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

Optimal. Leaf size=1088 \[ \text{result too large to display} \]

[Out]

-Sqrt[c + d*x^8]/(2*a*c*x^2) + (Sqrt[d]*x^2*Sqrt[c + d*x^8])/(2*a*c*(Sqrt[c] + S
qrt[d]*x^4)) - (b*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]])/(8*a*(b*c - a*d)) - (b*Sqrt[(b*
c - a*d)/(Sqrt[-a]*Sqrt[b])]*ArcTan[(Sqrt[(b*c - a*d)/(Sqrt[-a]*Sqrt[b])]*x^2)/S
qrt[c + d*x^8]])/(8*a*(b*c - a*d)) - (d^(1/4)*(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
])/(2*a*c^(3/4)*Sqrt[c + d*x^8]) + (d^(1/4)*(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])
/(4*a*c^(3/4)*Sqrt[c + d*x^8]) + (Sqrt[b]*d^(1/4)*(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])/(8*a*c^(1/4)*(Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])*Sqrt[c + d*x^8]) + (Sqr
t[b]*d^(1/4)*(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])/(8*a*c^(1/4)*(Sqrt[b]*Sqrt[c]
+ Sqrt[-a]*Sqrt[d])*Sqrt[c + d*x^8]) + (Sqrt[b]*(Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt
[d])*(Sqrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*Ellipti
cPi[(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])/(16*a*c^(1/4)*(Sqrt[-a]*Sqrt[b]*Sqrt[c]
- a*Sqrt[d])*d^(1/4)*Sqrt[c + d*x^8]) - (Sqrt[b]*(Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqr
t[d])*(Sqrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*Ellipt
icPi[-(Sqrt[c]*(Sqrt[b] - (Sqrt[-a]*Sqrt[d])/Sqrt[c])^2)/(4*Sqrt[-a]*Sqrt[b]*Sqr
t[d]), 2*ArcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(16*a*c^(1/4)*(Sqrt[-a]*Sqrt[b]*Sq
rt[c] + a*Sqrt[d])*d^(1/4)*Sqrt[c + d*x^8])

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Rubi [A]  time = 3.02527, antiderivative size = 1088, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{\sqrt{d} \sqrt{d x^8+c} x^2}{2 a c \left (\sqrt{d} x^4+\sqrt{c}\right )}-\frac{b \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 )}{8 a (b c-a d)}-\frac{b \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 )}{8 a (b c-a d)}-\frac{\sqrt [4]{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 )}{2 a c^{3/4} \sqrt{d x^8+c}}+\frac{\sqrt [4]{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 )}{4 a c^{3/4} \sqrt{d x^8+c}}+\frac{\sqrt{b} \sqrt [4]{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 )}{8 a \sqrt [4]{c} \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) \sqrt{d x^8+c}}+\frac{\sqrt{b} \sqrt [4]{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 )}{8 a \sqrt [4]{c} \left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right ) \sqrt{d x^8+c}}+\frac{\sqrt{b} \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) \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 )}{16 a \sqrt [4]{c} \left (\sqrt{-a} \sqrt{b} \sqrt{c}-a \sqrt{d}\right ) \sqrt [4]{d} \sqrt{d x^8+c}}-\frac{\sqrt{b} \left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right ) \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 )}{16 a \sqrt [4]{c} \left (\sqrt{d} a+\sqrt{-a} \sqrt{b} \sqrt{c}\right ) \sqrt [4]{d} \sqrt{d x^8+c}}-\frac{\sqrt{d x^8+c}}{2 a c x^2} \]

Warning: Unable to verify antiderivative.

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

[Out]

-Sqrt[c + d*x^8]/(2*a*c*x^2) + (Sqrt[d]*x^2*Sqrt[c + d*x^8])/(2*a*c*(Sqrt[c] + S
qrt[d]*x^4)) - (b*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]])/(8*a*(b*c - a*d)) - (b*Sqrt[(b*
c - a*d)/(Sqrt[-a]*Sqrt[b])]*ArcTan[(Sqrt[(b*c - a*d)/(Sqrt[-a]*Sqrt[b])]*x^2)/S
qrt[c + d*x^8]])/(8*a*(b*c - a*d)) - (d^(1/4)*(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
])/(2*a*c^(3/4)*Sqrt[c + d*x^8]) + (d^(1/4)*(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])
/(4*a*c^(3/4)*Sqrt[c + d*x^8]) + (Sqrt[b]*d^(1/4)*(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])/(8*a*c^(1/4)*(Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])*Sqrt[c + d*x^8]) + (Sqr
t[b]*d^(1/4)*(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])/(8*a*c^(1/4)*(Sqrt[b]*Sqrt[c]
+ Sqrt[-a]*Sqrt[d])*Sqrt[c + d*x^8]) + (Sqrt[b]*(Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt
[d])*(Sqrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*Ellipti
cPi[(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])/(16*a*c^(1/4)*(Sqrt[-a]*Sqrt[b]*Sqrt[c]
- a*Sqrt[d])*d^(1/4)*Sqrt[c + d*x^8]) - (Sqrt[b]*(Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqr
t[d])*(Sqrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*Ellipt
icPi[-(Sqrt[c]*(Sqrt[b] - (Sqrt[-a]*Sqrt[d])/Sqrt[c])^2)/(4*Sqrt[-a]*Sqrt[b]*Sqr
t[d]), 2*ArcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(16*a*c^(1/4)*(Sqrt[-a]*Sqrt[b]*Sq
rt[c] + a*Sqrt[d])*d^(1/4)*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(1/x**3/(b*x**8+a)/(d*x**8+c)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 0.478021, size = 344, normalized size = 0.32 \[ \frac{\frac{49 x^8 (b c-a d) F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}{\left (a+b x^8\right ) \left (2 x^8 \left (2 b c F_1\left (\frac{7}{4};\frac{1}{2},2;\frac{11}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )+a d F_1\left (\frac{7}{4};\frac{3}{2},1;\frac{11}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )\right )-7 a c F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )\right )}-\frac{33 b d x^{16} F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}{\left (a+b x^8\right ) \left (2 x^8 \left (2 b c F_1\left (\frac{11}{4};\frac{1}{2},2;\frac{15}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )+a d F_1\left (\frac{11}{4};\frac{3}{2},1;\frac{15}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )\right )-11 a c F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )\right )}-\frac{21 \left (c+d x^8\right )}{a c}}{42 x^2 \sqrt{c+d x^8}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((-21*(c + d*x^8))/(a*c) + (49*(b*c - a*d)*x^8*AppellF1[3/4, 1/2, 1, 7/4, -((d*x
^8)/c), -((b*x^8)/a)])/((a + b*x^8)*(-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
*b*d*x^16*AppellF1[7/4, 1/2, 1, 11/4, -((d*x^8)/c), -((b*x^8)/a)])/((a + b*x^8)*
(-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)]))))/(42*x^2*Sqrt[c + d*x^8])

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Maple [F]  time = 0.106, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3} \left ( b{x}^{8}+a \right ) }{\frac{1}{\sqrt{d{x}^{8}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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