3.701 \(\int \frac{x^{17}}{\left (a+b x^6\right )^2 \sqrt{c+d x^6}} \, dx\)

Optimal. Leaf size=123 \[ -\frac{a^2 \sqrt{c+d x^6}}{6 b^2 \left (a+b x^6\right ) (b c-a d)}+\frac{a (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right )}{6 b^{5/2} (b c-a d)^{3/2}}+\frac{\sqrt{c+d x^6}}{3 b^2 d} \]

[Out]

Sqrt[c + d*x^6]/(3*b^2*d) - (a^2*Sqrt[c + d*x^6])/(6*b^2*(b*c - a*d)*(a + b*x^6)
) + (a*(4*b*c - 3*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^6])/Sqrt[b*c - a*d]])/(6*b^
(5/2)*(b*c - a*d)^(3/2))

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Rubi [A]  time = 0.421695, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{a^2 \sqrt{c+d x^6}}{6 b^2 \left (a+b x^6\right ) (b c-a d)}+\frac{a (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right )}{6 b^{5/2} (b c-a d)^{3/2}}+\frac{\sqrt{c+d x^6}}{3 b^2 d} \]

Antiderivative was successfully verified.

[In]  Int[x^17/((a + b*x^6)^2*Sqrt[c + d*x^6]),x]

[Out]

Sqrt[c + d*x^6]/(3*b^2*d) - (a^2*Sqrt[c + d*x^6])/(6*b^2*(b*c - a*d)*(a + b*x^6)
) + (a*(4*b*c - 3*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^6])/Sqrt[b*c - a*d]])/(6*b^
(5/2)*(b*c - a*d)^(3/2))

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Rubi in Sympy [A]  time = 35.273, size = 104, normalized size = 0.85 \[ \frac{a^{2} \sqrt{c + d x^{6}}}{6 b^{2} \left (a + b x^{6}\right ) \left (a d - b c\right )} - \frac{a \left (3 a d - 4 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{6}}}{\sqrt{a d - b c}} \right )}}{6 b^{\frac{5}{2}} \left (a d - b c\right )^{\frac{3}{2}}} + \frac{\sqrt{c + d x^{6}}}{3 b^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**17/(b*x**6+a)**2/(d*x**6+c)**(1/2),x)

[Out]

a**2*sqrt(c + d*x**6)/(6*b**2*(a + b*x**6)*(a*d - b*c)) - a*(3*a*d - 4*b*c)*atan
(sqrt(b)*sqrt(c + d*x**6)/sqrt(a*d - b*c))/(6*b**(5/2)*(a*d - b*c)**(3/2)) + sqr
t(c + d*x**6)/(3*b**2*d)

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Mathematica [A]  time = 0.380275, size = 107, normalized size = 0.87 \[ \frac{1}{6} \left (\frac{\sqrt{c+d x^6} \left (\frac{a^2}{\left (a+b x^6\right ) (a d-b c)}+\frac{2}{d}\right )}{b^2}+\frac{a (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right )}{b^{5/2} (b c-a d)^{3/2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^17/((a + b*x^6)^2*Sqrt[c + d*x^6]),x]

[Out]

((Sqrt[c + d*x^6]*(2/d + a^2/((-(b*c) + a*d)*(a + b*x^6))))/b^2 + (a*(4*b*c - 3*
a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^6])/Sqrt[b*c - a*d]])/(b^(5/2)*(b*c - a*d)^(3
/2)))/6

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Maple [F]  time = 0.117, size = 0, normalized size = 0. \[ \int{\frac{{x}^{17}}{ \left ( b{x}^{6}+a \right ) ^{2}}{\frac{1}{\sqrt{d{x}^{6}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^17/(b*x^6+a)^2/(d*x^6+c)^(1/2),x)

[Out]

int(x^17/(b*x^6+a)^2/(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^17/((b*x^6 + a)^2*sqrt(d*x^6 + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.234718, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (2 \,{\left (b^{2} c - a b d\right )} x^{6} + 2 \, a b c - 3 \, a^{2} d\right )} \sqrt{d x^{6} + c} \sqrt{b^{2} c - a b d} +{\left ({\left (4 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x^{6} + 4 \, a^{2} b c d - 3 \, a^{3} d^{2}\right )} \log \left (\frac{{\left (b d x^{6} + 2 \, b c - a d\right )} \sqrt{b^{2} c - a b d} + 2 \, \sqrt{d x^{6} + c}{\left (b^{2} c - a b d\right )}}{b x^{6} + a}\right )}{12 \,{\left ({\left (b^{4} c d - a b^{3} d^{2}\right )} x^{6} + a b^{3} c d - a^{2} b^{2} d^{2}\right )} \sqrt{b^{2} c - a b d}}, \frac{{\left (2 \,{\left (b^{2} c - a b d\right )} x^{6} + 2 \, a b c - 3 \, a^{2} d\right )} \sqrt{d x^{6} + c} \sqrt{-b^{2} c + a b d} +{\left ({\left (4 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x^{6} + 4 \, a^{2} b c d - 3 \, a^{3} d^{2}\right )} \arctan \left (-\frac{b c - a d}{\sqrt{d x^{6} + c} \sqrt{-b^{2} c + a b d}}\right )}{6 \,{\left ({\left (b^{4} c d - a b^{3} d^{2}\right )} x^{6} + a b^{3} c d - a^{2} b^{2} d^{2}\right )} \sqrt{-b^{2} c + a b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^17/((b*x^6 + a)^2*sqrt(d*x^6 + c)),x, algorithm="fricas")

[Out]

[1/12*(2*(2*(b^2*c - a*b*d)*x^6 + 2*a*b*c - 3*a^2*d)*sqrt(d*x^6 + c)*sqrt(b^2*c
- a*b*d) + ((4*a*b^2*c*d - 3*a^2*b*d^2)*x^6 + 4*a^2*b*c*d - 3*a^3*d^2)*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)))/(((b^4*c*d - a*b^3*d^2)*x^6 + a*b^3*c*d - a^2*b^2*d^2)*sqrt(b^2*c - a
*b*d)), 1/6*((2*(b^2*c - a*b*d)*x^6 + 2*a*b*c - 3*a^2*d)*sqrt(d*x^6 + c)*sqrt(-b
^2*c + a*b*d) + ((4*a*b^2*c*d - 3*a^2*b*d^2)*x^6 + 4*a^2*b*c*d - 3*a^3*d^2)*arct
an(-(b*c - a*d)/(sqrt(d*x^6 + c)*sqrt(-b^2*c + a*b*d))))/(((b^4*c*d - a*b^3*d^2)
*x^6 + a*b^3*c*d - a^2*b^2*d^2)*sqrt(-b^2*c + a*b*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**17/(b*x**6+a)**2/(d*x**6+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.216553, size = 181, normalized size = 1.47 \[ -\frac{\sqrt{d x^{6} + c} a^{2} d}{6 \,{\left (b^{3} c - a b^{2} d\right )}{\left ({\left (d x^{6} + c\right )} b - b c + a d\right )}} - \frac{{\left (4 \, a b c - 3 \, a^{2} d\right )} \arctan \left (\frac{\sqrt{d x^{6} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{6 \,{\left (b^{3} c - a b^{2} d\right )} \sqrt{-b^{2} c + a b d}} + \frac{\sqrt{d x^{6} + c}}{3 \, b^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^17/((b*x^6 + a)^2*sqrt(d*x^6 + c)),x, algorithm="giac")

[Out]

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