3.654 \(\int \frac{x^{11}}{\left (a+b x^4\right )^2 \sqrt{c+d x^4}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.37161, 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 \[ -\frac{a^2 \sqrt{c+d x^4}}{4 b^2 \left (a+b x^4\right ) (b c-a d)}+\frac{a (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{4 b^{5/2} (b c-a d)^{3/2}}+\frac{\sqrt{c+d x^4}}{2 b^2 d} \]

Antiderivative was successfully verified.

[In]  Int[x^11/((a + b*x^4)^2*Sqrt[c + d*x^4]),x]

[Out]

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

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Rubi in Sympy [A]  time = 35.2634, size = 104, normalized size = 0.85 \[ \frac{a^{2} \sqrt{c + d x^{4}}}{4 b^{2} \left (a + b x^{4}\right ) \left (a d - b c\right )} - \frac{a \left (3 a d - 4 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{4}}}{\sqrt{a d - b c}} \right )}}{4 b^{\frac{5}{2}} \left (a d - b c\right )^{\frac{3}{2}}} + \frac{\sqrt{c + d x^{4}}}{2 b^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**11/(b*x**4+a)**2/(d*x**4+c)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.369529, size = 107, normalized size = 0.87 \[ \frac{1}{4} \left (\frac{\sqrt{c+d x^4} \left (\frac{a^2}{\left (a+b x^4\right ) (a d-b c)}+\frac{2}{d}\right )}{b^2}+\frac{a (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{b^{5/2} (b c-a d)^{3/2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^11/((a + b*x^4)^2*Sqrt[c + d*x^4]),x]

[Out]

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

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Maple [B]  time = 0.02, size = 876, normalized size = 7.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11/(b*x^4+a)^2/(d*x^4+c)^(1/2),x)

[Out]

1/2*(d*x^4+c)^(1/2)/b^2/d-1/8*a/b^3*(-a*b)^(1/2)/(a*d-b*c)/(x^2-1/b*(-a*b)^(1/2)
)*((x^2-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x^2-1/b*(-a*b)^(1/2))-(a*d-b*c
)/b)^(1/2)-1/8*a^2/b^3*d/(a*d-b*c)/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b+2*d*(
-a*b)^(1/2)/b*(x^2-1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x^2-1/b*(-a*b)^(1/
2))^2*d+2*d*(-a*b)^(1/2)/b*(x^2-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x^2-1/b*(
-a*b)^(1/2)))+1/8*a/b^3*(-a*b)^(1/2)/(a*d-b*c)/(x^2+1/b*(-a*b)^(1/2))*((x^2+1/b*
(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x^2+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)-1
/8*a^2/b^3*d/(a*d-b*c)/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/
b*(x^2+1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x^2+1/b*(-a*b)^(1/2))^2*d-2*d*
(-a*b)^(1/2)/b*(x^2+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x^2+1/b*(-a*b)^(1/2))
)+1/2*a/b^3/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b+2*d*(-a*b)^(1/2)/b*(x^2-1/b*
(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x^2-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2
)/b*(x^2-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x^2-1/b*(-a*b)^(1/2)))+1/2*a/b^3
/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(x^2+1/b*(-a*b)^(1/2
))+2*(-(a*d-b*c)/b)^(1/2)*((x^2+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x^2+1/
b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x^2+1/b*(-a*b)^(1/2)))

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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^4 + a)^2*sqrt(d*x^4 + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.234089, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (2 \,{\left (b^{2} c - a b d\right )} x^{4} + 2 \, a b c - 3 \, a^{2} d\right )} \sqrt{d x^{4} + c} \sqrt{b^{2} c - a b d} +{\left (4 \, a^{2} b c d - 3 \, a^{3} d^{2} +{\left (4 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x^{4}\right )} \log \left (\frac{{\left (b d x^{4} + 2 \, b c - a d\right )} \sqrt{b^{2} c - a b d} + 2 \, \sqrt{d x^{4} + c}{\left (b^{2} c - a b d\right )}}{b x^{4} + a}\right )}{8 \,{\left (a b^{3} c d - a^{2} b^{2} d^{2} +{\left (b^{4} c d - a b^{3} d^{2}\right )} x^{4}\right )} \sqrt{b^{2} c - a b d}}, \frac{{\left (2 \,{\left (b^{2} c - a b d\right )} x^{4} + 2 \, a b c - 3 \, a^{2} d\right )} \sqrt{d x^{4} + c} \sqrt{-b^{2} c + a b d} +{\left (4 \, a^{2} b c d - 3 \, a^{3} d^{2} +{\left (4 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x^{4}\right )} \arctan \left (-\frac{b c - a d}{\sqrt{d x^{4} + c} \sqrt{-b^{2} c + a b d}}\right )}{4 \,{\left (a b^{3} c d - a^{2} b^{2} d^{2} +{\left (b^{4} c d - a b^{3} d^{2}\right )} x^{4}\right )} \sqrt{-b^{2} c + a b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/((b*x^4 + a)^2*sqrt(d*x^4 + c)),x, algorithm="fricas")

[Out]

[1/8*(2*(2*(b^2*c - a*b*d)*x^4 + 2*a*b*c - 3*a^2*d)*sqrt(d*x^4 + c)*sqrt(b^2*c -
a*b*d) + (4*a^2*b*c*d - 3*a^3*d^2 + (4*a*b^2*c*d - 3*a^2*b*d^2)*x^4)*log(((b*d*
x^4 + 2*b*c - a*d)*sqrt(b^2*c - a*b*d) + 2*sqrt(d*x^4 + c)*(b^2*c - a*b*d))/(b*x
^4 + a)))/((a*b^3*c*d - a^2*b^2*d^2 + (b^4*c*d - a*b^3*d^2)*x^4)*sqrt(b^2*c - a*
b*d)), 1/4*((2*(b^2*c - a*b*d)*x^4 + 2*a*b*c - 3*a^2*d)*sqrt(d*x^4 + c)*sqrt(-b^
2*c + a*b*d) + (4*a^2*b*c*d - 3*a^3*d^2 + (4*a*b^2*c*d - 3*a^2*b*d^2)*x^4)*arcta
n(-(b*c - a*d)/(sqrt(d*x^4 + c)*sqrt(-b^2*c + a*b*d))))/((a*b^3*c*d - a^2*b^2*d^
2 + (b^4*c*d - a*b^3*d^2)*x^4)*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**11/(b*x**4+a)**2/(d*x**4+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.216983, size = 181, normalized size = 1.47 \[ -\frac{\sqrt{d x^{4} + c} a^{2} d}{4 \,{\left (b^{3} c - a b^{2} d\right )}{\left ({\left (d x^{4} + c\right )} b - b c + a d\right )}} - \frac{{\left (4 \, a b c - 3 \, a^{2} d\right )} \arctan \left (\frac{\sqrt{d x^{4} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{4 \,{\left (b^{3} c - a b^{2} d\right )} \sqrt{-b^{2} c + a b d}} + \frac{\sqrt{d x^{4} + c}}{2 \, b^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/((b*x^4 + a)^2*sqrt(d*x^4 + c)),x, algorithm="giac")

[Out]

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