3.644 \(\int \frac{1}{x^3 \left (a+b x^4\right ) \sqrt{c+d x^4}} \, dx\)

Optimal. Leaf size=80 \[ -\frac{b \tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{2 a^{3/2} \sqrt{b c-a d}}-\frac{\sqrt{c+d x^4}}{2 a c x^2} \]

[Out]

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

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Rubi [A]  time = 0.242398, antiderivative size = 80, 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 \[ -\frac{b \tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{2 a^{3/2} \sqrt{b c-a d}}-\frac{\sqrt{c+d x^4}}{2 a c x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 30.8796, size = 68, normalized size = 0.85 \[ - \frac{\sqrt{c + d x^{4}}}{2 a c x^{2}} - \frac{b \operatorname{atanh}{\left (\frac{x^{2} \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{4}}} \right )}}{2 a^{\frac{3}{2}} \sqrt{a d - b c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 1.13067, size = 129, normalized size = 1.61 \[ \frac{\sqrt{c+d x^4} \left (-\frac{b x^4 \sin ^{-1}\left (\frac{\sqrt{x^4 \left (\frac{b}{a}-\frac{d}{c}\right )}}{\sqrt{\frac{b x^4}{a}+1}}\right )}{\sqrt{\frac{b x^4}{a}+1} \sqrt{x^4 \left (\frac{b}{a}-\frac{d}{c}\right )} \sqrt{\frac{a \left (c+d x^4\right )}{c \left (a+b x^4\right )}}}-a\right )}{2 a^2 c x^2} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c + d*x^4]*(-a - (b*x^4*ArcSin[Sqrt[(b/a - d/c)*x^4]/Sqrt[1 + (b*x^4)/a]])
/(Sqrt[(b/a - d/c)*x^4]*Sqrt[1 + (b*x^4)/a]*Sqrt[(a*(c + d*x^4))/(c*(a + b*x^4))
])))/(2*a^2*c*x^2)

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Maple [B]  time = 0.022, size = 350, normalized size = 4.4 \[ -{\frac{1}{2\,ac{x}^{2}}\sqrt{d{x}^{4}+c}}+{\frac{b}{4\,a}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}+2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d+2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ({x}^{2}-{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-ab}}}{\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}}-{\frac{b}{4\,a}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}-2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d-2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ({x}^{2}+{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-ab}}}{\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )} \sqrt{d x^{4} + c} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((b*x^4 + a)*sqrt(d*x^4 + c)*x^3), x)

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Fricas [A]  time = 0.327371, size = 1, normalized size = 0.01 \[ \left [\frac{b c x^{2} \log \left (-\frac{4 \,{\left ({\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} x^{6} -{\left (a^{2} b c^{2} - a^{3} c d\right )} x^{2}\right )} \sqrt{d x^{4} + c} -{\left ({\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{8} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{4} + a^{2} c^{2}\right )} \sqrt{-a b c + a^{2} d}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}\right ) - 4 \, \sqrt{d x^{4} + c} \sqrt{-a b c + a^{2} d}}{8 \, \sqrt{-a b c + a^{2} d} a c x^{2}}, -\frac{b c x^{2} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{4} - a c}{2 \, \sqrt{d x^{4} + c} \sqrt{a b c - a^{2} d} x^{2}}\right ) + 2 \, \sqrt{d x^{4} + c} \sqrt{a b c - a^{2} d}}{4 \, \sqrt{a b c - a^{2} d} a c x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)*sqrt(d*x^4 + c)*x^3),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{3} \left (a + b x^{4}\right ) \sqrt{c + d x^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**3*(a + b*x**4)*sqrt(c + d*x**4)), x)

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GIAC/XCAS [A]  time = 0.223636, size = 86, normalized size = 1.08 \[ \frac{\frac{b c \arctan \left (\frac{a \sqrt{d + \frac{c}{x^{4}}}}{\sqrt{a b c - a^{2} d}}\right )}{\sqrt{a b c - a^{2} d} a} - \frac{\sqrt{d + \frac{c}{x^{4}}}}{a}}{2 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)*sqrt(d*x^4 + c)*x^3),x, algorithm="giac")

[Out]

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