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

Optimal. Leaf size=115 \[ \frac{b^2 \tan ^{-1}\left (\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right )}{4 a^{5/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^8} (2 a d+3 b c)}{12 a^2 c^2 x^4}-\frac{\sqrt{c+d x^8}}{12 a c x^{12}} \]

[Out]

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

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Rubi [A]  time = 0.470028, antiderivative size = 115, 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 \[ \frac{b^2 \tan ^{-1}\left (\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right )}{4 a^{5/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^8} (2 a d+3 b c)}{12 a^2 c^2 x^4}-\frac{\sqrt{c+d x^8}}{12 a c x^{12}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 68.671, size = 100, normalized size = 0.87 \[ - \frac{\sqrt{c + d x^{8}}}{12 a c x^{12}} + \frac{\sqrt{c + d x^{8}} \left (2 a d + 3 b c\right )}{12 a^{2} c^{2} x^{4}} + \frac{b^{2} \operatorname{atanh}{\left (\frac{x^{4} \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{8}}} \right )}}{4 a^{\frac{5}{2}} \sqrt{a d - b c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 1.29961, size = 149, normalized size = 1.3 \[ \frac{\sqrt{c+d x^8} \left (-a^2 c+\frac{3 b^2 c x^{16} \sin ^{-1}\left (\frac{\sqrt{x^8 \left (\frac{b}{a}-\frac{d}{c}\right )}}{\sqrt{\frac{b x^8}{a}+1}}\right )}{\sqrt{\frac{b x^8}{a}+1} \sqrt{x^8 \left (\frac{b}{a}-\frac{d}{c}\right )} \sqrt{\frac{a \left (c+d x^8\right )}{c \left (a+b x^8\right )}}}+a x^8 (2 a d+3 b c)\right )}{12 a^3 c^2 x^{12}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.103, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{13} \left ( b{x}^{8}+a \right ) }{\frac{1}{\sqrt{d{x}^{8}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.320611, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b^{2} c^{2} x^{12} \log \left (\frac{4 \,{\left ({\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} x^{12} -{\left (a^{2} b c^{2} - a^{3} c d\right )} x^{4}\right )} \sqrt{d x^{8} + c} +{\left ({\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{16} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{8} + a^{2} c^{2}\right )} \sqrt{-a b c + a^{2} d}}{b^{2} x^{16} + 2 \, a b x^{8} + a^{2}}\right ) + 4 \,{\left ({\left (3 \, b c + 2 \, a d\right )} x^{8} - a c\right )} \sqrt{d x^{8} + c} \sqrt{-a b c + a^{2} d}}{48 \, \sqrt{-a b c + a^{2} d} a^{2} c^{2} x^{12}}, \frac{3 \, b^{2} c^{2} x^{12} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{8} - a c}{2 \, \sqrt{d x^{8} + c} \sqrt{a b c - a^{2} d} x^{4}}\right ) + 2 \,{\left ({\left (3 \, b c + 2 \, a d\right )} x^{8} - a c\right )} \sqrt{d x^{8} + c} \sqrt{a b c - a^{2} d}}{24 \, \sqrt{a b c - a^{2} d} a^{2} c^{2} x^{12}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/48*(3*b^2*c^2*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)) + 4*((3*b*c + 2*a*d)*x^8 - a*c)*sqrt(d*x^8 + c)*sqrt(-a*b*c + a^2
*d))/(sqrt(-a*b*c + a^2*d)*a^2*c^2*x^12), 1/24*(3*b^2*c^2*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)) + 2*((3*b*c + 2*a
*d)*x^8 - a*c)*sqrt(d*x^8 + c)*sqrt(a*b*c - a^2*d))/(sqrt(a*b*c - a^2*d)*a^2*c^2
*x^12)]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.228, size = 144, normalized size = 1.25 \[ -\frac{b^{2} \arctan \left (\frac{a \sqrt{d + \frac{c}{x^{8}}}}{\sqrt{a b c - a^{2} d}}\right )}{4 \, \sqrt{a b c - a^{2} d} a^{2}} + \frac{3 \, a b c^{5} \sqrt{d + \frac{c}{x^{8}}} - a^{2} c^{4}{\left (d + \frac{c}{x^{8}}\right )}^{\frac{3}{2}} + 3 \, a^{2} c^{4} \sqrt{d + \frac{c}{x^{8}}} d}{12 \, a^{3} c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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