3.1105 \(\int (A+B x) (d+e x)^2 \left (b x+c x^2\right ) \, dx\)

Optimal. Leaf size=99 \[ \frac{1}{5} e x^5 (A c e+b B e+2 B c d)+\frac{1}{4} x^4 (A e (b e+2 c d)+B d (2 b e+c d))+\frac{1}{3} d x^3 (2 A b e+A c d+b B d)+\frac{1}{2} A b d^2 x^2+\frac{1}{6} B c e^2 x^6 \]

[Out]

(A*b*d^2*x^2)/2 + (d*(b*B*d + A*c*d + 2*A*b*e)*x^3)/3 + ((A*e*(2*c*d + b*e) + B*
d*(c*d + 2*b*e))*x^4)/4 + (e*(2*B*c*d + b*B*e + A*c*e)*x^5)/5 + (B*c*e^2*x^6)/6

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Rubi [A]  time = 0.294326, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{1}{5} e x^5 (A c e+b B e+2 B c d)+\frac{1}{4} x^4 (A e (b e+2 c d)+B d (2 b e+c d))+\frac{1}{3} d x^3 (2 A b e+A c d+b B d)+\frac{1}{2} A b d^2 x^2+\frac{1}{6} B c e^2 x^6 \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)*(d + e*x)^2*(b*x + c*x^2),x]

[Out]

(A*b*d^2*x^2)/2 + (d*(b*B*d + A*c*d + 2*A*b*e)*x^3)/3 + ((A*e*(2*c*d + b*e) + B*
d*(c*d + 2*b*e))*x^4)/4 + (e*(2*B*c*d + b*B*e + A*c*e)*x^5)/5 + (B*c*e^2*x^6)/6

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ A b d^{2} \int x\, dx + \frac{B c e^{2} x^{6}}{6} + \frac{d x^{3} \left (2 A b e + A c d + B b d\right )}{3} + \frac{e x^{5} \left (A c e + B b e + 2 B c d\right )}{5} + x^{4} \left (\frac{A b e^{2}}{4} + \frac{A c d e}{2} + \frac{B b d e}{2} + \frac{B c d^{2}}{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**2*(c*x**2+b*x),x)

[Out]

A*b*d**2*Integral(x, x) + B*c*e**2*x**6/6 + d*x**3*(2*A*b*e + A*c*d + B*b*d)/3 +
e*x**5*(A*c*e + B*b*e + 2*B*c*d)/5 + x**4*(A*b*e**2/4 + A*c*d*e/2 + B*b*d*e/2 +
B*c*d**2/4)

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Mathematica [A]  time = 0.0754354, size = 91, normalized size = 0.92 \[ \frac{1}{60} x^2 \left (12 e x^3 (A c e+b B e+2 B c d)+15 x^2 (A e (b e+2 c d)+B d (2 b e+c d))+20 d x (2 A b e+A c d+b B d)+30 A b d^2+10 B c e^2 x^4\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)*(d + e*x)^2*(b*x + c*x^2),x]

[Out]

(x^2*(30*A*b*d^2 + 20*d*(b*B*d + A*c*d + 2*A*b*e)*x + 15*(A*e*(2*c*d + b*e) + B*
d*(c*d + 2*b*e))*x^2 + 12*e*(2*B*c*d + b*B*e + A*c*e)*x^3 + 10*B*c*e^2*x^4))/60

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Maple [A]  time = 0.002, size = 104, normalized size = 1.1 \[{\frac{Bc{e}^{2}{x}^{6}}{6}}+{\frac{ \left ( \left ( A{e}^{2}+2\,Bde \right ) c+B{e}^{2}b \right ){x}^{5}}{5}}+{\frac{ \left ( \left ( 2\,Ade+B{d}^{2} \right ) c+ \left ( A{e}^{2}+2\,Bde \right ) b \right ){x}^{4}}{4}}+{\frac{ \left ( Ac{d}^{2}+ \left ( 2\,Ade+B{d}^{2} \right ) b \right ){x}^{3}}{3}}+{\frac{Ab{d}^{2}{x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^2*(c*x^2+b*x),x)

[Out]

1/6*B*c*e^2*x^6+1/5*((A*e^2+2*B*d*e)*c+B*e^2*b)*x^5+1/4*((2*A*d*e+B*d^2)*c+(A*e^
2+2*B*d*e)*b)*x^4+1/3*(A*c*d^2+(2*A*d*e+B*d^2)*b)*x^3+1/2*A*b*d^2*x^2

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Maxima [A]  time = 0.689431, size = 130, normalized size = 1.31 \[ \frac{1}{6} \, B c e^{2} x^{6} + \frac{1}{2} \, A b d^{2} x^{2} + \frac{1}{5} \,{\left (2 \, B c d e +{\left (B b + A c\right )} e^{2}\right )} x^{5} + \frac{1}{4} \,{\left (B c d^{2} + A b e^{2} + 2 \,{\left (B b + A c\right )} d e\right )} x^{4} + \frac{1}{3} \,{\left (2 \, A b d e +{\left (B b + A c\right )} d^{2}\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*(B*x + A)*(e*x + d)^2,x, algorithm="maxima")

[Out]

1/6*B*c*e^2*x^6 + 1/2*A*b*d^2*x^2 + 1/5*(2*B*c*d*e + (B*b + A*c)*e^2)*x^5 + 1/4*
(B*c*d^2 + A*b*e^2 + 2*(B*b + A*c)*d*e)*x^4 + 1/3*(2*A*b*d*e + (B*b + A*c)*d^2)*
x^3

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Fricas [A]  time = 0.315099, size = 1, normalized size = 0.01 \[ \frac{1}{6} x^{6} e^{2} c B + \frac{2}{5} x^{5} e d c B + \frac{1}{5} x^{5} e^{2} b B + \frac{1}{5} x^{5} e^{2} c A + \frac{1}{4} x^{4} d^{2} c B + \frac{1}{2} x^{4} e d b B + \frac{1}{2} x^{4} e d c A + \frac{1}{4} x^{4} e^{2} b A + \frac{1}{3} x^{3} d^{2} b B + \frac{1}{3} x^{3} d^{2} c A + \frac{2}{3} x^{3} e d b A + \frac{1}{2} x^{2} d^{2} b A \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*(B*x + A)*(e*x + d)^2,x, algorithm="fricas")

[Out]

1/6*x^6*e^2*c*B + 2/5*x^5*e*d*c*B + 1/5*x^5*e^2*b*B + 1/5*x^5*e^2*c*A + 1/4*x^4*
d^2*c*B + 1/2*x^4*e*d*b*B + 1/2*x^4*e*d*c*A + 1/4*x^4*e^2*b*A + 1/3*x^3*d^2*b*B
+ 1/3*x^3*d^2*c*A + 2/3*x^3*e*d*b*A + 1/2*x^2*d^2*b*A

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Sympy [A]  time = 0.141708, size = 121, normalized size = 1.22 \[ \frac{A b d^{2} x^{2}}{2} + \frac{B c e^{2} x^{6}}{6} + x^{5} \left (\frac{A c e^{2}}{5} + \frac{B b e^{2}}{5} + \frac{2 B c d e}{5}\right ) + x^{4} \left (\frac{A b e^{2}}{4} + \frac{A c d e}{2} + \frac{B b d e}{2} + \frac{B c d^{2}}{4}\right ) + x^{3} \left (\frac{2 A b d e}{3} + \frac{A c d^{2}}{3} + \frac{B b d^{2}}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(e*x+d)**2*(c*x**2+b*x),x)

[Out]

A*b*d**2*x**2/2 + B*c*e**2*x**6/6 + x**5*(A*c*e**2/5 + B*b*e**2/5 + 2*B*c*d*e/5)
+ x**4*(A*b*e**2/4 + A*c*d*e/2 + B*b*d*e/2 + B*c*d**2/4) + x**3*(2*A*b*d*e/3 +
A*c*d**2/3 + B*b*d**2/3)

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GIAC/XCAS [A]  time = 0.281504, size = 158, normalized size = 1.6 \[ \frac{1}{6} \, B c x^{6} e^{2} + \frac{2}{5} \, B c d x^{5} e + \frac{1}{4} \, B c d^{2} x^{4} + \frac{1}{5} \, B b x^{5} e^{2} + \frac{1}{5} \, A c x^{5} e^{2} + \frac{1}{2} \, B b d x^{4} e + \frac{1}{2} \, A c d x^{4} e + \frac{1}{3} \, B b d^{2} x^{3} + \frac{1}{3} \, A c d^{2} x^{3} + \frac{1}{4} \, A b x^{4} e^{2} + \frac{2}{3} \, A b d x^{3} e + \frac{1}{2} \, A b d^{2} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*(B*x + A)*(e*x + d)^2,x, algorithm="giac")

[Out]

1/6*B*c*x^6*e^2 + 2/5*B*c*d*x^5*e + 1/4*B*c*d^2*x^4 + 1/5*B*b*x^5*e^2 + 1/5*A*c*
x^5*e^2 + 1/2*B*b*d*x^4*e + 1/2*A*c*d*x^4*e + 1/3*B*b*d^2*x^3 + 1/3*A*c*d^2*x^3
+ 1/4*A*b*x^4*e^2 + 2/3*A*b*d*x^3*e + 1/2*A*b*d^2*x^2