3.1457 \(\int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx\)

Optimal. Leaf size=38 \[ \frac{(a+b x)^5 (b d-a e)}{5 b^2}+\frac{e (a+b x)^6}{6 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0573976, antiderivative size = 38, 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 \[ \frac{(a+b x)^5 (b d-a e)}{5 b^2}+\frac{e (a+b x)^6}{6 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 22.126, size = 31, normalized size = 0.82 \[ \frac{e \left (a + b x\right )^{6}}{6 b^{2}} - \frac{\left (a + b x\right )^{5} \left (a e - b d\right )}{5 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.0324812, size = 84, normalized size = 2.21 \[ \frac{1}{30} x \left (15 a^4 (2 d+e x)+20 a^3 b x (3 d+2 e x)+15 a^2 b^2 x^2 (4 d+3 e x)+6 a b^3 x^3 (5 d+4 e x)+b^4 x^4 (6 d+5 e x)\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x*(15*a^4*(2*d + e*x) + 20*a^3*b*x*(3*d + 2*e*x) + 15*a^2*b^2*x^2*(4*d + 3*e*x)
+ 6*a*b^3*x^3*(5*d + 4*e*x) + b^4*x^4*(6*d + 5*e*x)))/30

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Maple [B]  time = 0.001, size = 97, normalized size = 2.6 \[{\frac{e{b}^{4}{x}^{6}}{6}}+{\frac{ \left ( 4\,ea{b}^{3}+d{b}^{4} \right ){x}^{5}}{5}}+{\frac{ \left ( 6\,{a}^{2}e{b}^{2}+4\,ad{b}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ( 4\,{a}^{3}eb+6\,{a}^{2}d{b}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ({a}^{4}e+4\,{a}^{3}bd \right ){x}^{2}}{2}}+{a}^{4}dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.683606, size = 130, normalized size = 3.42 \[ \frac{1}{6} \, b^{4} e x^{6} + a^{4} d x + \frac{1}{5} \,{\left (b^{4} d + 4 \, a b^{3} e\right )} x^{5} + \frac{1}{2} \,{\left (2 \, a b^{3} d + 3 \, a^{2} b^{2} e\right )} x^{4} + \frac{2}{3} \,{\left (3 \, a^{2} b^{2} d + 2 \, a^{3} b e\right )} x^{3} + \frac{1}{2} \,{\left (4 \, a^{3} b d + a^{4} e\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.188762, size = 1, normalized size = 0.03 \[ \frac{1}{6} x^{6} e b^{4} + \frac{1}{5} x^{5} d b^{4} + \frac{4}{5} x^{5} e b^{3} a + x^{4} d b^{3} a + \frac{3}{2} x^{4} e b^{2} a^{2} + 2 x^{3} d b^{2} a^{2} + \frac{4}{3} x^{3} e b a^{3} + 2 x^{2} d b a^{3} + \frac{1}{2} x^{2} e a^{4} + x d a^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.148694, size = 100, normalized size = 2.63 \[ a^{4} d x + \frac{b^{4} e x^{6}}{6} + x^{5} \left (\frac{4 a b^{3} e}{5} + \frac{b^{4} d}{5}\right ) + x^{4} \left (\frac{3 a^{2} b^{2} e}{2} + a b^{3} d\right ) + x^{3} \left (\frac{4 a^{3} b e}{3} + 2 a^{2} b^{2} d\right ) + x^{2} \left (\frac{a^{4} e}{2} + 2 a^{3} b d\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.208318, size = 138, normalized size = 3.63 \[ \frac{1}{6} \, b^{4} x^{6} e + \frac{1}{5} \, b^{4} d x^{5} + \frac{4}{5} \, a b^{3} x^{5} e + a b^{3} d x^{4} + \frac{3}{2} \, a^{2} b^{2} x^{4} e + 2 \, a^{2} b^{2} d x^{3} + \frac{4}{3} \, a^{3} b x^{3} e + 2 \, a^{3} b d x^{2} + \frac{1}{2} \, a^{4} x^{2} e + a^{4} d x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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