Optimal. Leaf size=204 \[ \frac{e^3 (a+b x)^9 (-5 a B e+A b e+4 b B d)}{9 b^6}+\frac{e^2 (a+b x)^8 (b d-a e) (-5 a B e+2 A b e+3 b B d)}{4 b^6}+\frac{2 e (a+b x)^7 (b d-a e)^2 (-5 a B e+3 A b e+2 b B d)}{7 b^6}+\frac{(a+b x)^6 (b d-a e)^3 (-5 a B e+4 A b e+b B d)}{6 b^6}+\frac{(a+b x)^5 (A b-a B) (b d-a e)^4}{5 b^6}+\frac{B e^4 (a+b x)^{10}}{10 b^6} \]
[Out]
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Rubi [A] time = 1.19322, antiderivative size = 204, 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 \[ \frac{e^3 (a+b x)^9 (-5 a B e+A b e+4 b B d)}{9 b^6}+\frac{e^2 (a+b x)^8 (b d-a e) (-5 a B e+2 A b e+3 b B d)}{4 b^6}+\frac{2 e (a+b x)^7 (b d-a e)^2 (-5 a B e+3 A b e+2 b B d)}{7 b^6}+\frac{(a+b x)^6 (b d-a e)^3 (-5 a B e+4 A b e+b B d)}{6 b^6}+\frac{(a+b x)^5 (A b-a B) (b d-a e)^4}{5 b^6}+\frac{B e^4 (a+b x)^{10}}{10 b^6} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)*(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 156.935, size = 202, normalized size = 0.99 \[ \frac{B b^{4} \left (d + e x\right )^{10}}{10 e^{6}} + \frac{b^{3} \left (d + e x\right )^{9} \left (A b e + 4 B a e - 5 B b d\right )}{9 e^{6}} + \frac{b^{2} \left (d + e x\right )^{8} \left (a e - b d\right ) \left (2 A b e + 3 B a e - 5 B b d\right )}{4 e^{6}} + \frac{2 b \left (d + e x\right )^{7} \left (a e - b d\right )^{2} \left (3 A b e + 2 B a e - 5 B b d\right )}{7 e^{6}} + \frac{\left (d + e x\right )^{6} \left (a e - b d\right )^{3} \left (4 A b e + B a e - 5 B b d\right )}{6 e^{6}} + \frac{\left (d + e x\right )^{5} \left (A e - B d\right ) \left (a e - b d\right )^{4}}{5 e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)**4*(b**2*x**2+2*a*b*x+a**2)**2,x)
[Out]
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Mathematica [B] time = 0.343069, size = 512, normalized size = 2.51 \[ a^4 A d^4 x+\frac{1}{2} a^3 d^3 x^2 (4 A (a e+b d)+a B d)+\frac{2}{3} a^2 d^2 x^3 \left (A \left (3 a^2 e^2+8 a b d e+3 b^2 d^2\right )+2 a B d (a e+b d)\right )+\frac{1}{4} b^2 e^2 x^8 \left (3 a^2 B e^2+2 a b e (A e+4 B d)+b^2 d (2 A e+3 B d)\right )+\frac{2}{7} b e x^7 \left (2 a^3 B e^3+3 a^2 b e^2 (A e+4 B d)+4 a b^2 d e (2 A e+3 B d)+b^3 d^2 (3 A e+2 B d)\right )+\frac{1}{2} a d x^4 \left (a B d \left (3 a^2 e^2+8 a b d e+3 b^2 d^2\right )+2 A \left (a^3 e^3+6 a^2 b d e^2+6 a b^2 d^2 e+b^3 d^3\right )\right )+\frac{1}{6} x^6 \left (a^4 B e^4+4 a^3 b e^3 (A e+4 B d)+12 a^2 b^2 d e^2 (2 A e+3 B d)+8 a b^3 d^2 e (3 A e+2 B d)+b^4 d^3 (4 A e+B d)\right )+\frac{1}{5} x^5 \left (4 a B d \left (a^3 e^3+6 a^2 b d e^2+6 a b^2 d^2 e+b^3 d^3\right )+A \left (a^4 e^4+16 a^3 b d e^3+36 a^2 b^2 d^2 e^2+16 a b^3 d^3 e+b^4 d^4\right )\right )+\frac{1}{9} b^3 e^3 x^9 (4 a B e+A b e+4 b B d)+\frac{1}{10} b^4 B e^4 x^{10} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)*(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^2,x]
[Out]
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Maple [B] time = 0., size = 563, normalized size = 2.8 \[{\frac{B{e}^{4}{b}^{4}{x}^{10}}{10}}+{\frac{ \left ( \left ( A{e}^{4}+4\,Bd{e}^{3} \right ){b}^{4}+4\,B{e}^{4}a{b}^{3} \right ){x}^{9}}{9}}+{\frac{ \left ( \left ( 4\,Ad{e}^{3}+6\,B{d}^{2}{e}^{2} \right ){b}^{4}+4\, \left ( A{e}^{4}+4\,Bd{e}^{3} \right ) a{b}^{3}+6\,B{e}^{4}{a}^{2}{b}^{2} \right ){x}^{8}}{8}}+{\frac{ \left ( \left ( 6\,A{d}^{2}{e}^{2}+4\,B{d}^{3}e \right ){b}^{4}+4\, \left ( 4\,Ad{e}^{3}+6\,B{d}^{2}{e}^{2} \right ) a{b}^{3}+6\, \left ( A{e}^{4}+4\,Bd{e}^{3} \right ){a}^{2}{b}^{2}+4\,B{e}^{4}{a}^{3}b \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 4\,A{d}^{3}e+B{d}^{4} \right ){b}^{4}+4\, \left ( 6\,A{d}^{2}{e}^{2}+4\,B{d}^{3}e \right ) a{b}^{3}+6\, \left ( 4\,Ad{e}^{3}+6\,B{d}^{2}{e}^{2} \right ){a}^{2}{b}^{2}+4\, \left ( A{e}^{4}+4\,Bd{e}^{3} \right ){a}^{3}b+B{e}^{4}{a}^{4} \right ){x}^{6}}{6}}+{\frac{ \left ( A{d}^{4}{b}^{4}+4\, \left ( 4\,A{d}^{3}e+B{d}^{4} \right ) a{b}^{3}+6\, \left ( 6\,A{d}^{2}{e}^{2}+4\,B{d}^{3}e \right ){a}^{2}{b}^{2}+4\, \left ( 4\,Ad{e}^{3}+6\,B{d}^{2}{e}^{2} \right ){a}^{3}b+ \left ( A{e}^{4}+4\,Bd{e}^{3} \right ){a}^{4} \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,A{d}^{4}a{b}^{3}+6\, \left ( 4\,A{d}^{3}e+B{d}^{4} \right ){a}^{2}{b}^{2}+4\, \left ( 6\,A{d}^{2}{e}^{2}+4\,B{d}^{3}e \right ){a}^{3}b+ \left ( 4\,Ad{e}^{3}+6\,B{d}^{2}{e}^{2} \right ){a}^{4} \right ){x}^{4}}{4}}+{\frac{ \left ( 6\,A{d}^{4}{a}^{2}{b}^{2}+4\, \left ( 4\,A{d}^{3}e+B{d}^{4} \right ){a}^{3}b+ \left ( 6\,A{d}^{2}{e}^{2}+4\,B{d}^{3}e \right ){a}^{4} \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,A{d}^{4}{a}^{3}b+ \left ( 4\,A{d}^{3}e+B{d}^{4} \right ){a}^{4} \right ){x}^{2}}{2}}+A{d}^{4}{a}^{4}x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^2,x)
[Out]
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Maxima [A] time = 0.694579, size = 759, normalized size = 3.72 \[ \frac{1}{10} \, B b^{4} e^{4} x^{10} + A a^{4} d^{4} x + \frac{1}{9} \,{\left (4 \, B b^{4} d e^{3} +{\left (4 \, B a b^{3} + A b^{4}\right )} e^{4}\right )} x^{9} + \frac{1}{4} \,{\left (3 \, B b^{4} d^{2} e^{2} + 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{3} +{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{4}\right )} x^{8} + \frac{2}{7} \,{\left (2 \, B b^{4} d^{3} e + 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{2} + 4 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{3} +{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{4}\right )} x^{7} + \frac{1}{6} \,{\left (B b^{4} d^{4} + 4 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e + 12 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{2} + 8 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{3} +{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{4}\right )} x^{6} + \frac{1}{5} \,{\left (A a^{4} e^{4} +{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} + 8 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e + 12 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{2} + 4 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{3}\right )} x^{5} + \frac{1}{2} \,{\left (2 \, A a^{4} d e^{3} +{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{4} + 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{3} e + 3 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d^{2} e^{2}\right )} x^{4} + \frac{2}{3} \,{\left (3 \, A a^{4} d^{2} e^{2} +{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{4} + 2 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d^{3} e\right )} x^{3} + \frac{1}{2} \,{\left (4 \, A a^{4} d^{3} e +{\left (B a^{4} + 4 \, A a^{3} b\right )} d^{4}\right )} x^{2} \]
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 = 0.249142, size = 1, normalized size = 0. \[ \frac{1}{10} x^{10} e^{4} b^{4} B + \frac{4}{9} x^{9} e^{3} d b^{4} B + \frac{4}{9} x^{9} e^{4} b^{3} a B + \frac{1}{9} x^{9} e^{4} b^{4} A + \frac{3}{4} x^{8} e^{2} d^{2} b^{4} B + 2 x^{8} e^{3} d b^{3} a B + \frac{3}{4} x^{8} e^{4} b^{2} a^{2} B + \frac{1}{2} x^{8} e^{3} d b^{4} A + \frac{1}{2} x^{8} e^{4} b^{3} a A + \frac{4}{7} x^{7} e d^{3} b^{4} B + \frac{24}{7} x^{7} e^{2} d^{2} b^{3} a B + \frac{24}{7} x^{7} e^{3} d b^{2} a^{2} B + \frac{4}{7} x^{7} e^{4} b a^{3} B + \frac{6}{7} x^{7} e^{2} d^{2} b^{4} A + \frac{16}{7} x^{7} e^{3} d b^{3} a A + \frac{6}{7} x^{7} e^{4} b^{2} a^{2} A + \frac{1}{6} x^{6} d^{4} b^{4} B + \frac{8}{3} x^{6} e d^{3} b^{3} a B + 6 x^{6} e^{2} d^{2} b^{2} a^{2} B + \frac{8}{3} x^{6} e^{3} d b a^{3} B + \frac{1}{6} x^{6} e^{4} a^{4} B + \frac{2}{3} x^{6} e d^{3} b^{4} A + 4 x^{6} e^{2} d^{2} b^{3} a A + 4 x^{6} e^{3} d b^{2} a^{2} A + \frac{2}{3} x^{6} e^{4} b a^{3} A + \frac{4}{5} x^{5} d^{4} b^{3} a B + \frac{24}{5} x^{5} e d^{3} b^{2} a^{2} B + \frac{24}{5} x^{5} e^{2} d^{2} b a^{3} B + \frac{4}{5} x^{5} e^{3} d a^{4} B + \frac{1}{5} x^{5} d^{4} b^{4} A + \frac{16}{5} x^{5} e d^{3} b^{3} a A + \frac{36}{5} x^{5} e^{2} d^{2} b^{2} a^{2} A + \frac{16}{5} x^{5} e^{3} d b a^{3} A + \frac{1}{5} x^{5} e^{4} a^{4} A + \frac{3}{2} x^{4} d^{4} b^{2} a^{2} B + 4 x^{4} e d^{3} b a^{3} B + \frac{3}{2} x^{4} e^{2} d^{2} a^{4} B + x^{4} d^{4} b^{3} a A + 6 x^{4} e d^{3} b^{2} a^{2} A + 6 x^{4} e^{2} d^{2} b a^{3} A + x^{4} e^{3} d a^{4} A + \frac{4}{3} x^{3} d^{4} b a^{3} B + \frac{4}{3} x^{3} e d^{3} a^{4} B + 2 x^{3} d^{4} b^{2} a^{2} A + \frac{16}{3} x^{3} e d^{3} b a^{3} A + 2 x^{3} e^{2} d^{2} a^{4} A + \frac{1}{2} x^{2} d^{4} a^{4} B + 2 x^{2} d^{4} b a^{3} A + 2 x^{2} e d^{3} a^{4} A + x d^{4} a^{4} A \]
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 = 0.39202, size = 717, normalized size = 3.51 \[ A a^{4} d^{4} x + \frac{B b^{4} e^{4} x^{10}}{10} + x^{9} \left (\frac{A b^{4} e^{4}}{9} + \frac{4 B a b^{3} e^{4}}{9} + \frac{4 B b^{4} d e^{3}}{9}\right ) + x^{8} \left (\frac{A a b^{3} e^{4}}{2} + \frac{A b^{4} d e^{3}}{2} + \frac{3 B a^{2} b^{2} e^{4}}{4} + 2 B a b^{3} d e^{3} + \frac{3 B b^{4} d^{2} e^{2}}{4}\right ) + x^{7} \left (\frac{6 A a^{2} b^{2} e^{4}}{7} + \frac{16 A a b^{3} d e^{3}}{7} + \frac{6 A b^{4} d^{2} e^{2}}{7} + \frac{4 B a^{3} b e^{4}}{7} + \frac{24 B a^{2} b^{2} d e^{3}}{7} + \frac{24 B a b^{3} d^{2} e^{2}}{7} + \frac{4 B b^{4} d^{3} e}{7}\right ) + x^{6} \left (\frac{2 A a^{3} b e^{4}}{3} + 4 A a^{2} b^{2} d e^{3} + 4 A a b^{3} d^{2} e^{2} + \frac{2 A b^{4} d^{3} e}{3} + \frac{B a^{4} e^{4}}{6} + \frac{8 B a^{3} b d e^{3}}{3} + 6 B a^{2} b^{2} d^{2} e^{2} + \frac{8 B a b^{3} d^{3} e}{3} + \frac{B b^{4} d^{4}}{6}\right ) + x^{5} \left (\frac{A a^{4} e^{4}}{5} + \frac{16 A a^{3} b d e^{3}}{5} + \frac{36 A a^{2} b^{2} d^{2} e^{2}}{5} + \frac{16 A a b^{3} d^{3} e}{5} + \frac{A b^{4} d^{4}}{5} + \frac{4 B a^{4} d e^{3}}{5} + \frac{24 B a^{3} b d^{2} e^{2}}{5} + \frac{24 B a^{2} b^{2} d^{3} e}{5} + \frac{4 B a b^{3} d^{4}}{5}\right ) + x^{4} \left (A a^{4} d e^{3} + 6 A a^{3} b d^{2} e^{2} + 6 A a^{2} b^{2} d^{3} e + A a b^{3} d^{4} + \frac{3 B a^{4} d^{2} e^{2}}{2} + 4 B a^{3} b d^{3} e + \frac{3 B a^{2} b^{2} d^{4}}{2}\right ) + x^{3} \left (2 A a^{4} d^{2} e^{2} + \frac{16 A a^{3} b d^{3} e}{3} + 2 A a^{2} b^{2} d^{4} + \frac{4 B a^{4} d^{3} e}{3} + \frac{4 B a^{3} b d^{4}}{3}\right ) + x^{2} \left (2 A a^{4} d^{3} e + 2 A a^{3} b d^{4} + \frac{B a^{4} d^{4}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(e*x+d)**4*(b**2*x**2+2*a*b*x+a**2)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.282821, size = 913, normalized size = 4.48 \[ \frac{1}{10} \, B b^{4} x^{10} e^{4} + \frac{4}{9} \, B b^{4} d x^{9} e^{3} + \frac{3}{4} \, B b^{4} d^{2} x^{8} e^{2} + \frac{4}{7} \, B b^{4} d^{3} x^{7} e + \frac{1}{6} \, B b^{4} d^{4} x^{6} + \frac{4}{9} \, B a b^{3} x^{9} e^{4} + \frac{1}{9} \, A b^{4} x^{9} e^{4} + 2 \, B a b^{3} d x^{8} e^{3} + \frac{1}{2} \, A b^{4} d x^{8} e^{3} + \frac{24}{7} \, B a b^{3} d^{2} x^{7} e^{2} + \frac{6}{7} \, A b^{4} d^{2} x^{7} e^{2} + \frac{8}{3} \, B a b^{3} d^{3} x^{6} e + \frac{2}{3} \, A b^{4} d^{3} x^{6} e + \frac{4}{5} \, B a b^{3} d^{4} x^{5} + \frac{1}{5} \, A b^{4} d^{4} x^{5} + \frac{3}{4} \, B a^{2} b^{2} x^{8} e^{4} + \frac{1}{2} \, A a b^{3} x^{8} e^{4} + \frac{24}{7} \, B a^{2} b^{2} d x^{7} e^{3} + \frac{16}{7} \, A a b^{3} d x^{7} e^{3} + 6 \, B a^{2} b^{2} d^{2} x^{6} e^{2} + 4 \, A a b^{3} d^{2} x^{6} e^{2} + \frac{24}{5} \, B a^{2} b^{2} d^{3} x^{5} e + \frac{16}{5} \, A a b^{3} d^{3} x^{5} e + \frac{3}{2} \, B a^{2} b^{2} d^{4} x^{4} + A a b^{3} d^{4} x^{4} + \frac{4}{7} \, B a^{3} b x^{7} e^{4} + \frac{6}{7} \, A a^{2} b^{2} x^{7} e^{4} + \frac{8}{3} \, B a^{3} b d x^{6} e^{3} + 4 \, A a^{2} b^{2} d x^{6} e^{3} + \frac{24}{5} \, B a^{3} b d^{2} x^{5} e^{2} + \frac{36}{5} \, A a^{2} b^{2} d^{2} x^{5} e^{2} + 4 \, B a^{3} b d^{3} x^{4} e + 6 \, A a^{2} b^{2} d^{3} x^{4} e + \frac{4}{3} \, B a^{3} b d^{4} x^{3} + 2 \, A a^{2} b^{2} d^{4} x^{3} + \frac{1}{6} \, B a^{4} x^{6} e^{4} + \frac{2}{3} \, A a^{3} b x^{6} e^{4} + \frac{4}{5} \, B a^{4} d x^{5} e^{3} + \frac{16}{5} \, A a^{3} b d x^{5} e^{3} + \frac{3}{2} \, B a^{4} d^{2} x^{4} e^{2} + 6 \, A a^{3} b d^{2} x^{4} e^{2} + \frac{4}{3} \, B a^{4} d^{3} x^{3} e + \frac{16}{3} \, A a^{3} b d^{3} x^{3} e + \frac{1}{2} \, B a^{4} d^{4} x^{2} + 2 \, A a^{3} b d^{4} x^{2} + \frac{1}{5} \, A a^{4} x^{5} e^{4} + A a^{4} d x^{4} e^{3} + 2 \, A a^{4} d^{2} x^{3} e^{2} + 2 \, A a^{4} d^{3} x^{2} e + A a^{4} d^{4} x \]
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]