3.751 \(\int \frac{1}{x^7 \left (a+b x^8\right )^2 \sqrt{c+d x^8}} \, dx\)

Optimal. Leaf size=1108 \[ \text{result too large to display} \]

[Out]

-((7*b*c - 4*a*d)*Sqrt[c + d*x^8])/(24*a^2*c*(b*c - a*d)*x^6) + (b*Sqrt[c + d*x^
8])/(8*a*(b*c - a*d)*x^6*(a + b*x^8)) - (b*(7*b*c - 9*a*d)*ArcTan[(Sqrt[(Sqrt[-a
]*((b*c)/a - d))/Sqrt[b]]*x^2)/Sqrt[c + d*x^8]])/(32*a^3*(b*c - a*d)*Sqrt[-((b*c
- a*d)/(Sqrt[-a]*Sqrt[b]))]) - (b*(7*b*c - 9*a*d)*ArcTan[(Sqrt[(b*c - a*d)/(Sqr
t[-a]*Sqrt[b])]*x^2)/Sqrt[c + d*x^8]])/(32*a^3*(b*c - a*d)*Sqrt[(b*c - a*d)/(Sqr
t[-a]*Sqrt[b])]) + (b*d^(1/4)*(7*b*c - 9*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^2*c^(1/4)*(Sqrt[-a]*Sqrt[b]*Sqrt[c] - a*Sqrt[d])*(b*c - a*d)*Sqrt[c + d
*x^8]) - (b*d^(1/4)*(7*b*c - 9*a*d)*(Sqrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sq
rt[c] + Sqrt[d]*x^4)^2]*EllipticF[2*ArcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(32*a^2
*c^(1/4)*(Sqrt[-a]*Sqrt[b]*Sqrt[c] + a*Sqrt[d])*(b*c - a*d)*Sqrt[c + d*x^8]) - (
d^(3/4)*(7*b*c - 4*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])/(48*a^2*c^(5/4)*(b*
c - a*d)*Sqrt[c + d*x^8]) - (b*(Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])*(7*b*c - 9*a
*d)*(Sqrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*Elliptic
Pi[-(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^3*c^(1/4)*(Sqrt[b]*Sqrt[c] - Sqrt[
-a]*Sqrt[d])*d^(1/4)*(b*c - a*d)*Sqrt[c + d*x^8]) - (b*(Sqrt[b]*Sqrt[c] - Sqrt[-
a]*Sqrt[d])*(7*b*c - 9*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]*Sqrt[d])^2/(4*Sqrt[-a]*Sq
rt[b]*Sqrt[c]*Sqrt[d]), 2*ArcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(64*a^3*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 = 4.58504, antiderivative size = 1108, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ -\frac{(7 b c-9 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 ) b}{32 a^3 (b c-a d) \sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}-\frac{(7 b c-9 a d) \tan ^{-1}\left (\frac{\sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} x^2}{\sqrt{d x^8+c}}\right ) b}{32 a^3 (b c-a d) \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}+\frac{\sqrt [4]{d} (7 b c-9 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 ) b}{32 a^2 \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} (7 b c-9 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 ) b}{32 a^2 \sqrt [4]{c} \left (\sqrt{d} a+\sqrt{-a} \sqrt{b} \sqrt{c}\right ) (b c-a d) \sqrt{d x^8+c}}-\frac{\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right ) (7 b c-9 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 ) b}{64 a^3 \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 ) (7 b c-9 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 ) b}{64 a^3 \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{\sqrt{d x^8+c} b}{8 a (b c-a d) x^6 \left (b x^8+a\right )}-\frac{d^{3/4} (7 b c-4 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 )}{48 a^2 c^{5/4} (b c-a d) \sqrt{d x^8+c}}-\frac{(7 b c-4 a d) \sqrt{d x^8+c}}{24 a^2 c (b c-a d) x^6} \]

Warning: Unable to verify antiderivative.

[In]  Int[1/(x^7*(a + b*x^8)^2*Sqrt[c + d*x^8]),x]

[Out]

-((7*b*c - 4*a*d)*Sqrt[c + d*x^8])/(24*a^2*c*(b*c - a*d)*x^6) + (b*Sqrt[c + d*x^
8])/(8*a*(b*c - a*d)*x^6*(a + b*x^8)) - (b*(7*b*c - 9*a*d)*ArcTan[(Sqrt[(Sqrt[-a
]*((b*c)/a - d))/Sqrt[b]]*x^2)/Sqrt[c + d*x^8]])/(32*a^3*(b*c - a*d)*Sqrt[-((b*c
- a*d)/(Sqrt[-a]*Sqrt[b]))]) - (b*(7*b*c - 9*a*d)*ArcTan[(Sqrt[(b*c - a*d)/(Sqr
t[-a]*Sqrt[b])]*x^2)/Sqrt[c + d*x^8]])/(32*a^3*(b*c - a*d)*Sqrt[(b*c - a*d)/(Sqr
t[-a]*Sqrt[b])]) + (b*d^(1/4)*(7*b*c - 9*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^2*c^(1/4)*(Sqrt[-a]*Sqrt[b]*Sqrt[c] - a*Sqrt[d])*(b*c - a*d)*Sqrt[c + d
*x^8]) - (b*d^(1/4)*(7*b*c - 9*a*d)*(Sqrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sq
rt[c] + Sqrt[d]*x^4)^2]*EllipticF[2*ArcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(32*a^2
*c^(1/4)*(Sqrt[-a]*Sqrt[b]*Sqrt[c] + a*Sqrt[d])*(b*c - a*d)*Sqrt[c + d*x^8]) - (
d^(3/4)*(7*b*c - 4*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])/(48*a^2*c^(5/4)*(b*
c - a*d)*Sqrt[c + d*x^8]) - (b*(Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])*(7*b*c - 9*a
*d)*(Sqrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*Elliptic
Pi[-(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^3*c^(1/4)*(Sqrt[b]*Sqrt[c] - Sqrt[
-a]*Sqrt[d])*d^(1/4)*(b*c - a*d)*Sqrt[c + d*x^8]) - (b*(Sqrt[b]*Sqrt[c] - Sqrt[-
a]*Sqrt[d])*(7*b*c - 9*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]*Sqrt[d])^2/(4*Sqrt[-a]*Sq
rt[b]*Sqrt[c]*Sqrt[d]), 2*ArcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(64*a^3*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(1/x**7/(b*x**8+a)**2/(d*x**8+c)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 1.283, size = 399, normalized size = 0.36 \[ \frac{\frac{25 a x^8 \left (4 a^2 d^2+20 a b c d-21 b^2 c^2\right ) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}{2 x^8 \left (2 b c F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )+a d F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )\right )-5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}+\frac{5 \left (c+d x^8\right ) \left (-4 a^2 d+4 a b \left (c-d x^8\right )+7 b^2 c x^8\right )}{c}+\frac{9 a b d x^{16} (4 a d-7 b c) F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}{2 x^8 \left (2 b c F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )+a d F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )\right )-9 a c F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}}{120 a^2 x^6 \left (a+b x^8\right ) \sqrt{c+d x^8} (a d-b c)} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/(x^7*(a + b*x^8)^2*Sqrt[c + d*x^8]),x]

[Out]

((5*(c + d*x^8)*(-4*a^2*d + 7*b^2*c*x^8 + 4*a*b*(c - d*x^8)))/c + (25*a*(-21*b^2
*c^2 + 20*a*b*c*d + 4*a^2*d^2)*x^8*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*b*d*(-7*b*c + 4*a*d)*x^16*
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)])))/(120*a^2*(-(b*c) + a*d)*x^6*(a + b*x^8)*Sqrt[c + d*x^8])

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Maple [F]  time = 0.132, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{7} \left ( b{x}^{8}+a \right ) ^{2}}{\frac{1}{\sqrt{d{x}^{8}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^7/(b*x^8+a)^2/(d*x^8+c)^(1/2),x)

[Out]

int(1/x^7/(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. \[ \int \frac{1}{{\left (b x^{8} + a\right )}^{2} \sqrt{d x^{8} + c} x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^8 + a)^2*sqrt(d*x^8 + c)*x^7),x, algorithm="maxima")

[Out]

integrate(1/((b*x^8 + a)^2*sqrt(d*x^8 + c)*x^7), 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)^2*sqrt(d*x^8 + c)*x^7),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**7/(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. \[ \int \frac{1}{{\left (b x^{8} + a\right )}^{2} \sqrt{d x^{8} + c} x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^8 + a)^2*sqrt(d*x^8 + c)*x^7),x, algorithm="giac")

[Out]

integrate(1/((b*x^8 + a)^2*sqrt(d*x^8 + c)*x^7), x)