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

Optimal. Leaf size=85 \[ \frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right )}{4 a \sqrt{b c-a d}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^8}}{\sqrt{c}}\right )}{4 a \sqrt{c}} \]

[Out]

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

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Rubi [A]  time = 0.211865, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right )}{4 a \sqrt{b c-a d}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^8}}{\sqrt{c}}\right )}{4 a \sqrt{c}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 23.2274, size = 71, normalized size = 0.84 \[ - \frac{\sqrt{b} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{8}}}{\sqrt{a d - b c}} \right )}}{4 a \sqrt{a d - b c}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt{c + d x^{8}}}{\sqrt{c}} \right )}}{4 a \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.322559, size = 162, normalized size = 1.91 \[ \frac{5 b d x^8 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^8},-\frac{a}{b x^8}\right )}{12 \left (a+b x^8\right ) \sqrt{c+d x^8} \left (-5 b d x^8 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^8},-\frac{a}{b x^8}\right )+2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^8},-\frac{a}{b x^8}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^8},-\frac{a}{b x^8}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(5*b*d*x^8*AppellF1[3/2, 1/2, 1, 5/2, -(c/(d*x^8)), -(a/(b*x^8))])/(12*(a + b*x^
8)*Sqrt[c + d*x^8]*(-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*Appel
lF1[5/2, 3/2, 1, 7/2, -(c/(d*x^8)), -(a/(b*x^8))]))

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Maple [F]  time = 0.076, size = 0, normalized size = 0. \[ \int{\frac{1}{x \left ( b{x}^{8}+a \right ) }{\frac{1}{\sqrt{d{x}^{8}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.242064, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{c} \sqrt{\frac{b}{b c - a d}} \log \left (\frac{b d x^{8} + 2 \, b c - a d + 2 \, \sqrt{d x^{8} + c}{\left (b c - a d\right )} \sqrt{\frac{b}{b c - a d}}}{b x^{8} + a}\right ) + \log \left (\frac{{\left (d x^{8} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{8} + c} c}{x^{8}}\right )}{8 \, a \sqrt{c}}, \frac{2 \, \sqrt{c} \sqrt{-\frac{b}{b c - a d}} \arctan \left (-\frac{{\left (b c - a d\right )} \sqrt{-\frac{b}{b c - a d}}}{\sqrt{d x^{8} + c} b}\right ) + \log \left (\frac{{\left (d x^{8} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{8} + c} c}{x^{8}}\right )}{8 \, a \sqrt{c}}, \frac{\sqrt{-c} \sqrt{\frac{b}{b c - a d}} \log \left (\frac{b d x^{8} + 2 \, b c - a d + 2 \, \sqrt{d x^{8} + c}{\left (b c - a d\right )} \sqrt{\frac{b}{b c - a d}}}{b x^{8} + a}\right ) + 2 \, \arctan \left (\frac{c}{\sqrt{d x^{8} + c} \sqrt{-c}}\right )}{8 \, a \sqrt{-c}}, \frac{\sqrt{-c} \sqrt{-\frac{b}{b c - a d}} \arctan \left (-\frac{{\left (b c - a d\right )} \sqrt{-\frac{b}{b c - a d}}}{\sqrt{d x^{8} + c} b}\right ) + \arctan \left (\frac{c}{\sqrt{d x^{8} + c} \sqrt{-c}}\right )}{4 \, a \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(sqrt(c)*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)) + log(((d*x^8 + 2*c)*sqrt(c) - 2*
sqrt(d*x^8 + c)*c)/x^8))/(a*sqrt(c)), 1/8*(2*sqrt(c)*sqrt(-b/(b*c - a*d))*arctan
(-(b*c - a*d)*sqrt(-b/(b*c - a*d))/(sqrt(d*x^8 + c)*b)) + log(((d*x^8 + 2*c)*sqr
t(c) - 2*sqrt(d*x^8 + c)*c)/x^8))/(a*sqrt(c)), 1/8*(sqrt(-c)*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*arctan(c/(sqrt(d*x^8 + c)*sqrt(-c))))/(a*sqrt(-c)), 1/4*(sqrt(
-c)*sqrt(-b/(b*c - a*d))*arctan(-(b*c - a*d)*sqrt(-b/(b*c - a*d))/(sqrt(d*x^8 +
c)*b)) + arctan(c/(sqrt(d*x^8 + c)*sqrt(-c))))/(a*sqrt(-c))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x*(a + b*x**8)*sqrt(c + d*x**8)), x)

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GIAC/XCAS [A]  time = 0.213331, size = 107, normalized size = 1.26 \[ -\frac{1}{4} \, d{\left (\frac{b \arctan \left (\frac{\sqrt{d x^{8} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a d} - \frac{\arctan \left (\frac{\sqrt{d x^{8} + c}}{\sqrt{-c}}\right )}{a \sqrt{-c} d}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*d*(b*arctan(sqrt(d*x^8 + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*a
*d) - arctan(sqrt(d*x^8 + c)/sqrt(-c))/(a*sqrt(-c)*d))