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

Optimal. Leaf size=55 \[ \frac{1}{3} b^2 d x^3+\frac{1}{5} c x^5 (2 b e+c d)+\frac{1}{4} b x^4 (b e+2 c d)+\frac{1}{6} c^2 e x^6 \]

[Out]

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

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Rubi [A]  time = 0.142794, antiderivative size = 55, 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 \[ \frac{1}{3} b^2 d x^3+\frac{1}{5} c x^5 (2 b e+c d)+\frac{1}{4} b x^4 (b e+2 c d)+\frac{1}{6} c^2 e x^6 \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 13.6996, size = 49, normalized size = 0.89 \[ \frac{b^{2} d x^{3}}{3} + \frac{b x^{4} \left (b e + 2 c d\right )}{4} + \frac{c^{2} e x^{6}}{6} + \frac{c x^{5} \left (2 b e + c d\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0174615, size = 50, normalized size = 0.91 \[ \frac{1}{60} x^3 \left (5 b^2 (4 d+3 e x)+6 b c x (5 d+4 e x)+2 c^2 x^2 (6 d+5 e x)\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x^3*(5*b^2*(4*d + 3*e*x) + 6*b*c*x*(5*d + 4*e*x) + 2*c^2*x^2*(6*d + 5*e*x)))/60

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Maple [A]  time = 0.002, size = 52, normalized size = 1. \[{\frac{{c}^{2}e{x}^{6}}{6}}+{\frac{ \left ( 2\,bce+d{c}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ({b}^{2}e+2\,bcd \right ){x}^{4}}{4}}+{\frac{{b}^{2}d{x}^{3}}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)*(c*x^2+b*x)^2,x)

[Out]

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

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Maxima [A]  time = 0.684868, size = 69, normalized size = 1.25 \[ \frac{1}{6} \, c^{2} e x^{6} + \frac{1}{3} \, b^{2} d x^{3} + \frac{1}{5} \,{\left (c^{2} d + 2 \, b c e\right )} x^{5} + \frac{1}{4} \,{\left (2 \, b c d + b^{2} e\right )} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.190432, size = 1, normalized size = 0.02 \[ \frac{1}{6} x^{6} e c^{2} + \frac{1}{5} x^{5} d c^{2} + \frac{2}{5} x^{5} e c b + \frac{1}{2} x^{4} d c b + \frac{1}{4} x^{4} e b^{2} + \frac{1}{3} x^{3} d b^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.122805, size = 54, normalized size = 0.98 \[ \frac{b^{2} d x^{3}}{3} + \frac{c^{2} e x^{6}}{6} + x^{5} \left (\frac{2 b c e}{5} + \frac{c^{2} d}{5}\right ) + x^{4} \left (\frac{b^{2} e}{4} + \frac{b c d}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.2039, size = 76, normalized size = 1.38 \[ \frac{1}{6} \, c^{2} x^{6} e + \frac{1}{5} \, c^{2} d x^{5} + \frac{2}{5} \, b c x^{5} e + \frac{1}{2} \, b c d x^{4} + \frac{1}{4} \, b^{2} x^{4} e + \frac{1}{3} \, b^{2} d x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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