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

Optimal. Leaf size=278 \[ \frac{x \sqrt [4]{a+b x^2}}{2 c \left (c+d x^2\right )}+\frac{\sqrt{a} \sqrt{b} \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{2 c d \left (a+b x^2\right )^{3/4}}-\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (b c-2 a d) \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c d x (b c-a d)}-\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (b c-2 a d) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c d x (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.604801, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381 \[ \frac{x \sqrt [4]{a+b x^2}}{2 c \left (c+d x^2\right )}+\frac{\sqrt{a} \sqrt{b} \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{2 c d \left (a+b x^2\right )^{3/4}}-\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (b c-2 a d) \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c d x (b c-a d)}-\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (b c-2 a d) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c d x (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 106.984, size = 233, normalized size = 0.84 \[ - \frac{\sqrt [4]{a} \sqrt{- \frac{b x^{2}}{a}} \left (a d - \frac{b c}{2}\right ) \Pi \left (- \frac{\sqrt{a} \sqrt{d}}{\sqrt{a d - b c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{a + b x^{2}}}{\sqrt [4]{a}} \right )}\middle | -1\right )}{2 c d x \left (a d - b c\right )} - \frac{\sqrt [4]{a} \sqrt{- \frac{b x^{2}}{a}} \left (a d - \frac{b c}{2}\right ) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d - b c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{a + b x^{2}}}{\sqrt [4]{a}} \right )}\middle | -1\right )}{2 c d x \left (a d - b c\right )} + \frac{\sqrt{a} \sqrt{b} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{2 c d \left (a + b x^{2}\right )^{\frac{3}{4}}} + \frac{x \sqrt [4]{a + b x^{2}}}{2 c \left (c + d x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.281755, size = 322, normalized size = 1.16 \[ \frac{x \left (-\frac{18 a^2 F_1\left (\frac{1}{2};\frac{3}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{x^2 \left (4 a d F_1\left (\frac{3}{2};\frac{3}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+3 b c F_1\left (\frac{3}{2};\frac{7}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )-6 a c F_1\left (\frac{1}{2};\frac{3}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}-\frac{5 a b x^2 F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{x^2 \left (4 a d F_1\left (\frac{5}{2};\frac{3}{4},2;\frac{7}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+3 b c F_1\left (\frac{5}{2};\frac{7}{4},1;\frac{7}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )-10 a c F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}+\frac{3 \left (a+b x^2\right )}{c}\right )}{6 \left (a+b x^2\right )^{3/4} \left (c+d x^2\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(x*((3*(a + b*x^2))/c - (18*a^2*AppellF1[1/2, 3/4, 1, 3/2, -((b*x^2)/a), -((d*x^
2)/c)])/(-6*a*c*AppellF1[1/2, 3/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + x^2*(4*
a*d*AppellF1[3/2, 3/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + 3*b*c*AppellF1[3/2,
7/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])) - (5*a*b*x^2*AppellF1[3/2, 3/4, 1, 5
/2, -((b*x^2)/a), -((d*x^2)/c)])/(-10*a*c*AppellF1[3/2, 3/4, 1, 5/2, -((b*x^2)/a
), -((d*x^2)/c)] + x^2*(4*a*d*AppellF1[5/2, 3/4, 2, 7/2, -((b*x^2)/a), -((d*x^2)
/c)] + 3*b*c*AppellF1[5/2, 7/4, 1, 7/2, -((b*x^2)/a), -((d*x^2)/c)]))))/(6*(a +
b*x^2)^(3/4)*(c + d*x^2))

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Maple [F]  time = 0.064, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( d{x}^{2}+c \right ) ^{2}}\sqrt [4]{b{x}^{2}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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