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

Optimal. Leaf size=87 \[ \frac{1}{5} x^5 \left (b^2 e^2+4 b c d e+c^2 d^2\right )+\frac{1}{3} b^2 d^2 x^3+\frac{1}{3} c e x^6 (b e+c d)+\frac{1}{2} b d x^4 (b e+c d)+\frac{1}{7} c^2 e^2 x^7 \]

[Out]

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

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Rubi [A]  time = 0.230635, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{1}{5} x^5 \left (b^2 e^2+4 b c d e+c^2 d^2\right )+\frac{1}{3} b^2 d^2 x^3+\frac{1}{3} c e x^6 (b e+c d)+\frac{1}{2} b d x^4 (b e+c d)+\frac{1}{7} c^2 e^2 x^7 \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 23.7966, size = 83, normalized size = 0.95 \[ \frac{b^{2} d^{2} x^{3}}{3} + \frac{b d x^{4} \left (b e + c d\right )}{2} + \frac{c^{2} e^{2} x^{7}}{7} + \frac{c e x^{6} \left (b e + c d\right )}{3} + x^{5} \left (\frac{b^{2} e^{2}}{5} + \frac{4 b c d e}{5} + \frac{c^{2} d^{2}}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0243661, size = 87, normalized size = 1. \[ \frac{1}{5} x^5 \left (b^2 e^2+4 b c d e+c^2 d^2\right )+\frac{1}{3} b^2 d^2 x^3+\frac{1}{3} c e x^6 (b e+c d)+\frac{1}{2} b d x^4 (b e+c d)+\frac{1}{7} c^2 e^2 x^7 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.002, size = 90, normalized size = 1. \[{\frac{{c}^{2}{e}^{2}{x}^{7}}{7}}+{\frac{ \left ( 2\,{e}^{2}bc+2\,de{c}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ({b}^{2}{e}^{2}+4\,bcde+{c}^{2}{d}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,{b}^{2}de+2\,{d}^{2}bc \right ){x}^{4}}{4}}+{\frac{{b}^{2}{d}^{2}{x}^{3}}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^2*(c*x^2+b*x)^2,x)

[Out]

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

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Maxima [A]  time = 0.695872, size = 115, normalized size = 1.32 \[ \frac{1}{7} \, c^{2} e^{2} x^{7} + \frac{1}{3} \, b^{2} d^{2} x^{3} + \frac{1}{3} \,{\left (c^{2} d e + b c e^{2}\right )} x^{6} + \frac{1}{5} \,{\left (c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2}\right )} x^{5} + \frac{1}{2} \,{\left (b c d^{2} + b^{2} d e\right )} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.189298, size = 1, normalized size = 0.01 \[ \frac{1}{7} x^{7} e^{2} c^{2} + \frac{1}{3} x^{6} e d c^{2} + \frac{1}{3} x^{6} e^{2} c b + \frac{1}{5} x^{5} d^{2} c^{2} + \frac{4}{5} x^{5} e d c b + \frac{1}{5} x^{5} e^{2} b^{2} + \frac{1}{2} x^{4} d^{2} c b + \frac{1}{2} x^{4} e d b^{2} + \frac{1}{3} x^{3} d^{2} b^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.147233, size = 94, normalized size = 1.08 \[ \frac{b^{2} d^{2} x^{3}}{3} + \frac{c^{2} e^{2} x^{7}}{7} + x^{6} \left (\frac{b c e^{2}}{3} + \frac{c^{2} d e}{3}\right ) + x^{5} \left (\frac{b^{2} e^{2}}{5} + \frac{4 b c d e}{5} + \frac{c^{2} d^{2}}{5}\right ) + x^{4} \left (\frac{b^{2} d e}{2} + \frac{b c d^{2}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.206943, size = 127, normalized size = 1.46 \[ \frac{1}{7} \, c^{2} x^{7} e^{2} + \frac{1}{3} \, c^{2} d x^{6} e + \frac{1}{5} \, c^{2} d^{2} x^{5} + \frac{1}{3} \, b c x^{6} e^{2} + \frac{4}{5} \, b c d x^{5} e + \frac{1}{2} \, b c d^{2} x^{4} + \frac{1}{5} \, b^{2} x^{5} e^{2} + \frac{1}{2} \, b^{2} d x^{4} e + \frac{1}{3} \, b^{2} d^{2} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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