Optimal. Leaf size=206 \[ \frac{b^3 (-4 a B e-A b e+5 b B d)}{7 e^6 (d+e x)^7}-\frac{b^2 (b d-a e) (-3 a B e-2 A b e+5 b B d)}{4 e^6 (d+e x)^8}+\frac{2 b (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{9 e^6 (d+e x)^9}-\frac{(b d-a e)^3 (-a B e-4 A b e+5 b B d)}{10 e^6 (d+e x)^{10}}+\frac{(b d-a e)^4 (B d-A e)}{11 e^6 (d+e x)^{11}}-\frac{b^4 B}{6 e^6 (d+e x)^6} \]
[Out]
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Rubi [A] time = 0.53302, antiderivative size = 206, 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 \[ \frac{b^3 (-4 a B e-A b e+5 b B d)}{7 e^6 (d+e x)^7}-\frac{b^2 (b d-a e) (-3 a B e-2 A b e+5 b B d)}{4 e^6 (d+e x)^8}+\frac{2 b (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{9 e^6 (d+e x)^9}-\frac{(b d-a e)^3 (-a B e-4 A b e+5 b B d)}{10 e^6 (d+e x)^{10}}+\frac{(b d-a e)^4 (B d-A e)}{11 e^6 (d+e x)^{11}}-\frac{b^4 B}{6 e^6 (d+e x)^6} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/(d + e*x)^12,x]
[Out]
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Rubi in Sympy [A] time = 156.911, size = 204, normalized size = 0.99 \[ - \frac{B b^{4}}{6 e^{6} \left (d + e x\right )^{6}} - \frac{b^{3} \left (A b e + 4 B a e - 5 B b d\right )}{7 e^{6} \left (d + e x\right )^{7}} - \frac{b^{2} \left (a e - b d\right ) \left (2 A b e + 3 B a e - 5 B b d\right )}{4 e^{6} \left (d + e x\right )^{8}} - \frac{2 b \left (a e - b d\right )^{2} \left (3 A b e + 2 B a e - 5 B b d\right )}{9 e^{6} \left (d + e x\right )^{9}} - \frac{\left (a e - b d\right )^{3} \left (4 A b e + B a e - 5 B b d\right )}{10 e^{6} \left (d + e x\right )^{10}} - \frac{\left (A e - B d\right ) \left (a e - b d\right )^{4}}{11 e^{6} \left (d + e x\right )^{11}} \]
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)**12,x)
[Out]
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Mathematica [A] time = 0.313964, size = 323, normalized size = 1.57 \[ -\frac{126 a^4 e^4 (10 A e+B (d+11 e x))+56 a^3 b e^3 \left (9 A e (d+11 e x)+2 B \left (d^2+11 d e x+55 e^2 x^2\right )\right )+21 a^2 b^2 e^2 \left (8 A e \left (d^2+11 d e x+55 e^2 x^2\right )+3 B \left (d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right )\right )+6 a b^3 e \left (7 A e \left (d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right )+4 B \left (d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right )\right )+b^4 \left (6 A e \left (d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right )+5 B \left (d^5+11 d^4 e x+55 d^3 e^2 x^2+165 d^2 e^3 x^3+330 d e^4 x^4+462 e^5 x^5\right )\right )}{13860 e^6 (d+e x)^{11}} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/(d + e*x)^12,x]
[Out]
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Maple [B] time = 0.012, size = 430, normalized size = 2.1 \[ -{\frac{4\,A{a}^{3}b{e}^{4}-12\,Ad{a}^{2}{b}^{2}{e}^{3}+12\,A{d}^{2}a{b}^{3}{e}^{2}-4\,A{d}^{3}{b}^{4}e+B{e}^{4}{a}^{4}-8\,Bd{a}^{3}b{e}^{3}+18\,B{d}^{2}{a}^{2}{b}^{2}{e}^{2}-16\,B{d}^{3}a{b}^{3}e+5\,{b}^{4}B{d}^{4}}{10\,{e}^{6} \left ( ex+d \right ) ^{10}}}-{\frac{{b}^{3} \left ( Abe+4\,aBe-5\,Bbd \right ) }{7\,{e}^{6} \left ( ex+d \right ) ^{7}}}-{\frac{{b}^{4}B}{6\,{e}^{6} \left ( ex+d \right ) ^{6}}}-{\frac{{b}^{2} \left ( 2\,Aab{e}^{2}-2\,Ad{b}^{2}e+3\,{a}^{2}B{e}^{2}-8\,Bdabe+5\,{b}^{2}B{d}^{2} \right ) }{4\,{e}^{6} \left ( ex+d \right ) ^{8}}}-{\frac{2\,b \left ( 3\,A{a}^{2}b{e}^{3}-6\,Aa{b}^{2}d{e}^{2}+3\,A{b}^{3}{d}^{2}e+2\,B{e}^{3}{a}^{3}-9\,B{a}^{2}bd{e}^{2}+12\,Ba{b}^{2}{d}^{2}e-5\,B{b}^{3}{d}^{3} \right ) }{9\,{e}^{6} \left ( ex+d \right ) ^{9}}}-{\frac{A{a}^{4}{e}^{5}-4\,Ad{a}^{3}b{e}^{4}+6\,A{d}^{2}{a}^{2}{b}^{2}{e}^{3}-4\,A{d}^{3}a{b}^{3}{e}^{2}+A{d}^{4}{b}^{4}e-B{a}^{4}d{e}^{4}+4\,B{d}^{2}{a}^{3}b{e}^{3}-6\,B{d}^{3}{a}^{2}{b}^{2}{e}^{2}+4\,B{d}^{4}a{b}^{3}e-{b}^{4}B{d}^{5}}{11\,{e}^{6} \left ( ex+d \right ) ^{11}}} \]
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)^12,x)
[Out]
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Maxima [A] time = 0.716729, size = 705, normalized size = 3.42 \[ -\frac{2310 \, B b^{4} e^{5} x^{5} + 5 \, B b^{4} d^{5} + 1260 \, A a^{4} e^{5} + 6 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 21 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + 56 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + 126 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 330 \,{\left (5 \, B b^{4} d e^{4} + 6 \,{\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 165 \,{\left (5 \, B b^{4} d^{2} e^{3} + 6 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 21 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 55 \,{\left (5 \, B b^{4} d^{3} e^{2} + 6 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 21 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} + 56 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 11 \,{\left (5 \, B b^{4} d^{4} e + 6 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 21 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} + 56 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + 126 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{13860 \,{\left (e^{17} x^{11} + 11 \, d e^{16} x^{10} + 55 \, d^{2} e^{15} x^{9} + 165 \, d^{3} e^{14} x^{8} + 330 \, d^{4} e^{13} x^{7} + 462 \, d^{5} e^{12} x^{6} + 462 \, d^{6} e^{11} x^{5} + 330 \, d^{7} e^{10} x^{4} + 165 \, d^{8} e^{9} x^{3} + 55 \, d^{9} e^{8} x^{2} + 11 \, d^{10} e^{7} x + d^{11} e^{6}\right )}} \]
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)^12,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.268679, size = 705, normalized size = 3.42 \[ -\frac{2310 \, B b^{4} e^{5} x^{5} + 5 \, B b^{4} d^{5} + 1260 \, A a^{4} e^{5} + 6 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 21 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + 56 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + 126 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 330 \,{\left (5 \, B b^{4} d e^{4} + 6 \,{\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 165 \,{\left (5 \, B b^{4} d^{2} e^{3} + 6 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 21 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 55 \,{\left (5 \, B b^{4} d^{3} e^{2} + 6 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 21 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} + 56 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 11 \,{\left (5 \, B b^{4} d^{4} e + 6 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 21 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} + 56 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + 126 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{13860 \,{\left (e^{17} x^{11} + 11 \, d e^{16} x^{10} + 55 \, d^{2} e^{15} x^{9} + 165 \, d^{3} e^{14} x^{8} + 330 \, d^{4} e^{13} x^{7} + 462 \, d^{5} e^{12} x^{6} + 462 \, d^{6} e^{11} x^{5} + 330 \, d^{7} e^{10} x^{4} + 165 \, d^{8} e^{9} x^{3} + 55 \, d^{9} e^{8} x^{2} + 11 \, d^{10} e^{7} x + d^{11} e^{6}\right )}} \]
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)^12,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)**12,x)
[Out]
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GIAC/XCAS [A] time = 0.3109, size = 594, normalized size = 2.88 \[ -\frac{{\left (2310 \, B b^{4} x^{5} e^{5} + 1650 \, B b^{4} d x^{4} e^{4} + 825 \, B b^{4} d^{2} x^{3} e^{3} + 275 \, B b^{4} d^{3} x^{2} e^{2} + 55 \, B b^{4} d^{4} x e + 5 \, B b^{4} d^{5} + 7920 \, B a b^{3} x^{4} e^{5} + 1980 \, A b^{4} x^{4} e^{5} + 3960 \, B a b^{3} d x^{3} e^{4} + 990 \, A b^{4} d x^{3} e^{4} + 1320 \, B a b^{3} d^{2} x^{2} e^{3} + 330 \, A b^{4} d^{2} x^{2} e^{3} + 264 \, B a b^{3} d^{3} x e^{2} + 66 \, A b^{4} d^{3} x e^{2} + 24 \, B a b^{3} d^{4} e + 6 \, A b^{4} d^{4} e + 10395 \, B a^{2} b^{2} x^{3} e^{5} + 6930 \, A a b^{3} x^{3} e^{5} + 3465 \, B a^{2} b^{2} d x^{2} e^{4} + 2310 \, A a b^{3} d x^{2} e^{4} + 693 \, B a^{2} b^{2} d^{2} x e^{3} + 462 \, A a b^{3} d^{2} x e^{3} + 63 \, B a^{2} b^{2} d^{3} e^{2} + 42 \, A a b^{3} d^{3} e^{2} + 6160 \, B a^{3} b x^{2} e^{5} + 9240 \, A a^{2} b^{2} x^{2} e^{5} + 1232 \, B a^{3} b d x e^{4} + 1848 \, A a^{2} b^{2} d x e^{4} + 112 \, B a^{3} b d^{2} e^{3} + 168 \, A a^{2} b^{2} d^{2} e^{3} + 1386 \, B a^{4} x e^{5} + 5544 \, A a^{3} b x e^{5} + 126 \, B a^{4} d e^{4} + 504 \, A a^{3} b d e^{4} + 1260 \, A a^{4} e^{5}\right )} e^{\left (-6\right )}}{13860 \,{\left (x e + d\right )}^{11}} \]
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)^12,x, algorithm="giac")
[Out]