Optimal. Leaf size=256 \[ \frac{c (d+e x)^3 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}-\frac{(d+e x)^2 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{2 e^7}+\frac{3 x \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6}-\frac{\left (a e^2-b d e+c d^2\right )^3}{e^7 (d+e x)}-\frac{3 (2 c d-b e) \log (d+e x) \left (a e^2-b d e+c d^2\right )^2}{e^7}-\frac{3 c^2 (d+e x)^4 (2 c d-b e)}{4 e^7}+\frac{c^3 (d+e x)^5}{5 e^7} \]
[Out]
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Rubi [A] time = 0.819677, antiderivative size = 256, 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 \[ \frac{c (d+e x)^3 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}-\frac{(d+e x)^2 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{2 e^7}+\frac{3 x \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6}-\frac{\left (a e^2-b d e+c d^2\right )^3}{e^7 (d+e x)}-\frac{3 (2 c d-b e) \log (d+e x) \left (a e^2-b d e+c d^2\right )^2}{e^7}-\frac{3 c^2 (d+e x)^4 (2 c d-b e)}{4 e^7}+\frac{c^3 (d+e x)^5}{5 e^7} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)^3/(d + e*x)^2,x]
[Out]
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Rubi in Sympy [A] time = 110.19, size = 250, normalized size = 0.98 \[ \frac{c^{3} \left (d + e x\right )^{5}}{5 e^{7}} + \frac{3 c^{2} \left (d + e x\right )^{4} \left (b e - 2 c d\right )}{4 e^{7}} + \frac{c \left (d + e x\right )^{3} \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{e^{7}} + \frac{3 x \left (a e^{2} - b d e + c d^{2}\right ) \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{e^{6}} + \frac{\left (d + e x\right )^{2} \left (b e - 2 c d\right ) \left (6 a c e^{2} + b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right )}{2 e^{7}} + \frac{3 \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2} \log{\left (d + e x \right )}}{e^{7}} - \frac{\left (a e^{2} - b d e + c d^{2}\right )^{3}}{e^{7} \left (d + e x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x+a)**3/(e*x+d)**2,x)
[Out]
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Mathematica [A] time = 0.197126, size = 255, normalized size = 1. \[ \frac{20 e x \left (3 c e^2 \left (a^2 e^2-4 a b d e+3 b^2 d^2\right )+b^2 e^3 (3 a e-2 b d)+3 c^2 d^2 e (3 a e-4 b d)+5 c^3 d^4\right )+10 e^2 x^2 (b e-c d) \left (c e (6 a e-5 b d)+b^2 e^2+4 c^2 d^2\right )+20 c e^3 x^3 \left (c e (a e-2 b d)+b^2 e^2+c^2 d^2\right )-\frac{20 \left (e (a e-b d)+c d^2\right )^3}{d+e x}-60 (2 c d-b e) \log (d+e x) \left (e (a e-b d)+c d^2\right )^2+5 c^2 e^4 x^4 (3 b e-2 c d)+4 c^3 e^5 x^5}{20 e^7} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)^3/(d + e*x)^2,x]
[Out]
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Maple [B] time = 0.017, size = 585, normalized size = 2.3 \[ -2\,{\frac{b{x}^{3}{c}^{2}d}{{e}^{3}}}+3\,{\frac{ab{x}^{2}c}{{e}^{2}}}-3\,{\frac{a{b}^{2}{d}^{2}}{{e}^{3} \left ( ex+d \right ) }}-3\,{\frac{{b}^{2}c{d}^{4}}{{e}^{5} \left ( ex+d \right ) }}+3\,{\frac{b{c}^{2}{d}^{5}}{{e}^{6} \left ( ex+d \right ) }}-6\,{\frac{\ln \left ( ex+d \right ) a{b}^{2}d}{{e}^{3}}}-{\frac{d{c}^{3}{x}^{4}}{2\,{e}^{3}}}+{\frac{{b}^{3}{d}^{3}}{{e}^{4} \left ( ex+d \right ) }}+3\,{\frac{\ln \left ( ex+d \right ){a}^{2}b}{{e}^{2}}}+3\,{\frac{\ln \left ( ex+d \right ){b}^{3}{d}^{2}}{{e}^{4}}}-3\,{\frac{a{c}^{2}{d}^{4}}{{e}^{5} \left ( ex+d \right ) }}+{\frac{{x}^{2}{b}^{3}}{2\,{e}^{2}}}+{\frac{3\,b{x}^{4}{c}^{2}}{4\,{e}^{2}}}+{\frac{{b}^{2}{x}^{3}c}{{e}^{2}}}+15\,{\frac{\ln \left ( ex+d \right ) b{c}^{2}{d}^{4}}{{e}^{6}}}+3\,{\frac{{a}^{2}bd}{{e}^{2} \left ( ex+d \right ) }}-3\,{\frac{{b}^{2}{x}^{2}cd}{{e}^{3}}}+{\frac{9\,b{x}^{2}{c}^{2}{d}^{2}}{2\,{e}^{4}}}+9\,{\frac{c{b}^{2}{d}^{2}x}{{e}^{4}}}-12\,{\frac{{d}^{3}b{c}^{2}x}{{e}^{5}}}-12\,{\frac{cabdx}{{e}^{3}}}+18\,{\frac{\ln \left ( ex+d \right ) abc{d}^{2}}{{e}^{4}}}+6\,{\frac{abc{d}^{3}}{{e}^{4} \left ( ex+d \right ) }}+{\frac{{c}^{3}{x}^{5}}{5\,{e}^{2}}}-6\,{\frac{cd\ln \left ( ex+d \right ){a}^{2}}{{e}^{3}}}-3\,{\frac{{c}^{2}{x}^{2}ad}{{e}^{3}}}+9\,{\frac{a{c}^{2}{d}^{2}x}{{e}^{4}}}+{\frac{{x}^{3}{c}^{3}{d}^{2}}{{e}^{4}}}-2\,{\frac{{x}^{2}{c}^{3}{d}^{3}}{{e}^{5}}}+5\,{\frac{{c}^{3}{d}^{4}x}{{e}^{6}}}-6\,{\frac{{d}^{5}\ln \left ( ex+d \right ){c}^{3}}{{e}^{7}}}-{\frac{{c}^{3}{d}^{6}}{{e}^{7} \left ( ex+d \right ) }}-12\,{\frac{{c}^{2}{d}^{3}\ln \left ( ex+d \right ) a}{{e}^{5}}}-3\,{\frac{{a}^{2}c{d}^{2}}{{e}^{3} \left ( ex+d \right ) }}+{\frac{{c}^{2}{x}^{3}a}{{e}^{2}}}+3\,{\frac{{a}^{2}cx}{{e}^{2}}}-12\,{\frac{\ln \left ( ex+d \right ){b}^{2}c{d}^{3}}{{e}^{5}}}+3\,{\frac{a{b}^{2}x}{{e}^{2}}}-2\,{\frac{d{b}^{3}x}{{e}^{3}}}-{\frac{{a}^{3}}{e \left ( ex+d \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)^3/(e*x+d)^2,x)
[Out]
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Maxima [A] time = 0.810456, size = 554, normalized size = 2.16 \[ -\frac{c^{3} d^{6} - 3 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} + a^{3} e^{6} + 3 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 3 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4}}{e^{8} x + d e^{7}} + \frac{4 \, c^{3} e^{4} x^{5} - 5 \,{\left (2 \, c^{3} d e^{3} - 3 \, b c^{2} e^{4}\right )} x^{4} + 20 \,{\left (c^{3} d^{2} e^{2} - 2 \, b c^{2} d e^{3} +{\left (b^{2} c + a c^{2}\right )} e^{4}\right )} x^{3} - 10 \,{\left (4 \, c^{3} d^{3} e - 9 \, b c^{2} d^{2} e^{2} + 6 \,{\left (b^{2} c + a c^{2}\right )} d e^{3} -{\left (b^{3} + 6 \, a b c\right )} e^{4}\right )} x^{2} + 20 \,{\left (5 \, c^{3} d^{4} - 12 \, b c^{2} d^{3} e + 9 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} - 2 \,{\left (b^{3} + 6 \, a b c\right )} d e^{3} + 3 \,{\left (a b^{2} + a^{2} c\right )} e^{4}\right )} x}{20 \, e^{6}} - \frac{3 \,{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \,{\left (a b^{2} + a^{2} c\right )} d e^{4}\right )} \log \left (e x + d\right )}{e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3/(e*x + d)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.226898, size = 783, normalized size = 3.06 \[ \frac{4 \, c^{3} e^{6} x^{6} - 20 \, c^{3} d^{6} + 60 \, b c^{2} d^{5} e + 60 \, a^{2} b d e^{5} - 20 \, a^{3} e^{6} - 60 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} + 20 \,{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} - 60 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} - 3 \,{\left (2 \, c^{3} d e^{5} - 5 \, b c^{2} e^{6}\right )} x^{5} + 5 \,{\left (2 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} + 4 \,{\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} - 10 \,{\left (2 \, c^{3} d^{3} e^{3} - 5 \, b c^{2} d^{2} e^{4} + 4 \,{\left (b^{2} c + a c^{2}\right )} d e^{5} -{\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 30 \,{\left (2 \, c^{3} d^{4} e^{2} - 5 \, b c^{2} d^{3} e^{3} + 4 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} -{\left (b^{3} + 6 \, a b c\right )} d e^{5} + 2 \,{\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 20 \,{\left (5 \, c^{3} d^{5} e - 12 \, b c^{2} d^{4} e^{2} + 9 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - 2 \,{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 3 \,{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x - 60 \,{\left (2 \, c^{3} d^{6} - 5 \, b c^{2} d^{5} e - a^{2} b d e^{5} + 4 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 2 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} +{\left (2 \, c^{3} d^{5} e - 5 \, b c^{2} d^{4} e^{2} - a^{2} b e^{6} + 4 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 2 \,{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x\right )} \log \left (e x + d\right )}{20 \,{\left (e^{8} x + d e^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3/(e*x + d)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 9.7502, size = 403, normalized size = 1.57 \[ \frac{c^{3} x^{5}}{5 e^{2}} - \frac{a^{3} e^{6} - 3 a^{2} b d e^{5} + 3 a^{2} c d^{2} e^{4} + 3 a b^{2} d^{2} e^{4} - 6 a b c d^{3} e^{3} + 3 a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} + 3 b^{2} c d^{4} e^{2} - 3 b c^{2} d^{5} e + c^{3} d^{6}}{d e^{7} + e^{8} x} + \frac{x^{4} \left (3 b c^{2} e - 2 c^{3} d\right )}{4 e^{3}} + \frac{x^{3} \left (a c^{2} e^{2} + b^{2} c e^{2} - 2 b c^{2} d e + c^{3} d^{2}\right )}{e^{4}} + \frac{x^{2} \left (6 a b c e^{3} - 6 a c^{2} d e^{2} + b^{3} e^{3} - 6 b^{2} c d e^{2} + 9 b c^{2} d^{2} e - 4 c^{3} d^{3}\right )}{2 e^{5}} + \frac{x \left (3 a^{2} c e^{4} + 3 a b^{2} e^{4} - 12 a b c d e^{3} + 9 a c^{2} d^{2} e^{2} - 2 b^{3} d e^{3} + 9 b^{2} c d^{2} e^{2} - 12 b c^{2} d^{3} e + 5 c^{3} d^{4}\right )}{e^{6}} + \frac{3 \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2} \log{\left (d + e x \right )}}{e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x+a)**3/(e*x+d)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.206775, size = 730, normalized size = 2.85 \[ \frac{1}{20} \,{\left (4 \, c^{3} - \frac{15 \,{\left (2 \, c^{3} d e - b c^{2} e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac{20 \,{\left (5 \, c^{3} d^{2} e^{2} - 5 \, b c^{2} d e^{3} + b^{2} c e^{4} + a c^{2} e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac{10 \,{\left (20 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} + 12 \, b^{2} c d e^{5} + 12 \, a c^{2} d e^{5} - b^{3} e^{6} - 6 \, a b c e^{6}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}} + \frac{60 \,{\left (5 \, c^{3} d^{4} e^{4} - 10 \, b c^{2} d^{3} e^{5} + 6 \, b^{2} c d^{2} e^{6} + 6 \, a c^{2} d^{2} e^{6} - b^{3} d e^{7} - 6 \, a b c d e^{7} + a b^{2} e^{8} + a^{2} c e^{8}\right )} e^{\left (-4\right )}}{{\left (x e + d\right )}^{4}}\right )}{\left (x e + d\right )}^{5} e^{\left (-7\right )} + 3 \,{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, b^{2} c d^{3} e^{2} + 4 \, a c^{2} d^{3} e^{2} - b^{3} d^{2} e^{3} - 6 \, a b c d^{2} e^{3} + 2 \, a b^{2} d e^{4} + 2 \, a^{2} c d e^{4} - a^{2} b e^{5}\right )} e^{\left (-7\right )}{\rm ln}\left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) -{\left (\frac{c^{3} d^{6} e^{5}}{x e + d} - \frac{3 \, b c^{2} d^{5} e^{6}}{x e + d} + \frac{3 \, b^{2} c d^{4} e^{7}}{x e + d} + \frac{3 \, a c^{2} d^{4} e^{7}}{x e + d} - \frac{b^{3} d^{3} e^{8}}{x e + d} - \frac{6 \, a b c d^{3} e^{8}}{x e + d} + \frac{3 \, a b^{2} d^{2} e^{9}}{x e + d} + \frac{3 \, a^{2} c d^{2} e^{9}}{x e + d} - \frac{3 \, a^{2} b d e^{10}}{x e + d} + \frac{a^{3} e^{11}}{x e + d}\right )} e^{\left (-12\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3/(e*x + d)^2,x, algorithm="giac")
[Out]