3.750 \(\int \frac{x}{\left (a+b x^8\right )^2 \sqrt{c+d x^8}} \, dx\)

Optimal. Leaf size=1050 \[ \text{result too large to display} \]

[Out]

(b*x^2*Sqrt[c + d*x^8])/(8*a*(b*c - a*d)*(a + b*x^8)) + ((3*b*c - 5*a*d)*ArcTan[
(Sqrt[(Sqrt[-a]*((b*c)/a - d))/Sqrt[b]]*x^2)/Sqrt[c + d*x^8]])/(32*a^2*(b*c - a*
d)*Sqrt[-((b*c - a*d)/(Sqrt[-a]*Sqrt[b]))]) + ((3*b*c - 5*a*d)*ArcTan[(Sqrt[(b*c
- a*d)/(Sqrt[-a]*Sqrt[b])]*x^2)/Sqrt[c + d*x^8]])/(32*a^2*(b*c - a*d)*Sqrt[(b*c
- a*d)/(Sqrt[-a]*Sqrt[b])]) + (d^(3/4)*(Sqrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)
/(Sqrt[c] + Sqrt[d]*x^4)^2]*EllipticF[2*ArcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(16
*a*c^(1/4)*(b*c - a*d)*Sqrt[c + d*x^8]) - (d^(1/4)*(3*b*c - 5*a*d)*(Sqrt[c] + Sq
rt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*EllipticF[2*ArcTan[(d^(1/
4)*x^2)/c^(1/4)], 1/2])/(32*a*c^(1/4)*(Sqrt[-a]*Sqrt[b]*Sqrt[c] - a*Sqrt[d])*(b*
c - a*d)*Sqrt[c + d*x^8]) + (d^(1/4)*(3*b*c - 5*a*d)*(Sqrt[c] + Sqrt[d]*x^4)*Sqr
t[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*EllipticF[2*ArcTan[(d^(1/4)*x^2)/c^(1/4
)], 1/2])/(32*a*c^(1/4)*(Sqrt[-a]*Sqrt[b]*Sqrt[c] + a*Sqrt[d])*(b*c - a*d)*Sqrt[
c + d*x^8]) + ((Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])*(3*b*c - 5*a*d)*(Sqrt[c] + S
qrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*EllipticPi[-(Sqrt[b]*Sqr
t[c] - Sqrt[-a]*Sqrt[d])^2/(4*Sqrt[-a]*Sqrt[b]*Sqrt[c]*Sqrt[d]), 2*ArcTan[(d^(1/
4)*x^2)/c^(1/4)], 1/2])/(64*a^2*c^(1/4)*(Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])*d^(
1/4)*(b*c - a*d)*Sqrt[c + d*x^8]) + ((Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])*(3*b*c
- 5*a*d)*(Sqrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*El
lipticPi[(Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])^2/(4*Sqrt[-a]*Sqrt[b]*Sqrt[c]*Sqrt
[d]), 2*ArcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(64*a^2*c^(1/4)*(Sqrt[b]*Sqrt[c] +
Sqrt[-a]*Sqrt[d])*d^(1/4)*(b*c - a*d)*Sqrt[c + d*x^8])

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Rubi [A]  time = 3.17293, antiderivative size = 1050, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{b \sqrt{d x^8+c} x^2}{8 a (b c-a d) \left (b x^8+a\right )}+\frac{(3 b c-5 a d) \tan ^{-1}\left (\frac{\sqrt{\frac{\sqrt{-a} \left (\frac{b c}{a}-d\right )}{\sqrt{b}}} x^2}{\sqrt{d x^8+c}}\right )}{32 a^2 (b c-a d) \sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}+\frac{(3 b c-5 a d) \tan ^{-1}\left (\frac{\sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} x^2}{\sqrt{d x^8+c}}\right )}{32 a^2 (b c-a d) \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}-\frac{\sqrt [4]{d} (3 b c-5 a d) \left (\sqrt{d} x^4+\sqrt{c}\right ) \sqrt{\frac{d x^8+c}{\left (\sqrt{d} x^4+\sqrt{c}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{32 a \sqrt [4]{c} \left (\sqrt{-a} \sqrt{b} \sqrt{c}-a \sqrt{d}\right ) (b c-a d) \sqrt{d x^8+c}}+\frac{\sqrt [4]{d} (3 b c-5 a d) \left (\sqrt{d} x^4+\sqrt{c}\right ) \sqrt{\frac{d x^8+c}{\left (\sqrt{d} x^4+\sqrt{c}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{32 a \sqrt [4]{c} \left (\sqrt{d} a+\sqrt{-a} \sqrt{b} \sqrt{c}\right ) (b c-a d) \sqrt{d x^8+c}}+\frac{d^{3/4} \left (\sqrt{d} x^4+\sqrt{c}\right ) \sqrt{\frac{d x^8+c}{\left (\sqrt{d} x^4+\sqrt{c}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{16 a \sqrt [4]{c} (b c-a d) \sqrt{d x^8+c}}+\frac{\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right ) (3 b c-5 a d) \left (\sqrt{d} x^4+\sqrt{c}\right ) \sqrt{\frac{d x^8+c}{\left (\sqrt{d} x^4+\sqrt{c}\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]{d} x^2}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{64 a^2 \sqrt [4]{c} \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) \sqrt [4]{d} (b c-a d) \sqrt{d x^8+c}}+\frac{\left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) (3 b c-5 a d) \left (\sqrt{d} x^4+\sqrt{c}\right ) \sqrt{\frac{d x^8+c}{\left (\sqrt{d} x^4+\sqrt{c}\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]{d} x^2}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{64 a^2 \sqrt [4]{c} \left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right ) \sqrt [4]{d} (b c-a d) \sqrt{d x^8+c}} \]

Warning: Unable to verify antiderivative.

[In]  Int[x/((a + b*x^8)^2*Sqrt[c + d*x^8]),x]

[Out]

(b*x^2*Sqrt[c + d*x^8])/(8*a*(b*c - a*d)*(a + b*x^8)) + ((3*b*c - 5*a*d)*ArcTan[
(Sqrt[(Sqrt[-a]*((b*c)/a - d))/Sqrt[b]]*x^2)/Sqrt[c + d*x^8]])/(32*a^2*(b*c - a*
d)*Sqrt[-((b*c - a*d)/(Sqrt[-a]*Sqrt[b]))]) + ((3*b*c - 5*a*d)*ArcTan[(Sqrt[(b*c
- a*d)/(Sqrt[-a]*Sqrt[b])]*x^2)/Sqrt[c + d*x^8]])/(32*a^2*(b*c - a*d)*Sqrt[(b*c
- a*d)/(Sqrt[-a]*Sqrt[b])]) + (d^(3/4)*(Sqrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)
/(Sqrt[c] + Sqrt[d]*x^4)^2]*EllipticF[2*ArcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(16
*a*c^(1/4)*(b*c - a*d)*Sqrt[c + d*x^8]) - (d^(1/4)*(3*b*c - 5*a*d)*(Sqrt[c] + Sq
rt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*EllipticF[2*ArcTan[(d^(1/
4)*x^2)/c^(1/4)], 1/2])/(32*a*c^(1/4)*(Sqrt[-a]*Sqrt[b]*Sqrt[c] - a*Sqrt[d])*(b*
c - a*d)*Sqrt[c + d*x^8]) + (d^(1/4)*(3*b*c - 5*a*d)*(Sqrt[c] + Sqrt[d]*x^4)*Sqr
t[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*EllipticF[2*ArcTan[(d^(1/4)*x^2)/c^(1/4
)], 1/2])/(32*a*c^(1/4)*(Sqrt[-a]*Sqrt[b]*Sqrt[c] + a*Sqrt[d])*(b*c - a*d)*Sqrt[
c + d*x^8]) + ((Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])*(3*b*c - 5*a*d)*(Sqrt[c] + S
qrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*EllipticPi[-(Sqrt[b]*Sqr
t[c] - Sqrt[-a]*Sqrt[d])^2/(4*Sqrt[-a]*Sqrt[b]*Sqrt[c]*Sqrt[d]), 2*ArcTan[(d^(1/
4)*x^2)/c^(1/4)], 1/2])/(64*a^2*c^(1/4)*(Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])*d^(
1/4)*(b*c - a*d)*Sqrt[c + d*x^8]) + ((Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])*(3*b*c
- 5*a*d)*(Sqrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*El
lipticPi[(Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])^2/(4*Sqrt[-a]*Sqrt[b]*Sqrt[c]*Sqrt
[d]), 2*ArcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(64*a^2*c^(1/4)*(Sqrt[b]*Sqrt[c] +
Sqrt[-a]*Sqrt[d])*d^(1/4)*(b*c - a*d)*Sqrt[c + d*x^8])

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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(x/(b*x**8+a)**2/(d*x**8+c)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 0.527045, size = 343, normalized size = 0.33 \[ \frac{x^2 \left (\frac{9 b c d x^8 F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}{2 x^8 \left (2 b c F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )+a d F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )\right )-9 a c F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}+\frac{25 c (3 b c-4 a d) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}{2 x^8 \left (2 b c F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )+a d F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )\right )-5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}-\frac{5 b \left (c+d x^8\right )}{a}\right )}{40 \left (a+b x^8\right ) \sqrt{c+d x^8} (a d-b c)} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x/((a + b*x^8)^2*Sqrt[c + d*x^8]),x]

[Out]

(x^2*((-5*b*(c + d*x^8))/a + (25*c*(3*b*c - 4*a*d)*AppellF1[1/4, 1/2, 1, 5/4, -(
(d*x^8)/c), -((b*x^8)/a)])/(-5*a*c*AppellF1[1/4, 1/2, 1, 5/4, -((d*x^8)/c), -((b
*x^8)/a)] + 2*x^8*(2*b*c*AppellF1[5/4, 1/2, 2, 9/4, -((d*x^8)/c), -((b*x^8)/a)]
+ a*d*AppellF1[5/4, 3/2, 1, 9/4, -((d*x^8)/c), -((b*x^8)/a)])) + (9*b*c*d*x^8*Ap
pellF1[5/4, 1/2, 1, 9/4, -((d*x^8)/c), -((b*x^8)/a)])/(-9*a*c*AppellF1[5/4, 1/2,
1, 9/4, -((d*x^8)/c), -((b*x^8)/a)] + 2*x^8*(2*b*c*AppellF1[9/4, 1/2, 2, 13/4,
-((d*x^8)/c), -((b*x^8)/a)] + a*d*AppellF1[9/4, 3/2, 1, 13/4, -((d*x^8)/c), -((b
*x^8)/a)]))))/(40*(-(b*c) + a*d)*(a + b*x^8)*Sqrt[c + d*x^8])

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Maple [F]  time = 0.065, size = 0, normalized size = 0. \[ \int{\frac{x}{ \left ( b{x}^{8}+a \right ) ^{2}}{\frac{1}{\sqrt{d{x}^{8}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(b*x^8+a)^2/(d*x^8+c)^(1/2),x)

[Out]

int(x/(b*x^8+a)^2/(d*x^8+c)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{{\left (b x^{8} + a\right )}^{2} \sqrt{d x^{8} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/((b*x^8 + a)^2*sqrt(d*x^8 + c)),x, algorithm="maxima")

[Out]

integrate(x/((b*x^8 + a)^2*sqrt(d*x^8 + 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(x/((b*x^8 + a)^2*sqrt(d*x^8 + c)),x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b*x**8+a)**2/(d*x**8+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{{\left (b x^{8} + a\right )}^{2} \sqrt{d x^{8} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/((b*x^8 + a)^2*sqrt(d*x^8 + c)),x, algorithm="giac")

[Out]

integrate(x/((b*x^8 + a)^2*sqrt(d*x^8 + c)), x)