3.638 \(\int \frac{x^3}{\left (a+b x^4\right ) \sqrt{c+d x^4}} \, dx\)

Optimal. Leaf size=51 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{2 \sqrt{b} \sqrt{b c-a d}} \]

[Out]

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

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Rubi [A]  time = 0.131971, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{2 \sqrt{b} \sqrt{b c-a d}} \]

Antiderivative was successfully verified.

[In]  Int[x^3/((a + b*x^4)*Sqrt[c + d*x^4]),x]

[Out]

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

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Rubi in Sympy [A]  time = 15.0644, size = 42, normalized size = 0.82 \[ \frac{\operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{4}}}{\sqrt{a d - b c}} \right )}}{2 \sqrt{b} \sqrt{a d - b c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(b*x**4+a)/(d*x**4+c)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0404209, size = 51, normalized size = 1. \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{2 \sqrt{b} \sqrt{b c-a d}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/((a + b*x^4)*Sqrt[c + d*x^4]),x]

[Out]

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

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Maple [B]  time = 0.009, size = 316, normalized size = 6.2 \[ -{\frac{1}{4\,b}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}+2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d+2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ({x}^{2}-{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}}-{\frac{1}{4\,b}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}-2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d-2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ({x}^{2}+{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(b*x^4+a)/(d*x^4+c)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.239609, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (\frac{{\left (b d x^{4} + 2 \, b c - a d\right )} \sqrt{b^{2} c - a b d} - 2 \, \sqrt{d x^{4} + c}{\left (b^{2} c - a b d\right )}}{b x^{4} + a}\right )}{4 \, \sqrt{b^{2} c - a b d}}, -\frac{\arctan \left (-\frac{b c - a d}{\sqrt{d x^{4} + c} \sqrt{-b^{2} c + a b d}}\right )}{2 \, \sqrt{-b^{2} c + a b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/((b*x^4 + a)*sqrt(d*x^4 + c)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\left (a + b x^{4}\right ) \sqrt{c + d x^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(b*x**4+a)/(d*x**4+c)**(1/2),x)

[Out]

Integral(x**3/((a + b*x**4)*sqrt(c + d*x**4)), x)

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GIAC/XCAS [A]  time = 0.211, size = 54, normalized size = 1.06 \[ \frac{\arctan \left (\frac{\sqrt{d x^{4} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{2 \, \sqrt{-b^{2} c + a b d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/((b*x^4 + a)*sqrt(d*x^4 + c)),x, algorithm="giac")

[Out]

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