3.722 \(\int \frac{1}{x^9 \left (a+b x^8\right ) \sqrt{c+d x^8}} \, dx\)

Optimal. Leaf size=117 \[ -\frac{b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right )}{4 a^2 \sqrt{b c-a d}}+\frac{(a d+2 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^8}}{\sqrt{c}}\right )}{8 a^2 c^{3/2}}-\frac{\sqrt{c+d x^8}}{8 a c x^8} \]

[Out]

-Sqrt[c + d*x^8]/(8*a*c*x^8) + ((2*b*c + a*d)*ArcTanh[Sqrt[c + d*x^8]/Sqrt[c]])/
(8*a^2*c^(3/2)) - (b^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^8])/Sqrt[b*c - a*d]])/(
4*a^2*Sqrt[b*c - a*d])

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Rubi [A]  time = 0.354749, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right )}{4 a^2 \sqrt{b c-a d}}+\frac{(a d+2 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^8}}{\sqrt{c}}\right )}{8 a^2 c^{3/2}}-\frac{\sqrt{c+d x^8}}{8 a c x^8} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[c + d*x^8]/(8*a*c*x^8) + ((2*b*c + a*d)*ArcTanh[Sqrt[c + d*x^8]/Sqrt[c]])/
(8*a^2*c^(3/2)) - (b^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^8])/Sqrt[b*c - a*d]])/(
4*a^2*Sqrt[b*c - a*d])

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Rubi in Sympy [A]  time = 46.3184, size = 100, normalized size = 0.85 \[ - \frac{\sqrt{c + d x^{8}}}{8 a c x^{8}} + \frac{b^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{8}}}{\sqrt{a d - b c}} \right )}}{4 a^{2} \sqrt{a d - b c}} + \frac{\left (\frac{a d}{2} + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{8}}}{\sqrt{c}} \right )}}{4 a^{2} c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(c + d*x**8)/(8*a*c*x**8) + b**(3/2)*atan(sqrt(b)*sqrt(c + d*x**8)/sqrt(a*d
- b*c))/(4*a**2*sqrt(a*d - b*c)) + (a*d/2 + b*c)*atanh(sqrt(c + d*x**8)/sqrt(c)
)/(4*a**2*c**(3/2))

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Mathematica [C]  time = 0.790837, size = 410, normalized size = 3.5 \[ \frac{\frac{5 b d x^8 \left (a \left (3 c+2 d x^8\right )+b x^8 \left (c+3 d x^8\right )\right ) F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^8},-\frac{a}{b x^8}\right )-3 \left (a+b x^8\right ) \left (c+d x^8\right ) \left (2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^8},-\frac{a}{b x^8}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^8},-\frac{a}{b x^8}\right )\right )}{a c \left (-5 b d x^8 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^8},-\frac{a}{b x^8}\right )+2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^8},-\frac{a}{b x^8}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^8},-\frac{a}{b x^8}\right )\right )}+\frac{6 b d x^{16} F_1\left (1;\frac{1}{2},1;2;-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}{x^8 \left (2 b c F_1\left (2;\frac{1}{2},2;3;-\frac{d x^8}{c},-\frac{b x^8}{a}\right )+a d F_1\left (2;\frac{3}{2},1;3;-\frac{d x^8}{c},-\frac{b x^8}{a}\right )\right )-4 a c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}}{24 x^8 \left (a+b x^8\right ) \sqrt{c+d x^8}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((6*b*d*x^16*AppellF1[1, 1/2, 1, 2, -((d*x^8)/c), -((b*x^8)/a)])/(-4*a*c*AppellF
1[1, 1/2, 1, 2, -((d*x^8)/c), -((b*x^8)/a)] + x^8*(2*b*c*AppellF1[2, 1/2, 2, 3,
-((d*x^8)/c), -((b*x^8)/a)] + a*d*AppellF1[2, 3/2, 1, 3, -((d*x^8)/c), -((b*x^8)
/a)])) + (5*b*d*x^8*(a*(3*c + 2*d*x^8) + b*x^8*(c + 3*d*x^8))*AppellF1[3/2, 1/2,
1, 5/2, -(c/(d*x^8)), -(a/(b*x^8))] - 3*(a + b*x^8)*(c + d*x^8)*(2*a*d*AppellF1
[5/2, 1/2, 2, 7/2, -(c/(d*x^8)), -(a/(b*x^8))] + b*c*AppellF1[5/2, 3/2, 1, 7/2,
-(c/(d*x^8)), -(a/(b*x^8))]))/(a*c*(-5*b*d*x^8*AppellF1[3/2, 1/2, 1, 5/2, -(c/(d
*x^8)), -(a/(b*x^8))] + 2*a*d*AppellF1[5/2, 1/2, 2, 7/2, -(c/(d*x^8)), -(a/(b*x^
8))] + b*c*AppellF1[5/2, 3/2, 1, 7/2, -(c/(d*x^8)), -(a/(b*x^8))])))/(24*x^8*(a
+ b*x^8)*Sqrt[c + d*x^8])

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Maple [F]  time = 0.122, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{9} \left ( b{x}^{8}+a \right ) }{\frac{1}{\sqrt{d{x}^{8}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(1/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. \[ \int \frac{1}{{\left (b x^{8} + a\right )} \sqrt{d x^{8} + c} x^{9}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^8 + a)*sqrt(d*x^8 + c)*x^9),x, algorithm="maxima")

[Out]

integrate(1/((b*x^8 + a)*sqrt(d*x^8 + c)*x^9), x)

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Fricas [A]  time = 0.255531, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, b c^{\frac{3}{2}} x^{8} \sqrt{\frac{b}{b c - a d}} \log \left (\frac{b d x^{8} + 2 \, b c - a d - 2 \, \sqrt{d x^{8} + c}{\left (b c - a d\right )} \sqrt{\frac{b}{b c - a d}}}{b x^{8} + a}\right ) +{\left (2 \, b c + a d\right )} x^{8} \log \left (\frac{{\left (d x^{8} + 2 \, c\right )} \sqrt{c} + 2 \, \sqrt{d x^{8} + c} c}{x^{8}}\right ) - 2 \, \sqrt{d x^{8} + c} a \sqrt{c}}{16 \, a^{2} c^{\frac{3}{2}} x^{8}}, -\frac{4 \, b c^{\frac{3}{2}} x^{8} \sqrt{-\frac{b}{b c - a d}} \arctan \left (-\frac{{\left (b c - a d\right )} \sqrt{-\frac{b}{b c - a d}}}{\sqrt{d x^{8} + c} b}\right ) -{\left (2 \, b c + a d\right )} x^{8} \log \left (\frac{{\left (d x^{8} + 2 \, c\right )} \sqrt{c} + 2 \, \sqrt{d x^{8} + c} c}{x^{8}}\right ) + 2 \, \sqrt{d x^{8} + c} a \sqrt{c}}{16 \, a^{2} c^{\frac{3}{2}} x^{8}}, \frac{b \sqrt{-c} c x^{8} \sqrt{\frac{b}{b c - a d}} \log \left (\frac{b d x^{8} + 2 \, b c - a d - 2 \, \sqrt{d x^{8} + c}{\left (b c - a d\right )} \sqrt{\frac{b}{b c - a d}}}{b x^{8} + a}\right ) -{\left (2 \, b c + a d\right )} x^{8} \arctan \left (\frac{c}{\sqrt{d x^{8} + c} \sqrt{-c}}\right ) - \sqrt{d x^{8} + c} a \sqrt{-c}}{8 \, a^{2} \sqrt{-c} c x^{8}}, -\frac{2 \, b \sqrt{-c} c x^{8} \sqrt{-\frac{b}{b c - a d}} \arctan \left (-\frac{{\left (b c - a d\right )} \sqrt{-\frac{b}{b c - a d}}}{\sqrt{d x^{8} + c} b}\right ) +{\left (2 \, b c + a d\right )} x^{8} \arctan \left (\frac{c}{\sqrt{d x^{8} + c} \sqrt{-c}}\right ) + \sqrt{d x^{8} + c} a \sqrt{-c}}{8 \, a^{2} \sqrt{-c} c x^{8}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^8 + a)*sqrt(d*x^8 + c)*x^9),x, algorithm="fricas")

[Out]

[1/16*(2*b*c^(3/2)*x^8*sqrt(b/(b*c - a*d))*log((b*d*x^8 + 2*b*c - a*d - 2*sqrt(d
*x^8 + c)*(b*c - a*d)*sqrt(b/(b*c - a*d)))/(b*x^8 + a)) + (2*b*c + a*d)*x^8*log(
((d*x^8 + 2*c)*sqrt(c) + 2*sqrt(d*x^8 + c)*c)/x^8) - 2*sqrt(d*x^8 + c)*a*sqrt(c)
)/(a^2*c^(3/2)*x^8), -1/16*(4*b*c^(3/2)*x^8*sqrt(-b/(b*c - a*d))*arctan(-(b*c -
a*d)*sqrt(-b/(b*c - a*d))/(sqrt(d*x^8 + c)*b)) - (2*b*c + a*d)*x^8*log(((d*x^8 +
2*c)*sqrt(c) + 2*sqrt(d*x^8 + c)*c)/x^8) + 2*sqrt(d*x^8 + c)*a*sqrt(c))/(a^2*c^
(3/2)*x^8), 1/8*(b*sqrt(-c)*c*x^8*sqrt(b/(b*c - a*d))*log((b*d*x^8 + 2*b*c - a*d
- 2*sqrt(d*x^8 + c)*(b*c - a*d)*sqrt(b/(b*c - a*d)))/(b*x^8 + a)) - (2*b*c + a*
d)*x^8*arctan(c/(sqrt(d*x^8 + c)*sqrt(-c))) - sqrt(d*x^8 + c)*a*sqrt(-c))/(a^2*s
qrt(-c)*c*x^8), -1/8*(2*b*sqrt(-c)*c*x^8*sqrt(-b/(b*c - a*d))*arctan(-(b*c - a*d
)*sqrt(-b/(b*c - a*d))/(sqrt(d*x^8 + c)*b)) + (2*b*c + a*d)*x^8*arctan(c/(sqrt(d
*x^8 + c)*sqrt(-c))) + sqrt(d*x^8 + c)*a*sqrt(-c))/(a^2*sqrt(-c)*c*x^8)]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.218919, size = 159, normalized size = 1.36 \[ \frac{1}{8} \, d^{2}{\left (\frac{2 \, b^{2} \arctan \left (\frac{\sqrt{d x^{8} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a^{2} d^{2}} - \frac{{\left (2 \, b c + a d\right )} \arctan \left (\frac{\sqrt{d x^{8} + c}}{\sqrt{-c}}\right )}{a^{2} \sqrt{-c} c d^{2}} - \frac{\sqrt{d x^{8} + c}}{a c d^{2} x^{8}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^8 + a)*sqrt(d*x^8 + c)*x^9),x, algorithm="giac")

[Out]

1/8*d^2*(2*b^2*arctan(sqrt(d*x^8 + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b
*d)*a^2*d^2) - (2*b*c + a*d)*arctan(sqrt(d*x^8 + c)/sqrt(-c))/(a^2*sqrt(-c)*c*d^
2) - sqrt(d*x^8 + c)/(a*c*d^2*x^8))