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

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

[Out]

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

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Rubi [A]  time = 0.222582, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{(b c-2 a d) \tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{4 a^{3/2} (b c-a d)^{3/2}}+\frac{b x^2 \sqrt{c+d x^4}}{4 a \left (a+b x^4\right ) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 27.1056, size = 87, normalized size = 0.84 \[ - \frac{b x^{2} \sqrt{c + d x^{4}}}{4 a \left (a + b x^{4}\right ) \left (a d - b c\right )} + \frac{\left (2 a d - b c\right ) \operatorname{atanh}{\left (\frac{x^{2} \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{4}}} \right )}}{4 a^{\frac{3}{2}} \left (a d - b c\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.176686, size = 104, normalized size = 1. \[ \frac{\frac{\sqrt{a} b x^2 \sqrt{c+d x^4}}{\left (a+b x^4\right ) (b c-a d)}+\frac{(b c-2 a d) \tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{(b c-a d)^{3/2}}}{4 a^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.009, size = 867, normalized size = 8.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8/a/(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/a*d*(-a*b)^(1/2)/(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/(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/a*d*(-a*b)^(1/2)/(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/(-a*b)^(1/2)/(-(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/(-a*b)^(1/2)/(-(a*d-b*c)/b)^(1/2)*l
n((-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. \[ \int \frac{x}{{\left (b x^{4} + a\right )}^{2} \sqrt{d x^{4} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x/((b*x^4 + a)^2*sqrt(d*x^4 + c)), x)

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Fricas [A]  time = 0.377797, size = 1, normalized size = 0.01 \[ \left [\frac{4 \, \sqrt{d x^{4} + c} \sqrt{-a b c + a^{2} d} b x^{2} +{\left ({\left (b^{2} c - 2 \, a b d\right )} x^{4} + a b c - 2 \, a^{2} d\right )} \log \left (\frac{4 \,{\left ({\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} x^{6} -{\left (a^{2} b c^{2} - a^{3} c d\right )} x^{2}\right )} \sqrt{d x^{4} + c} +{\left ({\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{8} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{4} + a^{2} c^{2}\right )} \sqrt{-a b c + a^{2} d}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}\right )}{16 \,{\left ({\left (a b^{2} c - a^{2} b d\right )} x^{4} + a^{2} b c - a^{3} d\right )} \sqrt{-a b c + a^{2} d}}, \frac{2 \, \sqrt{d x^{4} + c} \sqrt{a b c - a^{2} d} b x^{2} +{\left ({\left (b^{2} c - 2 \, a b d\right )} x^{4} + a b c - 2 \, a^{2} d\right )} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{4} - a c}{2 \, \sqrt{d x^{4} + c} \sqrt{a b c - a^{2} d} x^{2}}\right )}{8 \,{\left ({\left (a b^{2} c - a^{2} b d\right )} x^{4} + a^{2} b c - a^{3} d\right )} \sqrt{a b c - a^{2} d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(4*sqrt(d*x^4 + c)*sqrt(-a*b*c + a^2*d)*b*x^2 + ((b^2*c - 2*a*b*d)*x^4 + a
*b*c - 2*a^2*d)*log((4*((a*b^2*c^2 - 3*a^2*b*c*d + 2*a^3*d^2)*x^6 - (a^2*b*c^2 -
a^3*c*d)*x^2)*sqrt(d*x^4 + c) + ((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^8 - 2*(3*a
*b*c^2 - 4*a^2*c*d)*x^4 + a^2*c^2)*sqrt(-a*b*c + a^2*d))/(b^2*x^8 + 2*a*b*x^4 +
a^2)))/(((a*b^2*c - a^2*b*d)*x^4 + a^2*b*c - a^3*d)*sqrt(-a*b*c + a^2*d)), 1/8*(
2*sqrt(d*x^4 + c)*sqrt(a*b*c - a^2*d)*b*x^2 + ((b^2*c - 2*a*b*d)*x^4 + a*b*c - 2
*a^2*d)*arctan(1/2*((b*c - 2*a*d)*x^4 - a*c)/(sqrt(d*x^4 + c)*sqrt(a*b*c - a^2*d
)*x^2)))/(((a*b^2*c - a^2*b*d)*x^4 + a^2*b*c - a^3*d)*sqrt(a*b*c - a^2*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/(b*x**4+a)**2/(d*x**4+c)**(1/2),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.230097, size = 320, normalized size = 3.08 \[ -\frac{1}{4} \, d^{\frac{3}{2}}{\left (\frac{{\left (b c - 2 \, a d\right )} \arctan \left (\frac{{\left (\sqrt{d} x^{2} - \sqrt{d x^{4} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt{a b c d - a^{2} d^{2}}}\right )}{{\left (a b c d - a^{2} d^{2}\right )}^{\frac{3}{2}}} + \frac{2 \,{\left ({\left (\sqrt{d} x^{2} - \sqrt{d x^{4} + c}\right )}^{2} b c - 2 \,{\left (\sqrt{d} x^{2} - \sqrt{d x^{4} + c}\right )}^{2} a d - b c^{2}\right )}}{{\left ({\left (\sqrt{d} x^{2} - \sqrt{d x^{4} + c}\right )}^{4} b - 2 \,{\left (\sqrt{d} x^{2} - \sqrt{d x^{4} + c}\right )}^{2} b c + 4 \,{\left (\sqrt{d} x^{2} - \sqrt{d x^{4} + c}\right )}^{2} a d + b c^{2}\right )}{\left (a b c d - a^{2} d^{2}\right )}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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