3.754 \(\int \frac{1}{x^3 \left (a+b x^8\right )^2 \sqrt{c+d x^8}} \, dx\)

Optimal. Leaf size=1299 \[ \text{result too large to display} \]

[Out]

-((5*b*c - 4*a*d)*Sqrt[c + d*x^8])/(8*a^2*c*(b*c - a*d)*x^2) + (Sqrt[d]*(5*b*c -
4*a*d)*x^2*Sqrt[c + d*x^8])/(8*a^2*c*(b*c - a*d)*(Sqrt[c] + Sqrt[d]*x^4)) + (b*
Sqrt[c + d*x^8])/(8*a*(b*c - a*d)*x^2*(a + b*x^8)) - (b*(5*b*c - 7*a*d)*Sqrt[-((
b*c - a*d)/(Sqrt[-a]*Sqrt[b]))]*ArcTan[(Sqrt[-((b*c - a*d)/(Sqrt[-a]*Sqrt[b]))]*
x^2)/Sqrt[c + d*x^8]])/(32*a^2*(b*c - a*d)^2) - (b*(5*b*c - 7*a*d)*Sqrt[(b*c - a
*d)/(Sqrt[-a]*Sqrt[b])]*ArcTan[(Sqrt[(b*c - a*d)/(Sqrt[-a]*Sqrt[b])]*x^2)/Sqrt[c
+ d*x^8]])/(32*a^2*(b*c - a*d)^2) - (d^(1/4)*(5*b*c - 4*a*d)*(Sqrt[c] + Sqrt[d]
*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*EllipticE[2*ArcTan[(d^(1/4)*x^
2)/c^(1/4)], 1/2])/(8*a^2*c^(3/4)*(b*c - a*d)*Sqrt[c + d*x^8]) + (Sqrt[b]*d^(1/4
)*(5*b*c - 7*a*d)*(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])/(32*a^2*c^(1/4)*(Sqrt[b]*
Sqrt[c] - Sqrt[-a]*Sqrt[d])*(b*c - a*d)*Sqrt[c + d*x^8]) + (Sqrt[b]*d^(1/4)*(5*b
*c - 7*a*d)*(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])/(32*a^2*c^(1/4)*(Sqrt[b]*Sqrt[c
] + Sqrt[-a]*Sqrt[d])*(b*c - a*d)*Sqrt[c + d*x^8]) + (d^(1/4)*(5*b*c - 4*a*d)*(S
qrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*EllipticF[2*Ar
cTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(16*a^2*c^(3/4)*(b*c - a*d)*Sqrt[c + d*x^8])
+ (Sqrt[b]*(Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])*(5*b*c - 7*a*d)*(Sqrt[c] + Sqrt[
d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*EllipticPi[(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[-a]*Sqrt[b]*Sqrt[c] - a*Sqrt[d])*d^(1/4
)*(b*c - a*d)*Sqrt[c + d*x^8]) - (Sqrt[b]*(Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])*(
5*b*c - 7*a*d)*(Sqrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^
2]*EllipticPi[-(Sqrt[c]*(Sqrt[b] - (Sqrt[-a]*Sqrt[d])/Sqrt[c])^2)/(4*Sqrt[-a]*Sq
rt[b]*Sqrt[d]), 2*ArcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(64*a^2*c^(1/4)*(Sqrt[-a]
*Sqrt[b]*Sqrt[c] + a*Sqrt[d])*d^(1/4)*(b*c - a*d)*Sqrt[c + d*x^8])

_______________________________________________________________________________________

Rubi [A]  time = 5.42838, antiderivative size = 1299, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ \frac{\sqrt{d} (5 b c-4 a d) \sqrt{d x^8+c} x^2}{8 a^2 c (b c-a d) \left (\sqrt{d} x^4+\sqrt{c}\right )}-\frac{b (5 b c-7 a d) \sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} \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)^2}-\frac{b (5 b c-7 a d) \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} \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)^2}-\frac{\sqrt [4]{d} (5 b c-4 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}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 a^2 c^{3/4} (b c-a d) \sqrt{d x^8+c}}+\frac{\sqrt{b} \sqrt [4]{d} (5 b c-7 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^2 \sqrt [4]{c} \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) (b c-a d) \sqrt{d x^8+c}}+\frac{\sqrt{b} \sqrt [4]{d} (5 b c-7 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^2 \sqrt [4]{c} \left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right ) (b c-a d) \sqrt{d x^8+c}}+\frac{\sqrt [4]{d} (5 b c-4 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 )}{16 a^2 c^{3/4} (b c-a d) \sqrt{d x^8+c}}+\frac{\sqrt{b} \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) (5 b c-7 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{-a} \sqrt{b} \sqrt{c}-a \sqrt{d}\right ) \sqrt [4]{d} (b c-a d) \sqrt{d x^8+c}}-\frac{\sqrt{b} \left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right ) (5 b c-7 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{\sqrt{c} \left (\sqrt{b}-\frac{\sqrt{-a} \sqrt{d}}{\sqrt{c}}\right )^2}{4 \sqrt{-a} \sqrt{b} \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{d} a+\sqrt{-a} \sqrt{b} \sqrt{c}\right ) \sqrt [4]{d} (b c-a d) \sqrt{d x^8+c}}-\frac{(5 b c-4 a d) \sqrt{d x^8+c}}{8 a^2 c (b c-a d) x^2}+\frac{b \sqrt{d x^8+c}}{8 a (b c-a d) \left (b x^8+a\right ) x^2} \]

Warning: Unable to verify antiderivative.

[In]  Int[1/(x^3*(a + b*x^8)^2*Sqrt[c + d*x^8]),x]

[Out]

-((5*b*c - 4*a*d)*Sqrt[c + d*x^8])/(8*a^2*c*(b*c - a*d)*x^2) + (Sqrt[d]*(5*b*c -
4*a*d)*x^2*Sqrt[c + d*x^8])/(8*a^2*c*(b*c - a*d)*(Sqrt[c] + Sqrt[d]*x^4)) + (b*
Sqrt[c + d*x^8])/(8*a*(b*c - a*d)*x^2*(a + b*x^8)) - (b*(5*b*c - 7*a*d)*Sqrt[-((
b*c - a*d)/(Sqrt[-a]*Sqrt[b]))]*ArcTan[(Sqrt[-((b*c - a*d)/(Sqrt[-a]*Sqrt[b]))]*
x^2)/Sqrt[c + d*x^8]])/(32*a^2*(b*c - a*d)^2) - (b*(5*b*c - 7*a*d)*Sqrt[(b*c - a
*d)/(Sqrt[-a]*Sqrt[b])]*ArcTan[(Sqrt[(b*c - a*d)/(Sqrt[-a]*Sqrt[b])]*x^2)/Sqrt[c
+ d*x^8]])/(32*a^2*(b*c - a*d)^2) - (d^(1/4)*(5*b*c - 4*a*d)*(Sqrt[c] + Sqrt[d]
*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*EllipticE[2*ArcTan[(d^(1/4)*x^
2)/c^(1/4)], 1/2])/(8*a^2*c^(3/4)*(b*c - a*d)*Sqrt[c + d*x^8]) + (Sqrt[b]*d^(1/4
)*(5*b*c - 7*a*d)*(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])/(32*a^2*c^(1/4)*(Sqrt[b]*
Sqrt[c] - Sqrt[-a]*Sqrt[d])*(b*c - a*d)*Sqrt[c + d*x^8]) + (Sqrt[b]*d^(1/4)*(5*b
*c - 7*a*d)*(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])/(32*a^2*c^(1/4)*(Sqrt[b]*Sqrt[c
] + Sqrt[-a]*Sqrt[d])*(b*c - a*d)*Sqrt[c + d*x^8]) + (d^(1/4)*(5*b*c - 4*a*d)*(S
qrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*EllipticF[2*Ar
cTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(16*a^2*c^(3/4)*(b*c - a*d)*Sqrt[c + d*x^8])
+ (Sqrt[b]*(Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])*(5*b*c - 7*a*d)*(Sqrt[c] + Sqrt[
d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*EllipticPi[(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[-a]*Sqrt[b]*Sqrt[c] - a*Sqrt[d])*d^(1/4
)*(b*c - a*d)*Sqrt[c + d*x^8]) - (Sqrt[b]*(Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])*(
5*b*c - 7*a*d)*(Sqrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^
2]*EllipticPi[-(Sqrt[c]*(Sqrt[b] - (Sqrt[-a]*Sqrt[d])/Sqrt[c])^2)/(4*Sqrt[-a]*Sq
rt[b]*Sqrt[d]), 2*ArcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(64*a^2*c^(1/4)*(Sqrt[-a]
*Sqrt[b]*Sqrt[c] + a*Sqrt[d])*d^(1/4)*(b*c - a*d)*Sqrt[c + d*x^8])

_______________________________________________________________________________________

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/x**3/(b*x**8+a)**2/(d*x**8+c)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [C]  time = 1.53944, size = 399, normalized size = 0.31 \[ \frac{-\frac{49 a x^8 \left (4 a^2 d^2-12 a b c d+5 b^2 c^2\right ) F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}{2 x^8 \left (2 b c F_1\left (\frac{7}{4};\frac{1}{2},2;\frac{11}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )+a d F_1\left (\frac{7}{4};\frac{3}{2},1;\frac{11}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )\right )-7 a c F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}+\frac{21 \left (c+d x^8\right ) \left (-4 a^2 d+4 a b \left (c-d x^8\right )+5 b^2 c x^8\right )}{c}+\frac{33 a b d x^{16} (5 b c-4 a d) F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}{2 x^8 \left (2 b c F_1\left (\frac{11}{4};\frac{1}{2},2;\frac{15}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )+a d F_1\left (\frac{11}{4};\frac{3}{2},1;\frac{15}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )\right )-11 a c F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}}{168 a^2 x^2 \left (a+b x^8\right ) \sqrt{c+d x^8} (a d-b c)} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/(x^3*(a + b*x^8)^2*Sqrt[c + d*x^8]),x]

[Out]

((21*(c + d*x^8)*(-4*a^2*d + 5*b^2*c*x^8 + 4*a*b*(c - d*x^8)))/c - (49*a*(5*b^2*
c^2 - 12*a*b*c*d + 4*a^2*d^2)*x^8*AppellF1[3/4, 1/2, 1, 7/4, -((d*x^8)/c), -((b*
x^8)/a)])/(-7*a*c*AppellF1[3/4, 1/2, 1, 7/4, -((d*x^8)/c), -((b*x^8)/a)] + 2*x^8
*(2*b*c*AppellF1[7/4, 1/2, 2, 11/4, -((d*x^8)/c), -((b*x^8)/a)] + a*d*AppellF1[7
/4, 3/2, 1, 11/4, -((d*x^8)/c), -((b*x^8)/a)])) + (33*a*b*d*(5*b*c - 4*a*d)*x^16
*AppellF1[7/4, 1/2, 1, 11/4, -((d*x^8)/c), -((b*x^8)/a)])/(-11*a*c*AppellF1[7/4,
1/2, 1, 11/4, -((d*x^8)/c), -((b*x^8)/a)] + 2*x^8*(2*b*c*AppellF1[11/4, 1/2, 2,
15/4, -((d*x^8)/c), -((b*x^8)/a)] + a*d*AppellF1[11/4, 3/2, 1, 15/4, -((d*x^8)/
c), -((b*x^8)/a)])))/(168*a^2*(-(b*c) + a*d)*x^2*(a + b*x^8)*Sqrt[c + d*x^8])

_______________________________________________________________________________________

Maple [F]  time = 0.124, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3} \left ( b{x}^{8}+a \right ) ^{2}}{\frac{1}{\sqrt{d{x}^{8}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(b*x^8+a)^2/(d*x^8+c)^(1/2),x)

[Out]

int(1/x^3/(b*x^8+a)^2/(d*x^8+c)^(1/2),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{8} + a\right )}^{2} \sqrt{d x^{8} + c} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^8 + a)^2*sqrt(d*x^8 + c)*x^3),x, algorithm="maxima")

[Out]

integrate(1/((b*x^8 + a)^2*sqrt(d*x^8 + c)*x^3), x)

_______________________________________________________________________________________

Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^8 + a)^2*sqrt(d*x^8 + c)*x^3),x, algorithm="fricas")

[Out]

Exception raised: TypeError

_______________________________________________________________________________________

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(1/x**3/(b*x**8+a)**2/(d*x**8+c)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{8} + a\right )}^{2} \sqrt{d x^{8} + c} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^8 + a)^2*sqrt(d*x^8 + c)*x^3),x, algorithm="giac")

[Out]

integrate(1/((b*x^8 + a)^2*sqrt(d*x^8 + c)*x^3), x)