Optimal. Leaf size=189 \[ -\frac{b^3 \log (d+e x) (-4 a B e-A b e+5 b B d)}{e^6}-\frac{2 b^2 (b d-a e) (-3 a B e-2 A b e+5 b B d)}{e^6 (d+e x)}+\frac{b (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{e^6 (d+e x)^2}-\frac{(b d-a e)^3 (-a B e-4 A b e+5 b B d)}{3 e^6 (d+e x)^3}+\frac{(b d-a e)^4 (B d-A e)}{4 e^6 (d+e x)^4}+\frac{b^4 B x}{e^5} \]
[Out]
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Rubi [A] time = 0.552933, 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 \[ -\frac{b^3 \log (d+e x) (-4 a B e-A b e+5 b B d)}{e^6}-\frac{2 b^2 (b d-a e) (-3 a B e-2 A b e+5 b B d)}{e^6 (d+e x)}+\frac{b (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{e^6 (d+e x)^2}-\frac{(b d-a e)^3 (-a B e-4 A b e+5 b B d)}{3 e^6 (d+e x)^3}+\frac{(b d-a e)^4 (B d-A e)}{4 e^6 (d+e x)^4}+\frac{b^4 B x}{e^5} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/(d + e*x)^5,x]
[Out]
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Rubi in Sympy [A] time = 117.903, size = 189, normalized size = 1. \[ \frac{B b^{4} x}{e^{5}} + \frac{b^{3} \left (A b e + 4 B a e - 5 B b d\right ) \log{\left (d + e x \right )}}{e^{6}} - \frac{2 b^{2} \left (a e - b d\right ) \left (2 A b e + 3 B a e - 5 B b d\right )}{e^{6} \left (d + e x\right )} - \frac{b \left (a e - b d\right )^{2} \left (3 A b e + 2 B a e - 5 B b d\right )}{e^{6} \left (d + e x\right )^{2}} - \frac{\left (a e - b d\right )^{3} \left (4 A b e + B a e - 5 B b d\right )}{3 e^{6} \left (d + e x\right )^{3}} - \frac{\left (A e - B d\right ) \left (a e - b d\right )^{4}}{4 e^{6} \left (d + e x\right )^{4}} \]
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)**5,x)
[Out]
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Mathematica [A] time = 0.33023, size = 338, normalized size = 1.79 \[ -\frac{a^4 e^4 (3 A e+B (d+4 e x))+4 a^3 b e^3 \left (A e (d+4 e x)+B \left (d^2+4 d e x+6 e^2 x^2\right )\right )+6 a^2 b^2 e^2 \left (A e \left (d^2+4 d e x+6 e^2 x^2\right )+3 B \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )\right )-4 a b^3 e \left (B d \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )-3 A e \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )\right )+12 b^3 (d+e x)^4 \log (d+e x) (-4 a B e-A b e+5 b B d)+b^4 \left (-\left (A d e \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )-B \left (77 d^5+248 d^4 e x+252 d^3 e^2 x^2+48 d^2 e^3 x^3-48 d e^4 x^4-12 e^5 x^5\right )\right )\right )}{12 e^6 (d+e x)^4} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/(d + e*x)^5,x]
[Out]
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Maple [B] time = 0.017, size = 641, normalized size = 3.4 \[ \text{result too large to display} \]
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)^5,x)
[Out]
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Maxima [A] time = 0.708456, size = 594, normalized size = 3.14 \[ \frac{B b^{4} x}{e^{5}} - \frac{77 \, B b^{4} d^{5} + 3 \, A a^{4} e^{5} - 25 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + 2 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} +{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 24 \,{\left (5 \, B b^{4} d^{2} e^{3} - 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} +{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 12 \,{\left (25 \, B b^{4} d^{3} e^{2} - 9 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 3 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} +{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 4 \,{\left (65 \, B b^{4} d^{4} e - 22 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} + 2 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} +{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{12 \,{\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\right )}} - \frac{{\left (5 \, B b^{4} d -{\left (4 \, B a b^{3} + A b^{4}\right )} e\right )} \log \left (e x + d\right )}{e^{6}} \]
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)^5,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.281768, size = 813, normalized size = 4.3 \[ \frac{12 \, B b^{4} e^{5} x^{5} + 48 \, B b^{4} d e^{4} x^{4} - 77 \, B b^{4} d^{5} - 3 \, A a^{4} e^{5} + 25 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e - 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 2 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} -{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} - 24 \,{\left (2 \, B b^{4} d^{2} e^{3} - 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} +{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} - 12 \,{\left (21 \, B b^{4} d^{3} e^{2} - 9 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 3 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} +{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} - 4 \,{\left (62 \, B b^{4} d^{4} e - 22 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} + 2 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} +{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x - 12 \,{\left (5 \, B b^{4} d^{5} -{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e +{\left (5 \, B b^{4} d e^{4} -{\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 4 \,{\left (5 \, B b^{4} d^{2} e^{3} -{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4}\right )} x^{3} + 6 \,{\left (5 \, B b^{4} d^{3} e^{2} -{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3}\right )} x^{2} + 4 \,{\left (5 \, B b^{4} d^{4} e -{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2}\right )} x\right )} \log \left (e x + d\right )}{12 \,{\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\right )}} \]
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)^5,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
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)**5,x)
[Out]
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GIAC/XCAS [A] time = 0.294417, size = 880, normalized size = 4.66 \[ \text{result too large to display} \]
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)^5,x, algorithm="giac")
[Out]