Optimal. Leaf size=120 \[ -\frac{b (d+e x)^7 (-2 a B e-A b e+3 b B d)}{7 e^4}+\frac{(d+e x)^6 (b d-a e) (-a B e-2 A b e+3 b B d)}{6 e^4}-\frac{(d+e x)^5 (b d-a e)^2 (B d-A e)}{5 e^4}+\frac{b^2 B (d+e x)^8}{8 e^4} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.578631, antiderivative size = 120, 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 \[ -\frac{b (d+e x)^7 (-2 a B e-A b e+3 b B d)}{7 e^4}+\frac{(d+e x)^6 (b d-a e) (-a B e-2 A b e+3 b B d)}{6 e^4}-\frac{(d+e x)^5 (b d-a e)^2 (B d-A e)}{5 e^4}+\frac{b^2 B (d+e x)^8}{8 e^4} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^2*(A + B*x)*(d + e*x)^4,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 56.4106, size = 112, normalized size = 0.93 \[ \frac{B b^{2} \left (d + e x\right )^{8}}{8 e^{4}} + \frac{b \left (d + e x\right )^{7} \left (A b e + 2 B a e - 3 B b d\right )}{7 e^{4}} + \frac{\left (d + e x\right )^{6} \left (a e - b d\right ) \left (2 A b e + B a e - 3 B b d\right )}{6 e^{4}} + \frac{\left (d + e x\right )^{5} \left (A e - B d\right ) \left (a e - b d\right )^{2}}{5 e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**2*(B*x+A)*(e*x+d)**4,x)
[Out]
_______________________________________________________________________________________
Mathematica [B] time = 0.162112, size = 283, normalized size = 2.36 \[ \frac{1}{5} e x^5 \left (a^2 e^2 (A e+4 B d)+4 a b d e (2 A e+3 B d)+2 b^2 d^2 (3 A e+2 B d)\right )+\frac{1}{4} d x^4 \left (2 a^2 e^2 (2 A e+3 B d)+4 a b d e (3 A e+2 B d)+b^2 d^2 (4 A e+B d)\right )+\frac{1}{3} d^2 x^3 \left (A \left (6 a^2 e^2+8 a b d e+b^2 d^2\right )+2 a B d (2 a e+b d)\right )+\frac{1}{6} e^2 x^6 \left (a^2 B e^2+2 a b e (A e+4 B d)+2 b^2 d (2 A e+3 B d)\right )+a^2 A d^4 x+\frac{1}{2} a d^3 x^2 (4 a A e+a B d+2 A b d)+\frac{1}{7} b e^3 x^7 (2 a B e+A b e+4 b B d)+\frac{1}{8} b^2 B e^4 x^8 \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^2*(A + B*x)*(d + e*x)^4,x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.003, size = 305, normalized size = 2.5 \[{\frac{{b}^{2}B{e}^{4}{x}^{8}}{8}}+{\frac{ \left ( \left ({b}^{2}A+2\,Bba \right ){e}^{4}+4\,{b}^{2}Bd{e}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 2\,Aab+B{a}^{2} \right ){e}^{4}+4\, \left ({b}^{2}A+2\,Bba \right ) d{e}^{3}+6\,{b}^{2}B{d}^{2}{e}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ({a}^{2}A{e}^{4}+4\, \left ( 2\,Aab+B{a}^{2} \right ) d{e}^{3}+6\, \left ({b}^{2}A+2\,Bba \right ){d}^{2}{e}^{2}+4\,{b}^{2}B{d}^{3}e \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,{a}^{2}Ad{e}^{3}+6\, \left ( 2\,Aab+B{a}^{2} \right ){d}^{2}{e}^{2}+4\, \left ({b}^{2}A+2\,Bba \right ){d}^{3}e+{b}^{2}B{d}^{4} \right ){x}^{4}}{4}}+{\frac{ \left ( 6\,{a}^{2}A{d}^{2}{e}^{2}+4\, \left ( 2\,Aab+B{a}^{2} \right ){d}^{3}e+ \left ({b}^{2}A+2\,Bba \right ){d}^{4} \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,{a}^{2}A{d}^{3}e+ \left ( 2\,Aab+B{a}^{2} \right ){d}^{4} \right ){x}^{2}}{2}}+{a}^{2}A{d}^{4}x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^2*(B*x+A)*(e*x+d)^4,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.33565, size = 410, normalized size = 3.42 \[ \frac{1}{8} \, B b^{2} e^{4} x^{8} + A a^{2} d^{4} x + \frac{1}{7} \,{\left (4 \, B b^{2} d e^{3} +{\left (2 \, B a b + A b^{2}\right )} e^{4}\right )} x^{7} + \frac{1}{6} \,{\left (6 \, B b^{2} d^{2} e^{2} + 4 \,{\left (2 \, B a b + A b^{2}\right )} d e^{3} +{\left (B a^{2} + 2 \, A a b\right )} e^{4}\right )} x^{6} + \frac{1}{5} \,{\left (4 \, B b^{2} d^{3} e + A a^{2} e^{4} + 6 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e^{2} + 4 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (B b^{2} d^{4} + 4 \, A a^{2} d e^{3} + 4 \,{\left (2 \, B a b + A b^{2}\right )} d^{3} e + 6 \,{\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (6 \, A a^{2} d^{2} e^{2} +{\left (2 \, B a b + A b^{2}\right )} d^{4} + 4 \,{\left (B a^{2} + 2 \, A a b\right )} d^{3} e\right )} x^{3} + \frac{1}{2} \,{\left (4 \, A a^{2} d^{3} e +{\left (B a^{2} + 2 \, A a b\right )} d^{4}\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^2*(e*x + d)^4,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.189521, size = 1, normalized size = 0.01 \[ \frac{1}{8} x^{8} e^{4} b^{2} B + \frac{4}{7} x^{7} e^{3} d b^{2} B + \frac{2}{7} x^{7} e^{4} b a B + \frac{1}{7} x^{7} e^{4} b^{2} A + x^{6} e^{2} d^{2} b^{2} B + \frac{4}{3} x^{6} e^{3} d b a B + \frac{1}{6} x^{6} e^{4} a^{2} B + \frac{2}{3} x^{6} e^{3} d b^{2} A + \frac{1}{3} x^{6} e^{4} b a A + \frac{4}{5} x^{5} e d^{3} b^{2} B + \frac{12}{5} x^{5} e^{2} d^{2} b a B + \frac{4}{5} x^{5} e^{3} d a^{2} B + \frac{6}{5} x^{5} e^{2} d^{2} b^{2} A + \frac{8}{5} x^{5} e^{3} d b a A + \frac{1}{5} x^{5} e^{4} a^{2} A + \frac{1}{4} x^{4} d^{4} b^{2} B + 2 x^{4} e d^{3} b a B + \frac{3}{2} x^{4} e^{2} d^{2} a^{2} B + x^{4} e d^{3} b^{2} A + 3 x^{4} e^{2} d^{2} b a A + x^{4} e^{3} d a^{2} A + \frac{2}{3} x^{3} d^{4} b a B + \frac{4}{3} x^{3} e d^{3} a^{2} B + \frac{1}{3} x^{3} d^{4} b^{2} A + \frac{8}{3} x^{3} e d^{3} b a A + 2 x^{3} e^{2} d^{2} a^{2} A + \frac{1}{2} x^{2} d^{4} a^{2} B + x^{2} d^{4} b a A + 2 x^{2} e d^{3} a^{2} A + x d^{4} a^{2} A \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^2*(e*x + d)^4,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 0.251048, size = 384, normalized size = 3.2 \[ A a^{2} d^{4} x + \frac{B b^{2} e^{4} x^{8}}{8} + x^{7} \left (\frac{A b^{2} e^{4}}{7} + \frac{2 B a b e^{4}}{7} + \frac{4 B b^{2} d e^{3}}{7}\right ) + x^{6} \left (\frac{A a b e^{4}}{3} + \frac{2 A b^{2} d e^{3}}{3} + \frac{B a^{2} e^{4}}{6} + \frac{4 B a b d e^{3}}{3} + B b^{2} d^{2} e^{2}\right ) + x^{5} \left (\frac{A a^{2} e^{4}}{5} + \frac{8 A a b d e^{3}}{5} + \frac{6 A b^{2} d^{2} e^{2}}{5} + \frac{4 B a^{2} d e^{3}}{5} + \frac{12 B a b d^{2} e^{2}}{5} + \frac{4 B b^{2} d^{3} e}{5}\right ) + x^{4} \left (A a^{2} d e^{3} + 3 A a b d^{2} e^{2} + A b^{2} d^{3} e + \frac{3 B a^{2} d^{2} e^{2}}{2} + 2 B a b d^{3} e + \frac{B b^{2} d^{4}}{4}\right ) + x^{3} \left (2 A a^{2} d^{2} e^{2} + \frac{8 A a b d^{3} e}{3} + \frac{A b^{2} d^{4}}{3} + \frac{4 B a^{2} d^{3} e}{3} + \frac{2 B a b d^{4}}{3}\right ) + x^{2} \left (2 A a^{2} d^{3} e + A a b d^{4} + \frac{B a^{2} d^{4}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**2*(B*x+A)*(e*x+d)**4,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.212054, size = 489, normalized size = 4.08 \[ \frac{1}{8} \, B b^{2} x^{8} e^{4} + \frac{4}{7} \, B b^{2} d x^{7} e^{3} + B b^{2} d^{2} x^{6} e^{2} + \frac{4}{5} \, B b^{2} d^{3} x^{5} e + \frac{1}{4} \, B b^{2} d^{4} x^{4} + \frac{2}{7} \, B a b x^{7} e^{4} + \frac{1}{7} \, A b^{2} x^{7} e^{4} + \frac{4}{3} \, B a b d x^{6} e^{3} + \frac{2}{3} \, A b^{2} d x^{6} e^{3} + \frac{12}{5} \, B a b d^{2} x^{5} e^{2} + \frac{6}{5} \, A b^{2} d^{2} x^{5} e^{2} + 2 \, B a b d^{3} x^{4} e + A b^{2} d^{3} x^{4} e + \frac{2}{3} \, B a b d^{4} x^{3} + \frac{1}{3} \, A b^{2} d^{4} x^{3} + \frac{1}{6} \, B a^{2} x^{6} e^{4} + \frac{1}{3} \, A a b x^{6} e^{4} + \frac{4}{5} \, B a^{2} d x^{5} e^{3} + \frac{8}{5} \, A a b d x^{5} e^{3} + \frac{3}{2} \, B a^{2} d^{2} x^{4} e^{2} + 3 \, A a b d^{2} x^{4} e^{2} + \frac{4}{3} \, B a^{2} d^{3} x^{3} e + \frac{8}{3} \, A a b d^{3} x^{3} e + \frac{1}{2} \, B a^{2} d^{4} x^{2} + A a b d^{4} x^{2} + \frac{1}{5} \, A a^{2} x^{5} e^{4} + A a^{2} d x^{4} e^{3} + 2 \, A a^{2} d^{2} x^{3} e^{2} + 2 \, A a^{2} d^{3} x^{2} e + A a^{2} d^{4} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^2*(e*x + d)^4,x, algorithm="giac")
[Out]