3.747 \(\int \frac{1}{x^5 \left (a+b x^8\right )^2 \sqrt{c+d x^8}} \, dx\)

Optimal. Leaf size=149 \[ -\frac{b (3 b c-4 a d) \tan ^{-1}\left (\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right )}{8 a^{5/2} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^8} (3 b c-2 a d)}{8 a^2 c x^4 (b c-a d)}+\frac{b \sqrt{c+d x^8}}{8 a x^4 \left (a+b x^8\right ) (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.555401, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{b (3 b c-4 a d) \tan ^{-1}\left (\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right )}{8 a^{5/2} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^8} (3 b c-2 a d)}{8 a^2 c x^4 (b c-a d)}+\frac{b \sqrt{c+d x^8}}{8 a x^4 \left (a+b x^8\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*(a + b*x^8)^2*Sqrt[c + d*x^8]),x]

[Out]

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

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Rubi in Sympy [A]  time = 69.2035, size = 129, normalized size = 0.87 \[ - \frac{b \sqrt{c + d x^{8}}}{8 a x^{4} \left (a + b x^{8}\right ) \left (a d - b c\right )} - \frac{\sqrt{c + d x^{8}} \left (2 a d - 3 b c\right )}{8 a^{2} c x^{4} \left (a d - b c\right )} - \frac{b \left (4 a d - 3 b c\right ) \operatorname{atanh}{\left (\frac{x^{4} \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{8}}} \right )}}{8 a^{\frac{5}{2}} \left (a d - b c\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**5/(b*x**8+a)**2/(d*x**8+c)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 1.96041, size = 172, normalized size = 1.15 \[ \frac{1}{8} \sqrt{c+d x^8} \left (\frac{b x^{12} (4 a d-3 b c) \sin ^{-1}\left (\frac{\sqrt{x^8 \left (\frac{b}{a}-\frac{d}{c}\right )}}{\sqrt{\frac{b x^8}{a}+1}}\right )}{a^4 c^2 \sqrt{\frac{b x^8}{a}+1} \left (\frac{x^8 (b c-a d)}{a c}\right )^{3/2} \sqrt{\frac{a \left (c+d x^8\right )}{c \left (a+b x^8\right )}}}+\frac{\frac{b^2 x^8}{\left (a+b x^8\right ) (a d-b c)}-\frac{2}{c}}{a^2 x^4}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^5*(a + b*x^8)^2*Sqrt[c + d*x^8]),x]

[Out]

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

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Maple [F]  time = 0.108, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{5} \left ( b{x}^{8}+a \right ) ^{2}}{\frac{1}{\sqrt{d{x}^{8}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(b*x^8+a)^2/(d*x^8+c)^(1/2),x)

[Out]

int(1/x^5/(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. \[ \int \frac{1}{{\left (b x^{8} + a\right )}^{2} \sqrt{d x^{8} + c} x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^8 + a)^2*sqrt(d*x^8 + c)*x^5),x, algorithm="maxima")

[Out]

integrate(1/((b*x^8 + a)^2*sqrt(d*x^8 + c)*x^5), x)

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Fricas [A]  time = 0.391966, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \,{\left ({\left (3 \, b^{2} c - 2 \, a b d\right )} x^{8} + 2 \, a b c - 2 \, a^{2} d\right )} \sqrt{d x^{8} + c} \sqrt{-a b c + a^{2} d} -{\left ({\left (3 \, b^{3} c^{2} - 4 \, a b^{2} c d\right )} x^{12} +{\left (3 \, a b^{2} c^{2} - 4 \, a^{2} b c d\right )} x^{4}\right )} \log \left (-\frac{4 \,{\left ({\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} x^{12} -{\left (a^{2} b c^{2} - a^{3} c d\right )} x^{4}\right )} \sqrt{d x^{8} + c} -{\left ({\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{16} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{8} + a^{2} c^{2}\right )} \sqrt{-a b c + a^{2} d}}{b^{2} x^{16} + 2 \, a b x^{8} + a^{2}}\right )}{32 \,{\left ({\left (a^{2} b^{2} c^{2} - a^{3} b c d\right )} x^{12} +{\left (a^{3} b c^{2} - a^{4} c d\right )} x^{4}\right )} \sqrt{-a b c + a^{2} d}}, -\frac{2 \,{\left ({\left (3 \, b^{2} c - 2 \, a b d\right )} x^{8} + 2 \, a b c - 2 \, a^{2} d\right )} \sqrt{d x^{8} + c} \sqrt{a b c - a^{2} d} +{\left ({\left (3 \, b^{3} c^{2} - 4 \, a b^{2} c d\right )} x^{12} +{\left (3 \, a b^{2} c^{2} - 4 \, a^{2} b c d\right )} x^{4}\right )} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{8} - a c}{2 \, \sqrt{d x^{8} + c} \sqrt{a b c - a^{2} d} x^{4}}\right )}{16 \,{\left ({\left (a^{2} b^{2} c^{2} - a^{3} b c d\right )} x^{12} +{\left (a^{3} b c^{2} - a^{4} c d\right )} x^{4}\right )} \sqrt{a b c - a^{2} d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^8 + a)^2*sqrt(d*x^8 + c)*x^5),x, algorithm="fricas")

[Out]

[-1/32*(4*((3*b^2*c - 2*a*b*d)*x^8 + 2*a*b*c - 2*a^2*d)*sqrt(d*x^8 + c)*sqrt(-a*
b*c + a^2*d) - ((3*b^3*c^2 - 4*a*b^2*c*d)*x^12 + (3*a*b^2*c^2 - 4*a^2*b*c*d)*x^4
)*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^2*b^2*c^2 - a^3*b*c*d)*x^12 + (a^3*b*c^2 - a^4*c*d)*x^4)*sqrt(-a*b*c + a^2*d))
, -1/16*(2*((3*b^2*c - 2*a*b*d)*x^8 + 2*a*b*c - 2*a^2*d)*sqrt(d*x^8 + c)*sqrt(a*
b*c - a^2*d) + ((3*b^3*c^2 - 4*a*b^2*c*d)*x^12 + (3*a*b^2*c^2 - 4*a^2*b*c*d)*x^4
)*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^2*b^2*c^2 - a^3*b*c*d)*x^12 + (a^3*b*c^2 - a^4*c*d)*x^4)*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**5/(b*x**8+a)**2/(d*x**8+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.234678, size = 181, normalized size = 1.21 \[ -\frac{b^{2} c \sqrt{d + \frac{c}{x^{8}}}}{8 \,{\left (a^{2} b c - a^{3} d\right )}{\left (b c + a{\left (d + \frac{c}{x^{8}}\right )} - a d\right )}} + \frac{{\left (3 \, b^{2} c - 4 \, a b d\right )} \arctan \left (\frac{a \sqrt{d + \frac{c}{x^{8}}}}{\sqrt{a b c - a^{2} d}}\right )}{8 \,{\left (a^{2} b c - a^{3} d\right )} \sqrt{a b c - a^{2} d}} - \frac{\sqrt{d + \frac{c}{x^{8}}}}{4 \, a^{2} c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^8 + a)^2*sqrt(d*x^8 + c)*x^5),x, algorithm="giac")

[Out]

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