3.1117 \(\int (A+B x) (d+e x) \left (b x+c x^2\right )^2 \, dx\)

Optimal. Leaf size=100 \[ \frac{1}{5} x^5 \left (2 b c (A e+B d)+A c^2 d+b^2 B e\right )+\frac{1}{3} A b^2 d x^3+\frac{1}{6} c x^6 (A c e+2 b B e+B c d)+\frac{1}{4} b x^4 (A b e+2 A c d+b B d)+\frac{1}{7} B c^2 e x^7 \]

[Out]

(A*b^2*d*x^3)/3 + (b*(b*B*d + 2*A*c*d + A*b*e)*x^4)/4 + ((A*c^2*d + b^2*B*e + 2*
b*c*(B*d + A*e))*x^5)/5 + (c*(B*c*d + 2*b*B*e + A*c*e)*x^6)/6 + (B*c^2*e*x^7)/7

_______________________________________________________________________________________

Rubi [A]  time = 0.293278, antiderivative size = 100, 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 \[ \frac{1}{5} x^5 \left (2 b c (A e+B d)+A c^2 d+b^2 B e\right )+\frac{1}{3} A b^2 d x^3+\frac{1}{6} c x^6 (A c e+2 b B e+B c d)+\frac{1}{4} b x^4 (A b e+2 A c d+b B d)+\frac{1}{7} B c^2 e x^7 \]

Antiderivative was successfully verified.

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

[Out]

(A*b^2*d*x^3)/3 + (b*(b*B*d + 2*A*c*d + A*b*e)*x^4)/4 + ((A*c^2*d + b^2*B*e + 2*
b*c*(B*d + A*e))*x^5)/5 + (c*(B*c*d + 2*b*B*e + A*c*e)*x^6)/6 + (B*c^2*e*x^7)/7

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 35.7705, size = 110, normalized size = 1.1 \[ \frac{A b^{2} d x^{3}}{3} + \frac{B c^{2} e x^{7}}{7} + \frac{b x^{4} \left (A b e + 2 A c d + B b d\right )}{4} + \frac{c x^{6} \left (A c e + 2 B b e + B c d\right )}{6} + x^{5} \left (\frac{2 A b c e}{5} + \frac{A c^{2} d}{5} + \frac{B b^{2} e}{5} + \frac{2 B b c d}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*b**2*d*x**3/3 + B*c**2*e*x**7/7 + b*x**4*(A*b*e + 2*A*c*d + B*b*d)/4 + c*x**6*
(A*c*e + 2*B*b*e + B*c*d)/6 + x**5*(2*A*b*c*e/5 + A*c**2*d/5 + B*b**2*e/5 + 2*B*
b*c*d/5)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0540157, size = 101, normalized size = 1.01 \[ \frac{1}{5} x^5 \left (2 A b c e+A c^2 d+b^2 B e+2 b B c d\right )+\frac{1}{3} A b^2 d x^3+\frac{1}{6} c x^6 (A c e+2 b B e+B c d)+\frac{1}{4} b x^4 (A b e+2 A c d+b B d)+\frac{1}{7} B c^2 e x^7 \]

Antiderivative was successfully verified.

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

[Out]

(A*b^2*d*x^3)/3 + (b*(b*B*d + 2*A*c*d + A*b*e)*x^4)/4 + ((2*b*B*c*d + A*c^2*d +
b^2*B*e + 2*A*b*c*e)*x^5)/5 + (c*(B*c*d + 2*b*B*e + A*c*e)*x^6)/6 + (B*c^2*e*x^7
)/7

_______________________________________________________________________________________

Maple [A]  time = 0.001, size = 97, normalized size = 1. \[{\frac{B{c}^{2}e{x}^{7}}{7}}+{\frac{ \left ( \left ( Ae+Bd \right ){c}^{2}+2\,Bebc \right ){x}^{6}}{6}}+{\frac{ \left ( A{c}^{2}d+{b}^{2}Be+2\,bc \left ( Ae+Bd \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,Abcd+{b}^{2} \left ( Ae+Bd \right ) \right ){x}^{4}}{4}}+{\frac{A{b}^{2}d{x}^{3}}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*B*c^2*e*x^7+1/6*((A*e+B*d)*c^2+2*B*e*b*c)*x^6+1/5*(A*c^2*d+b^2*B*e+2*b*c*(A*
e+B*d))*x^5+1/4*(2*A*b*c*d+b^2*(A*e+B*d))*x^4+1/3*A*b^2*d*x^3

_______________________________________________________________________________________

Maxima [A]  time = 0.683752, size = 139, normalized size = 1.39 \[ \frac{1}{7} \, B c^{2} e x^{7} + \frac{1}{3} \, A b^{2} d x^{3} + \frac{1}{6} \,{\left (B c^{2} d +{\left (2 \, B b c + A c^{2}\right )} e\right )} x^{6} + \frac{1}{5} \,{\left ({\left (2 \, B b c + A c^{2}\right )} d +{\left (B b^{2} + 2 \, A b c\right )} e\right )} x^{5} + \frac{1}{4} \,{\left (A b^{2} e +{\left (B b^{2} + 2 \, A b c\right )} d\right )} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*B*c^2*e*x^7 + 1/3*A*b^2*d*x^3 + 1/6*(B*c^2*d + (2*B*b*c + A*c^2)*e)*x^6 + 1/
5*((2*B*b*c + A*c^2)*d + (B*b^2 + 2*A*b*c)*e)*x^5 + 1/4*(A*b^2*e + (B*b^2 + 2*A*
b*c)*d)*x^4

_______________________________________________________________________________________

Fricas [A]  time = 0.254715, size = 1, normalized size = 0.01 \[ \frac{1}{7} x^{7} e c^{2} B + \frac{1}{6} x^{6} d c^{2} B + \frac{1}{3} x^{6} e c b B + \frac{1}{6} x^{6} e c^{2} A + \frac{2}{5} x^{5} d c b B + \frac{1}{5} x^{5} e b^{2} B + \frac{1}{5} x^{5} d c^{2} A + \frac{2}{5} x^{5} e c b A + \frac{1}{4} x^{4} d b^{2} B + \frac{1}{2} x^{4} d c b A + \frac{1}{4} x^{4} e b^{2} A + \frac{1}{3} x^{3} d b^{2} A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*x^7*e*c^2*B + 1/6*x^6*d*c^2*B + 1/3*x^6*e*c*b*B + 1/6*x^6*e*c^2*A + 2/5*x^5*
d*c*b*B + 1/5*x^5*e*b^2*B + 1/5*x^5*d*c^2*A + 2/5*x^5*e*c*b*A + 1/4*x^4*d*b^2*B
+ 1/2*x^4*d*c*b*A + 1/4*x^4*e*b^2*A + 1/3*x^3*d*b^2*A

_______________________________________________________________________________________

Sympy [A]  time = 0.144314, size = 121, normalized size = 1.21 \[ \frac{A b^{2} d x^{3}}{3} + \frac{B c^{2} e x^{7}}{7} + x^{6} \left (\frac{A c^{2} e}{6} + \frac{B b c e}{3} + \frac{B c^{2} d}{6}\right ) + x^{5} \left (\frac{2 A b c e}{5} + \frac{A c^{2} d}{5} + \frac{B b^{2} e}{5} + \frac{2 B b c d}{5}\right ) + x^{4} \left (\frac{A b^{2} e}{4} + \frac{A b c d}{2} + \frac{B b^{2} d}{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*b**2*d*x**3/3 + B*c**2*e*x**7/7 + x**6*(A*c**2*e/6 + B*b*c*e/3 + B*c**2*d/6) +
x**5*(2*A*b*c*e/5 + A*c**2*d/5 + B*b**2*e/5 + 2*B*b*c*d/5) + x**4*(A*b**2*e/4 +
A*b*c*d/2 + B*b**2*d/4)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.278814, size = 166, normalized size = 1.66 \[ \frac{1}{7} \, B c^{2} x^{7} e + \frac{1}{6} \, B c^{2} d x^{6} + \frac{1}{3} \, B b c x^{6} e + \frac{1}{6} \, A c^{2} x^{6} e + \frac{2}{5} \, B b c d x^{5} + \frac{1}{5} \, A c^{2} d x^{5} + \frac{1}{5} \, B b^{2} x^{5} e + \frac{2}{5} \, A b c x^{5} e + \frac{1}{4} \, B b^{2} d x^{4} + \frac{1}{2} \, A b c d x^{4} + \frac{1}{4} \, A b^{2} x^{4} e + \frac{1}{3} \, A b^{2} d x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*B*c^2*x^7*e + 1/6*B*c^2*d*x^6 + 1/3*B*b*c*x^6*e + 1/6*A*c^2*x^6*e + 2/5*B*b*
c*d*x^5 + 1/5*A*c^2*d*x^5 + 1/5*B*b^2*x^5*e + 2/5*A*b*c*x^5*e + 1/4*B*b^2*d*x^4
+ 1/2*A*b*c*d*x^4 + 1/4*A*b^2*x^4*e + 1/3*A*b^2*d*x^3