Optimal. Leaf size=417 \[ \frac{\sqrt{a} d^3 \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{b} \left (a d^4+b c^4\right )}-\frac{c^2 d \log \left (a+b x^4\right )}{4 \left (a d^4+b c^4\right )}+\frac{c^2 d \log (c+d x)}{a d^4+b c^4}+\frac{c \left (\sqrt{a} d^2+\sqrt{b} c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \left (a d^4+b c^4\right )}-\frac{c \left (\sqrt{a} d^2+\sqrt{b} c^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \left (a d^4+b c^4\right )}-\frac{c \left (\sqrt{b} c^2-\sqrt{a} d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \left (a d^4+b c^4\right )}+\frac{c \left (\sqrt{b} c^2-\sqrt{a} d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \left (a d^4+b c^4\right )} \]
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Rubi [A] time = 1.17695, antiderivative size = 417, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 11, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.55 \[ \frac{\sqrt{a} d^3 \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{b} \left (a d^4+b c^4\right )}-\frac{c^2 d \log \left (a+b x^4\right )}{4 \left (a d^4+b c^4\right )}+\frac{c^2 d \log (c+d x)}{a d^4+b c^4}+\frac{c \left (\sqrt{a} d^2+\sqrt{b} c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \left (a d^4+b c^4\right )}-\frac{c \left (\sqrt{a} d^2+\sqrt{b} c^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \left (a d^4+b c^4\right )}-\frac{c \left (\sqrt{b} c^2-\sqrt{a} d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \left (a d^4+b c^4\right )}+\frac{c \left (\sqrt{b} c^2-\sqrt{a} d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \left (a d^4+b c^4\right )} \]
Antiderivative was successfully verified.
[In] Int[x^2/((c + d*x)*(a + b*x^4)),x]
[Out]
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Rubi in Sympy [A] time = 148.362, size = 381, normalized size = 0.91 \[ \frac{\sqrt{a} d^{3} \operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2 \sqrt{b} \left (a d^{4} + b c^{4}\right )} - \frac{c^{2} d \log{\left (a + b x^{4} \right )}}{4 \left (a d^{4} + b c^{4}\right )} + \frac{c^{2} d \log{\left (c + d x \right )}}{a d^{4} + b c^{4}} + \frac{\sqrt{2} c \left (\sqrt{a} d^{2} - \sqrt{b} c^{2}\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{4 \sqrt [4]{a} \sqrt [4]{b} \left (a d^{4} + b c^{4}\right )} - \frac{\sqrt{2} c \left (\sqrt{a} d^{2} - \sqrt{b} c^{2}\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{4 \sqrt [4]{a} \sqrt [4]{b} \left (a d^{4} + b c^{4}\right )} + \frac{\sqrt{2} c \left (\sqrt{a} d^{2} + \sqrt{b} c^{2}\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} b^{\frac{3}{4}} x + \sqrt{a} \sqrt{b} + b x^{2} \right )}}{8 \sqrt [4]{a} \sqrt [4]{b} \left (a d^{4} + b c^{4}\right )} - \frac{\sqrt{2} c \left (\sqrt{a} d^{2} + \sqrt{b} c^{2}\right ) \log{\left (\sqrt{2} \sqrt [4]{a} b^{\frac{3}{4}} x + \sqrt{a} \sqrt{b} + b x^{2} \right )}}{8 \sqrt [4]{a} \sqrt [4]{b} \left (a d^{4} + b c^{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(d*x+c)/(b*x**4+a),x)
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Mathematica [A] time = 0.486246, size = 370, normalized size = 0.89 \[ \frac{-2 \left (2 a^{3/4} d^3-\sqrt{2} \sqrt{a} \sqrt [4]{b} c d^2+\sqrt{2} b^{3/4} c^3\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+2 \left (-2 a^{3/4} d^3-\sqrt{2} \sqrt{a} \sqrt [4]{b} c d^2+\sqrt{2} b^{3/4} c^3\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )+\sqrt [4]{b} c \left (\sqrt{2} \left (\sqrt{a} d^2+\sqrt{b} c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )-\sqrt{2} \sqrt{b} c^2 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )-2 \sqrt [4]{a} \sqrt [4]{b} c d \log \left (a+b x^4\right )+8 \sqrt [4]{a} \sqrt [4]{b} c d \log (c+d x)-\sqrt{2} \sqrt{a} d^2 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )\right )}{8 \sqrt [4]{a} \sqrt{b} \left (a d^4+b c^4\right )} \]
Antiderivative was successfully verified.
[In] Integrate[x^2/((c + d*x)*(a + b*x^4)),x]
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Maple [A] time = 0.019, size = 422, normalized size = 1. \[ -{\frac{c{d}^{2}\sqrt{2}}{4\,a{d}^{4}+4\,b{c}^{4}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }-{\frac{c{d}^{2}\sqrt{2}}{8\,a{d}^{4}+8\,b{c}^{4}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }-{\frac{c{d}^{2}\sqrt{2}}{4\,a{d}^{4}+4\,b{c}^{4}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{a{d}^{3}}{2\,a{d}^{4}+2\,b{c}^{4}}\arctan \left ({x}^{2}\sqrt{{\frac{b}{a}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{{c}^{3}\sqrt{2}}{8\,a{d}^{4}+8\,b{c}^{4}}\ln \left ({1 \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{{c}^{3}\sqrt{2}}{4\,a{d}^{4}+4\,b{c}^{4}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{{c}^{3}\sqrt{2}}{4\,a{d}^{4}+4\,b{c}^{4}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{{c}^{2}d\ln \left ( b{x}^{4}+a \right ) }{4\,a{d}^{4}+4\,b{c}^{4}}}+{\frac{{c}^{2}d\ln \left ( dx+c \right ) }{a{d}^{4}+b{c}^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(d*x+c)/(b*x^4+a),x)
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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^2/((b*x^4 + a)*(d*x + c)),x, algorithm="maxima")
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((b*x^4 + a)*(d*x + c)),x, algorithm="fricas")
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(d*x+c)/(b*x**4+a),x)
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GIAC/XCAS [A] time = 0.294233, size = 575, normalized size = 1.38 \[ \frac{c^{2} d^{2}{\rm ln}\left ({\left | d x + c \right |}\right )}{b c^{4} d + a d^{5}} - \frac{c^{2} d{\rm ln}\left ({\left | b x^{4} + a \right |}\right )}{4 \,{\left (b c^{4} + a d^{4}\right )}} - \frac{{\left (\sqrt{2} a^{2} b^{3} d - \left (a b^{3}\right )^{\frac{3}{4}} a b c\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{2 \,{\left (\sqrt{2} a^{2} b^{4} c^{2} + \sqrt{2} \sqrt{a b} a^{2} b^{3} d^{2} - 2 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b^{3} c d\right )}} + \frac{{\left (\sqrt{2} a^{2} b^{3} d + \left (a b^{3}\right )^{\frac{3}{4}} a b c\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{2 \,{\left (\sqrt{2} a^{2} b^{4} c^{2} + \sqrt{2} \sqrt{a b} a^{2} b^{3} d^{2} + 2 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b^{3} c d\right )}} - \frac{{\left (\left (a b^{3}\right )^{\frac{1}{4}} a b c d^{2} + \left (a b^{3}\right )^{\frac{3}{4}} c^{3}\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{4 \,{\left (\sqrt{2} a b^{3} c^{4} + \sqrt{2} a^{2} b^{2} d^{4}\right )}} + \frac{{\left (\left (a b^{3}\right )^{\frac{1}{4}} a b c d^{2} + \left (a b^{3}\right )^{\frac{3}{4}} c^{3}\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{4 \,{\left (\sqrt{2} a b^{3} c^{4} + \sqrt{2} a^{2} b^{2} d^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((b*x^4 + a)*(d*x + c)),x, algorithm="giac")
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