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

Optimal. Leaf size=915 \[ \text{result too large to display} \]

[Out]

-Sqrt[c + d*x^8]/(6*a*c*x^6) - (b*ArcTan[(Sqrt[(Sqrt[-a]*((b*c)/a - d))/Sqrt[b]]
*x^2)/Sqrt[c + d*x^8]])/(8*a^2*Sqrt[-((b*c - a*d)/(Sqrt[-a]*Sqrt[b]))]) - (b*Arc
Tan[(Sqrt[(b*c - a*d)/(Sqrt[-a]*Sqrt[b])]*x^2)/Sqrt[c + d*x^8]])/(8*a^2*Sqrt[(b*
c - a*d)/(Sqrt[-a]*Sqrt[b])]) + (b*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*a*c^(1/4)*(Sqrt[-a]*Sqrt[b]*Sqrt[c] - a*Sqrt[d])*Sqrt[c + d*x^8]) - (b*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*a*c^(1/4)*(Sqrt[-a]*Sqrt[b]*Sqrt[c] +
a*Sqrt[d])*Sqrt[c + d*x^8]) - (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])/(12*
a*c^(5/4)*Sqrt[c + d*x^8]) - (b*(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]*Sq
rt[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*a^2*c^(1/4)*(Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])*d^
(1/4)*Sqrt[c + d*x^8]) - (b*(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*a^2*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 = 2.19853, antiderivative size = 915, 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{b \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 a^2 \sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}-\frac{b \tan ^{-1}\left (\frac{\sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} x^2}{\sqrt{d x^8+c}}\right )}{8 a^2 \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}-\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 )}{12 a c^{5/4} \sqrt{d x^8+c}}+\frac{b \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 a \sqrt [4]{c} \left (\sqrt{-a} \sqrt{b} \sqrt{c}-a \sqrt{d}\right ) \sqrt{d x^8+c}}-\frac{b \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 a \sqrt [4]{c} \left (\sqrt{d} a+\sqrt{-a} \sqrt{b} \sqrt{c}\right ) \sqrt{d x^8+c}}-\frac{b \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 a^2 \sqrt [4]{c} \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) \sqrt [4]{d} \sqrt{d x^8+c}}-\frac{b \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 a^2 \sqrt [4]{c} \left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right ) \sqrt [4]{d} \sqrt{d x^8+c}}-\frac{\sqrt{d x^8+c}}{6 a c x^6} \]

Warning: Unable to verify antiderivative.

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

[Out]

-Sqrt[c + d*x^8]/(6*a*c*x^6) - (b*ArcTan[(Sqrt[(Sqrt[-a]*((b*c)/a - d))/Sqrt[b]]
*x^2)/Sqrt[c + d*x^8]])/(8*a^2*Sqrt[-((b*c - a*d)/(Sqrt[-a]*Sqrt[b]))]) - (b*Arc
Tan[(Sqrt[(b*c - a*d)/(Sqrt[-a]*Sqrt[b])]*x^2)/Sqrt[c + d*x^8]])/(8*a^2*Sqrt[(b*
c - a*d)/(Sqrt[-a]*Sqrt[b])]) + (b*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*a*c^(1/4)*(Sqrt[-a]*Sqrt[b]*Sqrt[c] - a*Sqrt[d])*Sqrt[c + d*x^8]) - (b*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*a*c^(1/4)*(Sqrt[-a]*Sqrt[b]*Sqrt[c] +
a*Sqrt[d])*Sqrt[c + d*x^8]) - (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])/(12*
a*c^(5/4)*Sqrt[c + d*x^8]) - (b*(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]*Sq
rt[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*a^2*c^(1/4)*(Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])*d^
(1/4)*Sqrt[c + d*x^8]) - (b*(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*a^2*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(1/x**7/(b*x**8+a)/(d*x**8+c)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 0.482024, size = 344, normalized size = 0.38 \[ \frac{\frac{25 x^8 (a d+3 b c) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}{\left (a+b x^8\right ) \left (2 x^8 \left (2 b c F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )+a d F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )\right )-5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )\right )}+\frac{9 b d x^{16} F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}{\left (a+b x^8\right ) \left (2 x^8 \left (2 b c F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )+a d F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )\right )-9 a c F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )\right )}-\frac{5 \left (c+d x^8\right )}{a c}}{30 x^6 \sqrt{c+d x^8}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((-5*(c + d*x^8))/(a*c) + (25*(3*b*c + a*d)*x^8*AppellF1[1/4, 1/2, 1, 5/4, -((d*
x^8)/c), -((b*x^8)/a)])/((a + b*x^8)*(-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*b
*d*x^16*AppellF1[5/4, 1/2, 1, 9/4, -((d*x^8)/c), -((b*x^8)/a)])/((a + b*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*Appel
lF1[9/4, 1/2, 2, 13/4, -((d*x^8)/c), -((b*x^8)/a)] + a*d*AppellF1[9/4, 3/2, 1, 1
3/4, -((d*x^8)/c), -((b*x^8)/a)]))))/(30*x^6*Sqrt[c + d*x^8])

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Maple [F]  time = 0.109, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{7} \left ( b{x}^{8}+a \right ) }{\frac{1}{\sqrt{d{x}^{8}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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