Optimal. Leaf size=120 \[ \frac{b (-2 a B e-A b e+3 b B d)}{5 e^4 (d+e x)^5}-\frac{(b d-a e) (-a B e-2 A b e+3 b B d)}{6 e^4 (d+e x)^6}+\frac{(b d-a e)^2 (B d-A e)}{7 e^4 (d+e x)^7}-\frac{b^2 B}{4 e^4 (d+e x)^4} \]
[Out]
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Rubi [A] time = 0.218106, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{b (-2 a B e-A b e+3 b B d)}{5 e^4 (d+e x)^5}-\frac{(b d-a e) (-a B e-2 A b e+3 b B d)}{6 e^4 (d+e x)^6}+\frac{(b d-a e)^2 (B d-A e)}{7 e^4 (d+e x)^7}-\frac{b^2 B}{4 e^4 (d+e x)^4} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2))/(d + e*x)^8,x]
[Out]
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Rubi in Sympy [A] time = 67.2985, size = 114, normalized size = 0.95 \[ - \frac{B b^{2}}{4 e^{4} \left (d + e x\right )^{4}} - \frac{b \left (A b e + 2 B a e - 3 B b d\right )}{5 e^{4} \left (d + e x\right )^{5}} - \frac{\left (a e - b d\right ) \left (2 A b e + B a e - 3 B b d\right )}{6 e^{4} \left (d + e x\right )^{6}} - \frac{\left (A e - B d\right ) \left (a e - b d\right )^{2}}{7 e^{4} \left (d + e x\right )^{7}} \]
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)/(e*x+d)**8,x)
[Out]
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Mathematica [A] time = 0.104265, size = 129, normalized size = 1.08 \[ -\frac{10 a^2 e^2 (6 A e+B (d+7 e x))+4 a b e \left (5 A e (d+7 e x)+2 B \left (d^2+7 d e x+21 e^2 x^2\right )\right )+b^2 \left (4 A e \left (d^2+7 d e x+21 e^2 x^2\right )+3 B \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )\right )}{420 e^4 (d+e x)^7} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2))/(d + e*x)^8,x]
[Out]
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Maple [A] time = 0.009, size = 166, normalized size = 1.4 \[ -{\frac{b \left ( Abe+2\,aBe-3\,Bbd \right ) }{5\,{e}^{4} \left ( ex+d \right ) ^{5}}}-{\frac{{b}^{2}B}{4\,{e}^{4} \left ( ex+d \right ) ^{4}}}-{\frac{A{a}^{2}{e}^{3}-2\,Adab{e}^{2}+A{d}^{2}{b}^{2}e-Bd{a}^{2}{e}^{2}+2\,B{d}^{2}abe-{b}^{2}B{d}^{3}}{7\,{e}^{4} \left ( ex+d \right ) ^{7}}}-{\frac{2\,abA{e}^{2}-2\,Ad{b}^{2}e+{a}^{2}B{e}^{2}-4\,Bdabe+3\,{b}^{2}B{d}^{2}}{6\,{e}^{4} \left ( ex+d \right ) ^{6}}} \]
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)/(e*x+d)^8,x)
[Out]
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Maxima [A] time = 0.702108, size = 304, normalized size = 2.53 \[ -\frac{105 \, B b^{2} e^{3} x^{3} + 3 \, B b^{2} d^{3} + 60 \, A a^{2} e^{3} + 4 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e + 10 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{2} + 21 \,{\left (3 \, B b^{2} d e^{2} + 4 \,{\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 7 \,{\left (3 \, B b^{2} d^{2} e + 4 \,{\left (2 \, B a b + A b^{2}\right )} d e^{2} + 10 \,{\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x}{420 \,{\left (e^{11} x^{7} + 7 \, d e^{10} x^{6} + 21 \, d^{2} e^{9} x^{5} + 35 \, d^{3} e^{8} x^{4} + 35 \, d^{4} e^{7} x^{3} + 21 \, d^{5} e^{6} x^{2} + 7 \, d^{6} e^{5} x + d^{7} e^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)*(B*x + A)/(e*x + d)^8,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.275887, size = 304, normalized size = 2.53 \[ -\frac{105 \, B b^{2} e^{3} x^{3} + 3 \, B b^{2} d^{3} + 60 \, A a^{2} e^{3} + 4 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e + 10 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{2} + 21 \,{\left (3 \, B b^{2} d e^{2} + 4 \,{\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 7 \,{\left (3 \, B b^{2} d^{2} e + 4 \,{\left (2 \, B a b + A b^{2}\right )} d e^{2} + 10 \,{\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x}{420 \,{\left (e^{11} x^{7} + 7 \, d e^{10} x^{6} + 21 \, d^{2} e^{9} x^{5} + 35 \, d^{3} e^{8} x^{4} + 35 \, d^{4} e^{7} x^{3} + 21 \, d^{5} e^{6} x^{2} + 7 \, d^{6} e^{5} x + d^{7} e^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)*(B*x + A)/(e*x + d)^8,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)/(e*x+d)**8,x)
[Out]
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GIAC/XCAS [A] time = 0.27649, size = 216, normalized size = 1.8 \[ -\frac{{\left (105 \, B b^{2} x^{3} e^{3} + 63 \, B b^{2} d x^{2} e^{2} + 21 \, B b^{2} d^{2} x e + 3 \, B b^{2} d^{3} + 168 \, B a b x^{2} e^{3} + 84 \, A b^{2} x^{2} e^{3} + 56 \, B a b d x e^{2} + 28 \, A b^{2} d x e^{2} + 8 \, B a b d^{2} e + 4 \, A b^{2} d^{2} e + 70 \, B a^{2} x e^{3} + 140 \, A a b x e^{3} + 10 \, B a^{2} d e^{2} + 20 \, A a b d e^{2} + 60 \, A a^{2} e^{3}\right )} e^{\left (-4\right )}}{420 \,{\left (x e + d\right )}^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)*(B*x + A)/(e*x + d)^8,x, algorithm="giac")
[Out]