3.656 \(\int \frac{x^3}{\left (a+b x^4\right )^2 \sqrt{c+d x^4}} \, dx\)

Optimal. Leaf size=87 \[ \frac{d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{4 \sqrt{b} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^4}}{4 \left (a+b x^4\right ) (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.188381, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{4 \sqrt{b} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^4}}{4 \left (a+b x^4\right ) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 20.4234, size = 70, normalized size = 0.8 \[ \frac{\sqrt{c + d x^{4}}}{4 \left (a + b x^{4}\right ) \left (a d - b c\right )} + \frac{d \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{4}}}{\sqrt{a d - b c}} \right )}}{4 \sqrt{b} \left (a d - b c\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.144848, size = 84, normalized size = 0.97 \[ \frac{\frac{\sqrt{c+d x^4}}{a+b x^4}-\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{\sqrt{b} \sqrt{b c-a d}}}{4 a d-4 b c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.01, size = 541, normalized size = 6.2 \[ -{\frac{1}{8\,ab \left ( ad-bc \right ) }\sqrt{-ab}\sqrt{ \left ({x}^{2}-{\frac{1}{b}\sqrt{-ab}} \right ) ^{2}d+2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \left ({x}^{2}-{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}}-{\frac{d}{8\, \left ( ad-bc \right ) b}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}+2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d+2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ({x}^{2}-{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}}+{\frac{1}{8\,ab \left ( ad-bc \right ) }\sqrt{-ab}\sqrt{ \left ({x}^{2}+{\frac{1}{b}\sqrt{-ab}} \right ) ^{2}d-2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \left ({x}^{2}+{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}}-{\frac{d}{8\, \left ( ad-bc \right ) b}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}-2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d-2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ({x}^{2}+{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(-a*b)^(1/2)/a/b/(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/b*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)^(1/2)/a/
b/(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/b*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)))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238405, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (b d x^{4} + a d\right )} \log \left (\frac{{\left (b d x^{4} + 2 \, b c - a d\right )} \sqrt{b^{2} c - a b d} - 2 \, \sqrt{d x^{4} + c}{\left (b^{2} c - a b d\right )}}{b x^{4} + a}\right ) + 2 \, \sqrt{d x^{4} + c} \sqrt{b^{2} c - a b d}}{8 \,{\left ({\left (b^{2} c - a b d\right )} x^{4} + a b c - a^{2} d\right )} \sqrt{b^{2} c - a b d}}, \frac{{\left (b d x^{4} + a d\right )} \arctan \left (-\frac{b c - a d}{\sqrt{d x^{4} + c} \sqrt{-b^{2} c + a b d}}\right ) - \sqrt{d x^{4} + c} \sqrt{-b^{2} c + a b d}}{4 \,{\left ({\left (b^{2} c - a b d\right )} x^{4} + a b c - a^{2} d\right )} \sqrt{-b^{2} c + a b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/((b*x^4 + a)^2*sqrt(d*x^4 + c)),x, algorithm="fricas")

[Out]

[-1/8*((b*d*x^4 + a*d)*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*sqrt(d*x^4 + c)*sqrt(b^2*c - a*b*d
))/(((b^2*c - a*b*d)*x^4 + a*b*c - a^2*d)*sqrt(b^2*c - a*b*d)), 1/4*((b*d*x^4 +
a*d)*arctan(-(b*c - a*d)/(sqrt(d*x^4 + c)*sqrt(-b^2*c + a*b*d))) - sqrt(d*x^4 +
c)*sqrt(-b^2*c + a*b*d))/(((b^2*c - a*b*d)*x^4 + a*b*c - a^2*d)*sqrt(-b^2*c + a*
b*d))]

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Sympy [F(-2)]  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)**2/(d*x**4+c)**(1/2),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.214091, size = 124, normalized size = 1.43 \[ -\frac{1}{4} \, d{\left (\frac{\arctan \left (\frac{\sqrt{d x^{4} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d}{\left (b c - a d\right )}} + \frac{\sqrt{d x^{4} + c}}{{\left ({\left (d x^{4} + c\right )} b - b c + a d\right )}{\left (b c - a d\right )}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/((b*x^4 + a)^2*sqrt(d*x^4 + c)),x, algorithm="giac")

[Out]

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