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

Optimal. Leaf size=969 \[ \text{result too large to display} \]

[Out]

(b*x)/(2*a*(b*c - a*d)*Sqrt[a + b*x^4]) - (d*ArcTan[(Sqrt[-((Sqrt[-c]*(b - (a*d)
/c))/Sqrt[d])]*x)/Sqrt[a + b*x^4]])/(4*c*(b*c - a*d)*Sqrt[(b*c - a*d)/(Sqrt[-c]*
Sqrt[d])]) - (d*ArcTan[(Sqrt[(Sqrt[-c]*(b - (a*d)/c))/Sqrt[d]]*x)/Sqrt[a + b*x^4
]])/(4*c*(b*c - a*d)*Sqrt[-((b*c - a*d)/(Sqrt[-c]*Sqrt[d]))]) + (b^(3/4)*(Sqrt[a
] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2*ArcTan[
(b^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(5/4)*(b*c - a*d)*Sqrt[a + b*x^4]) - (b^(1/4)*d
*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2
*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(1/4)*(Sqrt[b]*c - Sqrt[a]*Sqrt[-c]*Sqr
t[d])*(b*c - a*d)*Sqrt[a + b*x^4]) - (b^(1/4)*d*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a
+ b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2
])/(4*a^(1/4)*(Sqrt[b]*c + Sqrt[a]*Sqrt[-c]*Sqrt[d])*(b*c - a*d)*Sqrt[a + b*x^4]
) + ((Sqrt[b]*Sqrt[-c] + Sqrt[a]*Sqrt[d])*d*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*
x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticPi[-(Sqrt[b]*Sqrt[-c] - Sqrt[a]*Sqrt[d])
^2/(4*Sqrt[a]*Sqrt[b]*Sqrt[-c]*Sqrt[d]), 2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(8
*a^(1/4)*b^(1/4)*c*(Sqrt[b]*Sqrt[-c] - Sqrt[a]*Sqrt[d])*(b*c - a*d)*Sqrt[a + b*x
^4]) + ((Sqrt[b]*Sqrt[-c] - Sqrt[a]*Sqrt[d])*d*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a +
b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticPi[(Sqrt[b]*Sqrt[-c] + Sqrt[a]*Sqrt[d
])^2/(4*Sqrt[a]*Sqrt[b]*Sqrt[-c]*Sqrt[d]), 2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/
(8*a^(1/4)*b^(1/4)*c*(Sqrt[b]*Sqrt[-c] + Sqrt[a]*Sqrt[d])*(b*c - a*d)*Sqrt[a + b
*x^4])

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Rubi [A]  time = 2.40571, antiderivative size = 969, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{b x}{2 a (b c-a d) \sqrt{b x^4+a}}-\frac{d \tan ^{-1}\left (\frac{\sqrt{-\frac{\sqrt{-c} \left (b-\frac{a d}{c}\right )}{\sqrt{d}}} x}{\sqrt{b x^4+a}}\right )}{4 c (b c-a d) \sqrt{\frac{b c-a d}{\sqrt{-c} \sqrt{d}}}}-\frac{d \tan ^{-1}\left (\frac{\sqrt{\frac{\sqrt{-c} \left (b-\frac{a d}{c}\right )}{\sqrt{d}}} x}{\sqrt{b x^4+a}}\right )}{4 c (b c-a d) \sqrt{-\frac{b c-a d}{\sqrt{-c} \sqrt{d}}}}-\frac{\sqrt [4]{b} d \left (\sqrt{b} x^2+\sqrt{a}\right ) \sqrt{\frac{b x^4+a}{\left (\sqrt{b} x^2+\sqrt{a}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} \left (\sqrt{b} c-\sqrt{a} \sqrt{-c} \sqrt{d}\right ) (b c-a d) \sqrt{b x^4+a}}-\frac{\sqrt [4]{b} d \left (\sqrt{b} x^2+\sqrt{a}\right ) \sqrt{\frac{b x^4+a}{\left (\sqrt{b} x^2+\sqrt{a}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} \left (\sqrt{b} c+\sqrt{a} \sqrt{-c} \sqrt{d}\right ) (b c-a d) \sqrt{b x^4+a}}+\frac{b^{3/4} \left (\sqrt{b} x^2+\sqrt{a}\right ) \sqrt{\frac{b x^4+a}{\left (\sqrt{b} x^2+\sqrt{a}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 a^{5/4} (b c-a d) \sqrt{b x^4+a}}+\frac{\left (\sqrt{b} \sqrt{-c}+\sqrt{a} \sqrt{d}\right ) d \left (\sqrt{b} x^2+\sqrt{a}\right ) \sqrt{\frac{b x^4+a}{\left (\sqrt{b} x^2+\sqrt{a}\right )^2}} \Pi \left (-\frac{\left (\sqrt{b} \sqrt{-c}-\sqrt{a} \sqrt{d}\right )^2}{4 \sqrt{a} \sqrt{b} \sqrt{-c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{b} c \left (\sqrt{b} \sqrt{-c}-\sqrt{a} \sqrt{d}\right ) (b c-a d) \sqrt{b x^4+a}}+\frac{\left (\sqrt{b} \sqrt{-c}-\sqrt{a} \sqrt{d}\right ) d \left (\sqrt{b} x^2+\sqrt{a}\right ) \sqrt{\frac{b x^4+a}{\left (\sqrt{b} x^2+\sqrt{a}\right )^2}} \Pi \left (\frac{\left (\sqrt{b} \sqrt{-c}+\sqrt{a} \sqrt{d}\right )^2}{4 \sqrt{a} \sqrt{b} \sqrt{-c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{b} c \left (\sqrt{b} \sqrt{-c}+\sqrt{a} \sqrt{d}\right ) (b c-a d) \sqrt{b x^4+a}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(b*x)/(2*a*(b*c - a*d)*Sqrt[a + b*x^4]) - (d*ArcTan[(Sqrt[-((Sqrt[-c]*(b - (a*d)
/c))/Sqrt[d])]*x)/Sqrt[a + b*x^4]])/(4*c*(b*c - a*d)*Sqrt[(b*c - a*d)/(Sqrt[-c]*
Sqrt[d])]) - (d*ArcTan[(Sqrt[(Sqrt[-c]*(b - (a*d)/c))/Sqrt[d]]*x)/Sqrt[a + b*x^4
]])/(4*c*(b*c - a*d)*Sqrt[-((b*c - a*d)/(Sqrt[-c]*Sqrt[d]))]) + (b^(3/4)*(Sqrt[a
] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2*ArcTan[
(b^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(5/4)*(b*c - a*d)*Sqrt[a + b*x^4]) - (b^(1/4)*d
*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2
*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(1/4)*(Sqrt[b]*c - Sqrt[a]*Sqrt[-c]*Sqr
t[d])*(b*c - a*d)*Sqrt[a + b*x^4]) - (b^(1/4)*d*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a
+ b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2
])/(4*a^(1/4)*(Sqrt[b]*c + Sqrt[a]*Sqrt[-c]*Sqrt[d])*(b*c - a*d)*Sqrt[a + b*x^4]
) + ((Sqrt[b]*Sqrt[-c] + Sqrt[a]*Sqrt[d])*d*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*
x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticPi[-(Sqrt[b]*Sqrt[-c] - Sqrt[a]*Sqrt[d])
^2/(4*Sqrt[a]*Sqrt[b]*Sqrt[-c]*Sqrt[d]), 2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(8
*a^(1/4)*b^(1/4)*c*(Sqrt[b]*Sqrt[-c] - Sqrt[a]*Sqrt[d])*(b*c - a*d)*Sqrt[a + b*x
^4]) + ((Sqrt[b]*Sqrt[-c] - Sqrt[a]*Sqrt[d])*d*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a +
b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticPi[(Sqrt[b]*Sqrt[-c] + Sqrt[a]*Sqrt[d
])^2/(4*Sqrt[a]*Sqrt[b]*Sqrt[-c]*Sqrt[d]), 2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/
(8*a^(1/4)*b^(1/4)*c*(Sqrt[b]*Sqrt[-c] + Sqrt[a]*Sqrt[d])*(b*c - a*d)*Sqrt[a + b
*x^4])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [C]  time = 0.495144, size = 342, normalized size = 0.35 \[ \frac{x \left (\frac{9 b c d x^4 F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{\left (c+d x^4\right ) \left (2 x^4 \left (2 a d F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+b c F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )-9 a c F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )}+\frac{25 c (b c-2 a d) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{\left (c+d x^4\right ) \left (2 x^4 \left (2 a d F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+b c F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )-5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )}-\frac{5 b}{a}\right )}{10 \sqrt{a+b x^4} (a d-b c)} \]

Warning: Unable to verify antiderivative.

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

[Out]

(x*((-5*b)/a + (25*c*(b*c - 2*a*d)*AppellF1[1/4, 1/2, 1, 5/4, -((b*x^4)/a), -((d
*x^4)/c)])/((c + d*x^4)*(-5*a*c*AppellF1[1/4, 1/2, 1, 5/4, -((b*x^4)/a), -((d*x^
4)/c)] + 2*x^4*(2*a*d*AppellF1[5/4, 1/2, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + b
*c*AppellF1[5/4, 3/2, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)]))) + (9*b*c*d*x^4*Appe
llF1[5/4, 1/2, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])/((c + d*x^4)*(-9*a*c*AppellF
1[5/4, 1/2, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + 2*x^4*(2*a*d*AppellF1[9/4, 1/2
, 2, 13/4, -((b*x^4)/a), -((d*x^4)/c)] + b*c*AppellF1[9/4, 3/2, 1, 13/4, -((b*x^
4)/a), -((d*x^4)/c)])))))/(10*(-(b*c) + a*d)*Sqrt[a + b*x^4])

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Maple [C]  time = 0.043, size = 313, normalized size = 0.3 \[ -{\frac{bx}{2\,a \left ( ad-bc \right ) }{\frac{1}{\sqrt{ \left ({x}^{4}+{\frac{a}{b}} \right ) b}}}}-{\frac{b}{2\,a \left ( ad-bc \right ) }\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}+{\frac{1}{8}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{4}d+c \right ) }{\frac{1}{ \left ( ad-bc \right ){{\it \_alpha}}^{3}} \left ( -{1{\it Artanh} \left ({\frac{2\,{{\it \_alpha}}^{2}b{x}^{2}+2\,a}{2}{\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \right ){\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}}+2\,{\frac{{{\it \_alpha}}^{3}d}{c\sqrt{b{x}^{4}+a}}\sqrt{1-{\frac{i\sqrt{b}{x}^{2}}{\sqrt{a}}}}\sqrt{1+{\frac{i\sqrt{b}{x}^{2}}{\sqrt{a}}}}{\it EllipticPi} \left ( x\sqrt{{\frac{i\sqrt{b}}{\sqrt{a}}}},{\frac{i\sqrt{a}{{\it \_alpha}}^{2}d}{c\sqrt{b}}},{1\sqrt{{\frac{-i\sqrt{b}}{\sqrt{a}}}}{\frac{1}{\sqrt{{\frac{i\sqrt{b}}{\sqrt{a}}}}}}} \right ){\frac{1}{\sqrt{{\frac{i\sqrt{b}}{\sqrt{a}}}}}}} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*b/a*x/(a*d-b*c)/((x^4+a/b)*b)^(1/2)-1/2*b/a/(a*d-b*c)/(I/a^(1/2)*b^(1/2))^(
1/2)*(1-I/a^(1/2)*b^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*b^(1/2)*x^2)^(1/2)/(b*x^4+a)^(
1/2)*EllipticF(x*(I/a^(1/2)*b^(1/2))^(1/2),I)+1/8*sum(1/(a*d-b*c)/_alpha^3*(-1/(
(a*d-b*c)/d)^(1/2)*arctanh(1/2*(2*_alpha^2*b*x^2+2*a)/((a*d-b*c)/d)^(1/2)/(b*x^4
+a)^(1/2))+2/(I/a^(1/2)*b^(1/2))^(1/2)*_alpha^3*d/c*(1-I/a^(1/2)*b^(1/2)*x^2)^(1
/2)*(1+I/a^(1/2)*b^(1/2)*x^2)^(1/2)/(b*x^4+a)^(1/2)*EllipticPi(x*(I/a^(1/2)*b^(1
/2))^(1/2),I*a^(1/2)/b^(1/2)*_alpha^2/c*d,(-I/a^(1/2)*b^(1/2))^(1/2)/(I/a^(1/2)*
b^(1/2))^(1/2))),_alpha=RootOf(_Z^4*d+c))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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