\(\int (a+b x)^3 (a c+(b c+a d) x+b d x^2)^2 \, dx\) [1769]

Optimal result

Integrand size = 29, antiderivative size = 65 \[ \int (a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right )^2 \, dx=\frac {(b c-a d)^2 (a+b x)^6}{6 b^3}+\frac {2 d (b c-a d) (a+b x)^7}{7 b^3}+\frac {d^2 (a+b x)^8}{8 b^3} \]

[Out]

1/6*(-a*d+b*c)^2*(b*x+a)^6/b^3+2/7*d*(-a*d+b*c)*(b*x+a)^7/b^3+1/8*d^2*(b*x+a)^8/b^3

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {640, 45} \[ \int (a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right )^2 \, dx=\frac {2 d (a+b x)^7 (b c-a d)}{7 b^3}+\frac {(a+b x)^6 (b c-a d)^2}{6 b^3}+\frac {d^2 (a+b x)^8}{8 b^3} \]

[In]

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

[Out]

((b*c - a*d)^2*(a + b*x)^6)/(6*b^3) + (2*d*(b*c - a*d)*(a + b*x)^7)/(7*b^3) + (d^2*(a + b*x)^8)/(8*b^3)

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)^5 (c+d x)^2 \, dx \\ & = \int \left (\frac {(b c-a d)^2 (a+b x)^5}{b^2}+\frac {2 d (b c-a d) (a+b x)^6}{b^2}+\frac {d^2 (a+b x)^7}{b^2}\right ) \, dx \\ & = \frac {(b c-a d)^2 (a+b x)^6}{6 b^3}+\frac {2 d (b c-a d) (a+b x)^7}{7 b^3}+\frac {d^2 (a+b x)^8}{8 b^3} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(189\) vs. \(2(65)=130\).

Time = 0.02 (sec) , antiderivative size = 189, normalized size of antiderivative = 2.91 \[ \int (a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right )^2 \, dx=a^5 c^2 x+\frac {1}{2} a^4 c (5 b c+2 a d) x^2+\frac {1}{3} a^3 \left (10 b^2 c^2+10 a b c d+a^2 d^2\right ) x^3+\frac {5}{4} a^2 b \left (2 b^2 c^2+4 a b c d+a^2 d^2\right ) x^4+a b^2 \left (b^2 c^2+4 a b c d+2 a^2 d^2\right ) x^5+\frac {1}{6} b^3 \left (b^2 c^2+10 a b c d+10 a^2 d^2\right ) x^6+\frac {1}{7} b^4 d (2 b c+5 a d) x^7+\frac {1}{8} b^5 d^2 x^8 \]

[In]

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

[Out]

a^5*c^2*x + (a^4*c*(5*b*c + 2*a*d)*x^2)/2 + (a^3*(10*b^2*c^2 + 10*a*b*c*d + a^2*d^2)*x^3)/3 + (5*a^2*b*(2*b^2*
c^2 + 4*a*b*c*d + a^2*d^2)*x^4)/4 + a*b^2*(b^2*c^2 + 4*a*b*c*d + 2*a^2*d^2)*x^5 + (b^3*(b^2*c^2 + 10*a*b*c*d +
10*a^2*d^2)*x^6)/6 + (b^4*d*(2*b*c + 5*a*d)*x^7)/7 + (b^5*d^2*x^8)/8

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(194\) vs. \(2(59)=118\).

Time = 2.37 (sec) , antiderivative size = 195, normalized size of antiderivative = 3.00

method	result	size
norman	\(\frac {b^{5} d^{2} x^{8}}{8}+\left (\frac {5}{7} a \,b^{4} d^{2}+\frac {2}{7} b^{5} c d \right ) x^{7}+\left (\frac {5}{3} a^{2} b^{3} d^{2}+\frac {5}{3} a \,b^{4} c d +\frac {1}{6} b^{5} c^{2}\right ) x^{6}+\left (2 a^{3} b^{2} d^{2}+4 a^{2} b^{3} c d +c^{2} a \,b^{4}\right ) x^{5}+\left (\frac {5}{4} d^{2} a^{4} b +5 a^{3} b^{2} c d +\frac {5}{2} a^{2} c^{2} b^{3}\right ) x^{4}+\left (\frac {1}{3} d^{2} a^{5}+\frac {10}{3} a^{4} b c d +\frac {10}{3} a^{3} b^{2} c^{2}\right ) x^{3}+\left (a^{5} c d +\frac {5}{2} a^{4} b \,c^{2}\right ) x^{2}+a^{5} c^{2} x\)	\(195\)
risch	\(\frac {1}{8} b^{5} d^{2} x^{8}+\frac {5}{7} x^{7} a \,b^{4} d^{2}+\frac {2}{7} x^{7} b^{5} c d +\frac {5}{3} x^{6} a^{2} b^{3} d^{2}+\frac {5}{3} x^{6} a \,b^{4} c d +\frac {1}{6} x^{6} b^{5} c^{2}+2 a^{3} b^{2} d^{2} x^{5}+4 a^{2} b^{3} c d \,x^{5}+a \,b^{4} c^{2} x^{5}+\frac {5}{4} x^{4} d^{2} a^{4} b +5 x^{4} a^{3} b^{2} c d +\frac {5}{2} x^{4} a^{2} c^{2} b^{3}+\frac {1}{3} x^{3} d^{2} a^{5}+\frac {10}{3} x^{3} a^{4} b c d +\frac {10}{3} x^{3} a^{3} b^{2} c^{2}+x^{2} a^{5} c d +\frac {5}{2} x^{2} a^{4} b \,c^{2}+a^{5} c^{2} x\)	\(213\)
parallelrisch	\(\frac {1}{8} b^{5} d^{2} x^{8}+\frac {5}{7} x^{7} a \,b^{4} d^{2}+\frac {2}{7} x^{7} b^{5} c d +\frac {5}{3} x^{6} a^{2} b^{3} d^{2}+\frac {5}{3} x^{6} a \,b^{4} c d +\frac {1}{6} x^{6} b^{5} c^{2}+2 a^{3} b^{2} d^{2} x^{5}+4 a^{2} b^{3} c d \,x^{5}+a \,b^{4} c^{2} x^{5}+\frac {5}{4} x^{4} d^{2} a^{4} b +5 x^{4} a^{3} b^{2} c d +\frac {5}{2} x^{4} a^{2} c^{2} b^{3}+\frac {1}{3} x^{3} d^{2} a^{5}+\frac {10}{3} x^{3} a^{4} b c d +\frac {10}{3} x^{3} a^{3} b^{2} c^{2}+x^{2} a^{5} c d +\frac {5}{2} x^{2} a^{4} b \,c^{2}+a^{5} c^{2} x\)	\(213\)
gosper	\(\frac {x \left (21 b^{5} d^{2} x^{7}+120 x^{6} a \,b^{4} d^{2}+48 x^{6} b^{5} c d +280 x^{5} a^{2} b^{3} d^{2}+280 x^{5} a \,b^{4} c d +28 x^{5} b^{5} c^{2}+336 a^{3} b^{2} d^{2} x^{4}+672 a^{2} b^{3} c d \,x^{4}+168 c^{2} x^{4} a \,b^{4}+210 x^{3} d^{2} a^{4} b +840 x^{3} a^{3} b^{2} c d +420 x^{3} a^{2} c^{2} b^{3}+56 x^{2} d^{2} a^{5}+560 x^{2} a^{4} b c d +560 x^{2} a^{3} b^{2} c^{2}+168 x \,a^{5} c d +420 x \,a^{4} b \,c^{2}+168 a^{5} c^{2}\right )}{168}\)	\(214\)
default	\(\frac {b^{5} d^{2} x^{8}}{8}+\frac {\left (3 a \,b^{4} d^{2}+2 b^{4} d \left (a d +b c \right )\right ) x^{7}}{7}+\frac {\left (3 a^{2} b^{3} d^{2}+6 a \,b^{3} d \left (a d +b c \right )+b^{3} \left (\left (a d +b c \right )^{2}+2 a b c d \right )\right ) x^{6}}{6}+\frac {\left (a^{3} b^{2} d^{2}+6 a^{2} b^{2} d \left (a d +b c \right )+3 a \,b^{2} \left (\left (a d +b c \right )^{2}+2 a b c d \right )+2 b^{3} a c \left (a d +b c \right )\right ) x^{5}}{5}+\frac {\left (2 a^{3} b d \left (a d +b c \right )+3 a^{2} b \left (\left (a d +b c \right )^{2}+2 a b c d \right )+6 a^{2} b^{2} c \left (a d +b c \right )+a^{2} c^{2} b^{3}\right ) x^{4}}{4}+\frac {\left (a^{3} \left (\left (a d +b c \right )^{2}+2 a b c d \right )+6 a^{3} b c \left (a d +b c \right )+3 a^{3} b^{2} c^{2}\right ) x^{3}}{3}+\frac {\left (2 a^{4} c \left (a d +b c \right )+3 a^{4} b \,c^{2}\right ) x^{2}}{2}+a^{5} c^{2} x\)	\(315\)

[In]

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

[Out]

1/8*b^5*d^2*x^8+(5/7*a*b^4*d^2+2/7*b^5*c*d)*x^7+(5/3*a^2*b^3*d^2+5/3*a*b^4*c*d+1/6*b^5*c^2)*x^6+(2*a^3*b^2*d^2
+4*a^2*b^3*c*d+a*b^4*c^2)*x^5+(5/4*d^2*a^4*b+5*a^3*b^2*c*d+5/2*a^2*c^2*b^3)*x^4+(1/3*d^2*a^5+10/3*a^4*b*c*d+10
/3*a^3*b^2*c^2)*x^3+(a^5*c*d+5/2*a^4*b*c^2)*x^2+a^5*c^2*x

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (59) = 118\).

Time = 0.36 (sec) , antiderivative size = 197, normalized size of antiderivative = 3.03 \[ \int (a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right )^2 \, dx=\frac {1}{8} \, b^{5} d^{2} x^{8} + a^{5} c^{2} x + \frac {1}{7} \, {\left (2 \, b^{5} c d + 5 \, a b^{4} d^{2}\right )} x^{7} + \frac {1}{6} \, {\left (b^{5} c^{2} + 10 \, a b^{4} c d + 10 \, a^{2} b^{3} d^{2}\right )} x^{6} + {\left (a b^{4} c^{2} + 4 \, a^{2} b^{3} c d + 2 \, a^{3} b^{2} d^{2}\right )} x^{5} + \frac {5}{4} \, {\left (2 \, a^{2} b^{3} c^{2} + 4 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )} x^{4} + \frac {1}{3} \, {\left (10 \, a^{3} b^{2} c^{2} + 10 \, a^{4} b c d + a^{5} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (5 \, a^{4} b c^{2} + 2 \, a^{5} c d\right )} x^{2} \]

[In]

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

[Out]

1/8*b^5*d^2*x^8 + a^5*c^2*x + 1/7*(2*b^5*c*d + 5*a*b^4*d^2)*x^7 + 1/6*(b^5*c^2 + 10*a*b^4*c*d + 10*a^2*b^3*d^2
)*x^6 + (a*b^4*c^2 + 4*a^2*b^3*c*d + 2*a^3*b^2*d^2)*x^5 + 5/4*(2*a^2*b^3*c^2 + 4*a^3*b^2*c*d + a^4*b*d^2)*x^4
+ 1/3*(10*a^3*b^2*c^2 + 10*a^4*b*c*d + a^5*d^2)*x^3 + 1/2*(5*a^4*b*c^2 + 2*a^5*c*d)*x^2

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (56) = 112\).

Time = 0.04 (sec) , antiderivative size = 218, normalized size of antiderivative = 3.35 \[ \int (a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right )^2 \, dx=a^{5} c^{2} x + \frac {b^{5} d^{2} x^{8}}{8} + x^{7} \cdot \left (\frac {5 a b^{4} d^{2}}{7} + \frac {2 b^{5} c d}{7}\right ) + x^{6} \cdot \left (\frac {5 a^{2} b^{3} d^{2}}{3} + \frac {5 a b^{4} c d}{3} + \frac {b^{5} c^{2}}{6}\right ) + x^{5} \cdot \left (2 a^{3} b^{2} d^{2} + 4 a^{2} b^{3} c d + a b^{4} c^{2}\right ) + x^{4} \cdot \left (\frac {5 a^{4} b d^{2}}{4} + 5 a^{3} b^{2} c d + \frac {5 a^{2} b^{3} c^{2}}{2}\right ) + x^{3} \left (\frac {a^{5} d^{2}}{3} + \frac {10 a^{4} b c d}{3} + \frac {10 a^{3} b^{2} c^{2}}{3}\right ) + x^{2} \left (a^{5} c d + \frac {5 a^{4} b c^{2}}{2}\right ) \]

[In]

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

[Out]

a**5*c**2*x + b**5*d**2*x**8/8 + x**7*(5*a*b**4*d**2/7 + 2*b**5*c*d/7) + x**6*(5*a**2*b**3*d**2/3 + 5*a*b**4*c
*d/3 + b**5*c**2/6) + x**5*(2*a**3*b**2*d**2 + 4*a**2*b**3*c*d + a*b**4*c**2) + x**4*(5*a**4*b*d**2/4 + 5*a**3
*b**2*c*d + 5*a**2*b**3*c**2/2) + x**3*(a**5*d**2/3 + 10*a**4*b*c*d/3 + 10*a**3*b**2*c**2/3) + x**2*(a**5*c*d
+ 5*a**4*b*c**2/2)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (59) = 118\).

Time = 0.21 (sec) , antiderivative size = 197, normalized size of antiderivative = 3.03 \[ \int (a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right )^2 \, dx=\frac {1}{8} \, b^{5} d^{2} x^{8} + a^{5} c^{2} x + \frac {1}{7} \, {\left (2 \, b^{5} c d + 5 \, a b^{4} d^{2}\right )} x^{7} + \frac {1}{6} \, {\left (b^{5} c^{2} + 10 \, a b^{4} c d + 10 \, a^{2} b^{3} d^{2}\right )} x^{6} + {\left (a b^{4} c^{2} + 4 \, a^{2} b^{3} c d + 2 \, a^{3} b^{2} d^{2}\right )} x^{5} + \frac {5}{4} \, {\left (2 \, a^{2} b^{3} c^{2} + 4 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )} x^{4} + \frac {1}{3} \, {\left (10 \, a^{3} b^{2} c^{2} + 10 \, a^{4} b c d + a^{5} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (5 \, a^{4} b c^{2} + 2 \, a^{5} c d\right )} x^{2} \]

[In]

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

[Out]

1/8*b^5*d^2*x^8 + a^5*c^2*x + 1/7*(2*b^5*c*d + 5*a*b^4*d^2)*x^7 + 1/6*(b^5*c^2 + 10*a*b^4*c*d + 10*a^2*b^3*d^2
)*x^6 + (a*b^4*c^2 + 4*a^2*b^3*c*d + 2*a^3*b^2*d^2)*x^5 + 5/4*(2*a^2*b^3*c^2 + 4*a^3*b^2*c*d + a^4*b*d^2)*x^4
+ 1/3*(10*a^3*b^2*c^2 + 10*a^4*b*c*d + a^5*d^2)*x^3 + 1/2*(5*a^4*b*c^2 + 2*a^5*c*d)*x^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 212 vs. \(2 (59) = 118\).

Time = 0.26 (sec) , antiderivative size = 212, normalized size of antiderivative = 3.26 \[ \int (a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right )^2 \, dx=\frac {1}{8} \, b^{5} d^{2} x^{8} + \frac {2}{7} \, b^{5} c d x^{7} + \frac {5}{7} \, a b^{4} d^{2} x^{7} + \frac {1}{6} \, b^{5} c^{2} x^{6} + \frac {5}{3} \, a b^{4} c d x^{6} + \frac {5}{3} \, a^{2} b^{3} d^{2} x^{6} + a b^{4} c^{2} x^{5} + 4 \, a^{2} b^{3} c d x^{5} + 2 \, a^{3} b^{2} d^{2} x^{5} + \frac {5}{2} \, a^{2} b^{3} c^{2} x^{4} + 5 \, a^{3} b^{2} c d x^{4} + \frac {5}{4} \, a^{4} b d^{2} x^{4} + \frac {10}{3} \, a^{3} b^{2} c^{2} x^{3} + \frac {10}{3} \, a^{4} b c d x^{3} + \frac {1}{3} \, a^{5} d^{2} x^{3} + \frac {5}{2} \, a^{4} b c^{2} x^{2} + a^{5} c d x^{2} + a^{5} c^{2} x \]

[In]

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

[Out]

1/8*b^5*d^2*x^8 + 2/7*b^5*c*d*x^7 + 5/7*a*b^4*d^2*x^7 + 1/6*b^5*c^2*x^6 + 5/3*a*b^4*c*d*x^6 + 5/3*a^2*b^3*d^2*
x^6 + a*b^4*c^2*x^5 + 4*a^2*b^3*c*d*x^5 + 2*a^3*b^2*d^2*x^5 + 5/2*a^2*b^3*c^2*x^4 + 5*a^3*b^2*c*d*x^4 + 5/4*a^
4*b*d^2*x^4 + 10/3*a^3*b^2*c^2*x^3 + 10/3*a^4*b*c*d*x^3 + 1/3*a^5*d^2*x^3 + 5/2*a^4*b*c^2*x^2 + a^5*c*d*x^2 +
a^5*c^2*x

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 181, normalized size of antiderivative = 2.78 \[ \int (a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right )^2 \, dx=x^3\,\left (\frac {a^5\,d^2}{3}+\frac {10\,a^4\,b\,c\,d}{3}+\frac {10\,a^3\,b^2\,c^2}{3}\right )+x^6\,\left (\frac {5\,a^2\,b^3\,d^2}{3}+\frac {5\,a\,b^4\,c\,d}{3}+\frac {b^5\,c^2}{6}\right )+a^5\,c^2\,x+\frac {b^5\,d^2\,x^8}{8}+\frac {a^4\,c\,x^2\,\left (2\,a\,d+5\,b\,c\right )}{2}+\frac {b^4\,d\,x^7\,\left (5\,a\,d+2\,b\,c\right )}{7}+\frac {5\,a^2\,b\,x^4\,\left (a^2\,d^2+4\,a\,b\,c\,d+2\,b^2\,c^2\right )}{4}+a\,b^2\,x^5\,\left (2\,a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right ) \]

[In]

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

[Out]

x^3*((a^5*d^2)/3 + (10*a^3*b^2*c^2)/3 + (10*a^4*b*c*d)/3) + x^6*((b^5*c^2)/6 + (5*a^2*b^3*d^2)/3 + (5*a*b^4*c*
d)/3) + a^5*c^2*x + (b^5*d^2*x^8)/8 + (a^4*c*x^2*(2*a*d + 5*b*c))/2 + (b^4*d*x^7*(5*a*d + 2*b*c))/7 + (5*a^2*b
*x^4*(a^2*d^2 + 2*b^2*c^2 + 4*a*b*c*d))/4 + a*b^2*x^5*(2*a^2*d^2 + b^2*c^2 + 4*a*b*c*d)