Optimal. Leaf size=110 \[ \frac{b^2 \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{a^{5/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^2} (2 a d+3 b c)}{3 a^2 c^2 x}-\frac{\sqrt{c+d x^2}}{3 a c x^3} \]
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Rubi [A] time = 0.373361, antiderivative size = 110, 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 \[ \frac{b^2 \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{a^{5/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^2} (2 a d+3 b c)}{3 a^2 c^2 x}-\frac{\sqrt{c+d x^2}}{3 a c x^3} \]
Antiderivative was successfully verified.
[In] Int[1/(x^4*(a + b*x^2)*Sqrt[c + d*x^2]),x]
[Out]
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Rubi in Sympy [A] time = 58.2618, size = 95, normalized size = 0.86 \[ - \frac{\sqrt{c + d x^{2}}}{3 a c x^{3}} + \frac{\sqrt{c + d x^{2}} \left (2 a d + 3 b c\right )}{3 a^{2} c^{2} x} + \frac{b^{2} \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{a^{\frac{5}{2}} \sqrt{a d - b c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**4/(b*x**2+a)/(d*x**2+c)**(1/2),x)
[Out]
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Mathematica [A] time = 0.17042, size = 96, normalized size = 0.87 \[ \frac{b^2 \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{a^{5/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^2} \left (-a c+2 a d x^2+3 b c x^2\right )}{3 a^2 c^2 x^3} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^4*(a + b*x^2)*Sqrt[c + d*x^2]),x]
[Out]
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Maple [B] time = 0.021, size = 379, normalized size = 3.5 \[ -{\frac{1}{3\,ac{x}^{3}}\sqrt{d{x}^{2}+c}}+{\frac{2\,d}{3\,a{c}^{2}x}\sqrt{d{x}^{2}+c}}+{\frac{b}{{a}^{2}cx}\sqrt{d{x}^{2}+c}}-{\frac{{b}^{2}}{2\,{a}^{2}}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ( x-{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-ab}}}{\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}}+{\frac{{b}^{2}}{2\,{a}^{2}}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-ab}}}{\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^4/(b*x^2+a)/(d*x^2+c)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )} \sqrt{d x^{2} + c} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)*sqrt(d*x^2 + c)*x^4),x, algorithm="maxima")
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Fricas [A] time = 0.324493, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b^{2} c^{2} x^{3} \log \left (\frac{{\left ({\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2}\right )} \sqrt{-a b c + a^{2} d} + 4 \,{\left ({\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} x^{3} -{\left (a^{2} b c^{2} - a^{3} c d\right )} x\right )} \sqrt{d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \, \sqrt{-a b c + a^{2} d}{\left ({\left (3 \, b c + 2 \, a d\right )} x^{2} - a c\right )} \sqrt{d x^{2} + c}}{12 \, \sqrt{-a b c + a^{2} d} a^{2} c^{2} x^{3}}, \frac{3 \, b^{2} c^{2} x^{3} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{2} - a c}{2 \, \sqrt{a b c - a^{2} d} \sqrt{d x^{2} + c} x}\right ) + 2 \, \sqrt{a b c - a^{2} d}{\left ({\left (3 \, b c + 2 \, a d\right )} x^{2} - a c\right )} \sqrt{d x^{2} + c}}{6 \, \sqrt{a b c - a^{2} d} a^{2} c^{2} x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)*sqrt(d*x^2 + c)*x^4),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{4} \left (a + b x^{2}\right ) \sqrt{c + d x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**4/(b*x**2+a)/(d*x**2+c)**(1/2),x)
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GIAC/XCAS [A] time = 0.739166, size = 263, normalized size = 2.39 \[ -\frac{1}{3} \, d^{\frac{5}{2}}{\left (\frac{3 \, b^{2} \arctan \left (\frac{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt{a b c d - a^{2} d^{2}}}\right )}{\sqrt{a b c d - a^{2} d^{2}} a^{2} d^{2}} + \frac{2 \,{\left (3 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} b - 6 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b c - 6 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a d + 3 \, b c^{2} + 2 \, a c d\right )}}{{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} - c\right )}^{3} a^{2} d^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)*sqrt(d*x^2 + c)*x^4),x, algorithm="giac")
[Out]