3.724 \(\int \frac{x^{11}}{\left (a+b x^8\right ) \sqrt{c+d x^8}} \, dx\)

Optimal. Leaf size=91 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{d} x^4}{\sqrt{c+d x^8}}\right )}{4 b \sqrt{d}}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right )}{4 b \sqrt{b c-a d}} \]

[Out]

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

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Rubi [A]  time = 0.254034, antiderivative size = 91, 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 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{d} x^4}{\sqrt{c+d x^8}}\right )}{4 b \sqrt{d}}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right )}{4 b \sqrt{b c-a d}} \]

Antiderivative was successfully verified.

[In]  Int[x^11/((a + b*x^8)*Sqrt[c + d*x^8]),x]

[Out]

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

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Rubi in Sympy [A]  time = 32.7051, size = 76, normalized size = 0.84 \[ - \frac{\sqrt{a} \operatorname{atanh}{\left (\frac{x^{4} \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{8}}} \right )}}{4 b \sqrt{a d - b c}} + \frac{\operatorname{atanh}{\left (\frac{\sqrt{d} x^{4}}{\sqrt{c + d x^{8}}} \right )}}{4 b \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**11/(b*x**8+a)/(d*x**8+c)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0761412, size = 90, normalized size = 0.99 \[ \frac{\frac{\log \left (\sqrt{d} \sqrt{c+d x^8}+d x^4\right )}{\sqrt{d}}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right )}{\sqrt{b c-a d}}}{4 b} \]

Antiderivative was successfully verified.

[In]  Integrate[x^11/((a + b*x^8)*Sqrt[c + d*x^8]),x]

[Out]

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

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \[ \int{\frac{{x}^{11}}{b{x}^{8}+a}{\frac{1}{\sqrt{d{x}^{8}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11/(b*x^8+a)/(d*x^8+c)^(1/2),x)

[Out]

int(x^11/(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^11/((b*x^8 + a)*sqrt(d*x^8 + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.312895, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{d} \sqrt{-\frac{a}{b c - a d}} \log \left (\frac{{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{16} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{8} + a^{2} c^{2} - 4 \,{\left ({\left (b^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{12} -{\left (a b c^{2} - a^{2} c d\right )} x^{4}\right )} \sqrt{d x^{8} + c} \sqrt{-\frac{a}{b c - a d}}}{b^{2} x^{16} + 2 \, a b x^{8} + a^{2}}\right ) + 2 \, \log \left (-2 \, \sqrt{d x^{8} + c} d x^{4} -{\left (2 \, d x^{8} + c\right )} \sqrt{d}\right )}{16 \, b \sqrt{d}}, \frac{\sqrt{-d} \sqrt{-\frac{a}{b c - a d}} \log \left (\frac{{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{16} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{8} + a^{2} c^{2} - 4 \,{\left ({\left (b^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{12} -{\left (a b c^{2} - a^{2} c d\right )} x^{4}\right )} \sqrt{d x^{8} + c} \sqrt{-\frac{a}{b c - a d}}}{b^{2} x^{16} + 2 \, a b x^{8} + a^{2}}\right ) + 4 \, \arctan \left (\frac{\sqrt{-d} x^{4}}{\sqrt{d x^{8} + c}}\right )}{16 \, b \sqrt{-d}}, -\frac{\sqrt{d} \sqrt{\frac{a}{b c - a d}} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{8} - a c}{2 \, \sqrt{d x^{8} + c}{\left (b c - a d\right )} x^{4} \sqrt{\frac{a}{b c - a d}}}\right ) - \log \left (-2 \, \sqrt{d x^{8} + c} d x^{4} -{\left (2 \, d x^{8} + c\right )} \sqrt{d}\right )}{8 \, b \sqrt{d}}, -\frac{\sqrt{-d} \sqrt{\frac{a}{b c - a d}} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{8} - a c}{2 \, \sqrt{d x^{8} + c}{\left (b c - a d\right )} x^{4} \sqrt{\frac{a}{b c - a d}}}\right ) - 2 \, \arctan \left (\frac{\sqrt{-d} x^{4}}{\sqrt{d x^{8} + c}}\right )}{8 \, b \sqrt{-d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/((b*x^8 + a)*sqrt(d*x^8 + c)),x, algorithm="fricas")

[Out]

[1/16*(sqrt(d)*sqrt(-a/(b*c - a*d))*log(((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 - 4*((b^2*c^2 - 3*a*b*c*d + 2*a^2*d^2)
*x^12 - (a*b*c^2 - a^2*c*d)*x^4)*sqrt(d*x^8 + c)*sqrt(-a/(b*c - a*d)))/(b^2*x^16
+ 2*a*b*x^8 + a^2)) + 2*log(-2*sqrt(d*x^8 + c)*d*x^4 - (2*d*x^8 + c)*sqrt(d)))/
(b*sqrt(d)), 1/16*(sqrt(-d)*sqrt(-a/(b*c - a*d))*log(((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 - 4*((b^2*c^2 - 3*a*b*c*d
+ 2*a^2*d^2)*x^12 - (a*b*c^2 - a^2*c*d)*x^4)*sqrt(d*x^8 + c)*sqrt(-a/(b*c - a*d
)))/(b^2*x^16 + 2*a*b*x^8 + a^2)) + 4*arctan(sqrt(-d)*x^4/sqrt(d*x^8 + c)))/(b*s
qrt(-d)), -1/8*(sqrt(d)*sqrt(a/(b*c - a*d))*arctan(1/2*((b*c - 2*a*d)*x^8 - a*c)
/(sqrt(d*x^8 + c)*(b*c - a*d)*x^4*sqrt(a/(b*c - a*d)))) - log(-2*sqrt(d*x^8 + c)
*d*x^4 - (2*d*x^8 + c)*sqrt(d)))/(b*sqrt(d)), -1/8*(sqrt(-d)*sqrt(a/(b*c - a*d))
*arctan(1/2*((b*c - 2*a*d)*x^8 - a*c)/(sqrt(d*x^8 + c)*(b*c - a*d)*x^4*sqrt(a/(b
*c - a*d)))) - 2*arctan(sqrt(-d)*x^4/sqrt(d*x^8 + c)))/(b*sqrt(-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**11/(b*x**8+a)/(d*x**8+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.23311, size = 107, normalized size = 1.18 \[ \frac{1}{4} \, c{\left (\frac{a \arctan \left (\frac{a \sqrt{d + \frac{c}{x^{8}}}}{\sqrt{a b c - a^{2} d}}\right )}{\sqrt{a b c - a^{2} d} b c} - \frac{\arctan \left (\frac{\sqrt{d + \frac{c}{x^{8}}}}{\sqrt{-d}}\right )}{b c \sqrt{-d}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/((b*x^8 + a)*sqrt(d*x^8 + c)),x, algorithm="giac")

[Out]

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