Optimal. Leaf size=255 \[ -\frac{x \left (-9 c^2 d e (2 b d-a e)+3 b c e^2 (3 b d-2 a e)-b^3 e^3+10 c^3 d^3\right )}{e^6}+\frac{3 \log (d+e x) \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}+\frac{3 c x^2 \left (-c e (3 b d-a e)+b^2 e^2+2 c^2 d^2\right )}{2 e^5}+\frac{3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7 (d+e x)}-\frac{\left (a e^2-b d e+c d^2\right )^3}{2 e^7 (d+e x)^2}-\frac{c^2 x^3 (c d-b e)}{e^4}+\frac{c^3 x^4}{4 e^3} \]
[Out]
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Rubi [A] time = 0.88491, antiderivative size = 255, 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 \[ -\frac{x \left (-9 c^2 d e (2 b d-a e)+3 b c e^2 (3 b d-2 a e)-b^3 e^3+10 c^3 d^3\right )}{e^6}+\frac{3 \log (d+e x) \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}+\frac{3 c x^2 \left (-c e (3 b d-a e)+b^2 e^2+2 c^2 d^2\right )}{2 e^5}+\frac{3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7 (d+e x)}-\frac{\left (a e^2-b d e+c d^2\right )^3}{2 e^7 (d+e x)^2}-\frac{c^2 x^3 (c d-b e)}{e^4}+\frac{c^3 x^4}{4 e^3} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)^3/(d + e*x)^3,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{c^{3} \left (d + e x\right )^{4}}{4 e^{7}} + \frac{c^{2} \left (d + e x\right )^{3} \left (b e - 2 c d\right )}{e^{7}} + \frac{3 c \left (d + e x\right )^{2} \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{2 e^{7}} + \left (b e - 2 c d\right ) \left (6 a c e^{2} + b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right ) \int \frac{1}{e^{6}}\, dx + \frac{3 \left (a e^{2} - b d e + c d^{2}\right ) \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) \log{\left (d + e x \right )}}{e^{7}} - \frac{3 \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2}}{e^{7} \left (d + e x\right )} - \frac{\left (a e^{2} - b d e + c d^{2}\right )^{3}}{2 e^{7} \left (d + e x\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x+a)**3/(e*x+d)**3,x)
[Out]
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Mathematica [A] time = 0.254098, size = 265, normalized size = 1.04 \[ \frac{12 \log (d+e x) \left (c e^2 \left (a^2 e^2-6 a b d e+6 b^2 d^2\right )+b^2 e^3 (a e-b d)+2 c^2 d^2 e (3 a e-5 b d)+5 c^3 d^4\right )+4 e x \left (9 c^2 d e (2 b d-a e)+3 b c e^2 (2 a e-3 b d)+b^3 e^3-10 c^3 d^3\right )+6 c e^2 x^2 \left (c e (a e-3 b d)+b^2 e^2+2 c^2 d^2\right )+\frac{12 (2 c d-b e) \left (e (a e-b d)+c d^2\right )^2}{d+e x}-\frac{2 \left (e (a e-b d)+c d^2\right )^3}{(d+e x)^2}+4 c^2 e^3 x^3 (b e-c d)+c^3 e^4 x^4}{4 e^7} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)^3/(d + e*x)^3,x]
[Out]
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Maple [B] time = 0.017, size = 624, normalized size = 2.5 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)^3/(e*x+d)^3,x)
[Out]
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Maxima [A] time = 0.838597, size = 563, normalized size = 2.21 \[ \frac{11 \, c^{3} d^{6} - 27 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} - a^{3} e^{6} + 21 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - 5 \,{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 9 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 6 \,{\left (2 \, c^{3} d^{5} e - 5 \, b c^{2} d^{4} e^{2} - a^{2} b e^{6} + 4 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 2 \,{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x}{2 \,{\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} + \frac{c^{3} e^{3} x^{4} - 4 \,{\left (c^{3} d e^{2} - b c^{2} e^{3}\right )} x^{3} + 6 \,{\left (2 \, c^{3} d^{2} e - 3 \, b c^{2} d e^{2} +{\left (b^{2} c + a c^{2}\right )} e^{3}\right )} x^{2} - 4 \,{\left (10 \, c^{3} d^{3} - 18 \, b c^{2} d^{2} e + 9 \,{\left (b^{2} c + a c^{2}\right )} d e^{2} -{\left (b^{3} + 6 \, a b c\right )} e^{3}\right )} x}{4 \, e^{6}} + \frac{3 \,{\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d e^{3} +{\left (a b^{2} + a^{2} c\right )} e^{4}\right )} \log \left (e x + d\right )}{e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3/(e*x + d)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.22994, size = 861, normalized size = 3.38 \[ \frac{c^{3} e^{6} x^{6} + 22 \, c^{3} d^{6} - 54 \, b c^{2} d^{5} e - 6 \, a^{2} b d e^{5} - 2 \, a^{3} e^{6} + 42 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - 10 \,{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 18 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} - 2 \,{\left (c^{3} d e^{5} - 2 \, b c^{2} e^{6}\right )} x^{5} +{\left (5 \, c^{3} d^{2} e^{4} - 10 \, b c^{2} d e^{5} + 6 \,{\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} - 4 \,{\left (5 \, c^{3} d^{3} e^{3} - 10 \, b c^{2} d^{2} e^{4} + 6 \,{\left (b^{2} c + a c^{2}\right )} d e^{5} -{\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} - 2 \,{\left (34 \, c^{3} d^{4} e^{2} - 63 \, b c^{2} d^{3} e^{3} + 33 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} - 4 \,{\left (b^{3} + 6 \, a b c\right )} d e^{5}\right )} x^{2} - 4 \,{\left (4 \, c^{3} d^{5} e - 3 \, b c^{2} d^{4} e^{2} + 3 \, a^{2} b e^{6} - 3 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} + 2 \,{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} - 6 \,{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x + 12 \,{\left (5 \, c^{3} d^{6} - 10 \, b c^{2} d^{5} e + 6 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} +{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} +{\left (5 \, c^{3} d^{4} e^{2} - 10 \, b c^{2} d^{3} e^{3} + 6 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} -{\left (b^{3} + 6 \, a b c\right )} d e^{5} +{\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 2 \,{\left (5 \, c^{3} d^{5} e - 10 \, b c^{2} d^{4} e^{2} + 6 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} +{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x\right )} \log \left (e x + d\right )}{4 \,{\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3/(e*x + d)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 30.3021, size = 457, normalized size = 1.79 \[ \frac{c^{3} x^{4}}{4 e^{3}} - \frac{a^{3} e^{6} + 3 a^{2} b d e^{5} - 9 a^{2} c d^{2} e^{4} - 9 a b^{2} d^{2} e^{4} + 30 a b c d^{3} e^{3} - 21 a c^{2} d^{4} e^{2} + 5 b^{3} d^{3} e^{3} - 21 b^{2} c d^{4} e^{2} + 27 b c^{2} d^{5} e - 11 c^{3} d^{6} + x \left (6 a^{2} b e^{6} - 12 a^{2} c d e^{5} - 12 a b^{2} d e^{5} + 36 a b c d^{2} e^{4} - 24 a c^{2} d^{3} e^{3} + 6 b^{3} d^{2} e^{4} - 24 b^{2} c d^{3} e^{3} + 30 b c^{2} d^{4} e^{2} - 12 c^{3} d^{5} e\right )}{2 d^{2} e^{7} + 4 d e^{8} x + 2 e^{9} x^{2}} + \frac{x^{3} \left (b c^{2} e - c^{3} d\right )}{e^{4}} + \frac{x^{2} \left (3 a c^{2} e^{2} + 3 b^{2} c e^{2} - 9 b c^{2} d e + 6 c^{3} d^{2}\right )}{2 e^{5}} + \frac{x \left (6 a b c e^{3} - 9 a c^{2} d e^{2} + b^{3} e^{3} - 9 b^{2} c d e^{2} + 18 b c^{2} d^{2} e - 10 c^{3} d^{3}\right )}{e^{6}} + \frac{3 \left (a e^{2} - b d e + c d^{2}\right ) \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) \log{\left (d + e x \right )}}{e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x+a)**3/(e*x+d)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.206, size = 583, normalized size = 2.29 \[ 3 \,{\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, b^{2} c d^{2} e^{2} + 6 \, a c^{2} d^{2} e^{2} - b^{3} d e^{3} - 6 \, a b c d e^{3} + a b^{2} e^{4} + a^{2} c e^{4}\right )} e^{\left (-7\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{4} \,{\left (c^{3} x^{4} e^{9} - 4 \, c^{3} d x^{3} e^{8} + 12 \, c^{3} d^{2} x^{2} e^{7} - 40 \, c^{3} d^{3} x e^{6} + 4 \, b c^{2} x^{3} e^{9} - 18 \, b c^{2} d x^{2} e^{8} + 72 \, b c^{2} d^{2} x e^{7} + 6 \, b^{2} c x^{2} e^{9} + 6 \, a c^{2} x^{2} e^{9} - 36 \, b^{2} c d x e^{8} - 36 \, a c^{2} d x e^{8} + 4 \, b^{3} x e^{9} + 24 \, a b c x e^{9}\right )} e^{\left (-12\right )} + \frac{{\left (11 \, c^{3} d^{6} - 27 \, b c^{2} d^{5} e + 21 \, b^{2} c d^{4} e^{2} + 21 \, a c^{2} d^{4} e^{2} - 5 \, b^{3} d^{3} e^{3} - 30 \, a b c d^{3} e^{3} + 9 \, a b^{2} d^{2} e^{4} + 9 \, a^{2} c d^{2} e^{4} - 3 \, a^{2} b d e^{5} - a^{3} e^{6} + 6 \,{\left (2 \, c^{3} d^{5} e - 5 \, b c^{2} d^{4} e^{2} + 4 \, b^{2} c d^{3} e^{3} + 4 \, a c^{2} d^{3} e^{3} - b^{3} d^{2} e^{4} - 6 \, a b c d^{2} e^{4} + 2 \, a b^{2} d e^{5} + 2 \, a^{2} c d e^{5} - a^{2} b e^{6}\right )} x\right )} e^{\left (-7\right )}}{2 \,{\left (x e + d\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3/(e*x + d)^3,x, algorithm="giac")
[Out]