3.718 \(\int \frac{x^{23}}{\left (a+b x^8\right ) \sqrt{c+d x^8}} \, dx\)

Optimal. Leaf size=104 \[ -\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right )}{4 b^{5/2} \sqrt{b c-a d}}-\frac{\sqrt{c+d x^8} (a d+b c)}{4 b^2 d^2}+\frac{\left (c+d x^8\right )^{3/2}}{12 b d^2} \]

[Out]

-((b*c + a*d)*Sqrt[c + d*x^8])/(4*b^2*d^2) + (c + d*x^8)^(3/2)/(12*b*d^2) - (a^2
*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^8])/Sqrt[b*c - a*d]])/(4*b^(5/2)*Sqrt[b*c - a*d])

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Rubi [A]  time = 0.289497, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right )}{4 b^{5/2} \sqrt{b c-a d}}-\frac{\sqrt{c+d x^8} (a d+b c)}{4 b^2 d^2}+\frac{\left (c+d x^8\right )^{3/2}}{12 b d^2} \]

Antiderivative was successfully verified.

[In]  Int[x^23/((a + b*x^8)*Sqrt[c + d*x^8]),x]

[Out]

-((b*c + a*d)*Sqrt[c + d*x^8])/(4*b^2*d^2) + (c + d*x^8)^(3/2)/(12*b*d^2) - (a^2
*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^8])/Sqrt[b*c - a*d]])/(4*b^(5/2)*Sqrt[b*c - a*d])

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Rubi in Sympy [A]  time = 31.6813, size = 88, normalized size = 0.85 \[ \frac{a^{2} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{8}}}{\sqrt{a d - b c}} \right )}}{4 b^{\frac{5}{2}} \sqrt{a d - b c}} + \frac{\left (c + d x^{8}\right )^{\frac{3}{2}}}{12 b d^{2}} - \frac{\sqrt{c + d x^{8}} \left (a d + b c\right )}{4 b^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**23/(b*x**8+a)/(d*x**8+c)**(1/2),x)

[Out]

a**2*atan(sqrt(b)*sqrt(c + d*x**8)/sqrt(a*d - b*c))/(4*b**(5/2)*sqrt(a*d - b*c))
+ (c + d*x**8)**(3/2)/(12*b*d**2) - sqrt(c + d*x**8)*(a*d + b*c)/(4*b**2*d**2)

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Mathematica [A]  time = 0.272246, size = 91, normalized size = 0.88 \[ \frac{\sqrt{c+d x^8} \left (-3 a d-2 b c+b d x^8\right )}{12 b^2 d^2}-\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right )}{4 b^{5/2} \sqrt{b c-a d}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^23/((a + b*x^8)*Sqrt[c + d*x^8]),x]

[Out]

(Sqrt[c + d*x^8]*(-2*b*c - 3*a*d + b*d*x^8))/(12*b^2*d^2) - (a^2*ArcTanh[(Sqrt[b
]*Sqrt[c + d*x^8])/Sqrt[b*c - a*d]])/(4*b^(5/2)*Sqrt[b*c - a*d])

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Maple [F]  time = 0.105, size = 0, normalized size = 0. \[ \int{\frac{{x}^{23}}{b{x}^{8}+a}{\frac{1}{\sqrt{d{x}^{8}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^23/(b*x^8+a)/(d*x^8+c)^(1/2),x)

[Out]

int(x^23/(b*x^8+a)/(d*x^8+c)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^23/((b*x^8 + a)*sqrt(d*x^8 + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.225562, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a^{2} d^{2} \log \left (\frac{{\left (b d x^{8} + 2 \, b c - a d\right )} \sqrt{b^{2} c - a b d} - 2 \, \sqrt{d x^{8} + c}{\left (b^{2} c - a b d\right )}}{b x^{8} + a}\right ) + 2 \,{\left (b d x^{8} - 2 \, b c - 3 \, a d\right )} \sqrt{d x^{8} + c} \sqrt{b^{2} c - a b d}}{24 \, \sqrt{b^{2} c - a b d} b^{2} d^{2}}, -\frac{3 \, a^{2} d^{2} \arctan \left (-\frac{b c - a d}{\sqrt{d x^{8} + c} \sqrt{-b^{2} c + a b d}}\right ) -{\left (b d x^{8} - 2 \, b c - 3 \, a d\right )} \sqrt{d x^{8} + c} \sqrt{-b^{2} c + a b d}}{12 \, \sqrt{-b^{2} c + a b d} b^{2} d^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^23/((b*x^8 + a)*sqrt(d*x^8 + c)),x, algorithm="fricas")

[Out]

[1/24*(3*a^2*d^2*log(((b*d*x^8 + 2*b*c - a*d)*sqrt(b^2*c - a*b*d) - 2*sqrt(d*x^8
+ c)*(b^2*c - a*b*d))/(b*x^8 + a)) + 2*(b*d*x^8 - 2*b*c - 3*a*d)*sqrt(d*x^8 + c
)*sqrt(b^2*c - a*b*d))/(sqrt(b^2*c - a*b*d)*b^2*d^2), -1/12*(3*a^2*d^2*arctan(-(
b*c - a*d)/(sqrt(d*x^8 + c)*sqrt(-b^2*c + a*b*d))) - (b*d*x^8 - 2*b*c - 3*a*d)*s
qrt(d*x^8 + c)*sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*b^2*d^2)]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.21367, size = 143, normalized size = 1.38 \[ \frac{a^{2} \arctan \left (\frac{\sqrt{d x^{8} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{4 \, \sqrt{-b^{2} c + a b d} b^{2}} + \frac{{\left (d x^{8} + c\right )}^{\frac{3}{2}} b^{2} d^{4} - 3 \, \sqrt{d x^{8} + c} b^{2} c d^{4} - 3 \, \sqrt{d x^{8} + c} a b d^{5}}{12 \, b^{3} d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^23/((b*x^8 + a)*sqrt(d*x^8 + c)),x, algorithm="giac")

[Out]

1/4*a^2*arctan(sqrt(d*x^8 + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*b^2
) + 1/12*((d*x^8 + c)^(3/2)*b^2*d^4 - 3*sqrt(d*x^8 + c)*b^2*c*d^4 - 3*sqrt(d*x^8
+ c)*a*b*d^5)/(b^3*d^6)