3.84 \(\int \frac{1}{\left (a+b x^4\right )^{5/2} \left (c+d x^4\right )} \, dx\)

Optimal. Leaf size=1029 \[ \text{result too large to display} \]

[Out]

(b*x)/(6*a*(b*c - a*d)*(a + b*x^4)^(3/2)) + (b*(5*b*c - 11*a*d)*x)/(12*a^2*(b*c
- a*d)^2*Sqrt[a + b*x^4]) + (d^2*ArcTan[(Sqrt[-((Sqrt[-c]*(b - (a*d)/c))/Sqrt[d]
)]*x)/Sqrt[a + b*x^4]])/(4*c*(b*c - a*d)^2*Sqrt[(b*c - a*d)/(Sqrt[-c]*Sqrt[d])])
+ (d^2*ArcTan[(Sqrt[(Sqrt[-c]*(b - (a*d)/c))/Sqrt[d]]*x)/Sqrt[a + b*x^4]])/(4*c
*(b*c - a*d)^2*Sqrt[-((b*c - a*d)/(Sqrt[-c]*Sqrt[d]))]) + (b^(1/4)*d^2*(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)^2*Sqrt[a + b*x^4]) + (b^(1/4)*d^2*(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)^2*Sqrt[a + b*x^4]) +
(b^(3/4)*(5*b*c - 11*a*d)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sq
rt[b]*x^2)^2]*EllipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(24*a^(9/4)*(b*c -
a*d)^2*Sqrt[a + b*x^4]) - ((Sqrt[b]*Sqrt[-c] + Sqrt[a]*Sqrt[d])*d^2*(Sqrt[a] + S
qrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticPi[-(Sqrt[b]*Sqr
t[-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)^2*Sqrt[a + b*x^4]) - ((Sqrt[b]*Sqrt[-c] - Sqrt[a]*Sqrt[d])*d^2*(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]*Sqr
t[d])*(b*c - a*d)^2*Sqrt[a + b*x^4])

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Rubi [A]  time = 4.38356, antiderivative size = 1029, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{-\frac{\sqrt{-c} \left (b-\frac{a d}{c}\right )}{\sqrt{d}}} x}{\sqrt{b x^4+a}}\right ) d^2}{4 c (b c-a d)^2 \sqrt{\frac{b c-a d}{\sqrt{-c} \sqrt{d}}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{\frac{\sqrt{-c} \left (b-\frac{a d}{c}\right )}{\sqrt{d}}} x}{\sqrt{b x^4+a}}\right ) d^2}{4 c (b c-a d)^2 \sqrt{-\frac{b c-a d}{\sqrt{-c} \sqrt{d}}}}+\frac{\sqrt [4]{b} \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 ) d^2}{4 \sqrt [4]{a} \left (\sqrt{b} c-\sqrt{a} \sqrt{-c} \sqrt{d}\right ) (b c-a d)^2 \sqrt{b x^4+a}}+\frac{\sqrt [4]{b} \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 ) d^2}{4 \sqrt [4]{a} \left (\sqrt{b} c+\sqrt{a} \sqrt{-c} \sqrt{d}\right ) (b c-a d)^2 \sqrt{b x^4+a}}-\frac{\left (\sqrt{b} \sqrt{-c}+\sqrt{a} \sqrt{d}\right ) \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 ) d^2}{8 \sqrt [4]{a} \sqrt [4]{b} c \left (\sqrt{b} \sqrt{-c}-\sqrt{a} \sqrt{d}\right ) (b c-a d)^2 \sqrt{b x^4+a}}-\frac{\left (\sqrt{b} \sqrt{-c}-\sqrt{a} \sqrt{d}\right ) \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 ) d^2}{8 \sqrt [4]{a} \sqrt [4]{b} c \left (\sqrt{b} \sqrt{-c}+\sqrt{a} \sqrt{d}\right ) (b c-a d)^2 \sqrt{b x^4+a}}+\frac{b^{3/4} (5 b c-11 a 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 )}{24 a^{9/4} (b c-a d)^2 \sqrt{b x^4+a}}+\frac{b (5 b c-11 a d) x}{12 a^2 (b c-a d)^2 \sqrt{b x^4+a}}+\frac{b x}{6 a (b c-a d) \left (b x^4+a\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]  Int[1/((a + b*x^4)^(5/2)*(c + d*x^4)),x]

[Out]

(b*x)/(6*a*(b*c - a*d)*(a + b*x^4)^(3/2)) + (b*(5*b*c - 11*a*d)*x)/(12*a^2*(b*c
- a*d)^2*Sqrt[a + b*x^4]) + (d^2*ArcTan[(Sqrt[-((Sqrt[-c]*(b - (a*d)/c))/Sqrt[d]
)]*x)/Sqrt[a + b*x^4]])/(4*c*(b*c - a*d)^2*Sqrt[(b*c - a*d)/(Sqrt[-c]*Sqrt[d])])
+ (d^2*ArcTan[(Sqrt[(Sqrt[-c]*(b - (a*d)/c))/Sqrt[d]]*x)/Sqrt[a + b*x^4]])/(4*c
*(b*c - a*d)^2*Sqrt[-((b*c - a*d)/(Sqrt[-c]*Sqrt[d]))]) + (b^(1/4)*d^2*(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)^2*Sqrt[a + b*x^4]) + (b^(1/4)*d^2*(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)^2*Sqrt[a + b*x^4]) +
(b^(3/4)*(5*b*c - 11*a*d)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sq
rt[b]*x^2)^2]*EllipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(24*a^(9/4)*(b*c -
a*d)^2*Sqrt[a + b*x^4]) - ((Sqrt[b]*Sqrt[-c] + Sqrt[a]*Sqrt[d])*d^2*(Sqrt[a] + S
qrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticPi[-(Sqrt[b]*Sqr
t[-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)^2*Sqrt[a + b*x^4]) - ((Sqrt[b]*Sqrt[-c] - Sqrt[a]*Sqrt[d])*d^2*(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]*Sqr
t[d])*(b*c - a*d)^2*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)**(5/2)/(d*x**4+c),x)

[Out]

Timed out

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Mathematica [C]  time = 1.64098, size = 406, normalized size = 0.39 \[ \frac{x \left (\frac{25 a c \left (12 a^2 d^2-11 a b c d+5 b^2 c^2\right ) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{\left (c+d x^4\right ) \left (5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )-2 x^4 \left (2 a d F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+b c F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )\right )}+\frac{5 b \left (-13 a^2 d+a b \left (7 c-11 d x^4\right )+5 b^2 c x^4\right )}{a+b x^4}+\frac{9 a b c d x^4 (11 a d-5 b c) F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{\left (c+d x^4\right ) \left (2 x^4 \left (2 a d F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+b c F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )-9 a c F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )}\right )}{60 a^2 \sqrt{a+b x^4} (b c-a d)^2} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/((a + b*x^4)^(5/2)*(c + d*x^4)),x]

[Out]

(x*((5*b*(-13*a^2*d + 5*b^2*c*x^4 + a*b*(7*c - 11*d*x^4)))/(a + b*x^4) + (25*a*c
*(5*b^2*c^2 - 11*a*b*c*d + 12*a^2*d^2)*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*a*b*c*d*(-5
*b*c + 11*a*d)*x^4*AppellF1[5/4, 1/2, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])/((c +
d*x^4)*(-9*a*c*AppellF1[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)])))))/(60*a^2*(b*c - a*d)^2*Sqrt[a +
b*x^4])

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Maple [C]  time = 0.047, size = 371, normalized size = 0.4 \[ -{\frac{x}{6\,ab \left ( ad-bc \right ) }\sqrt{b{x}^{4}+a} \left ({x}^{4}+{\frac{a}{b}} \right ) ^{-2}}-{\frac{bx \left ( 11\,ad-5\,bc \right ) }{12\,{a}^{2} \left ( ad-bc \right ) ^{2}}{\frac{1}{\sqrt{ \left ({x}^{4}+{\frac{a}{b}} \right ) b}}}}-{\frac{b \left ( 11\,ad-5\,bc \right ) }{12\,{a}^{2} \left ( ad-bc \right ) ^{2}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}+{\frac{d}{8}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{4}d+c \right ) }{\frac{1}{ \left ( ad-bc \right ) ^{2}{{\it \_alpha}}^{3}} \left ( -{1{\it Artanh} \left ({\frac{2\,{{\it \_alpha}}^{2}b{x}^{2}+2\,a}{2}{\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \right ){\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}}+2\,{\frac{{{\it \_alpha}}^{3}d}{c\sqrt{b{x}^{4}+a}}\sqrt{1-{\frac{i\sqrt{b}{x}^{2}}{\sqrt{a}}}}\sqrt{1+{\frac{i\sqrt{b}{x}^{2}}{\sqrt{a}}}}{\it EllipticPi} \left ( x\sqrt{{\frac{i\sqrt{b}}{\sqrt{a}}}},{\frac{i\sqrt{a}{{\it \_alpha}}^{2}d}{c\sqrt{b}}},{1\sqrt{{\frac{-i\sqrt{b}}{\sqrt{a}}}}{\frac{1}{\sqrt{{\frac{i\sqrt{b}}{\sqrt{a}}}}}}} \right ){\frac{1}{\sqrt{{\frac{i\sqrt{b}}{\sqrt{a}}}}}}} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^4+a)^(5/2)/(d*x^4+c),x)

[Out]

-1/6/a*x/b/(a*d-b*c)*(b*x^4+a)^(1/2)/(x^4+a/b)^2-1/12*b/a^2*x*(11*a*d-5*b*c)/(a*
d-b*c)^2/((x^4+a/b)*b)^(1/2)-1/12*b/a^2*(11*a*d-5*b*c)/(a*d-b*c)^2/(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*d*sum(1/(a*d-b*c)^2/_a
lpha^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. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{5}{2}}{\left (d x^{4} + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(5/2)*(d*x^4 + c)),x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)^(5/2)*(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)^(5/2)*(d*x^4 + c)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x^{4}\right )^{\frac{5}{2}} \left (c + d x^{4}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x**4+a)**(5/2)/(d*x**4+c),x)

[Out]

Integral(1/((a + b*x**4)**(5/2)*(c + d*x**4)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{5}{2}}{\left (d x^{4} + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(5/2)*(d*x^4 + c)),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(5/2)*(d*x^4 + c)), x)