Optimal. Leaf size=150 \[ -\frac{-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{2 e^5 (d+e x)^2}+\frac{2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^5 (d+e x)^3}-\frac{\left (a e^2-b d e+c d^2\right )^2}{4 e^5 (d+e x)^4}+\frac{2 c (2 c d-b e)}{e^5 (d+e x)}+\frac{c^2 \log (d+e x)}{e^5} \]
[Out]
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Rubi [A] time = 0.319162, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{2 e^5 (d+e x)^2}+\frac{2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^5 (d+e x)^3}-\frac{\left (a e^2-b d e+c d^2\right )^2}{4 e^5 (d+e x)^4}+\frac{2 c (2 c d-b e)}{e^5 (d+e x)}+\frac{c^2 \log (d+e x)}{e^5} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)^2/(d + e*x)^5,x]
[Out]
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Rubi in Sympy [A] time = 47.1556, size = 143, normalized size = 0.95 \[ \frac{c^{2} \log{\left (d + e x \right )}}{e^{5}} - \frac{2 c \left (b e - 2 c d\right )}{e^{5} \left (d + e x\right )} - \frac{2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}}{2 e^{5} \left (d + e x\right )^{2}} - \frac{2 \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )}{3 e^{5} \left (d + e x\right )^{3}} - \frac{\left (a e^{2} - b d e + c d^{2}\right )^{2}}{4 e^{5} \left (d + e x\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x+a)**2/(e*x+d)**5,x)
[Out]
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Mathematica [A] time = 0.137884, size = 170, normalized size = 1.13 \[ \frac{-e^2 \left (3 a^2 e^2+2 a b e (d+4 e x)+b^2 \left (d^2+4 d e x+6 e^2 x^2\right )\right )-2 c e \left (a e \left (d^2+4 d e x+6 e^2 x^2\right )+3 b \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )\right )+c^2 d \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )+12 c^2 (d+e x)^4 \log (d+e x)}{12 e^5 (d+e x)^4} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)^2/(d + e*x)^5,x]
[Out]
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Maple [A] time = 0.012, size = 287, normalized size = 1.9 \[ -{\frac{ac}{{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{{b}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}+3\,{\frac{bcd}{{e}^{4} \left ( ex+d \right ) ^{2}}}-3\,{\frac{{c}^{2}{d}^{2}}{{e}^{5} \left ( ex+d \right ) ^{2}}}-{\frac{2\,ab}{3\,{e}^{2} \left ( ex+d \right ) ^{3}}}+{\frac{4\,acd}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}+{\frac{2\,{b}^{2}d}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}-2\,{\frac{c{d}^{2}b}{{e}^{4} \left ( ex+d \right ) ^{3}}}+{\frac{4\,{c}^{2}{d}^{3}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}+{\frac{{c}^{2}\ln \left ( ex+d \right ) }{{e}^{5}}}-2\,{\frac{bc}{{e}^{4} \left ( ex+d \right ) }}+4\,{\frac{{c}^{2}d}{{e}^{5} \left ( ex+d \right ) }}-{\frac{{a}^{2}}{4\,e \left ( ex+d \right ) ^{4}}}+{\frac{bda}{2\,{e}^{2} \left ( ex+d \right ) ^{4}}}-{\frac{ac{d}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{4}}}-{\frac{{b}^{2}{d}^{2}}{4\,{e}^{3} \left ( ex+d \right ) ^{4}}}+{\frac{{d}^{3}bc}{2\,{e}^{4} \left ( ex+d \right ) ^{4}}}-{\frac{{c}^{2}{d}^{4}}{4\,{e}^{5} \left ( ex+d \right ) ^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)^2/(e*x+d)^5,x)
[Out]
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Maxima [A] time = 0.837216, size = 290, normalized size = 1.93 \[ \frac{25 \, c^{2} d^{4} - 6 \, b c d^{3} e - 2 \, a b d e^{3} - 3 \, a^{2} e^{4} -{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 24 \,{\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} x^{3} + 6 \,{\left (18 \, c^{2} d^{2} e^{2} - 6 \, b c d e^{3} -{\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} + 4 \,{\left (22 \, c^{2} d^{3} e - 6 \, b c d^{2} e^{2} - 2 \, a b e^{4} -{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x}{12 \,{\left (e^{9} x^{4} + 4 \, d e^{8} x^{3} + 6 \, d^{2} e^{7} x^{2} + 4 \, d^{3} e^{6} x + d^{4} e^{5}\right )}} + \frac{c^{2} \log \left (e x + d\right )}{e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2/(e*x + d)^5,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.210195, size = 355, normalized size = 2.37 \[ \frac{25 \, c^{2} d^{4} - 6 \, b c d^{3} e - 2 \, a b d e^{3} - 3 \, a^{2} e^{4} -{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 24 \,{\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} x^{3} + 6 \,{\left (18 \, c^{2} d^{2} e^{2} - 6 \, b c d e^{3} -{\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} + 4 \,{\left (22 \, c^{2} d^{3} e - 6 \, b c d^{2} e^{2} - 2 \, a b e^{4} -{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x + 12 \,{\left (c^{2} e^{4} x^{4} + 4 \, c^{2} d e^{3} x^{3} + 6 \, c^{2} d^{2} e^{2} x^{2} + 4 \, c^{2} d^{3} e x + c^{2} d^{4}\right )} \log \left (e x + d\right )}{12 \,{\left (e^{9} x^{4} + 4 \, d e^{8} x^{3} + 6 \, d^{2} e^{7} x^{2} + 4 \, d^{3} e^{6} x + d^{4} e^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2/(e*x + d)^5,x, algorithm="fricas")
[Out]
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Sympy [A] time = 57.6509, size = 238, normalized size = 1.59 \[ \frac{c^{2} \log{\left (d + e x \right )}}{e^{5}} - \frac{3 a^{2} e^{4} + 2 a b d e^{3} + 2 a c d^{2} e^{2} + b^{2} d^{2} e^{2} + 6 b c d^{3} e - 25 c^{2} d^{4} + x^{3} \left (24 b c e^{4} - 48 c^{2} d e^{3}\right ) + x^{2} \left (12 a c e^{4} + 6 b^{2} e^{4} + 36 b c d e^{3} - 108 c^{2} d^{2} e^{2}\right ) + x \left (8 a b e^{4} + 8 a c d e^{3} + 4 b^{2} d e^{3} + 24 b c d^{2} e^{2} - 88 c^{2} d^{3} e\right )}{12 d^{4} e^{5} + 48 d^{3} e^{6} x + 72 d^{2} e^{7} x^{2} + 48 d e^{8} x^{3} + 12 e^{9} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x+a)**2/(e*x+d)**5,x)
[Out]
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GIAC/XCAS [A] time = 0.209865, size = 410, normalized size = 2.73 \[ -c^{2} e^{\left (-5\right )}{\rm ln}\left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) + \frac{1}{12} \,{\left (\frac{48 \, c^{2} d e^{15}}{x e + d} - \frac{36 \, c^{2} d^{2} e^{15}}{{\left (x e + d\right )}^{2}} + \frac{16 \, c^{2} d^{3} e^{15}}{{\left (x e + d\right )}^{3}} - \frac{3 \, c^{2} d^{4} e^{15}}{{\left (x e + d\right )}^{4}} - \frac{24 \, b c e^{16}}{x e + d} + \frac{36 \, b c d e^{16}}{{\left (x e + d\right )}^{2}} - \frac{24 \, b c d^{2} e^{16}}{{\left (x e + d\right )}^{3}} + \frac{6 \, b c d^{3} e^{16}}{{\left (x e + d\right )}^{4}} - \frac{6 \, b^{2} e^{17}}{{\left (x e + d\right )}^{2}} - \frac{12 \, a c e^{17}}{{\left (x e + d\right )}^{2}} + \frac{8 \, b^{2} d e^{17}}{{\left (x e + d\right )}^{3}} + \frac{16 \, a c d e^{17}}{{\left (x e + d\right )}^{3}} - \frac{3 \, b^{2} d^{2} e^{17}}{{\left (x e + d\right )}^{4}} - \frac{6 \, a c d^{2} e^{17}}{{\left (x e + d\right )}^{4}} - \frac{8 \, a b e^{18}}{{\left (x e + d\right )}^{3}} + \frac{6 \, a b d e^{18}}{{\left (x e + d\right )}^{4}} - \frac{3 \, a^{2} e^{19}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-20\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2/(e*x + d)^5,x, algorithm="giac")
[Out]