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

Optimal. Leaf size=812 \[ \frac{\sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} \tan ^{-1}\left (\frac{\sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} x^2}{\sqrt{d x^8+c}}\right )}{8 (b c-a d)}+\frac{\sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} \tan ^{-1}\left (\frac{\sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} x^2}{\sqrt{d x^8+c}}\right )}{8 (b c-a d)}-\frac{\sqrt [4]{d} \left (\sqrt{d} x^4+\sqrt{c}\right ) \sqrt{\frac{d x^8+c}{\left (\sqrt{d} x^4+\sqrt{c}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 \sqrt{b} \sqrt [4]{c} \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) \sqrt{d x^8+c}}-\frac{\sqrt [4]{d} \left (\sqrt{d} x^4+\sqrt{c}\right ) \sqrt{\frac{d x^8+c}{\left (\sqrt{d} x^4+\sqrt{c}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 \sqrt{b} \sqrt [4]{c} \left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right ) \sqrt{d x^8+c}}-\frac{\left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) \left (\sqrt{d} x^4+\sqrt{c}\right ) \sqrt{\frac{d x^8+c}{\left (\sqrt{d} x^4+\sqrt{c}\right )^2}} \Pi \left (\frac{\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right )^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{16 \sqrt{b} \sqrt [4]{c} \left (\sqrt{-a} \sqrt{b} \sqrt{c}-a \sqrt{d}\right ) \sqrt [4]{d} \sqrt{d x^8+c}}+\frac{\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right ) \left (\sqrt{d} x^4+\sqrt{c}\right ) \sqrt{\frac{d x^8+c}{\left (\sqrt{d} x^4+\sqrt{c}\right )^2}} \Pi \left (-\frac{\sqrt{c} \left (\sqrt{b}-\frac{\sqrt{-a} \sqrt{d}}{\sqrt{c}}\right )^2}{4 \sqrt{-a} \sqrt{b} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{16 \sqrt{b} \sqrt [4]{c} \left (\sqrt{d} a+\sqrt{-a} \sqrt{b} \sqrt{c}\right ) \sqrt [4]{d} \sqrt{d x^8+c}} \]

[Out]

(Sqrt[-((b*c - a*d)/(Sqrt[-a]*Sqrt[b]))]*ArcTan[(Sqrt[-((b*c - a*d)/(Sqrt[-a]*Sq
rt[b]))]*x^2)/Sqrt[c + d*x^8]])/(8*(b*c - a*d)) + (Sqrt[(b*c - a*d)/(Sqrt[-a]*Sq
rt[b])]*ArcTan[(Sqrt[(b*c - a*d)/(Sqrt[-a]*Sqrt[b])]*x^2)/Sqrt[c + d*x^8]])/(8*(
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]*EllipticF[2*ArcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(8*Sqrt[b]*c^(1/4)*(
Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])*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])/(8*Sqrt[b]*c^(1/4)*(Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])*Sqr
t[c + d*x^8]) - ((Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])*(Sqrt[c] + Sqrt[d]*x^4)*Sq
rt[(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])/(16*Sqrt[b]*c^(1/4)*(Sqrt[-a]*Sqrt[b]*Sqrt[c] - a*Sqrt[d])*d^(1/4)*Sqrt
[c + d*x^8]) + ((Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])*(Sqrt[c] + Sqrt[d]*x^4)*Sqr
t[(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]*Sqrt[b]*Sqrt[d]), 2*ArcTan[(d^(1/4)*x^2)/c^(
1/4)], 1/2])/(16*Sqrt[b]*c^(1/4)*(Sqrt[-a]*Sqrt[b]*Sqrt[c] + a*Sqrt[d])*d^(1/4)*
Sqrt[c + d*x^8])

_______________________________________________________________________________________

Rubi [A]  time = 1.43874, antiderivative size = 812, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{\sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} \tan ^{-1}\left (\frac{\sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} x^2}{\sqrt{d x^8+c}}\right )}{8 (b c-a d)}+\frac{\sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} \tan ^{-1}\left (\frac{\sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} x^2}{\sqrt{d x^8+c}}\right )}{8 (b c-a d)}-\frac{\sqrt [4]{d} \left (\sqrt{d} x^4+\sqrt{c}\right ) \sqrt{\frac{d x^8+c}{\left (\sqrt{d} x^4+\sqrt{c}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 \sqrt{b} \sqrt [4]{c} \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) \sqrt{d x^8+c}}-\frac{\sqrt [4]{d} \left (\sqrt{d} x^4+\sqrt{c}\right ) \sqrt{\frac{d x^8+c}{\left (\sqrt{d} x^4+\sqrt{c}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 \sqrt{b} \sqrt [4]{c} \left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right ) \sqrt{d x^8+c}}-\frac{\left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) \left (\sqrt{d} x^4+\sqrt{c}\right ) \sqrt{\frac{d x^8+c}{\left (\sqrt{d} x^4+\sqrt{c}\right )^2}} \Pi \left (\frac{\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right )^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{16 \sqrt{b} \sqrt [4]{c} \left (\sqrt{-a} \sqrt{b} \sqrt{c}-a \sqrt{d}\right ) \sqrt [4]{d} \sqrt{d x^8+c}}+\frac{\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right ) \left (\sqrt{d} x^4+\sqrt{c}\right ) \sqrt{\frac{d x^8+c}{\left (\sqrt{d} x^4+\sqrt{c}\right )^2}} \Pi \left (-\frac{\sqrt{c} \left (\sqrt{b}-\frac{\sqrt{-a} \sqrt{d}}{\sqrt{c}}\right )^2}{4 \sqrt{-a} \sqrt{b} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{16 \sqrt{b} \sqrt [4]{c} \left (\sqrt{d} a+\sqrt{-a} \sqrt{b} \sqrt{c}\right ) \sqrt [4]{d} \sqrt{d x^8+c}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(Sqrt[-((b*c - a*d)/(Sqrt[-a]*Sqrt[b]))]*ArcTan[(Sqrt[-((b*c - a*d)/(Sqrt[-a]*Sq
rt[b]))]*x^2)/Sqrt[c + d*x^8]])/(8*(b*c - a*d)) + (Sqrt[(b*c - a*d)/(Sqrt[-a]*Sq
rt[b])]*ArcTan[(Sqrt[(b*c - a*d)/(Sqrt[-a]*Sqrt[b])]*x^2)/Sqrt[c + d*x^8]])/(8*(
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]*EllipticF[2*ArcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(8*Sqrt[b]*c^(1/4)*(
Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])*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])/(8*Sqrt[b]*c^(1/4)*(Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])*Sqr
t[c + d*x^8]) - ((Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])*(Sqrt[c] + Sqrt[d]*x^4)*Sq
rt[(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])/(16*Sqrt[b]*c^(1/4)*(Sqrt[-a]*Sqrt[b]*Sqrt[c] - a*Sqrt[d])*d^(1/4)*Sqrt
[c + d*x^8]) + ((Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])*(Sqrt[c] + Sqrt[d]*x^4)*Sqr
t[(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]*Sqrt[b]*Sqrt[d]), 2*ArcTan[(d^(1/4)*x^2)/c^(
1/4)], 1/2])/(16*Sqrt[b]*c^(1/4)*(Sqrt[-a]*Sqrt[b]*Sqrt[c] + a*Sqrt[d])*d^(1/4)*
Sqrt[c + d*x^8])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 143.284, size = 721, normalized size = 0.89 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-atan(x**2*sqrt((a*d - b*c)/(sqrt(b)*sqrt(-a)))/sqrt(c + d*x**8))/(8*sqrt(b)*sqr
t(-a)*sqrt((a*d - b*c)/(sqrt(b)*sqrt(-a)))) + atan(x**2*sqrt((-a*d + b*c)/(sqrt(
b)*sqrt(-a)))/sqrt(c + d*x**8))/(8*sqrt(b)*sqrt(-a)*sqrt((-a*d + b*c)/(sqrt(b)*s
qrt(-a)))) - d**(1/4)*sqrt((c + d*x**8)/(sqrt(c) + sqrt(d)*x**4)**2)*(sqrt(c) +
sqrt(d)*x**4)*elliptic_f(2*atan(d**(1/4)*x**2/c**(1/4)), 1/2)/(8*sqrt(b)*c**(1/4
)*sqrt(c + d*x**8)*(sqrt(b)*sqrt(c) + sqrt(d)*sqrt(-a))) - d**(1/4)*sqrt((c + d*
x**8)/(sqrt(c) + sqrt(d)*x**4)**2)*(sqrt(c) + sqrt(d)*x**4)*elliptic_f(2*atan(d*
*(1/4)*x**2/c**(1/4)), 1/2)/(8*sqrt(b)*c**(1/4)*sqrt(c + d*x**8)*(sqrt(b)*sqrt(c
) - sqrt(d)*sqrt(-a))) + sqrt((c + d*x**8)/(sqrt(c) + sqrt(d)*x**4)**2)*(sqrt(c)
+ sqrt(d)*x**4)*(sqrt(b)*sqrt(c) + sqrt(d)*sqrt(-a))*elliptic_pi(-sqrt(c)*(sqrt
(b) - sqrt(d)*sqrt(-a)/sqrt(c))**2/(4*sqrt(b)*sqrt(d)*sqrt(-a)), 2*atan(d**(1/4)
*x**2/c**(1/4)), 1/2)/(16*sqrt(b)*c**(1/4)*d**(1/4)*sqrt(c + d*x**8)*(a*sqrt(d)
+ sqrt(b)*sqrt(c)*sqrt(-a))) + sqrt((c + d*x**8)/(sqrt(c) + sqrt(d)*x**4)**2)*(s
qrt(c) + sqrt(d)*x**4)*(sqrt(b)*sqrt(c) - sqrt(d)*sqrt(-a))*elliptic_pi((sqrt(b)
*sqrt(c) + sqrt(d)*sqrt(-a))**2/(4*sqrt(b)*sqrt(c)*sqrt(d)*sqrt(-a)), 2*atan(d**
(1/4)*x**2/c**(1/4)), 1/2)/(16*sqrt(b)*c**(1/4)*d**(1/4)*sqrt(c + d*x**8)*(a*sqr
t(d) - sqrt(b)*sqrt(c)*sqrt(-a)))

_______________________________________________________________________________________

Mathematica [C]  time = 0.276226, size = 165, normalized size = 0.2 \[ -\frac{7 a c x^6 F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}{6 \left (a+b x^8\right ) \sqrt{c+d x^8} \left (2 x^8 \left (2 b c F_1\left (\frac{7}{4};\frac{1}{2},2;\frac{11}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )+a d F_1\left (\frac{7}{4};\frac{3}{2},1;\frac{11}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )\right )-7 a c F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-7*a*c*x^6*AppellF1[3/4, 1/2, 1, 7/4, -((d*x^8)/c), -((b*x^8)/a)])/(6*(a + b*x^
8)*Sqrt[c + d*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*A
ppellF1[7/4, 3/2, 1, 11/4, -((d*x^8)/c), -((b*x^8)/a)])))

_______________________________________________________________________________________

Maple [F]  time = 0.061, size = 0, normalized size = 0. \[ \int{\frac{{x}^{5}}{b{x}^{8}+a}{\frac{1}{\sqrt{d{x}^{8}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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)*sqrt(d*x^8 + c)),x, algorithm="fricas")

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**5/((a + b*x**8)*sqrt(c + d*x**8)), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5}}{{\left (b x^{8} + a\right )} \sqrt{d x^{8} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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