + d x^{4}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x**4+a)**(7/4)/(d*x**4+c),x)

[Out]

Integral(1/((a + b*x**4)**(7/4)*(c + d*x**4)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{7}{4}}{\left (d x^{4} + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(7/4)*(d*x^4 + c)),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(7/4)*(d*x^4 + c)), x)