3.222 \(\int (d+e x)^3 \left (b x+c x^2\right ) \, dx\)

Optimal. Leaf size=62 \[ -\frac{(d+e x)^5 (2 c d-b e)}{5 e^3}+\frac{d (d+e x)^4 (c d-b e)}{4 e^3}+\frac{c (d+e x)^6}{6 e^3} \]

[Out]

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

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Rubi [A]  time = 0.152017, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ -\frac{(d+e x)^5 (2 c d-b e)}{5 e^3}+\frac{d (d+e x)^4 (c d-b e)}{4 e^3}+\frac{c (d+e x)^6}{6 e^3} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^3*(b*x + c*x^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 16.6162, size = 53, normalized size = 0.85 \[ \frac{c \left (d + e x\right )^{6}}{6 e^{3}} - \frac{d \left (d + e x\right )^{4} \left (b e - c d\right )}{4 e^{3}} + \frac{\left (d + e x\right )^{5} \left (b e - 2 c d\right )}{5 e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**3*(c*x**2+b*x),x)

[Out]

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

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Mathematica [A]  time = 0.0268978, size = 67, normalized size = 1.08 \[ \frac{1}{60} x^2 \left (20 d^2 x (3 b e+c d)+12 e^2 x^3 (b e+3 c d)+45 d e x^2 (b e+c d)+30 b d^3+10 c e^3 x^4\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^3*(b*x + c*x^2),x]

[Out]

(x^2*(30*b*d^3 + 20*d^2*(c*d + 3*b*e)*x + 45*d*e*(c*d + b*e)*x^2 + 12*e^2*(3*c*d
+ b*e)*x^3 + 10*c*e^3*x^4))/60

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Maple [A]  time = 0.001, size = 76, normalized size = 1.2 \[{\frac{{e}^{3}c{x}^{6}}{6}}+{\frac{ \left ({e}^{3}b+3\,d{e}^{2}c \right ){x}^{5}}{5}}+{\frac{ \left ( 3\,d{e}^{2}b+3\,{d}^{2}ec \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,{d}^{2}eb+{d}^{3}c \right ){x}^{3}}{3}}+{\frac{{d}^{3}b{x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3*(c*x^2+b*x),x)

[Out]

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

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Maxima [A]  time = 0.748321, size = 99, normalized size = 1.6 \[ \frac{1}{6} \, c e^{3} x^{6} + \frac{1}{2} \, b d^{3} x^{2} + \frac{1}{5} \,{\left (3 \, c d e^{2} + b e^{3}\right )} x^{5} + \frac{3}{4} \,{\left (c d^{2} e + b d e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (c d^{3} + 3 \, b d^{2} e\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*(e*x + d)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.191979, size = 1, normalized size = 0.02 \[ \frac{1}{6} x^{6} e^{3} c + \frac{3}{5} x^{5} e^{2} d c + \frac{1}{5} x^{5} e^{3} b + \frac{3}{4} x^{4} e d^{2} c + \frac{3}{4} x^{4} e^{2} d b + \frac{1}{3} x^{3} d^{3} c + x^{3} e d^{2} b + \frac{1}{2} x^{2} d^{3} b \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*(e*x + d)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.131625, size = 80, normalized size = 1.29 \[ \frac{b d^{3} x^{2}}{2} + \frac{c e^{3} x^{6}}{6} + x^{5} \left (\frac{b e^{3}}{5} + \frac{3 c d e^{2}}{5}\right ) + x^{4} \left (\frac{3 b d e^{2}}{4} + \frac{3 c d^{2} e}{4}\right ) + x^{3} \left (b d^{2} e + \frac{c d^{3}}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**3*(c*x**2+b*x),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.20386, size = 100, normalized size = 1.61 \[ \frac{1}{6} \, c x^{6} e^{3} + \frac{3}{5} \, c d x^{5} e^{2} + \frac{3}{4} \, c d^{2} x^{4} e + \frac{1}{3} \, c d^{3} x^{3} + \frac{1}{5} \, b x^{5} e^{3} + \frac{3}{4} \, b d x^{4} e^{2} + b d^{2} x^{3} e + \frac{1}{2} \, b d^{3} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*(e*x + d)^3,x, algorithm="giac")

[Out]

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