3.728 \(\int \frac{x^9}{\left (a+b x^8\right ) \sqrt{c+d x^8}} \, dx\)

Optimal. Leaf size=887 \[ -\frac{\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 )}{8 b \sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}-\frac{\tan ^{-1}\left (\frac{\sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} x^2}{\sqrt{d x^8+c}}\right )}{8 b \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}+\frac{a \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 b \sqrt [4]{c} \left (\sqrt{-a} \sqrt{b} \sqrt{c}-a \sqrt{d}\right ) \sqrt{d x^8+c}}-\frac{a \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 b \sqrt [4]{c} \left (\sqrt{d} a+\sqrt{-a} \sqrt{b} \sqrt{c}\right ) \sqrt{d x^8+c}}+\frac{\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 b \sqrt [4]{c} \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{\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 b \sqrt [4]{c} \left (\sqrt{b} \sqrt{c}-\sqrt{-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{\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 b \sqrt [4]{c} \left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right ) \sqrt [4]{d} \sqrt{d x^8+c}} \]

[Out]

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

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Rubi [A]  time = 2.19942, antiderivative size = 887, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{\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 )}{8 b \sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}-\frac{\tan ^{-1}\left (\frac{\sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} x^2}{\sqrt{d x^8+c}}\right )}{8 b \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}+\frac{a \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 b \sqrt [4]{c} \left (\sqrt{-a} \sqrt{b} \sqrt{c}-a \sqrt{d}\right ) \sqrt{d x^8+c}}-\frac{a \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 b \sqrt [4]{c} \left (\sqrt{d} a+\sqrt{-a} \sqrt{b} \sqrt{c}\right ) \sqrt{d x^8+c}}+\frac{\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 b \sqrt [4]{c} \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{\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 b \sqrt [4]{c} \left (\sqrt{b} \sqrt{c}-\sqrt{-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{\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 b \sqrt [4]{c} \left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right ) \sqrt [4]{d} \sqrt{d x^8+c}} \]

Warning: Unable to verify antiderivative.

[In]  Int[x^9/((a + b*x^8)*Sqrt[c + d*x^8]),x]

[Out]

-ArcTan[(Sqrt[(Sqrt[-a]*((b*c)/a - d))/Sqrt[b]]*x^2)/Sqrt[c + d*x^8]]/(8*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*b*Sqrt[(b*c - a*d)/(Sqrt[-a]*Sqrt[b])]) + ((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*b*c^(1/4)*d^(1/4)*Sqrt[c + d*x^8]) + (a*d^(1/4)*(
Sqrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*EllipticF[2*A
rcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(8*b*c^(1/4)*(Sqrt[-a]*Sqrt[b]*Sqrt[c] - a*S
qrt[d])*Sqrt[c + d*x^8]) - (a*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*b*
c^(1/4)*(Sqrt[-a]*Sqrt[b]*Sqrt[c] + a*Sqrt[d])*Sqrt[c + d*x^8]) - ((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]*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*b*c^(1/4)*(Sqrt[
b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])*d^(1/4)*Sqrt[c + d*x^8]) - ((Sqrt[b]*Sqrt[c] - Sq
rt[-a]*Sqrt[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])/(16*b*c^(1/4)*(Sqrt[b]*Sqrt[c
] + Sqrt[-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(x**9/(b*x**8+a)/(d*x**8+c)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 0.274741, size = 165, normalized size = 0.19 \[ -\frac{9 a c x^{10} F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}{10 \left (a+b x^8\right ) \sqrt{c+d x^8} \left (2 x^8 \left (2 b c F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )+a d F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )\right )-9 a c F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^9/((a + b*x^8)*Sqrt[c + d*x^8]),x]

[Out]

(-9*a*c*x^10*AppellF1[5/4, 1/2, 1, 9/4, -((d*x^8)/c), -((b*x^8)/a)])/(10*(a + b*
x^8)*Sqrt[c + d*x^8]*(-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)])))

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Maple [F]  time = 0.073, size = 0, normalized size = 0. \[ \int{\frac{{x}^{9}}{b{x}^{8}+a}{\frac{1}{\sqrt{d{x}^{8}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9/(b*x^8+a)/(d*x^8+c)^(1/2),x)

[Out]

int(x^9/(b*x^8+a)/(d*x^8+c)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{9}}{{\left (b x^{8} + a\right )} \sqrt{d x^{8} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/((b*x^8 + a)*sqrt(d*x^8 + c)),x, algorithm="maxima")

[Out]

integrate(x^9/((b*x^8 + a)*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^9/((b*x^8 + a)*sqrt(d*x^8 + c)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{9}}{\left (a + b x^{8}\right ) \sqrt{c + d x^{8}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**9/(b*x**8+a)/(d*x**8+c)**(1/2),x)

[Out]

Integral(x**9/((a + b*x**8)*sqrt(c + d*x**8)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{9}}{{\left (b x^{8} + a\right )} \sqrt{d x^{8} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/((b*x^8 + a)*sqrt(d*x^8 + c)),x, algorithm="giac")

[Out]

integrate(x^9/((b*x^8 + a)*sqrt(d*x^8 + c)), x)