3.102 \(\int \frac{\sqrt [4]{a+b x^4}}{c+d x^4} \, dx\)

Optimal. Leaf size=166 \[ \frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c}+\frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c} \]

[Out]

(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) + (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)

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Rubi [A]  time = 0.29281, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c}+\frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c} \]

Antiderivative was successfully verified.

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

[Out]

(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) + (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)

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Rubi in Sympy [A]  time = 41.3291, size = 143, normalized size = 0.86 \[ \frac{\sqrt{\frac{a}{a + b x^{4}}} \sqrt{a + b x^{4}} \Pi \left (- \frac{\sqrt{- a d + b c}}{\sqrt{b} \sqrt{c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}\middle | -1\right )}{2 \sqrt [4]{b} c} + \frac{\sqrt{\frac{a}{a + b x^{4}}} \sqrt{a + b x^{4}} \Pi \left (\frac{\sqrt{- a d + b c}}{\sqrt{b} \sqrt{c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}\middle | -1\right )}{2 \sqrt [4]{b} c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(a/(a + b*x**4))*sqrt(a + b*x**4)*elliptic_pi(-sqrt(-a*d + b*c)/(sqrt(b)*sqr
t(c)), asin(b**(1/4)*x/(a + b*x**4)**(1/4)), -1)/(2*b**(1/4)*c) + 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)

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Mathematica [C]  time = 0.246518, size = 160, normalized size = 0.96 \[ \frac{5 a c x \sqrt [4]{a+b x^4} F_1\left (\frac{1}{4};-\frac{1}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{\left (c+d x^4\right ) \left (x^4 \left (b c F_1\left (\frac{5}{4};\frac{3}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )-4 a d F_1\left (\frac{5}{4};-\frac{1}{4},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )+5 a c F_1\left (\frac{1}{4};-\frac{1}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.055, size = 0, normalized size = 0. \[ \int{\frac{1}{d{x}^{4}+c}\sqrt [4]{b{x}^{4}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt [4]{a + b x^{4}}}{c + d x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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