Optimal. Leaf size=189 \[ -\frac{b^3 x (-4 a B e-A b e+4 b B d)}{e^5}+\frac{2 b^2 (b d-a e) \log (d+e x) (-3 a B e-2 A b e+5 b B d)}{e^6}+\frac{2 b (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{e^6 (d+e x)}-\frac{(b d-a e)^3 (-a B e-4 A b e+5 b B d)}{2 e^6 (d+e x)^2}+\frac{(b d-a e)^4 (B d-A e)}{3 e^6 (d+e x)^3}+\frac{b^4 B x^2}{2 e^4} \]
[Out]
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Rubi [A] time = 0.529179, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ -\frac{b^3 x (-4 a B e-A b e+4 b B d)}{e^5}+\frac{2 b^2 (b d-a e) \log (d+e x) (-3 a B e-2 A b e+5 b B d)}{e^6}+\frac{2 b (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{e^6 (d+e x)}-\frac{(b d-a e)^3 (-a B e-4 A b e+5 b B d)}{2 e^6 (d+e x)^2}+\frac{(b d-a e)^4 (B d-A e)}{3 e^6 (d+e x)^3}+\frac{b^4 B x^2}{2 e^4} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/(d + e*x)^4,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{B b^{4} \int x\, dx}{e^{4}} + \frac{b^{3} x \left (A b e + 4 B a e - 4 B b d\right )}{e^{5}} + \frac{2 b^{2} \left (a e - b d\right ) \left (2 A b e + 3 B a e - 5 B b d\right ) \log{\left (d + e x \right )}}{e^{6}} - \frac{2 b \left (a e - b d\right )^{2} \left (3 A b e + 2 B a e - 5 B b d\right )}{e^{6} \left (d + e x\right )} - \frac{\left (a e - b d\right )^{3} \left (4 A b e + B a e - 5 B b d\right )}{2 e^{6} \left (d + e x\right )^{2}} - \frac{\left (A e - B d\right ) \left (a e - b d\right )^{4}}{3 e^{6} \left (d + e x\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**2/(e*x+d)**4,x)
[Out]
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Mathematica [A] time = 0.30098, size = 351, normalized size = 1.86 \[ \frac{-a^4 e^4 (2 A e+B (d+3 e x))-4 a^3 b e^3 \left (A e (d+3 e x)+2 B \left (d^2+3 d e x+3 e^2 x^2\right )\right )+6 a^2 b^2 e^2 \left (B d \left (11 d^2+27 d e x+18 e^2 x^2\right )-2 A e \left (d^2+3 d e x+3 e^2 x^2\right )\right )+4 a b^3 e \left (A d e \left (11 d^2+27 d e x+18 e^2 x^2\right )-2 B \left (13 d^4+27 d^3 e x+9 d^2 e^2 x^2-9 d e^3 x^3-3 e^4 x^4\right )\right )+12 b^2 (d+e x)^3 (b d-a e) \log (d+e x) (-3 a B e-2 A b e+5 b B d)+b^4 \left (2 A e \left (-13 d^4-27 d^3 e x-9 d^2 e^2 x^2+9 d e^3 x^3+3 e^4 x^4\right )+B \left (47 d^5+81 d^4 e x-9 d^3 e^2 x^2-63 d^2 e^3 x^3-15 d e^4 x^4+3 e^5 x^5\right )\right )}{6 e^6 (d+e x)^3} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/(d + e*x)^4,x]
[Out]
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Maple [B] time = 0.018, size = 626, normalized size = 3.3 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^4,x)
[Out]
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Maxima [A] time = 0.707303, size = 582, normalized size = 3.08 \[ \frac{47 \, B b^{4} d^{5} - 2 \, A a^{4} e^{5} - 26 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 22 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} -{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 12 \,{\left (5 \, B b^{4} d^{3} e^{2} - 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 3 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} -{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 3 \,{\left (35 \, B b^{4} d^{4} e - 20 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 18 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} - 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} -{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{6 \,{\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} + \frac{B b^{4} e x^{2} - 2 \,{\left (4 \, B b^{4} d -{\left (4 \, B a b^{3} + A b^{4}\right )} e\right )} x}{2 \, e^{5}} + \frac{2 \,{\left (5 \, B b^{4} d^{2} - 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e +{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{2}\right )} \log \left (e x + d\right )}{e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(B*x + A)/(e*x + d)^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.290983, size = 878, normalized size = 4.65 \[ \frac{3 \, B b^{4} e^{5} x^{5} + 47 \, B b^{4} d^{5} - 2 \, A a^{4} e^{5} - 26 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 22 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} -{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} - 3 \,{\left (5 \, B b^{4} d e^{4} - 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} - 9 \,{\left (7 \, B b^{4} d^{2} e^{3} - 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4}\right )} x^{3} - 3 \,{\left (3 \, B b^{4} d^{3} e^{2} + 6 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} - 12 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} + 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 3 \,{\left (27 \, B b^{4} d^{4} e - 18 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 18 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} - 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} -{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x + 12 \,{\left (5 \, B b^{4} d^{5} - 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e +{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} +{\left (5 \, B b^{4} d^{2} e^{3} - 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} +{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 3 \,{\left (5 \, B b^{4} d^{3} e^{2} - 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} +{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4}\right )} x^{2} + 3 \,{\left (5 \, B b^{4} d^{4} e - 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} +{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3}\right )} x\right )} \log \left (e x + d\right )}{6 \,{\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(B*x + A)/(e*x + d)^4,x, algorithm="fricas")
[Out]
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Sympy [A] time = 115.793, size = 483, normalized size = 2.56 \[ \frac{B b^{4} x^{2}}{2 e^{4}} + \frac{2 b^{2} \left (a e - b d\right ) \left (2 A b e + 3 B a e - 5 B b d\right ) \log{\left (d + e x \right )}}{e^{6}} - \frac{2 A a^{4} e^{5} + 4 A a^{3} b d e^{4} + 12 A a^{2} b^{2} d^{2} e^{3} - 44 A a b^{3} d^{3} e^{2} + 26 A b^{4} d^{4} e + B a^{4} d e^{4} + 8 B a^{3} b d^{2} e^{3} - 66 B a^{2} b^{2} d^{3} e^{2} + 104 B a b^{3} d^{4} e - 47 B b^{4} d^{5} + x^{2} \left (36 A a^{2} b^{2} e^{5} - 72 A a b^{3} d e^{4} + 36 A b^{4} d^{2} e^{3} + 24 B a^{3} b e^{5} - 108 B a^{2} b^{2} d e^{4} + 144 B a b^{3} d^{2} e^{3} - 60 B b^{4} d^{3} e^{2}\right ) + x \left (12 A a^{3} b e^{5} + 36 A a^{2} b^{2} d e^{4} - 108 A a b^{3} d^{2} e^{3} + 60 A b^{4} d^{3} e^{2} + 3 B a^{4} e^{5} + 24 B a^{3} b d e^{4} - 162 B a^{2} b^{2} d^{2} e^{3} + 240 B a b^{3} d^{3} e^{2} - 105 B b^{4} d^{4} e\right )}{6 d^{3} e^{6} + 18 d^{2} e^{7} x + 18 d e^{8} x^{2} + 6 e^{9} x^{3}} + \frac{x \left (A b^{4} e + 4 B a b^{3} e - 4 B b^{4} d\right )}{e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**2/(e*x+d)**4,x)
[Out]
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GIAC/XCAS [A] time = 0.286035, size = 560, normalized size = 2.96 \[ 2 \,{\left (5 \, B b^{4} d^{2} - 8 \, B a b^{3} d e - 2 \, A b^{4} d e + 3 \, B a^{2} b^{2} e^{2} + 2 \, A a b^{3} e^{2}\right )} e^{\left (-6\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{2} \,{\left (B b^{4} x^{2} e^{4} - 8 \, B b^{4} d x e^{3} + 8 \, B a b^{3} x e^{4} + 2 \, A b^{4} x e^{4}\right )} e^{\left (-8\right )} + \frac{{\left (47 \, B b^{4} d^{5} - 104 \, B a b^{3} d^{4} e - 26 \, A b^{4} d^{4} e + 66 \, B a^{2} b^{2} d^{3} e^{2} + 44 \, A a b^{3} d^{3} e^{2} - 8 \, B a^{3} b d^{2} e^{3} - 12 \, A a^{2} b^{2} d^{2} e^{3} - B a^{4} d e^{4} - 4 \, A a^{3} b d e^{4} - 2 \, A a^{4} e^{5} + 12 \,{\left (5 \, B b^{4} d^{3} e^{2} - 12 \, B a b^{3} d^{2} e^{3} - 3 \, A b^{4} d^{2} e^{3} + 9 \, B a^{2} b^{2} d e^{4} + 6 \, A a b^{3} d e^{4} - 2 \, B a^{3} b e^{5} - 3 \, A a^{2} b^{2} e^{5}\right )} x^{2} + 3 \,{\left (35 \, B b^{4} d^{4} e - 80 \, B a b^{3} d^{3} e^{2} - 20 \, A b^{4} d^{3} e^{2} + 54 \, B a^{2} b^{2} d^{2} e^{3} + 36 \, A a b^{3} d^{2} e^{3} - 8 \, B a^{3} b d e^{4} - 12 \, A a^{2} b^{2} d e^{4} - B a^{4} e^{5} - 4 \, A a^{3} b e^{5}\right )} x\right )} e^{\left (-6\right )}}{6 \,{\left (x e + d\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(B*x + A)/(e*x + d)^4,x, algorithm="giac")
[Out]