3.1897 \(\int (a+b x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right ) \, dx\)

Optimal. Leaf size=92 \[ \frac{e^2 (a+b x)^6 (b d-a e)}{2 b^4}+\frac{3 e (a+b x)^5 (b d-a e)^2}{5 b^4}+\frac{(a+b x)^4 (b d-a e)^3}{4 b^4}+\frac{e^3 (a+b x)^7}{7 b^4} \]

[Out]

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

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Rubi [A]  time = 0.193952, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{e^2 (a+b x)^6 (b d-a e)}{2 b^4}+\frac{3 e (a+b x)^5 (b d-a e)^2}{5 b^4}+\frac{(a+b x)^4 (b d-a e)^3}{4 b^4}+\frac{e^3 (a+b x)^7}{7 b^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 51.1299, size = 80, normalized size = 0.87 \[ \frac{e^{3} \left (a + b x\right )^{7}}{7 b^{4}} - \frac{e^{2} \left (a + b x\right )^{6} \left (a e - b d\right )}{2 b^{4}} + \frac{3 e \left (a + b x\right )^{5} \left (a e - b d\right )^{2}}{5 b^{4}} - \frac{\left (a + b x\right )^{4} \left (a e - b d\right )^{3}}{4 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0458913, size = 161, normalized size = 1.75 \[ a^3 d^3 x+\frac{3}{5} b e x^5 \left (a^2 e^2+3 a b d e+b^2 d^2\right )+a d x^3 \left (a^2 e^2+3 a b d e+b^2 d^2\right )+\frac{3}{2} a^2 d^2 x^2 (a e+b d)+\frac{1}{4} x^4 \left (a^3 e^3+9 a^2 b d e^2+9 a b^2 d^2 e+b^3 d^3\right )+\frac{1}{2} b^2 e^2 x^6 (a e+b d)+\frac{1}{7} b^3 e^3 x^7 \]

Antiderivative was successfully verified.

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

[Out]

a^3*d^3*x + (3*a^2*d^2*(b*d + a*e)*x^2)/2 + a*d*(b^2*d^2 + 3*a*b*d*e + a^2*e^2)*
x^3 + ((b^3*d^3 + 9*a*b^2*d^2*e + 9*a^2*b*d*e^2 + a^3*e^3)*x^4)/4 + (3*b*e*(b^2*
d^2 + 3*a*b*d*e + a^2*e^2)*x^5)/5 + (b^2*e^2*(b*d + a*e)*x^6)/2 + (b^3*e^3*x^7)/
7

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Maple [B]  time = 0.001, size = 244, normalized size = 2.7 \[{\frac{{b}^{3}{e}^{3}{x}^{7}}{7}}+{\frac{ \left ( \left ( a{e}^{3}+3\,bd{e}^{2} \right ){b}^{2}+2\,a{b}^{2}{e}^{3} \right ){x}^{6}}{6}}+{\frac{ \left ( \left ( 3\,ad{e}^{2}+3\,b{d}^{2}e \right ){b}^{2}+2\, \left ( a{e}^{3}+3\,bd{e}^{2} \right ) ab+b{e}^{3}{a}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( \left ( 3\,a{d}^{2}e+b{d}^{3} \right ){b}^{2}+2\, \left ( 3\,ad{e}^{2}+3\,b{d}^{2}e \right ) ab+ \left ( a{e}^{3}+3\,bd{e}^{2} \right ){a}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( a{d}^{3}{b}^{2}+2\, \left ( 3\,a{d}^{2}e+b{d}^{3} \right ) ab+ \left ( 3\,ad{e}^{2}+3\,b{d}^{2}e \right ){a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,{a}^{2}{d}^{3}b+ \left ( 3\,a{d}^{2}e+b{d}^{3} \right ){a}^{2} \right ){x}^{2}}{2}}+{a}^{3}{d}^{3}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(e*x+d)^3*(b^2*x^2+2*a*b*x+a^2),x)

[Out]

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

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Maxima [A]  time = 0.711596, size = 225, normalized size = 2.45 \[ \frac{1}{7} \, b^{3} e^{3} x^{7} + a^{3} d^{3} x + \frac{1}{2} \,{\left (b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{6} + \frac{3}{5} \,{\left (b^{3} d^{2} e + 3 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (b^{3} d^{3} + 9 \, a b^{2} d^{2} e + 9 \, a^{2} b d e^{2} + a^{3} e^{3}\right )} x^{4} +{\left (a b^{2} d^{3} + 3 \, a^{2} b d^{2} e + a^{3} d e^{2}\right )} x^{3} + \frac{3}{2} \,{\left (a^{2} b d^{3} + a^{3} d^{2} e\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*b^3*e^3*x^7 + a^3*d^3*x + 1/2*(b^3*d*e^2 + a*b^2*e^3)*x^6 + 3/5*(b^3*d^2*e +
3*a*b^2*d*e^2 + a^2*b*e^3)*x^5 + 1/4*(b^3*d^3 + 9*a*b^2*d^2*e + 9*a^2*b*d*e^2 +
a^3*e^3)*x^4 + (a*b^2*d^3 + 3*a^2*b*d^2*e + a^3*d*e^2)*x^3 + 3/2*(a^2*b*d^3 + a
^3*d^2*e)*x^2

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Fricas [A]  time = 0.251737, size = 1, normalized size = 0.01 \[ \frac{1}{7} x^{7} e^{3} b^{3} + \frac{1}{2} x^{6} e^{2} d b^{3} + \frac{1}{2} x^{6} e^{3} b^{2} a + \frac{3}{5} x^{5} e d^{2} b^{3} + \frac{9}{5} x^{5} e^{2} d b^{2} a + \frac{3}{5} x^{5} e^{3} b a^{2} + \frac{1}{4} x^{4} d^{3} b^{3} + \frac{9}{4} x^{4} e d^{2} b^{2} a + \frac{9}{4} x^{4} e^{2} d b a^{2} + \frac{1}{4} x^{4} e^{3} a^{3} + x^{3} d^{3} b^{2} a + 3 x^{3} e d^{2} b a^{2} + x^{3} e^{2} d a^{3} + \frac{3}{2} x^{2} d^{3} b a^{2} + \frac{3}{2} x^{2} e d^{2} a^{3} + x d^{3} a^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*x^7*e^3*b^3 + 1/2*x^6*e^2*d*b^3 + 1/2*x^6*e^3*b^2*a + 3/5*x^5*e*d^2*b^3 + 9/
5*x^5*e^2*d*b^2*a + 3/5*x^5*e^3*b*a^2 + 1/4*x^4*d^3*b^3 + 9/4*x^4*e*d^2*b^2*a +
9/4*x^4*e^2*d*b*a^2 + 1/4*x^4*e^3*a^3 + x^3*d^3*b^2*a + 3*x^3*e*d^2*b*a^2 + x^3*
e^2*d*a^3 + 3/2*x^2*d^3*b*a^2 + 3/2*x^2*e*d^2*a^3 + x*d^3*a^3

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Sympy [A]  time = 0.190508, size = 190, normalized size = 2.07 \[ a^{3} d^{3} x + \frac{b^{3} e^{3} x^{7}}{7} + x^{6} \left (\frac{a b^{2} e^{3}}{2} + \frac{b^{3} d e^{2}}{2}\right ) + x^{5} \left (\frac{3 a^{2} b e^{3}}{5} + \frac{9 a b^{2} d e^{2}}{5} + \frac{3 b^{3} d^{2} e}{5}\right ) + x^{4} \left (\frac{a^{3} e^{3}}{4} + \frac{9 a^{2} b d e^{2}}{4} + \frac{9 a b^{2} d^{2} e}{4} + \frac{b^{3} d^{3}}{4}\right ) + x^{3} \left (a^{3} d e^{2} + 3 a^{2} b d^{2} e + a b^{2} d^{3}\right ) + x^{2} \left (\frac{3 a^{3} d^{2} e}{2} + \frac{3 a^{2} b d^{3}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**3*d**3*x + b**3*e**3*x**7/7 + x**6*(a*b**2*e**3/2 + b**3*d*e**2/2) + x**5*(3*
a**2*b*e**3/5 + 9*a*b**2*d*e**2/5 + 3*b**3*d**2*e/5) + x**4*(a**3*e**3/4 + 9*a**
2*b*d*e**2/4 + 9*a*b**2*d**2*e/4 + b**3*d**3/4) + x**3*(a**3*d*e**2 + 3*a**2*b*d
**2*e + a*b**2*d**3) + x**2*(3*a**3*d**2*e/2 + 3*a**2*b*d**3/2)

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GIAC/XCAS [A]  time = 0.276239, size = 248, normalized size = 2.7 \[ \frac{1}{7} \, b^{3} x^{7} e^{3} + \frac{1}{2} \, b^{3} d x^{6} e^{2} + \frac{3}{5} \, b^{3} d^{2} x^{5} e + \frac{1}{4} \, b^{3} d^{3} x^{4} + \frac{1}{2} \, a b^{2} x^{6} e^{3} + \frac{9}{5} \, a b^{2} d x^{5} e^{2} + \frac{9}{4} \, a b^{2} d^{2} x^{4} e + a b^{2} d^{3} x^{3} + \frac{3}{5} \, a^{2} b x^{5} e^{3} + \frac{9}{4} \, a^{2} b d x^{4} e^{2} + 3 \, a^{2} b d^{2} x^{3} e + \frac{3}{2} \, a^{2} b d^{3} x^{2} + \frac{1}{4} \, a^{3} x^{4} e^{3} + a^{3} d x^{3} e^{2} + \frac{3}{2} \, a^{3} d^{2} x^{2} e + a^{3} d^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*b^3*x^7*e^3 + 1/2*b^3*d*x^6*e^2 + 3/5*b^3*d^2*x^5*e + 1/4*b^3*d^3*x^4 + 1/2*
a*b^2*x^6*e^3 + 9/5*a*b^2*d*x^5*e^2 + 9/4*a*b^2*d^2*x^4*e + a*b^2*d^3*x^3 + 3/5*
a^2*b*x^5*e^3 + 9/4*a^2*b*d*x^4*e^2 + 3*a^2*b*d^2*x^3*e + 3/2*a^2*b*d^3*x^2 + 1/
4*a^3*x^4*e^3 + a^3*d*x^3*e^2 + 3/2*a^3*d^2*x^2*e + a^3*d^3*x