Optimal. Leaf size=151 \[ \frac{e \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{2 c^3}-\frac{(2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^3 \sqrt{b^2-4 a c}}+\frac{e^2 x (3 c d-b e)}{c^2}+\frac{e^3 x^2}{2 c} \]
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Rubi [A] time = 0.395879, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{e \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{2 c^3}-\frac{(2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^3 \sqrt{b^2-4 a c}}+\frac{e^2 x (3 c d-b e)}{c^2}+\frac{e^3 x^2}{2 c} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^3/(a + b*x + c*x^2),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - e^{2} \left (b e - 3 c d\right ) \int \frac{1}{c^{2}}\, dx + \frac{e^{3} \int x\, dx}{c} + \frac{e \left (- a c e^{2} + b^{2} e^{2} - 3 b c d e + 3 c^{2} d^{2}\right ) \log{\left (a + b x + c x^{2} \right )}}{2 c^{3}} + \frac{\left (b e - 2 c d\right ) \left (- 3 a c e^{2} + b^{2} e^{2} - b c d e + c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{c^{3} \sqrt{- 4 a c + b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**3/(c*x**2+b*x+a),x)
[Out]
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Mathematica [A] time = 0.233673, size = 148, normalized size = 0.98 \[ \frac{e \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \log (a+x (b+c x))+\frac{2 (2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+2 c e^2 x (3 c d-b e)+c^2 e^3 x^2}{2 c^3} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^3/(a + b*x + c*x^2),x]
[Out]
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Maple [B] time = 0.007, size = 366, normalized size = 2.4 \[{\frac{{e}^{3}{x}^{2}}{2\,c}}-{\frac{b{e}^{3}x}{{c}^{2}}}+3\,{\frac{d{e}^{2}x}{c}}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ) a{e}^{3}}{2\,{c}^{2}}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ){b}^{2}{e}^{3}}{2\,{c}^{3}}}-{\frac{3\,d\ln \left ( c{x}^{2}+bx+a \right ){e}^{2}b}{2\,{c}^{2}}}+{\frac{3\,\ln \left ( c{x}^{2}+bx+a \right ){d}^{2}e}{2\,c}}+3\,{\frac{ab{e}^{3}}{{c}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-6\,{\frac{ad{e}^{2}}{c\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+2\,{\frac{{d}^{3}}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{{b}^{3}{e}^{3}}{{c}^{3}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+3\,{\frac{{b}^{2}d{e}^{2}}{{c}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-3\,{\frac{b{d}^{2}e}{c\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^3/(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)^3/(c*x^2 + b*x + a),x, algorithm="maxima")
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Fricas [A] time = 0.229412, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \,{\left (b^{2} c - 2 \, a c^{2}\right )} d e^{2} -{\left (b^{3} - 3 \, a b c\right )} e^{3}\right )} \log \left (-\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x -{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) +{\left (c^{2} e^{3} x^{2} + 2 \,{\left (3 \, c^{2} d e^{2} - b c e^{3}\right )} x +{\left (3 \, c^{2} d^{2} e - 3 \, b c d e^{2} +{\left (b^{2} - a c\right )} e^{3}\right )} \log \left (c x^{2} + b x + a\right )\right )} \sqrt{b^{2} - 4 \, a c}}{2 \, \sqrt{b^{2} - 4 \, a c} c^{3}}, \frac{2 \,{\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \,{\left (b^{2} c - 2 \, a c^{2}\right )} d e^{2} -{\left (b^{3} - 3 \, a b c\right )} e^{3}\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) +{\left (c^{2} e^{3} x^{2} + 2 \,{\left (3 \, c^{2} d e^{2} - b c e^{3}\right )} x +{\left (3 \, c^{2} d^{2} e - 3 \, b c d e^{2} +{\left (b^{2} - a c\right )} e^{3}\right )} \log \left (c x^{2} + b x + a\right )\right )} \sqrt{-b^{2} + 4 \, a c}}{2 \, \sqrt{-b^{2} + 4 \, a c} c^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3/(c*x^2 + b*x + a),x, algorithm="fricas")
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Sympy [A] time = 11.3155, size = 892, normalized size = 5.91 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**3/(c*x**2+b*x+a),x)
[Out]
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GIAC/XCAS [A] time = 0.206756, size = 217, normalized size = 1.44 \[ \frac{c x^{2} e^{3} + 6 \, c d x e^{2} - 2 \, b x e^{3}}{2 \, c^{2}} + \frac{{\left (3 \, c^{2} d^{2} e - 3 \, b c d e^{2} + b^{2} e^{3} - a c e^{3}\right )}{\rm ln}\left (c x^{2} + b x + a\right )}{2 \, c^{3}} + \frac{{\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, b^{2} c d e^{2} - 6 \, a c^{2} d e^{2} - b^{3} e^{3} + 3 \, a b c e^{3}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3/(c*x^2 + b*x + a),x, algorithm="giac")
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