3.753 \(\int \frac{x^5}{\left (a+b x^8\right )^2 \sqrt{c+d x^8}} \, dx\)

Optimal. Leaf size=1218 \[ \text{result too large to display} \]

[Out]

-(Sqrt[d]*x^2*Sqrt[c + d*x^8])/(8*a*(b*c - a*d)*(Sqrt[c] + Sqrt[d]*x^4)) + (b*x^
6*Sqrt[c + d*x^8])/(8*a*(b*c - a*d)*(a + b*x^8)) + ((b*c - 3*a*d)*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]])/(32*a*(b*c - a*d)^2) + ((b*c - 3*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*(b*c - a*d)^2) + (c^(1/4)*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])/(8*
a*(b*c - a*d)*Sqrt[c + d*x^8]) - (c^(1/4)*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])/(16*a*(b*c - a*d)*Sqrt[c + d*x^8]) - (d^(1/4)*(b*c - 3*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*Sqrt[b]*c^(1/4)*(Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d
])*(b*c - a*d)*Sqrt[c + d*x^8]) - (d^(1/4)*(b*c - 3*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*Sqrt[b]*c^(1/4)*(Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])*(b*c -
a*d)*Sqrt[c + d*x^8]) - ((Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])*(b*c - 3*a*d)*(Sqr
t[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*Sqrt[b]*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[c] + Sqrt[-a]*
Sqrt[d])*(b*c - 3*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*Sqr
t[-a]*Sqrt[b]*Sqrt[d]), 2*ArcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(64*a*Sqrt[b]*c^(
1/4)*(Sqrt[-a]*Sqrt[b]*Sqrt[c] + a*Sqrt[d])*d^(1/4)*(b*c - a*d)*Sqrt[c + d*x^8])

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Rubi [A]  time = 4.10498, antiderivative size = 1218, 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{b \sqrt{d x^8+c} x^6}{8 a (b c-a d) \left (b x^8+a\right )}-\frac{\sqrt{d} \sqrt{d x^8+c} x^2}{8 a (b c-a d) \left (\sqrt{d} x^4+\sqrt{c}\right )}+\frac{(b c-3 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 (b c-a d)^2}+\frac{(b c-3 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 (b c-a d)^2}+\frac{\sqrt [4]{c} \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 )}{8 a (b c-a d) \sqrt{d x^8+c}}-\frac{\sqrt [4]{d} (b c-3 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{b} \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} (b c-3 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{b} \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]{c} \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 )}{16 a (b c-a d) \sqrt{d x^8+c}}-\frac{\left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) (b c-3 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 \sqrt{b} \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{\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right ) (b c-3 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 \sqrt{b} \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}} \]

Warning: Unable to verify antiderivative.

[In]  Int[x^5/((a + b*x^8)^2*Sqrt[c + d*x^8]),x]

[Out]

-(Sqrt[d]*x^2*Sqrt[c + d*x^8])/(8*a*(b*c - a*d)*(Sqrt[c] + Sqrt[d]*x^4)) + (b*x^
6*Sqrt[c + d*x^8])/(8*a*(b*c - a*d)*(a + b*x^8)) + ((b*c - 3*a*d)*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]])/(32*a*(b*c - a*d)^2) + ((b*c - 3*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*(b*c - a*d)^2) + (c^(1/4)*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])/(8*
a*(b*c - a*d)*Sqrt[c + d*x^8]) - (c^(1/4)*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])/(16*a*(b*c - a*d)*Sqrt[c + d*x^8]) - (d^(1/4)*(b*c - 3*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*Sqrt[b]*c^(1/4)*(Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d
])*(b*c - a*d)*Sqrt[c + d*x^8]) - (d^(1/4)*(b*c - 3*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*Sqrt[b]*c^(1/4)*(Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])*(b*c -
a*d)*Sqrt[c + d*x^8]) - ((Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])*(b*c - 3*a*d)*(Sqr
t[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*Sqrt[b]*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[c] + Sqrt[-a]*
Sqrt[d])*(b*c - 3*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*Sqr
t[-a]*Sqrt[b]*Sqrt[d]), 2*ArcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(64*a*Sqrt[b]*c^(
1/4)*(Sqrt[-a]*Sqrt[b]*Sqrt[c] + 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**5/(b*x**8+a)**2/(d*x**8+c)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 0.511804, size = 342, normalized size = 0.28 \[ \frac{x^6 \left (-\frac{33 b c d x^8 F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}{2 x^8 \left (2 b c F_1\left (\frac{11}{4};\frac{1}{2},2;\frac{15}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )+a d F_1\left (\frac{11}{4};\frac{3}{2},1;\frac{15}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )\right )-11 a c F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}+\frac{49 c (b c-4 a d) F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}{2 x^8 \left (2 b c F_1\left (\frac{7}{4};\frac{1}{2},2;\frac{11}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )+a d F_1\left (\frac{7}{4};\frac{3}{2},1;\frac{11}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )\right )-7 a c F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}-\frac{21 b \left (c+d x^8\right )}{a}\right )}{168 \left (a+b x^8\right ) \sqrt{c+d x^8} (a d-b c)} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^5/((a + b*x^8)^2*Sqrt[c + d*x^8]),x]

[Out]

(x^6*((-21*b*(c + d*x^8))/a + (49*c*(b*c - 4*a*d)*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*b*c*d*x^8*
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*(-(b*c) + a*d)*(a + b*x^8)*Sqrt[c + d*x^8])

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Maple [F]  time = 0.067, size = 0, normalized size = 0. \[ \int{\frac{{x}^{5}}{ \left ( b{x}^{8}+a \right ) ^{2}}{\frac{1}{\sqrt{d{x}^{8}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(b*x^8+a)^2/(d*x^8+c)^(1/2),x)

[Out]

int(x^5/(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. \[ \int \frac{x^{5}}{{\left (b x^{8} + a\right )}^{2} \sqrt{d x^{8} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/((b*x^8 + a)^2*sqrt(d*x^8 + c)),x, algorithm="maxima")

[Out]

integrate(x^5/((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^5/((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**5/(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. \[ \int \frac{x^{5}}{{\left (b x^{8} + a\right )}^{2} \sqrt{d x^{8} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/((b*x^8 + a)^2*sqrt(d*x^8 + c)),x, algorithm="giac")

[Out]

integrate(x^5/((b*x^8 + a)^2*sqrt(d*x^8 + c)), x)