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

Optimal. Leaf size=104 \[ -\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{2 b^{5/2} \sqrt{b c-a d}}-\frac{\sqrt{c+d x^4} (a d+b c)}{2 b^2 d^2}+\frac{\left (c+d x^4\right )^{3/2}}{6 b d^2} \]

[Out]

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

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Rubi [A]  time = 0.285243, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{2 b^{5/2} \sqrt{b c-a d}}-\frac{\sqrt{c+d x^4} (a d+b c)}{2 b^2 d^2}+\frac{\left (c+d x^4\right )^{3/2}}{6 b d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 30.4491, size = 88, normalized size = 0.85 \[ \frac{a^{2} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{4}}}{\sqrt{a d - b c}} \right )}}{2 b^{\frac{5}{2}} \sqrt{a d - b c}} + \frac{\left (c + d x^{4}\right )^{\frac{3}{2}}}{6 b d^{2}} - \frac{\sqrt{c + d x^{4}} \left (a d + b c\right )}{2 b^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.266707, size = 91, normalized size = 0.88 \[ \frac{\sqrt{c+d x^4} \left (-3 a d-2 b c+b d x^4\right )}{6 b^2 d^2}-\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{2 b^{5/2} \sqrt{b c-a d}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.035, size = 378, normalized size = 3.6 \[{\frac{{x}^{4}}{6\,bd}\sqrt{d{x}^{4}+c}}-{\frac{c}{3\,b{d}^{2}}\sqrt{d{x}^{4}+c}}-{\frac{a}{2\,{b}^{2}d}\sqrt{d{x}^{4}+c}}-{\frac{{a}^{2}}{4\,{b}^{3}}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}+2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d+2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ({x}^{2}-{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}}-{\frac{{a}^{2}}{4\,{b}^{3}}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}-2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d-2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ({x}^{2}+{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6/b*(d*x^4+c)^(1/2)/d*x^4-1/3/b*(d*x^4+c)^(1/2)/d^2*c-1/2/b^2*a/d*(d*x^4+c)^(1
/2)-1/4*a^2/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)))-1/4*a
^2/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)*sqrt(d*x^4 + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.239126, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a^{2} d^{2} \log \left (\frac{{\left (b d x^{4} + 2 \, b c - a d\right )} \sqrt{b^{2} c - a b d} - 2 \, \sqrt{d x^{4} + c}{\left (b^{2} c - a b d\right )}}{b x^{4} + a}\right ) + 2 \,{\left (b d x^{4} - 2 \, b c - 3 \, a d\right )} \sqrt{d x^{4} + c} \sqrt{b^{2} c - a b d}}{12 \, \sqrt{b^{2} c - a b d} b^{2} d^{2}}, -\frac{3 \, a^{2} d^{2} \arctan \left (-\frac{b c - a d}{\sqrt{d x^{4} + c} \sqrt{-b^{2} c + a b d}}\right ) -{\left (b d x^{4} - 2 \, b c - 3 \, a d\right )} \sqrt{d x^{4} + c} \sqrt{-b^{2} c + a b d}}{6 \, \sqrt{-b^{2} c + a b d} b^{2} d^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{11}}{\left (a + b x^{4}\right ) \sqrt{c + d x^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**11/((a + b*x**4)*sqrt(c + d*x**4)), x)

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GIAC/XCAS [A]  time = 0.21654, size = 143, normalized size = 1.38 \[ \frac{a^{2} \arctan \left (\frac{\sqrt{d x^{4} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{2 \, \sqrt{-b^{2} c + a b d} b^{2}} + \frac{{\left (d x^{4} + c\right )}^{\frac{3}{2}} b^{2} d^{4} - 3 \, \sqrt{d x^{4} + c} b^{2} c d^{4} - 3 \, \sqrt{d x^{4} + c} a b d^{5}}{6 \, b^{3} d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*a^2*arctan(sqrt(d*x^4 + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*b^2
) + 1/6*((d*x^4 + c)^(3/2)*b^2*d^4 - 3*sqrt(d*x^4 + c)*b^2*c*d^4 - 3*sqrt(d*x^4
+ c)*a*b*d^5)/(b^3*d^6)