\(\int (a+b x)^3 (c+d x)^3 \, dx\) [1260]

Optimal result

Integrand size = 15, antiderivative size = 92 \[ \int (a+b x)^3 (c+d x)^3 \, dx=\frac {(b c-a d)^3 (a+b x)^4}{4 b^4}+\frac {3 d (b c-a d)^2 (a+b x)^5}{5 b^4}+\frac {d^2 (b c-a d) (a+b x)^6}{2 b^4}+\frac {d^3 (a+b x)^7}{7 b^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45} \[ \int (a+b x)^3 (c+d x)^3 \, dx=\frac {d^2 (a+b x)^6 (b c-a d)}{2 b^4}+\frac {3 d (a+b x)^5 (b c-a d)^2}{5 b^4}+\frac {(a+b x)^4 (b c-a d)^3}{4 b^4}+\frac {d^3 (a+b x)^7}{7 b^4} \]

[In]

Int[(a + b*x)^3*(c + d*x)^3,x]

[Out]

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

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])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(b c-a d)^3 (a+b x)^3}{b^3}+\frac {3 d (b c-a d)^2 (a+b x)^4}{b^3}+\frac {3 d^2 (b c-a d) (a+b x)^5}{b^3}+\frac {d^3 (a+b x)^6}{b^3}\right ) \, dx \\ & = \frac {(b c-a d)^3 (a+b x)^4}{4 b^4}+\frac {3 d (b c-a d)^2 (a+b x)^5}{5 b^4}+\frac {d^2 (b c-a d) (a+b x)^6}{2 b^4}+\frac {d^3 (a+b x)^7}{7 b^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.75 \[ \int (a+b x)^3 (c+d x)^3 \, dx=a^3 c^3 x+\frac {3}{2} a^2 c^2 (b c+a d) x^2+a c \left (b^2 c^2+3 a b c d+a^2 d^2\right ) x^3+\frac {1}{4} \left (b^3 c^3+9 a b^2 c^2 d+9 a^2 b c d^2+a^3 d^3\right ) x^4+\frac {3}{5} b d \left (b^2 c^2+3 a b c d+a^2 d^2\right ) x^5+\frac {1}{2} b^2 d^2 (b c+a d) x^6+\frac {1}{7} b^3 d^3 x^7 \]

[In]

Integrate[(a + b*x)^3*(c + d*x)^3,x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(171\) vs. \(2(84)=168\).

Time = 0.18 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.87

method	result	size
norman	\(\frac {b^{3} d^{3} x^{7}}{7}+\left (\frac {1}{2} a \,b^{2} d^{3}+\frac {1}{2} b^{3} c \,d^{2}\right ) x^{6}+\left (\frac {3}{5} a^{2} b \,d^{3}+\frac {9}{5} a \,b^{2} c \,d^{2}+\frac {3}{5} b^{3} c^{2} d \right ) x^{5}+\left (\frac {1}{4} a^{3} d^{3}+\frac {9}{4} a^{2} b c \,d^{2}+\frac {9}{4} a \,b^{2} c^{2} d +\frac {1}{4} b^{3} c^{3}\right ) x^{4}+\left (a^{3} c \,d^{2}+3 a^{2} b \,c^{2} d +a \,b^{2} c^{3}\right ) x^{3}+\left (\frac {3}{2} a^{3} c^{2} d +\frac {3}{2} a^{2} b \,c^{3}\right ) x^{2}+a^{3} c^{3} x\)	\(172\)
default	\(\frac {b^{3} d^{3} x^{7}}{7}+\frac {\left (3 a \,b^{2} d^{3}+3 b^{3} c \,d^{2}\right ) x^{6}}{6}+\frac {\left (3 a^{2} b \,d^{3}+9 a \,b^{2} c \,d^{2}+3 b^{3} c^{2} d \right ) x^{5}}{5}+\frac {\left (a^{3} d^{3}+9 a^{2} b c \,d^{2}+9 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) x^{4}}{4}+\frac {\left (3 a^{3} c \,d^{2}+9 a^{2} b \,c^{2} d +3 a \,b^{2} c^{3}\right ) x^{3}}{3}+\frac {\left (3 a^{3} c^{2} d +3 a^{2} b \,c^{3}\right ) x^{2}}{2}+a^{3} c^{3} x\)	\(177\)
gosper	\(\frac {1}{7} b^{3} d^{3} x^{7}+\frac {1}{2} x^{6} a \,b^{2} d^{3}+\frac {1}{2} x^{6} b^{3} c \,d^{2}+\frac {3}{5} x^{5} a^{2} b \,d^{3}+\frac {9}{5} x^{5} a \,b^{2} c \,d^{2}+\frac {3}{5} x^{5} b^{3} c^{2} d +\frac {1}{4} x^{4} a^{3} d^{3}+\frac {9}{4} x^{4} a^{2} b c \,d^{2}+\frac {9}{4} x^{4} a \,b^{2} c^{2} d +\frac {1}{4} x^{4} b^{3} c^{3}+a^{3} c \,d^{2} x^{3}+3 a^{2} b \,c^{2} d \,x^{3}+a \,b^{2} c^{3} x^{3}+\frac {3}{2} x^{2} a^{3} c^{2} d +\frac {3}{2} x^{2} a^{2} b \,c^{3}+a^{3} c^{3} x\)	\(189\)
risch	\(\frac {1}{7} b^{3} d^{3} x^{7}+\frac {1}{2} x^{6} a \,b^{2} d^{3}+\frac {1}{2} x^{6} b^{3} c \,d^{2}+\frac {3}{5} x^{5} a^{2} b \,d^{3}+\frac {9}{5} x^{5} a \,b^{2} c \,d^{2}+\frac {3}{5} x^{5} b^{3} c^{2} d +\frac {1}{4} x^{4} a^{3} d^{3}+\frac {9}{4} x^{4} a^{2} b c \,d^{2}+\frac {9}{4} x^{4} a \,b^{2} c^{2} d +\frac {1}{4} x^{4} b^{3} c^{3}+a^{3} c \,d^{2} x^{3}+3 a^{2} b \,c^{2} d \,x^{3}+a \,b^{2} c^{3} x^{3}+\frac {3}{2} x^{2} a^{3} c^{2} d +\frac {3}{2} x^{2} a^{2} b \,c^{3}+a^{3} c^{3} x\)	\(189\)
parallelrisch	\(\frac {1}{7} b^{3} d^{3} x^{7}+\frac {1}{2} x^{6} a \,b^{2} d^{3}+\frac {1}{2} x^{6} b^{3} c \,d^{2}+\frac {3}{5} x^{5} a^{2} b \,d^{3}+\frac {9}{5} x^{5} a \,b^{2} c \,d^{2}+\frac {3}{5} x^{5} b^{3} c^{2} d +\frac {1}{4} x^{4} a^{3} d^{3}+\frac {9}{4} x^{4} a^{2} b c \,d^{2}+\frac {9}{4} x^{4} a \,b^{2} c^{2} d +\frac {1}{4} x^{4} b^{3} c^{3}+a^{3} c \,d^{2} x^{3}+3 a^{2} b \,c^{2} d \,x^{3}+a \,b^{2} c^{3} x^{3}+\frac {3}{2} x^{2} a^{3} c^{2} d +\frac {3}{2} x^{2} a^{2} b \,c^{3}+a^{3} c^{3} x\)	\(189\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.82 \[ \int (a+b x)^3 (c+d x)^3 \, dx=\frac {1}{7} \, b^{3} d^{3} x^{7} + a^{3} c^{3} x + \frac {1}{2} \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{6} + \frac {3}{5} \, {\left (b^{3} c^{2} d + 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{5} + \frac {1}{4} \, {\left (b^{3} c^{3} + 9 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{4} + {\left (a b^{2} c^{3} + 3 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} x^{3} + \frac {3}{2} \, {\left (a^{2} b c^{3} + a^{3} c^{2} d\right )} x^{2} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 190 vs. \(2 (80) = 160\).

Time = 0.04 (sec) , antiderivative size = 190, normalized size of antiderivative = 2.07 \[ \int (a+b x)^3 (c+d x)^3 \, dx=a^{3} c^{3} x + \frac {b^{3} d^{3} x^{7}}{7} + x^{6} \left (\frac {a b^{2} d^{3}}{2} + \frac {b^{3} c d^{2}}{2}\right ) + x^{5} \cdot \left (\frac {3 a^{2} b d^{3}}{5} + \frac {9 a b^{2} c d^{2}}{5} + \frac {3 b^{3} c^{2} d}{5}\right ) + x^{4} \left (\frac {a^{3} d^{3}}{4} + \frac {9 a^{2} b c d^{2}}{4} + \frac {9 a b^{2} c^{2} d}{4} + \frac {b^{3} c^{3}}{4}\right ) + x^{3} \left (a^{3} c d^{2} + 3 a^{2} b c^{2} d + a b^{2} c^{3}\right ) + x^{2} \cdot \left (\frac {3 a^{3} c^{2} d}{2} + \frac {3 a^{2} b c^{3}}{2}\right ) \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.82 \[ \int (a+b x)^3 (c+d x)^3 \, dx=\frac {1}{7} \, b^{3} d^{3} x^{7} + a^{3} c^{3} x + \frac {1}{2} \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{6} + \frac {3}{5} \, {\left (b^{3} c^{2} d + 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{5} + \frac {1}{4} \, {\left (b^{3} c^{3} + 9 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{4} + {\left (a b^{2} c^{3} + 3 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} x^{3} + \frac {3}{2} \, {\left (a^{2} b c^{3} + a^{3} c^{2} d\right )} x^{2} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 188 vs. \(2 (84) = 168\).

Time = 0.30 (sec) , antiderivative size = 188, normalized size of antiderivative = 2.04 \[ \int (a+b x)^3 (c+d x)^3 \, dx=\frac {1}{7} \, b^{3} d^{3} x^{7} + \frac {1}{2} \, b^{3} c d^{2} x^{6} + \frac {1}{2} \, a b^{2} d^{3} x^{6} + \frac {3}{5} \, b^{3} c^{2} d x^{5} + \frac {9}{5} \, a b^{2} c d^{2} x^{5} + \frac {3}{5} \, a^{2} b d^{3} x^{5} + \frac {1}{4} \, b^{3} c^{3} x^{4} + \frac {9}{4} \, a b^{2} c^{2} d x^{4} + \frac {9}{4} \, a^{2} b c d^{2} x^{4} + \frac {1}{4} \, a^{3} d^{3} x^{4} + a b^{2} c^{3} x^{3} + 3 \, a^{2} b c^{2} d x^{3} + a^{3} c d^{2} x^{3} + \frac {3}{2} \, a^{2} b c^{3} x^{2} + \frac {3}{2} \, a^{3} c^{2} d x^{2} + a^{3} c^{3} x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.65 \[ \int (a+b x)^3 (c+d x)^3 \, dx=x^4\,\left (\frac {a^3\,d^3}{4}+\frac {9\,a^2\,b\,c\,d^2}{4}+\frac {9\,a\,b^2\,c^2\,d}{4}+\frac {b^3\,c^3}{4}\right )+a^3\,c^3\,x+\frac {b^3\,d^3\,x^7}{7}+a\,c\,x^3\,\left (a^2\,d^2+3\,a\,b\,c\,d+b^2\,c^2\right )+\frac {3\,b\,d\,x^5\,\left (a^2\,d^2+3\,a\,b\,c\,d+b^2\,c^2\right )}{5}+\frac {3\,a^2\,c^2\,x^2\,\left (a\,d+b\,c\right )}{2}+\frac {b^2\,d^2\,x^6\,\left (a\,d+b\,c\right )}{2} \]

[In]

int((a + b*x)^3*(c + d*x)^3,x)

[Out]

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