Optimal. Leaf size=229 \[ -\frac{\left (-a^2 c^2 e+3 a b^2 c e-2 a b c^2 d+b^4 (-e)+b^3 c d\right ) \log \left (a+b x+c x^2\right )}{2 c^5}-\frac{\left (-5 a^2 b c^2 e+2 a^2 c^3 d+5 a b^3 c e-4 a b^2 c^2 d+b^5 (-e)+b^4 c d\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^5 \sqrt{b^2-4 a c}}-\frac{x^2 \left (a c e+b^2 (-e)+b c d\right )}{2 c^3}+\frac{x \left (2 a b c e-a c^2 d+b^3 (-e)+b^2 c d\right )}{c^4}+\frac{x^3 (c d-b e)}{3 c^2}+\frac{e x^4}{4 c} \]
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Rubi [A] time = 0.808237, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ -\frac{\left (-a^2 c^2 e+3 a b^2 c e-2 a b c^2 d+b^4 (-e)+b^3 c d\right ) \log \left (a+b x+c x^2\right )}{2 c^5}-\frac{\left (-5 a^2 b c^2 e+2 a^2 c^3 d+5 a b^3 c e-4 a b^2 c^2 d+b^5 (-e)+b^4 c d\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^5 \sqrt{b^2-4 a c}}-\frac{x^2 \left (a c e+b^2 (-e)+b c d\right )}{2 c^3}+\frac{x \left (2 a b c e-a c^2 d+b^3 (-e)+b^2 c d\right )}{c^4}+\frac{x^3 (c d-b e)}{3 c^2}+\frac{e x^4}{4 c} \]
Antiderivative was successfully verified.
[In] Int[(x^4*(d + e*x))/(a + b*x + c*x^2),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \left (- 2 a b c e + a c^{2} d + b^{3} e - b^{2} c d\right ) \int \frac{1}{c^{4}}\, dx + \frac{e x^{4}}{4 c} - \frac{x^{3} \left (b e - c d\right )}{3 c^{2}} + \frac{\left (- a c e + b^{2} e - b c d\right ) \int x\, dx}{c^{3}} + \frac{\left (a^{2} c^{2} e - 3 a b^{2} c e + 2 a b c^{2} d + b^{4} e - b^{3} c d\right ) \log{\left (a + b x + c x^{2} \right )}}{2 c^{5}} + \frac{\left (5 a^{2} b c^{2} e - 2 a^{2} c^{3} d - 5 a b^{3} c e + 4 a b^{2} c^{2} d + b^{5} e - b^{4} c d\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{c^{5} \sqrt{- 4 a c + b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4*(e*x+d)/(c*x**2+b*x+a),x)
[Out]
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Mathematica [A] time = 0.25541, size = 222, normalized size = 0.97 \[ \frac{6 \left (a^2 c^2 e-3 a b^2 c e+2 a b c^2 d+b^4 e-b^3 c d\right ) \log (a+x (b+c x))+\frac{12 \left (-5 a^2 b c^2 e+2 a^2 c^3 d+5 a b^3 c e-4 a b^2 c^2 d+b^5 (-e)+b^4 c d\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}-6 c^2 x^2 \left (a c e+b^2 (-e)+b c d\right )-12 c x \left (-2 a b c e+a c^2 d+b^3 e-b^2 c d\right )+4 c^3 x^3 (c d-b e)+3 c^4 e x^4}{12 c^5} \]
Antiderivative was successfully verified.
[In] Integrate[(x^4*(d + e*x))/(a + b*x + c*x^2),x]
[Out]
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Maple [B] time = 0.009, size = 445, normalized size = 1.9 \[{\frac{e{x}^{4}}{4\,c}}-{\frac{b{x}^{3}e}{3\,{c}^{2}}}+{\frac{d{x}^{3}}{3\,c}}-{\frac{ae{x}^{2}}{2\,{c}^{2}}}+{\frac{{b}^{2}e{x}^{2}}{2\,{c}^{3}}}-{\frac{{x}^{2}bd}{2\,{c}^{2}}}+2\,{\frac{beax}{{c}^{3}}}-{\frac{adx}{{c}^{2}}}-{\frac{{b}^{3}ex}{{c}^{4}}}+{\frac{{b}^{2}dx}{{c}^{3}}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ){a}^{2}e}{2\,{c}^{3}}}-{\frac{3\,\ln \left ( c{x}^{2}+bx+a \right ) a{b}^{2}e}{2\,{c}^{4}}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ) abd}{{c}^{3}}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ){b}^{4}e}{2\,{c}^{5}}}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ){b}^{3}d}{2\,{c}^{4}}}-5\,{\frac{{a}^{2}be}{{c}^{3}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+2\,{\frac{{a}^{2}d}{{c}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+5\,{\frac{a{b}^{3}e}{{c}^{4}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-4\,{\frac{a{b}^{2}d}{{c}^{3}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{{b}^{5}e}{{c}^{5}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{d{b}^{4}}{{c}^{4}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4*(e*x+d)/(c*x^2+b*x+a),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((e*x + d)*x^4/(c*x^2 + b*x + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.296476, size = 1, normalized size = 0. \[ \left [-\frac{6 \,{\left ({\left (b^{4} c - 4 \, a b^{2} c^{2} + 2 \, a^{2} c^{3}\right )} d -{\left (b^{5} - 5 \, a b^{3} c + 5 \, a^{2} b c^{2}\right )} e\right )} \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x +{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) -{\left (3 \, c^{4} e x^{4} + 4 \,{\left (c^{4} d - b c^{3} e\right )} x^{3} - 6 \,{\left (b c^{3} d -{\left (b^{2} c^{2} - a c^{3}\right )} e\right )} x^{2} + 12 \,{\left ({\left (b^{2} c^{2} - a c^{3}\right )} d -{\left (b^{3} c - 2 \, a b c^{2}\right )} e\right )} x - 6 \,{\left ({\left (b^{3} c - 2 \, a b c^{2}\right )} d -{\left (b^{4} - 3 \, a b^{2} c + a^{2} c^{2}\right )} e\right )} \log \left (c x^{2} + b x + a\right )\right )} \sqrt{b^{2} - 4 \, a c}}{12 \, \sqrt{b^{2} - 4 \, a c} c^{5}}, \frac{12 \,{\left ({\left (b^{4} c - 4 \, a b^{2} c^{2} + 2 \, a^{2} c^{3}\right )} d -{\left (b^{5} - 5 \, a b^{3} c + 5 \, a^{2} b c^{2}\right )} e\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) +{\left (3 \, c^{4} e x^{4} + 4 \,{\left (c^{4} d - b c^{3} e\right )} x^{3} - 6 \,{\left (b c^{3} d -{\left (b^{2} c^{2} - a c^{3}\right )} e\right )} x^{2} + 12 \,{\left ({\left (b^{2} c^{2} - a c^{3}\right )} d -{\left (b^{3} c - 2 \, a b c^{2}\right )} e\right )} x - 6 \,{\left ({\left (b^{3} c - 2 \, a b c^{2}\right )} d -{\left (b^{4} - 3 \, a b^{2} c + a^{2} c^{2}\right )} e\right )} \log \left (c x^{2} + b x + a\right )\right )} \sqrt{-b^{2} + 4 \, a c}}{12 \, \sqrt{-b^{2} + 4 \, a c} c^{5}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)*x^4/(c*x^2 + b*x + a),x, algorithm="fricas")
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Sympy [A] time = 10.639, size = 1088, normalized size = 4.75 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4*(e*x+d)/(c*x**2+b*x+a),x)
[Out]
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GIAC/XCAS [A] time = 0.272872, size = 333, normalized size = 1.45 \[ \frac{3 \, c^{3} x^{4} e + 4 \, c^{3} d x^{3} - 4 \, b c^{2} x^{3} e - 6 \, b c^{2} d x^{2} + 6 \, b^{2} c x^{2} e - 6 \, a c^{2} x^{2} e + 12 \, b^{2} c d x - 12 \, a c^{2} d x - 12 \, b^{3} x e + 24 \, a b c x e}{12 \, c^{4}} - \frac{{\left (b^{3} c d - 2 \, a b c^{2} d - b^{4} e + 3 \, a b^{2} c e - a^{2} c^{2} e\right )}{\rm ln}\left (c x^{2} + b x + a\right )}{2 \, c^{5}} + \frac{{\left (b^{4} c d - 4 \, a b^{2} c^{2} d + 2 \, a^{2} c^{3} d - b^{5} e + 5 \, a b^{3} c e - 5 \, a^{2} b c^{2} e\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)*x^4/(c*x^2 + b*x + a),x, algorithm="giac")
[Out]