3.749 \(\int \frac{x^9}{\left (a+b x^8\right )^2 \sqrt{c+d x^8}} \, dx\)

Optimal. Leaf size=1042 \[ \text{result too large to display} \]

[Out]

-(x^2*Sqrt[c + d*x^8])/(8*(b*c - a*d)*(a + b*x^8)) + ((b*c + a*d)*ArcTan[(Sqrt[(
Sqrt[-a]*((b*c)/a - d))/Sqrt[b]]*x^2)/Sqrt[c + d*x^8]])/(32*a*b*(b*c - a*d)*Sqrt
[-((b*c - a*d)/(Sqrt[-a]*Sqrt[b]))]) - ((b*c + a*d)*ArcTan[(Sqrt[(b*c - a*d)/(Sq
rt[-a]*Sqrt[b])]*x^2)/Sqrt[c + d*x^8]])/(32*(-a)^(3/2)*b^(3/2)*((b*c - a*d)/(Sqr
t[-a]*Sqrt[b]))^(3/2)) - (d^(3/4)*(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*b*c^(
1/4)*(b*c - a*d)*Sqrt[c + d*x^8]) - (d^(1/4)*(b*c + 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*b*c^(1/4)*(Sqrt[-a]*Sqrt[b]*Sqrt[c] - a*Sqrt[d])*(b*c - a*d)*S
qrt[c + d*x^8]) + (d^(1/4)*(b*c + 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*
b*c^(1/4)*(Sqrt[-a]*Sqrt[b]*Sqrt[c] + a*Sqrt[d])*(b*c - a*d)*Sqrt[c + d*x^8]) +
((Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])*(b*c + 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]*Sq
rt[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*b*c^(1/4)*(Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])*d^(1/4)*(b*c - a*d)*S
qrt[c + d*x^8]) + ((Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])*(b*c + a*d)*(Sqrt[c] + S
qrt[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*b*c^(1/4)*(Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])*d^(1
/4)*(b*c - a*d)*Sqrt[c + d*x^8])

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Rubi [A]  time = 4.12689, antiderivative size = 1042, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{\sqrt{d x^8+c} x^2}{8 (b c-a d) \left (b x^8+a\right )}+\frac{(b c+a d) \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 )}{32 a b (b c-a d) \sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}-\frac{(b c+a d) \tan ^{-1}\left (\frac{\sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} x^2}{\sqrt{d x^8+c}}\right )}{32 (-a)^{3/2} b^{3/2} \left (\frac{b c-a d}{\sqrt{-a} \sqrt{b}}\right )^{3/2}}-\frac{\sqrt [4]{d} (b c+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 b \sqrt [4]{c} \left (\sqrt{-a} \sqrt{b} \sqrt{c}-a \sqrt{d}\right ) (b c-a d) \sqrt{d x^8+c}}+\frac{\sqrt [4]{d} (b c+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 b \sqrt [4]{c} \left (\sqrt{d} a+\sqrt{-a} \sqrt{b} \sqrt{c}\right ) (b c-a d) \sqrt{d x^8+c}}-\frac{d^{3/4} \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 b \sqrt [4]{c} (b c-a d) \sqrt{d x^8+c}}+\frac{\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right ) (b c+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 b \sqrt [4]{c} \left (\sqrt{b} \sqrt{c}-\sqrt{-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+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 b \sqrt [4]{c} \left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right ) \sqrt [4]{d} (b c-a d) \sqrt{d x^8+c}} \]

Warning: Unable to verify antiderivative.

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

[Out]

-(x^2*Sqrt[c + d*x^8])/(8*(b*c - a*d)*(a + b*x^8)) + ((b*c + a*d)*ArcTan[(Sqrt[(
Sqrt[-a]*((b*c)/a - d))/Sqrt[b]]*x^2)/Sqrt[c + d*x^8]])/(32*a*b*(b*c - a*d)*Sqrt
[-((b*c - a*d)/(Sqrt[-a]*Sqrt[b]))]) - ((b*c + a*d)*ArcTan[(Sqrt[(b*c - a*d)/(Sq
rt[-a]*Sqrt[b])]*x^2)/Sqrt[c + d*x^8]])/(32*(-a)^(3/2)*b^(3/2)*((b*c - a*d)/(Sqr
t[-a]*Sqrt[b]))^(3/2)) - (d^(3/4)*(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*b*c^(
1/4)*(b*c - a*d)*Sqrt[c + d*x^8]) - (d^(1/4)*(b*c + 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*b*c^(1/4)*(Sqrt[-a]*Sqrt[b]*Sqrt[c] - a*Sqrt[d])*(b*c - a*d)*S
qrt[c + d*x^8]) + (d^(1/4)*(b*c + 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*
b*c^(1/4)*(Sqrt[-a]*Sqrt[b]*Sqrt[c] + a*Sqrt[d])*(b*c - a*d)*Sqrt[c + d*x^8]) +
((Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])*(b*c + 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]*Sq
rt[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*b*c^(1/4)*(Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])*d^(1/4)*(b*c - a*d)*S
qrt[c + d*x^8]) + ((Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])*(b*c + a*d)*(Sqrt[c] + S
qrt[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*b*c^(1/4)*(Sqrt[b]*Sqrt[c] + Sqrt[-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**9/(b*x**8+a)**2/(d*x**8+c)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 0.310513, size = 333, normalized size = 0.32 \[ \frac{x^2 \left (\frac{25 a c^2 F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}{2 x^8 \left (2 b c F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )+a d F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )\right )-5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}-\frac{9 a c d x^8 F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}{2 x^8 \left (2 b c F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )+a d F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )\right )-9 a c F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}+5 \left (c+d x^8\right )\right )}{40 \left (a+b x^8\right ) \sqrt{c+d x^8} (a d-b c)} \]

Warning: Unable to verify antiderivative.

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

[Out]

(x^2*(5*(c + d*x^8) + (25*a*c^2*AppellF1[1/4, 1/2, 1, 5/4, -((d*x^8)/c), -((b*x^
8)/a)])/(-5*a*c*AppellF1[1/4, 1/2, 1, 5/4, -((d*x^8)/c), -((b*x^8)/a)] + 2*x^8*(
2*b*c*AppellF1[5/4, 1/2, 2, 9/4, -((d*x^8)/c), -((b*x^8)/a)] + a*d*AppellF1[5/4,
3/2, 1, 9/4, -((d*x^8)/c), -((b*x^8)/a)])) - (9*a*c*d*x^8*AppellF1[5/4, 1/2, 1,
9/4, -((d*x^8)/c), -((b*x^8)/a)])/(-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)]))))/(40*(
-(b*c) + a*d)*(a + b*x^8)*Sqrt[c + d*x^8])

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Maple [F]  time = 0.074, size = 0, normalized size = 0. \[ \int{\frac{{x}^{9}}{ \left ( b{x}^{8}+a \right ) ^{2}}{\frac{1}{\sqrt{d{x}^{8}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(x^9/(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. \[ \int \frac{x^{9}}{{\left (b x^{8} + a\right )}^{2} \sqrt{d x^{8} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/((b*x^8 + a)^2*sqrt(d*x^8 + c)),x, algorithm="maxima")

[Out]

integrate(x^9/((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^9/((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**9/(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. \[ \int \frac{x^{9}}{{\left (b x^{8} + a\right )}^{2} \sqrt{d x^{8} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/((b*x^8 + a)^2*sqrt(d*x^8 + c)),x, algorithm="giac")

[Out]

integrate(x^9/((b*x^8 + a)^2*sqrt(d*x^8 + c)), x)