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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]