3.321 \(\int \frac{\sqrt [4]{a+b x^2}}{c+d x^2} \, dx\)

Optimal. Leaf size=199 \[ -\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{d x}-\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{d x}+\frac{2 \sqrt{a} \sqrt{b} \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{d \left (a+b x^2\right )^{3/4}} \]

[Out]

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

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Rubi [A]  time = 0.445002, antiderivative size = 199, 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 \[ -\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{d x}-\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{d x}+\frac{2 \sqrt{a} \sqrt{b} \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{d \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 82.7589, size = 172, normalized size = 0.86 \[ - \frac{\sqrt [4]{a} \sqrt{- \frac{b x^{2}}{a}} \Pi \left (- \frac{\sqrt{a} \sqrt{d}}{\sqrt{a d - b c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{a + b x^{2}}}{\sqrt [4]{a}} \right )}\middle | -1\right )}{d x} - \frac{\sqrt [4]{a} \sqrt{- \frac{b x^{2}}{a}} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d - b c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{a + b x^{2}}}{\sqrt [4]{a}} \right )}\middle | -1\right )}{d x} + \frac{2 \sqrt{a} \sqrt{b} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{d \left (a + b x^{2}\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.250046, size = 160, normalized size = 0.8 \[ \frac{6 a c x \sqrt [4]{a+b x^2} F_1\left (\frac{1}{2};-\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{\left (c+d x^2\right ) \left (x^2 \left (b c F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )-4 a d F_1\left (\frac{3}{2};-\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )+6 a c F_1\left (\frac{1}{2};-\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(6*a*c*x*(a + b*x^2)^(1/4)*AppellF1[1/2, -1/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c
)])/((c + d*x^2)*(6*a*c*AppellF1[1/2, -1/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)]
+ x^2*(-4*a*d*AppellF1[3/2, -1/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + b*c*Appe
llF1[3/2, 3/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))

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Maple [F]  time = 0.075, size = 0, normalized size = 0. \[ \int{\frac{1}{d{x}^{2}+c}\sqrt [4]{b{x}^{2}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^2 + a)^(1/4)/(d*x^2 + 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^2 + a)^(1/4)/(d*x^2 + c),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt [4]{a + b x^{2}}}{c + d x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a + b*x**2)**(1/4)/(c + d*x**2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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