3.1236 \(\int (a+b x)^4 (c+d x) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0452104, 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 \[ \frac{(a+b x)^5 (b c-a d)}{5 b^2}+\frac{d (a+b x)^6}{6 b^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^4*(c + d*x),x]

[Out]

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

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Rubi in Sympy [A]  time = 12.0252, size = 31, normalized size = 0.82 \[ \frac{d \left (a + b x\right )^{6}}{6 b^{2}} - \frac{\left (a + b x\right )^{5} \left (a d - b c\right )}{5 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.0273502, size = 84, normalized size = 2.21 \[ \frac{1}{30} x \left (15 a^4 (2 c+d x)+20 a^3 b x (3 c+2 d x)+15 a^2 b^2 x^2 (4 c+3 d x)+6 a b^3 x^3 (5 c+4 d x)+b^4 x^4 (6 c+5 d x)\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^4*(d*x+c),x)

[Out]

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

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Maxima [A]  time = 1.35712, size = 130, normalized size = 3.42 \[ \frac{1}{6} \, b^{4} d x^{6} + a^{4} c x + \frac{1}{5} \,{\left (b^{4} c + 4 \, a b^{3} d\right )} x^{5} + \frac{1}{2} \,{\left (2 \, a b^{3} c + 3 \, a^{2} b^{2} d\right )} x^{4} + \frac{2}{3} \,{\left (3 \, a^{2} b^{2} c + 2 \, a^{3} b d\right )} x^{3} + \frac{1}{2} \,{\left (4 \, a^{3} b c + a^{4} d\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.176899, size = 1, normalized size = 0.03 \[ \frac{1}{6} x^{6} d b^{4} + \frac{1}{5} x^{5} c b^{4} + \frac{4}{5} x^{5} d b^{3} a + x^{4} c b^{3} a + \frac{3}{2} x^{4} d b^{2} a^{2} + 2 x^{3} c b^{2} a^{2} + \frac{4}{3} x^{3} d b a^{3} + 2 x^{2} c b a^{3} + \frac{1}{2} x^{2} d a^{4} + x c a^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.137085, size = 100, normalized size = 2.63 \[ a^{4} c x + \frac{b^{4} d x^{6}}{6} + x^{5} \left (\frac{4 a b^{3} d}{5} + \frac{b^{4} c}{5}\right ) + x^{4} \left (\frac{3 a^{2} b^{2} d}{2} + a b^{3} c\right ) + x^{3} \left (\frac{4 a^{3} b d}{3} + 2 a^{2} b^{2} c\right ) + x^{2} \left (\frac{a^{4} d}{2} + 2 a^{3} b c\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**4*(d*x+c),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.220299, size = 131, normalized size = 3.45 \[ \frac{1}{6} \, b^{4} d x^{6} + \frac{1}{5} \, b^{4} c x^{5} + \frac{4}{5} \, a b^{3} d x^{5} + a b^{3} c x^{4} + \frac{3}{2} \, a^{2} b^{2} d x^{4} + 2 \, a^{2} b^{2} c x^{3} + \frac{4}{3} \, a^{3} b d x^{3} + 2 \, a^{3} b c x^{2} + \frac{1}{2} \, a^{4} d x^{2} + a^{4} c x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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