3.689 \(\int \frac{x^{14}}{\left (a+b x^6\right ) \sqrt{c+d x^6}} \, dx\)

Optimal. Leaf size=123 \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right )}{3 b^2 \sqrt{b c-a d}}-\frac{(2 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} x^3}{\sqrt{c+d x^6}}\right )}{6 b^2 d^{3/2}}+\frac{x^3 \sqrt{c+d x^6}}{6 b d} \]

[Out]

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

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Rubi [A]  time = 0.399147, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right )}{3 b^2 \sqrt{b c-a d}}-\frac{(2 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} x^3}{\sqrt{c+d x^6}}\right )}{6 b^2 d^{3/2}}+\frac{x^3 \sqrt{c+d x^6}}{6 b d} \]

Antiderivative was successfully verified.

[In]  Int[x^14/((a + b*x^6)*Sqrt[c + d*x^6]),x]

[Out]

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

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Rubi in Sympy [A]  time = 45.7396, size = 107, normalized size = 0.87 \[ \frac{a^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{x^{3} \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{6}}} \right )}}{3 b^{2} \sqrt{a d - b c}} + \frac{x^{3} \sqrt{c + d x^{6}}}{6 b d} - \frac{\left (2 a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x^{3}}{\sqrt{c + d x^{6}}} \right )}}{6 b^{2} d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**14/(b*x**6+a)/(d*x**6+c)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.248955, size = 118, normalized size = 0.96 \[ \frac{\frac{2 a^{3/2} \tan ^{-1}\left (\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right )}{\sqrt{b c-a d}}-\frac{(2 a d+b c) \log \left (\sqrt{d} \sqrt{c+d x^6}+d x^3\right )}{d^{3/2}}+\frac{b x^3 \sqrt{c+d x^6}}{d}}{6 b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^14/((a + b*x^6)*Sqrt[c + d*x^6]),x]

[Out]

((b*x^3*Sqrt[c + d*x^6])/d + (2*a^(3/2)*ArcTan[(Sqrt[b*c - a*d]*x^3)/(Sqrt[a]*Sq
rt[c + d*x^6])])/Sqrt[b*c - a*d] - ((b*c + 2*a*d)*Log[d*x^3 + Sqrt[d]*Sqrt[c + d
*x^6]])/d^(3/2))/(6*b^2)

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Maple [F]  time = 0.115, size = 0, normalized size = 0. \[ \int{\frac{{x}^{14}}{b{x}^{6}+a}{\frac{1}{\sqrt{d{x}^{6}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^14/(b*x^6+a)/(d*x^6+c)^(1/2),x)

[Out]

int(x^14/(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^14/((b*x^6 + a)*sqrt(d*x^6 + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.36286, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^14/((b*x^6 + a)*sqrt(d*x^6 + c)),x, algorithm="fricas")

[Out]

[1/12*(2*sqrt(d*x^6 + c)*b*sqrt(d)*x^3 + a*d^(3/2)*sqrt(-a/(b*c - a*d))*log(((b^
2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^12 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^6 + a^2*c^2 +
4*((b^2*c^2 - 3*a*b*c*d + 2*a^2*d^2)*x^9 - (a*b*c^2 - a^2*c*d)*x^3)*sqrt(d*x^6 +
c)*sqrt(-a/(b*c - a*d)))/(b^2*x^12 + 2*a*b*x^6 + a^2)) + (b*c + 2*a*d)*log(2*sq
rt(d*x^6 + c)*d*x^3 - (2*d*x^6 + c)*sqrt(d)))/(b^2*d^(3/2)), 1/12*(2*sqrt(d*x^6
+ c)*b*sqrt(-d)*x^3 + a*sqrt(-d)*d*sqrt(-a/(b*c - a*d))*log(((b^2*c^2 - 8*a*b*c*
d + 8*a^2*d^2)*x^12 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^6 + a^2*c^2 + 4*((b^2*c^2 - 3*
a*b*c*d + 2*a^2*d^2)*x^9 - (a*b*c^2 - a^2*c*d)*x^3)*sqrt(d*x^6 + c)*sqrt(-a/(b*c
- a*d)))/(b^2*x^12 + 2*a*b*x^6 + a^2)) - 2*(b*c + 2*a*d)*arctan(sqrt(-d)*x^3/sq
rt(d*x^6 + c)))/(b^2*sqrt(-d)*d), 1/12*(2*sqrt(d*x^6 + c)*b*sqrt(d)*x^3 + 2*a*d^
(3/2)*sqrt(a/(b*c - a*d))*arctan(1/2*((b*c - 2*a*d)*x^6 - a*c)/(sqrt(d*x^6 + c)*
(b*c - a*d)*x^3*sqrt(a/(b*c - a*d)))) + (b*c + 2*a*d)*log(2*sqrt(d*x^6 + c)*d*x^
3 - (2*d*x^6 + c)*sqrt(d)))/(b^2*d^(3/2)), 1/6*(sqrt(d*x^6 + c)*b*sqrt(-d)*x^3 +
a*sqrt(-d)*d*sqrt(a/(b*c - a*d))*arctan(1/2*((b*c - 2*a*d)*x^6 - a*c)/(sqrt(d*x
^6 + c)*(b*c - a*d)*x^3*sqrt(a/(b*c - a*d)))) - (b*c + 2*a*d)*arctan(sqrt(-d)*x^
3/sqrt(d*x^6 + c)))/(b^2*sqrt(-d)*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**14/(b*x**6+a)/(d*x**6+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{14}}{{\left (b x^{6} + a\right )} \sqrt{d x^{6} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^14/((b*x^6 + a)*sqrt(d*x^6 + c)),x, algorithm="giac")

[Out]

integrate(x^14/((b*x^6 + a)*sqrt(d*x^6 + c)), x)