3.685 \(\int \frac{x^{11}}{\left (a+b x^6\right ) \sqrt{c+d x^6}} \, dx\)

Optimal. Leaf size=74 \[ \frac{a \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right )}{3 b^{3/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^6}}{3 b d} \]

[Out]

Sqrt[c + d*x^6]/(3*b*d) + (a*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^6])/Sqrt[b*c - a*d]])
/(3*b^(3/2)*Sqrt[b*c - a*d])

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Rubi [A]  time = 0.183051, antiderivative size = 74, 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 \[ \frac{a \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right )}{3 b^{3/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^6}}{3 b d} \]

Antiderivative was successfully verified.

[In]  Int[x^11/((a + b*x^6)*Sqrt[c + d*x^6]),x]

[Out]

Sqrt[c + d*x^6]/(3*b*d) + (a*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^6])/Sqrt[b*c - a*d]])
/(3*b^(3/2)*Sqrt[b*c - a*d])

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Rubi in Sympy [A]  time = 19.3827, size = 60, normalized size = 0.81 \[ - \frac{a \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{6}}}{\sqrt{a d - b c}} \right )}}{3 b^{\frac{3}{2}} \sqrt{a d - b c}} + \frac{\sqrt{c + d x^{6}}}{3 b d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**11/(b*x**6+a)/(d*x**6+c)**(1/2),x)

[Out]

-a*atan(sqrt(b)*sqrt(c + d*x**6)/sqrt(a*d - b*c))/(3*b**(3/2)*sqrt(a*d - b*c)) +
sqrt(c + d*x**6)/(3*b*d)

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Mathematica [A]  time = 0.0927724, size = 74, normalized size = 1. \[ \frac{a \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right )}{3 b^{3/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^6}}{3 b d} \]

Antiderivative was successfully verified.

[In]  Integrate[x^11/((a + b*x^6)*Sqrt[c + d*x^6]),x]

[Out]

Sqrt[c + d*x^6]/(3*b*d) + (a*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^6])/Sqrt[b*c - a*d]])
/(3*b^(3/2)*Sqrt[b*c - a*d])

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Maple [F]  time = 0.081, size = 0, normalized size = 0. \[ \int{\frac{{x}^{11}}{b{x}^{6}+a}{\frac{1}{\sqrt{d{x}^{6}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11/(b*x^6+a)/(d*x^6+c)^(1/2),x)

[Out]

int(x^11/(b*x^6+a)/(d*x^6+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^6 + a)*sqrt(d*x^6 + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.218021, size = 1, normalized size = 0.01 \[ \left [\frac{a d \log \left (\frac{{\left (b d x^{6} + 2 \, b c - a d\right )} \sqrt{b^{2} c - a b d} + 2 \, \sqrt{d x^{6} + c}{\left (b^{2} c - a b d\right )}}{b x^{6} + a}\right ) + 2 \, \sqrt{d x^{6} + c} \sqrt{b^{2} c - a b d}}{6 \, \sqrt{b^{2} c - a b d} b d}, \frac{a d \arctan \left (-\frac{b c - a d}{\sqrt{d x^{6} + c} \sqrt{-b^{2} c + a b d}}\right ) + \sqrt{d x^{6} + c} \sqrt{-b^{2} c + a b d}}{3 \, \sqrt{-b^{2} c + a b d} b d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/((b*x^6 + a)*sqrt(d*x^6 + c)),x, algorithm="fricas")

[Out]

[1/6*(a*d*log(((b*d*x^6 + 2*b*c - a*d)*sqrt(b^2*c - a*b*d) + 2*sqrt(d*x^6 + c)*(
b^2*c - a*b*d))/(b*x^6 + a)) + 2*sqrt(d*x^6 + c)*sqrt(b^2*c - a*b*d))/(sqrt(b^2*
c - a*b*d)*b*d), 1/3*(a*d*arctan(-(b*c - a*d)/(sqrt(d*x^6 + c)*sqrt(-b^2*c + a*b
*d))) + sqrt(d*x^6 + c)*sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*b*d)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{11}}{\left (a + b x^{6}\right ) \sqrt{c + d x^{6}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**11/(b*x**6+a)/(d*x**6+c)**(1/2),x)

[Out]

Integral(x**11/((a + b*x**6)*sqrt(c + d*x**6)), x)

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GIAC/XCAS [A]  time = 0.212741, size = 86, normalized size = 1.16 \[ -\frac{\frac{a d \arctan \left (\frac{\sqrt{d x^{6} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} b} - \frac{\sqrt{d x^{6} + c}}{b}}{3 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/((b*x^6 + a)*sqrt(d*x^6 + c)),x, algorithm="giac")

[Out]

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