Optimal. Leaf size=310 \[ -\frac{\left (\sqrt{b} d-3 \sqrt{a} f\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{3/4} b^{7/4}}+\frac{\left (\sqrt{b} d-3 \sqrt{a} f\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{3/4} b^{7/4}}-\frac{\left (3 \sqrt{a} f+\sqrt{b} d\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{3/4} b^{7/4}}+\frac{\left (3 \sqrt{a} f+\sqrt{b} d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{3/4} b^{7/4}}+\frac{e \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 \sqrt{a} b^{3/2}}-\frac{c+d x+e x^2+f x^3}{4 b \left (a+b x^4\right )} \]
[Out]
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Rubi [A] time = 0.581243, antiderivative size = 310, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357 \[ -\frac{\left (\sqrt{b} d-3 \sqrt{a} f\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{3/4} b^{7/4}}+\frac{\left (\sqrt{b} d-3 \sqrt{a} f\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{3/4} b^{7/4}}-\frac{\left (3 \sqrt{a} f+\sqrt{b} d\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{3/4} b^{7/4}}+\frac{\left (3 \sqrt{a} f+\sqrt{b} d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{3/4} b^{7/4}}+\frac{e \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 \sqrt{a} b^{3/2}}-\frac{c+d x+e x^2+f x^3}{4 b \left (a+b x^4\right )} \]
Antiderivative was successfully verified.
[In] Int[(x^3*(c + d*x + e*x^2 + f*x^3))/(a + b*x^4)^2,x]
[Out]
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Rubi in Sympy [A] time = 100.513, size = 291, normalized size = 0.94 \[ - \frac{c + d x + e x^{2} + f x^{3}}{4 b \left (a + b x^{4}\right )} + \frac{e \operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{4 \sqrt{a} b^{\frac{3}{2}}} + \frac{\sqrt{2} \left (3 \sqrt{a} f - \sqrt{b} d\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} b^{\frac{3}{4}} x + \sqrt{a} \sqrt{b} + b x^{2} \right )}}{32 a^{\frac{3}{4}} b^{\frac{7}{4}}} - \frac{\sqrt{2} \left (3 \sqrt{a} f - \sqrt{b} d\right ) \log{\left (\sqrt{2} \sqrt [4]{a} b^{\frac{3}{4}} x + \sqrt{a} \sqrt{b} + b x^{2} \right )}}{32 a^{\frac{3}{4}} b^{\frac{7}{4}}} - \frac{\sqrt{2} \left (3 \sqrt{a} f + \sqrt{b} d\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{16 a^{\frac{3}{4}} b^{\frac{7}{4}}} + \frac{\sqrt{2} \left (3 \sqrt{a} f + \sqrt{b} d\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{16 a^{\frac{3}{4}} b^{\frac{7}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(f*x**3+e*x**2+d*x+c)/(b*x**4+a)**2,x)
[Out]
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Mathematica [A] time = 0.531374, size = 294, normalized size = 0.95 \[ \frac{-\frac{2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (4 \sqrt [4]{a} \sqrt [4]{b} e+3 \sqrt{2} \sqrt{a} f+\sqrt{2} \sqrt{b} d\right )}{a^{3/4}}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (-4 \sqrt [4]{a} \sqrt [4]{b} e+3 \sqrt{2} \sqrt{a} f+\sqrt{2} \sqrt{b} d\right )}{a^{3/4}}+\frac{\sqrt{2} \left (3 \sqrt{a} f-\sqrt{b} d\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{a^{3/4}}+\frac{\sqrt{2} \left (\sqrt{b} d-3 \sqrt{a} f\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{a^{3/4}}-\frac{8 b^{3/4} (c+x (d+x (e+f x)))}{a+b x^4}}{32 b^{7/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^3*(c + d*x + e*x^2 + f*x^3))/(a + b*x^4)^2,x]
[Out]
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Maple [A] time = 0.021, size = 333, normalized size = 1.1 \[{\frac{1}{b{x}^{4}+a} \left ( -{\frac{f{x}^{3}}{4\,b}}-{\frac{e{x}^{2}}{4\,b}}-{\frac{dx}{4\,b}}-{\frac{c}{4\,b}} \right ) }+{\frac{d\sqrt{2}}{32\,ab}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{d\sqrt{2}}{16\,ab}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{d\sqrt{2}}{16\,ab}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{e}{4}\arctan \left ({x}^{2}\sqrt{{\frac{b}{a}}} \right ){\frac{1}{\sqrt{a{b}^{3}}}}}+{\frac{3\,f\sqrt{2}}{32\,{b}^{2}}\ln \left ({1 \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{3\,f\sqrt{2}}{16\,{b}^{2}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{3\,f\sqrt{2}}{16\,{b}^{2}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(f*x^3+e*x^2+d*x+c)/(b*x^4+a)^2,x)
[Out]
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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((f*x^3 + e*x^2 + d*x + c)*x^3/(b*x^4 + a)^2,x, algorithm="maxima")
[Out]
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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((f*x^3 + e*x^2 + d*x + c)*x^3/(b*x^4 + a)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 64.074, size = 508, normalized size = 1.64 \[ \operatorname{RootSum}{\left (65536 t^{4} a^{3} b^{7} + t^{2} \left (3072 a^{2} b^{4} d f + 2048 a^{2} b^{4} e^{2}\right ) + t \left (1152 a^{2} b^{2} e f^{2} - 128 a b^{3} d^{2} e\right ) + 81 a^{2} f^{4} + 18 a b d^{2} f^{2} - 48 a b d e^{2} f + 16 a b e^{4} + b^{2} d^{4}, \left ( t \mapsto t \log{\left (x + \frac{110592 t^{3} a^{4} b^{5} f^{3} - 12288 t^{3} a^{3} b^{6} d^{2} f + 32768 t^{3} a^{3} b^{6} d e^{2} + 13824 t^{2} a^{3} b^{4} d e f^{2} - 12288 t^{2} a^{3} b^{4} e^{3} f + 512 t^{2} a^{2} b^{5} d^{3} e + 3888 t a^{3} b^{2} d f^{4} + 5184 t a^{3} b^{2} e^{2} f^{3} - 576 t a^{2} b^{3} d^{3} f^{2} + 1728 t a^{2} b^{3} d^{2} e^{2} f + 512 t a^{2} b^{3} d e^{4} + 16 t a b^{4} d^{5} + 1458 a^{3} e f^{5} + 360 a^{2} b d e^{3} f^{2} - 192 a^{2} b e^{5} f + 30 a b^{2} d^{4} e f - 40 a b^{2} d^{3} e^{3}}{729 a^{3} f^{6} - 81 a^{2} b d^{2} f^{4} + 864 a^{2} b d e^{2} f^{3} - 576 a^{2} b e^{4} f^{2} - 9 a b^{2} d^{4} f^{2} + 96 a b^{2} d^{3} e^{2} f - 64 a b^{2} d^{2} e^{4} + b^{3} d^{6}} \right )} \right )\right )} - \frac{c + d x + e x^{2} + f x^{3}}{4 a b + 4 b^{2} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(f*x**3+e*x**2+d*x+c)/(b*x**4+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.232053, size = 409, normalized size = 1.32 \[ -\frac{f x^{3} + x^{2} e + d x + c}{4 \,{\left (b x^{4} + a\right )} b} + \frac{\sqrt{2}{\left (2 \, \sqrt{2} \sqrt{a b} b^{2} e + \left (a b^{3}\right )^{\frac{1}{4}} b^{2} d + 3 \, \left (a b^{3}\right )^{\frac{3}{4}} f\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{16 \, a b^{4}} + \frac{\sqrt{2}{\left (2 \, \sqrt{2} \sqrt{a b} b^{2} e + \left (a b^{3}\right )^{\frac{1}{4}} b^{2} d + 3 \, \left (a b^{3}\right )^{\frac{3}{4}} f\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{16 \, a b^{4}} + \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{2} d - 3 \, \left (a b^{3}\right )^{\frac{3}{4}} f\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{32 \, a b^{4}} - \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{2} d - 3 \, \left (a b^{3}\right )^{\frac{3}{4}} f\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{32 \, a b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^3 + e*x^2 + d*x + c)*x^3/(b*x^4 + a)^2,x, algorithm="giac")
[Out]