3.748 \(\int \frac{1}{x^{13} \left (a+b x^8\right )^2 \sqrt{c+d x^8}} \, dx\)

Optimal. Leaf size=208 \[ \frac{b^2 (5 b c-6 a d) \tan ^{-1}\left (\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right )}{8 a^{7/2} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^8} (5 b c-2 a d)}{24 a^2 c x^{12} (b c-a d)}+\frac{\sqrt{c+d x^8} \left (-4 a^2 d^2-8 a b c d+15 b^2 c^2\right )}{24 a^3 c^2 x^4 (b c-a d)}+\frac{b \sqrt{c+d x^8}}{8 a x^{12} \left (a+b x^8\right ) (b c-a d)} \]

[Out]

-((5*b*c - 2*a*d)*Sqrt[c + d*x^8])/(24*a^2*c*(b*c - a*d)*x^12) + ((15*b^2*c^2 -
8*a*b*c*d - 4*a^2*d^2)*Sqrt[c + d*x^8])/(24*a^3*c^2*(b*c - a*d)*x^4) + (b*Sqrt[c
+ d*x^8])/(8*a*(b*c - a*d)*x^12*(a + b*x^8)) + (b^2*(5*b*c - 6*a*d)*ArcTan[(Sqr
t[b*c - a*d]*x^4)/(Sqrt[a]*Sqrt[c + d*x^8])])/(8*a^(7/2)*(b*c - a*d)^(3/2))

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Rubi [A]  time = 0.915008, antiderivative size = 208, 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 \[ \frac{b^2 (5 b c-6 a d) \tan ^{-1}\left (\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right )}{8 a^{7/2} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^8} (5 b c-2 a d)}{24 a^2 c x^{12} (b c-a d)}+\frac{\sqrt{c+d x^8} \left (-4 a^2 d^2-8 a b c d+15 b^2 c^2\right )}{24 a^3 c^2 x^4 (b c-a d)}+\frac{b \sqrt{c+d x^8}}{8 a x^{12} \left (a+b x^8\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^13*(a + b*x^8)^2*Sqrt[c + d*x^8]),x]

[Out]

-((5*b*c - 2*a*d)*Sqrt[c + d*x^8])/(24*a^2*c*(b*c - a*d)*x^12) + ((15*b^2*c^2 -
8*a*b*c*d - 4*a^2*d^2)*Sqrt[c + d*x^8])/(24*a^3*c^2*(b*c - a*d)*x^4) + (b*Sqrt[c
+ d*x^8])/(8*a*(b*c - a*d)*x^12*(a + b*x^8)) + (b^2*(5*b*c - 6*a*d)*ArcTan[(Sqr
t[b*c - a*d]*x^4)/(Sqrt[a]*Sqrt[c + d*x^8])])/(8*a^(7/2)*(b*c - a*d)^(3/2))

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Rubi in Sympy [A]  time = 138.058, size = 184, normalized size = 0.88 \[ - \frac{b \sqrt{c + d x^{8}}}{8 a x^{12} \left (a + b x^{8}\right ) \left (a d - b c\right )} - \frac{\sqrt{c + d x^{8}} \left (2 a d - 5 b c\right )}{24 a^{2} c x^{12} \left (a d - b c\right )} + \frac{\sqrt{c + d x^{8}} \left (4 a^{2} d^{2} + 8 a b c d - 15 b^{2} c^{2}\right )}{24 a^{3} c^{2} x^{4} \left (a d - b c\right )} + \frac{b^{2} \left (6 a d - 5 b c\right ) \operatorname{atanh}{\left (\frac{x^{4} \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{8}}} \right )}}{8 a^{\frac{7}{2}} \left (a d - b c\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**13/(b*x**8+a)**2/(d*x**8+c)**(1/2),x)

[Out]

-b*sqrt(c + d*x**8)/(8*a*x**12*(a + b*x**8)*(a*d - b*c)) - sqrt(c + d*x**8)*(2*a
*d - 5*b*c)/(24*a**2*c*x**12*(a*d - b*c)) + sqrt(c + d*x**8)*(4*a**2*d**2 + 8*a*
b*c*d - 15*b**2*c**2)/(24*a**3*c**2*x**4*(a*d - b*c)) + b**2*(6*a*d - 5*b*c)*ata
nh(x**4*sqrt(a*d - b*c)/(sqrt(a)*sqrt(c + d*x**8)))/(8*a**(7/2)*(a*d - b*c)**(3/
2))

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Mathematica [A]  time = 1.86108, size = 195, normalized size = 0.94 \[ \frac{\sqrt{c+d x^8} \left (-\frac{2 a^2}{c}+\frac{3 a b^3 x^{16}}{\left (a+b x^8\right ) (b c-a d)}+\frac{3 b^2 x^{24} (5 b c-6 a d) \sin ^{-1}\left (\frac{\sqrt{x^8 \left (\frac{b}{a}-\frac{d}{c}\right )}}{\sqrt{\frac{b x^8}{a}+1}}\right )}{a c^2 \sqrt{\frac{b x^8}{a}+1} \left (\frac{x^8 (b c-a d)}{a c}\right )^{3/2} \sqrt{\frac{a \left (c+d x^8\right )}{c \left (a+b x^8\right )}}}+\frac{4 a x^8 (a d+3 b c)}{c^2}\right )}{24 a^4 x^{12}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^13*(a + b*x^8)^2*Sqrt[c + d*x^8]),x]

[Out]

(Sqrt[c + d*x^8]*((-2*a^2)/c + (4*a*(3*b*c + a*d)*x^8)/c^2 + (3*a*b^3*x^16)/((b*
c - a*d)*(a + b*x^8)) + (3*b^2*(5*b*c - 6*a*d)*x^24*ArcSin[Sqrt[(b/a - d/c)*x^8]
/Sqrt[1 + (b*x^8)/a]])/(a*c^2*(((b*c - a*d)*x^8)/(a*c))^(3/2)*Sqrt[1 + (b*x^8)/a
]*Sqrt[(a*(c + d*x^8))/(c*(a + b*x^8))])))/(24*a^4*x^12)

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Maple [F]  time = 0.12, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{13} \left ( b{x}^{8}+a \right ) ^{2}}{\frac{1}{\sqrt{d{x}^{8}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^13/(b*x^8+a)^2/(d*x^8+c)^(1/2),x)

[Out]

int(1/x^13/(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. \[ \int \frac{1}{{\left (b x^{8} + a\right )}^{2} \sqrt{d x^{8} + c} x^{13}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^8 + a)^2*sqrt(d*x^8 + c)*x^13),x, algorithm="maxima")

[Out]

integrate(1/((b*x^8 + a)^2*sqrt(d*x^8 + c)*x^13), x)

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Fricas [A]  time = 0.576477, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left ({\left (15 \, b^{3} c^{2} - 8 \, a b^{2} c d - 4 \, a^{2} b d^{2}\right )} x^{16} + 2 \,{\left (5 \, a b^{2} c^{2} - 3 \, a^{2} b c d - 2 \, a^{3} d^{2}\right )} x^{8} - 2 \, a^{2} b c^{2} + 2 \, a^{3} c d\right )} \sqrt{d x^{8} + c} \sqrt{-a b c + a^{2} d} + 3 \,{\left ({\left (5 \, b^{4} c^{3} - 6 \, a b^{3} c^{2} d\right )} x^{20} +{\left (5 \, a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d\right )} x^{12}\right )} \log \left (\frac{4 \,{\left ({\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} x^{12} -{\left (a^{2} b c^{2} - a^{3} c d\right )} x^{4}\right )} \sqrt{d x^{8} + c} +{\left ({\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{16} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{8} + a^{2} c^{2}\right )} \sqrt{-a b c + a^{2} d}}{b^{2} x^{16} + 2 \, a b x^{8} + a^{2}}\right )}{96 \,{\left ({\left (a^{3} b^{2} c^{3} - a^{4} b c^{2} d\right )} x^{20} +{\left (a^{4} b c^{3} - a^{5} c^{2} d\right )} x^{12}\right )} \sqrt{-a b c + a^{2} d}}, \frac{2 \,{\left ({\left (15 \, b^{3} c^{2} - 8 \, a b^{2} c d - 4 \, a^{2} b d^{2}\right )} x^{16} + 2 \,{\left (5 \, a b^{2} c^{2} - 3 \, a^{2} b c d - 2 \, a^{3} d^{2}\right )} x^{8} - 2 \, a^{2} b c^{2} + 2 \, a^{3} c d\right )} \sqrt{d x^{8} + c} \sqrt{a b c - a^{2} d} + 3 \,{\left ({\left (5 \, b^{4} c^{3} - 6 \, a b^{3} c^{2} d\right )} x^{20} +{\left (5 \, a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d\right )} x^{12}\right )} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{8} - a c}{2 \, \sqrt{d x^{8} + c} \sqrt{a b c - a^{2} d} x^{4}}\right )}{48 \,{\left ({\left (a^{3} b^{2} c^{3} - a^{4} b c^{2} d\right )} x^{20} +{\left (a^{4} b c^{3} - a^{5} c^{2} d\right )} x^{12}\right )} \sqrt{a b c - a^{2} d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^8 + a)^2*sqrt(d*x^8 + c)*x^13),x, algorithm="fricas")

[Out]

[1/96*(4*((15*b^3*c^2 - 8*a*b^2*c*d - 4*a^2*b*d^2)*x^16 + 2*(5*a*b^2*c^2 - 3*a^2
*b*c*d - 2*a^3*d^2)*x^8 - 2*a^2*b*c^2 + 2*a^3*c*d)*sqrt(d*x^8 + c)*sqrt(-a*b*c +
a^2*d) + 3*((5*b^4*c^3 - 6*a*b^3*c^2*d)*x^20 + (5*a*b^3*c^3 - 6*a^2*b^2*c^2*d)*
x^12)*log((4*((a*b^2*c^2 - 3*a^2*b*c*d + 2*a^3*d^2)*x^12 - (a^2*b*c^2 - a^3*c*d)
*x^4)*sqrt(d*x^8 + c) + ((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^16 - 2*(3*a*b*c^2 -
4*a^2*c*d)*x^8 + a^2*c^2)*sqrt(-a*b*c + a^2*d))/(b^2*x^16 + 2*a*b*x^8 + a^2)))/
(((a^3*b^2*c^3 - a^4*b*c^2*d)*x^20 + (a^4*b*c^3 - a^5*c^2*d)*x^12)*sqrt(-a*b*c +
a^2*d)), 1/48*(2*((15*b^3*c^2 - 8*a*b^2*c*d - 4*a^2*b*d^2)*x^16 + 2*(5*a*b^2*c^
2 - 3*a^2*b*c*d - 2*a^3*d^2)*x^8 - 2*a^2*b*c^2 + 2*a^3*c*d)*sqrt(d*x^8 + c)*sqrt
(a*b*c - a^2*d) + 3*((5*b^4*c^3 - 6*a*b^3*c^2*d)*x^20 + (5*a*b^3*c^3 - 6*a^2*b^2
*c^2*d)*x^12)*arctan(1/2*((b*c - 2*a*d)*x^8 - a*c)/(sqrt(d*x^8 + c)*sqrt(a*b*c -
a^2*d)*x^4)))/(((a^3*b^2*c^3 - a^4*b*c^2*d)*x^20 + (a^4*b*c^3 - a^5*c^2*d)*x^12
)*sqrt(a*b*c - a^2*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(1/x**13/(b*x**8+a)**2/(d*x**8+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.23661, size = 244, normalized size = 1.17 \[ \frac{b^{3} c \sqrt{d + \frac{c}{x^{8}}}}{8 \,{\left (a^{3} b c - a^{4} d\right )}{\left (b c + a{\left (d + \frac{c}{x^{8}}\right )} - a d\right )}} - \frac{{\left (5 \, b^{3} c - 6 \, a b^{2} d\right )} \arctan \left (\frac{a \sqrt{d + \frac{c}{x^{8}}}}{\sqrt{a b c - a^{2} d}}\right )}{8 \,{\left (a^{3} b c - a^{4} d\right )} \sqrt{a b c - a^{2} d}} + \frac{6 \, a^{3} b c^{5} \sqrt{d + \frac{c}{x^{8}}} - a^{4} c^{4}{\left (d + \frac{c}{x^{8}}\right )}^{\frac{3}{2}} + 3 \, a^{4} c^{4} \sqrt{d + \frac{c}{x^{8}}} d}{12 \, a^{6} c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^8 + a)^2*sqrt(d*x^8 + c)*x^13),x, algorithm="giac")

[Out]

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