3.1440 \(\int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right ) \, dx\)

Optimal. Leaf size=65 \[ -\frac{b (d+e x)^6 (b d-a e)}{3 e^3}+\frac{(d+e x)^5 (b d-a e)^2}{5 e^3}+\frac{b^2 (d+e x)^7}{7 e^3} \]

[Out]

((b*d - a*e)^2*(d + e*x)^5)/(5*e^3) - (b*(b*d - a*e)*(d + e*x)^6)/(3*e^3) + (b^2
*(d + e*x)^7)/(7*e^3)

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Rubi [A]  time = 0.205115, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{b (d+e x)^6 (b d-a e)}{3 e^3}+\frac{(d+e x)^5 (b d-a e)^2}{5 e^3}+\frac{b^2 (d+e x)^7}{7 e^3} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

((b*d - a*e)^2*(d + e*x)^5)/(5*e^3) - (b*(b*d - a*e)*(d + e*x)^6)/(3*e^3) + (b^2
*(d + e*x)^7)/(7*e^3)

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Rubi in Sympy [A]  time = 41.4816, size = 54, normalized size = 0.83 \[ \frac{b^{2} \left (d + e x\right )^{7}}{7 e^{3}} + \frac{b \left (d + e x\right )^{6} \left (a e - b d\right )}{3 e^{3}} + \frac{\left (d + e x\right )^{5} \left (a e - b d\right )^{2}}{5 e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**4*(b**2*x**2+2*a*b*x+a**2),x)

[Out]

b**2*(d + e*x)**7/(7*e**3) + b*(d + e*x)**6*(a*e - b*d)/(3*e**3) + (d + e*x)**5*
(a*e - b*d)**2/(5*e**3)

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Mathematica [B]  time = 0.0471108, size = 148, normalized size = 2.28 \[ \frac{1}{5} e^2 x^5 \left (a^2 e^2+8 a b d e+6 b^2 d^2\right )+d e x^4 \left (a^2 e^2+3 a b d e+b^2 d^2\right )+\frac{1}{3} d^2 x^3 \left (6 a^2 e^2+8 a b d e+b^2 d^2\right )+a^2 d^4 x+a d^3 x^2 (2 a e+b d)+\frac{1}{3} b e^3 x^6 (a e+2 b d)+\frac{1}{7} b^2 e^4 x^7 \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

a^2*d^4*x + a*d^3*(b*d + 2*a*e)*x^2 + (d^2*(b^2*d^2 + 8*a*b*d*e + 6*a^2*e^2)*x^3
)/3 + d*e*(b^2*d^2 + 3*a*b*d*e + a^2*e^2)*x^4 + (e^2*(6*b^2*d^2 + 8*a*b*d*e + a^
2*e^2)*x^5)/5 + (b*e^3*(2*b*d + a*e)*x^6)/3 + (b^2*e^4*x^7)/7

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Maple [B]  time = 0.002, size = 163, normalized size = 2.5 \[{\frac{{e}^{4}{b}^{2}{x}^{7}}{7}}+{\frac{ \left ( 2\,{e}^{4}ab+4\,d{e}^{3}{b}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ({a}^{2}{e}^{4}+8\,d{e}^{3}ab+6\,{d}^{2}{e}^{2}{b}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,d{e}^{3}{a}^{2}+12\,{d}^{2}{e}^{2}ab+4\,{d}^{3}e{b}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( 6\,{d}^{2}{e}^{2}{a}^{2}+8\,{d}^{3}eab+{d}^{4}{b}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,{d}^{3}e{a}^{2}+2\,{d}^{4}ab \right ){x}^{2}}{2}}+{d}^{4}{a}^{2}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^4*(b^2*x^2+2*a*b*x+a^2),x)

[Out]

1/7*e^4*b^2*x^7+1/6*(2*a*b*e^4+4*b^2*d*e^3)*x^6+1/5*(a^2*e^4+8*a*b*d*e^3+6*b^2*d
^2*e^2)*x^5+1/4*(4*a^2*d*e^3+12*a*b*d^2*e^2+4*b^2*d^3*e)*x^4+1/3*(6*a^2*d^2*e^2+
8*a*b*d^3*e+b^2*d^4)*x^3+1/2*(4*a^2*d^3*e+2*a*b*d^4)*x^2+d^4*a^2*x

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Maxima [A]  time = 0.685062, size = 211, normalized size = 3.25 \[ \frac{1}{7} \, b^{2} e^{4} x^{7} + a^{2} d^{4} x + \frac{1}{3} \,{\left (2 \, b^{2} d e^{3} + a b e^{4}\right )} x^{6} + \frac{1}{5} \,{\left (6 \, b^{2} d^{2} e^{2} + 8 \, a b d e^{3} + a^{2} e^{4}\right )} x^{5} +{\left (b^{2} d^{3} e + 3 \, a b d^{2} e^{2} + a^{2} d e^{3}\right )} x^{4} + \frac{1}{3} \,{\left (b^{2} d^{4} + 8 \, a b d^{3} e + 6 \, a^{2} d^{2} e^{2}\right )} x^{3} +{\left (a b d^{4} + 2 \, a^{2} d^{3} e\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)*(e*x + d)^4,x, algorithm="maxima")

[Out]

1/7*b^2*e^4*x^7 + a^2*d^4*x + 1/3*(2*b^2*d*e^3 + a*b*e^4)*x^6 + 1/5*(6*b^2*d^2*e
^2 + 8*a*b*d*e^3 + a^2*e^4)*x^5 + (b^2*d^3*e + 3*a*b*d^2*e^2 + a^2*d*e^3)*x^4 +
1/3*(b^2*d^4 + 8*a*b*d^3*e + 6*a^2*d^2*e^2)*x^3 + (a*b*d^4 + 2*a^2*d^3*e)*x^2

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Fricas [A]  time = 0.180502, size = 1, normalized size = 0.02 \[ \frac{1}{7} x^{7} e^{4} b^{2} + \frac{2}{3} x^{6} e^{3} d b^{2} + \frac{1}{3} x^{6} e^{4} b a + \frac{6}{5} x^{5} e^{2} d^{2} b^{2} + \frac{8}{5} x^{5} e^{3} d b a + \frac{1}{5} x^{5} e^{4} a^{2} + x^{4} e d^{3} b^{2} + 3 x^{4} e^{2} d^{2} b a + x^{4} e^{3} d a^{2} + \frac{1}{3} x^{3} d^{4} b^{2} + \frac{8}{3} x^{3} e d^{3} b a + 2 x^{3} e^{2} d^{2} a^{2} + x^{2} d^{4} b a + 2 x^{2} e d^{3} a^{2} + x d^{4} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)*(e*x + d)^4,x, algorithm="fricas")

[Out]

1/7*x^7*e^4*b^2 + 2/3*x^6*e^3*d*b^2 + 1/3*x^6*e^4*b*a + 6/5*x^5*e^2*d^2*b^2 + 8/
5*x^5*e^3*d*b*a + 1/5*x^5*e^4*a^2 + x^4*e*d^3*b^2 + 3*x^4*e^2*d^2*b*a + x^4*e^3*
d*a^2 + 1/3*x^3*d^4*b^2 + 8/3*x^3*e*d^3*b*a + 2*x^3*e^2*d^2*a^2 + x^2*d^4*b*a +
2*x^2*e*d^3*a^2 + x*d^4*a^2

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Sympy [A]  time = 0.1833, size = 168, normalized size = 2.58 \[ a^{2} d^{4} x + \frac{b^{2} e^{4} x^{7}}{7} + x^{6} \left (\frac{a b e^{4}}{3} + \frac{2 b^{2} d e^{3}}{3}\right ) + x^{5} \left (\frac{a^{2} e^{4}}{5} + \frac{8 a b d e^{3}}{5} + \frac{6 b^{2} d^{2} e^{2}}{5}\right ) + x^{4} \left (a^{2} d e^{3} + 3 a b d^{2} e^{2} + b^{2} d^{3} e\right ) + x^{3} \left (2 a^{2} d^{2} e^{2} + \frac{8 a b d^{3} e}{3} + \frac{b^{2} d^{4}}{3}\right ) + x^{2} \left (2 a^{2} d^{3} e + a b d^{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**4*(b**2*x**2+2*a*b*x+a**2),x)

[Out]

a**2*d**4*x + b**2*e**4*x**7/7 + x**6*(a*b*e**4/3 + 2*b**2*d*e**3/3) + x**5*(a**
2*e**4/5 + 8*a*b*d*e**3/5 + 6*b**2*d**2*e**2/5) + x**4*(a**2*d*e**3 + 3*a*b*d**2
*e**2 + b**2*d**3*e) + x**3*(2*a**2*d**2*e**2 + 8*a*b*d**3*e/3 + b**2*d**4/3) +
x**2*(2*a**2*d**3*e + a*b*d**4)

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GIAC/XCAS [A]  time = 0.210189, size = 221, normalized size = 3.4 \[ \frac{1}{7} \, b^{2} x^{7} e^{4} + \frac{2}{3} \, b^{2} d x^{6} e^{3} + \frac{6}{5} \, b^{2} d^{2} x^{5} e^{2} + b^{2} d^{3} x^{4} e + \frac{1}{3} \, b^{2} d^{4} x^{3} + \frac{1}{3} \, a b x^{6} e^{4} + \frac{8}{5} \, a b d x^{5} e^{3} + 3 \, a b d^{2} x^{4} e^{2} + \frac{8}{3} \, a b d^{3} x^{3} e + a b d^{4} x^{2} + \frac{1}{5} \, a^{2} x^{5} e^{4} + a^{2} d x^{4} e^{3} + 2 \, a^{2} d^{2} x^{3} e^{2} + 2 \, a^{2} d^{3} x^{2} e + a^{2} d^{4} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)*(e*x + d)^4,x, algorithm="giac")

[Out]

1/7*b^2*x^7*e^4 + 2/3*b^2*d*x^6*e^3 + 6/5*b^2*d^2*x^5*e^2 + b^2*d^3*x^4*e + 1/3*
b^2*d^4*x^3 + 1/3*a*b*x^6*e^4 + 8/5*a*b*d*x^5*e^3 + 3*a*b*d^2*x^4*e^2 + 8/3*a*b*
d^3*x^3*e + a*b*d^4*x^2 + 1/5*a^2*x^5*e^4 + a^2*d*x^4*e^3 + 2*a^2*d^2*x^3*e^2 +
2*a^2*d^3*x^2*e + a^2*d^4*x