\(\int (a+b x)^3 (a c+(b c+a d) x+b d x^2) \, dx\) [1759]

Optimal result

Integrand size = 27, antiderivative size = 38 \[ \int (a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right ) \, dx=\frac {(b c-a d) (a+b x)^5}{5 b^2}+\frac {d (a+b x)^6}{6 b^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {640, 45} \[ \int (a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right ) \, dx=\frac {(a+b x)^5 (b c-a d)}{5 b^2}+\frac {d (a+b x)^6}{6 b^2} \]

[In]

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

[Out]

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 640

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c/e)*x)^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&
IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = \int (a+b x)^4 (c+d x) \, dx \\ & = \int \left (\frac {(b c-a d) (a+b x)^4}{b}+\frac {d (a+b x)^5}{b}\right ) \, dx \\ & = \frac {(b c-a d) (a+b x)^5}{5 b^2}+\frac {d (a+b x)^6}{6 b^2} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(84\) vs. \(2(38)=76\).

Time = 0.01 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.21 \[ \int (a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right ) \, dx=\frac {1}{30} x \left (15 a^4 (2 c+d x)+20 a^3 b x (3 c+2 d x)+15 a^2 b^2 x^2 (4 c+3 d x)+6 a b^3 x^3 (5 c+4 d x)+b^4 x^4 (6 c+5 d x)\right ) \]

[In]

Integrate[(a + b*x)^3*(a*c + (b*c + a*d)*x + b*d*x^2),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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(93\) vs. \(2(34)=68\).

Time = 2.00 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.47

method	result	size
norman	\(\frac {b^{4} d \,x^{6}}{6}+\left (\frac {4}{5} a \,b^{3} d +\frac {1}{5} b^{4} c \right ) x^{5}+\left (\frac {3}{2} a^{2} b^{2} d +a \,b^{3} c \right ) x^{4}+\left (\frac {4}{3} d \,a^{3} b +2 a^{2} b^{2} c \right ) x^{3}+\left (\frac {1}{2} a^{4} d +2 a^{3} b c \right ) x^{2}+a^{4} c x\)	\(94\)
gosper	\(\frac {x \left (5 b^{4} d \,x^{5}+24 a \,b^{3} d \,x^{4}+6 b^{4} c \,x^{4}+45 x^{3} a^{2} b^{2} d +30 x^{3} a \,b^{3} c +40 a^{3} b d \,x^{2}+60 c \,x^{2} a^{2} b^{2}+15 a^{4} d x +60 a^{3} b c x +30 a^{4} c \right )}{30}\)	\(98\)
risch	\(\frac {1}{6} b^{4} d \,x^{6}+\frac {4}{5} x^{5} a \,b^{3} d +\frac {1}{5} x^{5} b^{4} c +\frac {3}{2} x^{4} a^{2} b^{2} d +x^{4} a \,b^{3} c +\frac {4}{3} a^{3} b d \,x^{3}+2 a^{2} b^{2} c \,x^{3}+\frac {1}{2} a^{4} d \,x^{2}+2 a^{3} b c \,x^{2}+a^{4} c x\)	\(98\)
parallelrisch	\(\frac {1}{6} b^{4} d \,x^{6}+\frac {4}{5} x^{5} a \,b^{3} d +\frac {1}{5} x^{5} b^{4} c +\frac {3}{2} x^{4} a^{2} b^{2} d +x^{4} a \,b^{3} c +\frac {4}{3} a^{3} b d \,x^{3}+2 a^{2} b^{2} c \,x^{3}+\frac {1}{2} a^{4} d \,x^{2}+2 a^{3} b c \,x^{2}+a^{4} c x\)	\(98\)
default	\(\frac {b^{4} d \,x^{6}}{6}+\frac {\left (3 a \,b^{3} d +b^{3} \left (a d +b c \right )\right ) x^{5}}{5}+\frac {\left (3 a^{2} b^{2} d +3 a \,b^{2} \left (a d +b c \right )+a \,b^{3} c \right ) x^{4}}{4}+\frac {\left (d \,a^{3} b +3 a^{2} b \left (a d +b c \right )+3 a^{2} b^{2} c \right ) x^{3}}{3}+\frac {\left (a^{3} \left (a d +b c \right )+3 a^{3} b c \right ) x^{2}}{2}+a^{4} c x\)	\(133\)

[In]

int((b*x+a)^3*(b*d*x^2+(a*d+b*c)*x+a*c),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (34) = 68\).

Time = 0.25 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.53 \[ \int (a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right ) \, dx=\frac {1}{6} \, b^{4} d x^{6} + a^{4} c x + \frac {1}{5} \, {\left (b^{4} c + 4 \, a b^{3} d\right )} x^{5} + \frac {1}{2} \, {\left (2 \, a b^{3} c + 3 \, a^{2} b^{2} d\right )} x^{4} + \frac {2}{3} \, {\left (3 \, a^{2} b^{2} c + 2 \, a^{3} b d\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{3} b c + a^{4} d\right )} x^{2} \]

[In]

integrate((b*x+a)^3*(a*c+(a*d+b*c)*x+b*d*x^2),x, algorithm="fricas")

[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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (32) = 64\).

Time = 0.03 (sec) , antiderivative size = 100, normalized size of antiderivative = 2.63 \[ \int (a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right ) \, dx=a^{4} c x + \frac {b^{4} d x^{6}}{6} + x^{5} \cdot \left (\frac {4 a b^{3} d}{5} + \frac {b^{4} c}{5}\right ) + x^{4} \cdot \left (\frac {3 a^{2} b^{2} d}{2} + a b^{3} c\right ) + x^{3} \cdot \left (\frac {4 a^{3} b d}{3} + 2 a^{2} b^{2} c\right ) + x^{2} \left (\frac {a^{4} d}{2} + 2 a^{3} b c\right ) \]

[In]

integrate((b*x+a)**3*(a*c+(a*d+b*c)*x+b*d*x**2),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)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (34) = 68\).

Time = 0.20 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.53 \[ \int (a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right ) \, dx=\frac {1}{6} \, b^{4} d x^{6} + a^{4} c x + \frac {1}{5} \, {\left (b^{4} c + 4 \, a b^{3} d\right )} x^{5} + \frac {1}{2} \, {\left (2 \, a b^{3} c + 3 \, a^{2} b^{2} d\right )} x^{4} + \frac {2}{3} \, {\left (3 \, a^{2} b^{2} c + 2 \, a^{3} b d\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{3} b c + a^{4} d\right )} x^{2} \]

[In]

integrate((b*x+a)^3*(a*c+(a*d+b*c)*x+b*d*x^2),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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (34) = 68\).

Time = 0.28 (sec) , antiderivative size = 97, normalized size of antiderivative = 2.55 \[ \int (a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right ) \, dx=\frac {1}{6} \, b^{4} d x^{6} + \frac {1}{5} \, b^{4} c x^{5} + \frac {4}{5} \, a b^{3} d x^{5} + a b^{3} c x^{4} + \frac {3}{2} \, a^{2} b^{2} d x^{4} + 2 \, a^{2} b^{2} c x^{3} + \frac {4}{3} \, a^{3} b d x^{3} + 2 \, a^{3} b c x^{2} + \frac {1}{2} \, a^{4} d x^{2} + a^{4} c x \]

[In]

integrate((b*x+a)^3*(a*c+(a*d+b*c)*x+b*d*x^2),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

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.32 \[ \int (a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right ) \, dx=x^5\,\left (\frac {c\,b^4}{5}+\frac {4\,a\,d\,b^3}{5}\right )+x^2\,\left (\frac {d\,a^4}{2}+2\,b\,c\,a^3\right )+\frac {b^4\,d\,x^6}{6}+a^4\,c\,x+\frac {2\,a^2\,b\,x^3\,\left (2\,a\,d+3\,b\,c\right )}{3}+\frac {a\,b^2\,x^4\,\left (3\,a\,d+2\,b\,c\right )}{2} \]

[In]

int((a + b*x)^3*(a*c + x*(a*d + b*c) + b*d*x^2),x)

[Out]

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