3.740 \(\int \frac{x^{15}}{\left (a+b x^8\right )^2 \sqrt{c+d x^8}} \, dx\)

Optimal. Leaf size=99 \[ \frac{a \sqrt{c+d x^8}}{8 b \left (a+b x^8\right ) (b c-a d)}-\frac{(2 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right )}{8 b^{3/2} (b c-a d)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.256272, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{a \sqrt{c+d x^8}}{8 b \left (a+b x^8\right ) (b c-a d)}-\frac{(2 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right )}{8 b^{3/2} (b c-a d)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^15/((a + b*x^8)^2*Sqrt[c + d*x^8]),x]

[Out]

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

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Rubi in Sympy [A]  time = 25.5303, size = 80, normalized size = 0.81 \[ - \frac{a \sqrt{c + d x^{8}}}{8 b \left (a + b x^{8}\right ) \left (a d - b c\right )} + \frac{\left (\frac{a d}{2} - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{8}}}{\sqrt{a d - b c}} \right )}}{4 b^{\frac{3}{2}} \left (a d - b c\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**15/(b*x**8+a)**2/(d*x**8+c)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.122729, size = 99, normalized size = 1. \[ \frac{a \sqrt{c+d x^8}}{8 b \left (a+b x^8\right ) (b c-a d)}-\frac{(2 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right )}{8 b^{3/2} (b c-a d)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^15/((a + b*x^8)^2*Sqrt[c + d*x^8]),x]

[Out]

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

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Maple [F]  time = 0.072, size = 0, normalized size = 0. \[ \int{\frac{{x}^{15}}{ \left ( b{x}^{8}+a \right ) ^{2}}{\frac{1}{\sqrt{d{x}^{8}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^15/(b*x^8+a)^2/(d*x^8+c)^(1/2),x)

[Out]

int(x^15/(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. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^15/((b*x^8 + a)^2*sqrt(d*x^8 + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.231824, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{d x^{8} + c} \sqrt{b^{2} c - a b d} a +{\left ({\left (2 \, b^{2} c - a b d\right )} x^{8} + 2 \, a b c - a^{2} d\right )} \log \left (\frac{{\left (b d x^{8} + 2 \, b c - a d\right )} \sqrt{b^{2} c - a b d} - 2 \, \sqrt{d x^{8} + c}{\left (b^{2} c - a b d\right )}}{b x^{8} + a}\right )}{16 \,{\left ({\left (b^{3} c - a b^{2} d\right )} x^{8} + a b^{2} c - a^{2} b d\right )} \sqrt{b^{2} c - a b d}}, \frac{\sqrt{d x^{8} + c} \sqrt{-b^{2} c + a b d} a -{\left ({\left (2 \, b^{2} c - a b d\right )} x^{8} + 2 \, a b c - a^{2} d\right )} \arctan \left (-\frac{b c - a d}{\sqrt{d x^{8} + c} \sqrt{-b^{2} c + a b d}}\right )}{8 \,{\left ({\left (b^{3} c - a b^{2} d\right )} x^{8} + a b^{2} c - a^{2} b d\right )} \sqrt{-b^{2} c + a b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^15/((b*x^8 + a)^2*sqrt(d*x^8 + c)),x, algorithm="fricas")

[Out]

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

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.216935, size = 157, normalized size = 1.59 \[ \frac{\frac{\sqrt{d x^{8} + c} a d^{2}}{{\left (b^{2} c - a b d\right )}{\left ({\left (d x^{8} + c\right )} b - b c + a d\right )}} + \frac{{\left (2 \, b c d - a d^{2}\right )} \arctan \left (\frac{\sqrt{d x^{8} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (b^{2} c - a b d\right )} \sqrt{-b^{2} c + a b d}}}{8 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^15/((b*x^8 + a)^2*sqrt(d*x^8 + c)),x, algorithm="giac")

[Out]

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