3.1237 \(\int (a+b x)^3 (c+d x) \, dx\)

Optimal. Leaf size=38 \[ \frac{(a+b x)^4 (b c-a d)}{4 b^2}+\frac{d (a+b x)^5}{5 b^2} \]

[Out]

((b*c - a*d)*(a + b*x)^4)/(4*b^2) + (d*(a + b*x)^5)/(5*b^2)

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Rubi [A]  time = 0.0391669, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{(a+b x)^4 (b c-a d)}{4 b^2}+\frac{d (a+b x)^5}{5 b^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^3*(c + d*x),x]

[Out]

((b*c - a*d)*(a + b*x)^4)/(4*b^2) + (d*(a + b*x)^5)/(5*b^2)

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Rubi in Sympy [A]  time = 10.5834, size = 31, normalized size = 0.82 \[ \frac{d \left (a + b x\right )^{5}}{5 b^{2}} - \frac{\left (a + b x\right )^{4} \left (a d - b c\right )}{4 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*(a + b*x)**5/(5*b**2) - (a + b*x)**4*(a*d - b*c)/(4*b**2)

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Mathematica [A]  time = 0.0163307, size = 67, normalized size = 1.76 \[ a^3 c x+\frac{1}{2} a^2 x^2 (a d+3 b c)+\frac{1}{4} b^2 x^4 (3 a d+b c)+a b x^3 (a d+b c)+\frac{1}{5} b^3 d x^5 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.003, size = 73, normalized size = 1.9 \[{\frac{{b}^{3}d{x}^{5}}{5}}+{\frac{ \left ( 3\,a{b}^{2}d+{b}^{3}c \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,{a}^{2}bd+3\,a{b}^{2}c \right ){x}^{3}}{3}}+{\frac{ \left ({a}^{3}d+3\,{a}^{2}bc \right ){x}^{2}}{2}}+{a}^{3}cx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3*(d*x+c),x)

[Out]

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

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Maxima [A]  time = 1.34244, size = 93, normalized size = 2.45 \[ \frac{1}{5} \, b^{3} d x^{5} + a^{3} c x + \frac{1}{4} \,{\left (b^{3} c + 3 \, a b^{2} d\right )} x^{4} +{\left (a b^{2} c + a^{2} b d\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} b c + a^{3} d\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.175011, size = 1, normalized size = 0.03 \[ \frac{1}{5} x^{5} d b^{3} + \frac{1}{4} x^{4} c b^{3} + \frac{3}{4} x^{4} d b^{2} a + x^{3} c b^{2} a + x^{3} d b a^{2} + \frac{3}{2} x^{2} c b a^{2} + \frac{1}{2} x^{2} d a^{3} + x c a^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.134906, size = 73, normalized size = 1.92 \[ a^{3} c x + \frac{b^{3} d x^{5}}{5} + x^{4} \left (\frac{3 a b^{2} d}{4} + \frac{b^{3} c}{4}\right ) + x^{3} \left (a^{2} b d + a b^{2} c\right ) + x^{2} \left (\frac{a^{3} d}{2} + \frac{3 a^{2} b c}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**3*(d*x+c),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.219049, size = 97, normalized size = 2.55 \[ \frac{1}{5} \, b^{3} d x^{5} + \frac{1}{4} \, b^{3} c x^{4} + \frac{3}{4} \, a b^{2} d x^{4} + a b^{2} c x^{3} + a^{2} b d x^{3} + \frac{3}{2} \, a^{2} b c x^{2} + \frac{1}{2} \, a^{3} d x^{2} + a^{3} c x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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