\(\int (a+b x)^7 (c+d x)^7 \, dx\) [1275]

Optimal result

Integrand size = 15, antiderivative size = 200 \[ \int (a+b x)^7 (c+d x)^7 \, dx=\frac {(b c-a d)^7 (a+b x)^8}{8 b^8}+\frac {7 d (b c-a d)^6 (a+b x)^9}{9 b^8}+\frac {21 d^2 (b c-a d)^5 (a+b x)^{10}}{10 b^8}+\frac {35 d^3 (b c-a d)^4 (a+b x)^{11}}{11 b^8}+\frac {35 d^4 (b c-a d)^3 (a+b x)^{12}}{12 b^8}+\frac {21 d^5 (b c-a d)^2 (a+b x)^{13}}{13 b^8}+\frac {d^6 (b c-a d) (a+b x)^{14}}{2 b^8}+\frac {d^7 (a+b x)^{15}}{15 b^8} \]

[Out]

1/8*(-a*d+b*c)^7*(b*x+a)^8/b^8+7/9*d*(-a*d+b*c)^6*(b*x+a)^9/b^8+21/10*d^2*(-a*d+b*c)^5*(b*x+a)^10/b^8+35/11*d^
3*(-a*d+b*c)^4*(b*x+a)^11/b^8+35/12*d^4*(-a*d+b*c)^3*(b*x+a)^12/b^8+21/13*d^5*(-a*d+b*c)^2*(b*x+a)^13/b^8+1/2*
d^6*(-a*d+b*c)*(b*x+a)^14/b^8+1/15*d^7*(b*x+a)^15/b^8

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 200, 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} \[ \int (a+b x)^7 (c+d x)^7 \, dx=\frac {d^6 (a+b x)^{14} (b c-a d)}{2 b^8}+\frac {21 d^5 (a+b x)^{13} (b c-a d)^2}{13 b^8}+\frac {35 d^4 (a+b x)^{12} (b c-a d)^3}{12 b^8}+\frac {35 d^3 (a+b x)^{11} (b c-a d)^4}{11 b^8}+\frac {21 d^2 (a+b x)^{10} (b c-a d)^5}{10 b^8}+\frac {7 d (a+b x)^9 (b c-a d)^6}{9 b^8}+\frac {(a+b x)^8 (b c-a d)^7}{8 b^8}+\frac {d^7 (a+b x)^{15}}{15 b^8} \]

[In]

Int[(a + b*x)^7*(c + d*x)^7,x]

[Out]

((b*c - a*d)^7*(a + b*x)^8)/(8*b^8) + (7*d*(b*c - a*d)^6*(a + b*x)^9)/(9*b^8) + (21*d^2*(b*c - a*d)^5*(a + b*x
)^10)/(10*b^8) + (35*d^3*(b*c - a*d)^4*(a + b*x)^11)/(11*b^8) + (35*d^4*(b*c - a*d)^3*(a + b*x)^12)/(12*b^8) +
(21*d^5*(b*c - a*d)^2*(a + b*x)^13)/(13*b^8) + (d^6*(b*c - a*d)*(a + b*x)^14)/(2*b^8) + (d^7*(a + b*x)^15)/(1
5*b^8)

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)^7 (a+b x)^7}{b^7}+\frac {7 d (b c-a d)^6 (a+b x)^8}{b^7}+\frac {21 d^2 (b c-a d)^5 (a+b x)^9}{b^7}+\frac {35 d^3 (b c-a d)^4 (a+b x)^{10}}{b^7}+\frac {35 d^4 (b c-a d)^3 (a+b x)^{11}}{b^7}+\frac {21 d^5 (b c-a d)^2 (a+b x)^{12}}{b^7}+\frac {7 d^6 (b c-a d) (a+b x)^{13}}{b^7}+\frac {d^7 (a+b x)^{14}}{b^7}\right ) \, dx \\ & = \frac {(b c-a d)^7 (a+b x)^8}{8 b^8}+\frac {7 d (b c-a d)^6 (a+b x)^9}{9 b^8}+\frac {21 d^2 (b c-a d)^5 (a+b x)^{10}}{10 b^8}+\frac {35 d^3 (b c-a d)^4 (a+b x)^{11}}{11 b^8}+\frac {35 d^4 (b c-a d)^3 (a+b x)^{12}}{12 b^8}+\frac {21 d^5 (b c-a d)^2 (a+b x)^{13}}{13 b^8}+\frac {d^6 (b c-a d) (a+b x)^{14}}{2 b^8}+\frac {d^7 (a+b x)^{15}}{15 b^8} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(785\) vs. \(2(200)=400\).

Time = 0.05 (sec) , antiderivative size = 785, normalized size of antiderivative = 3.92 \[ \int (a+b x)^7 (c+d x)^7 \, dx=a^7 c^7 x+\frac {7}{2} a^6 c^6 (b c+a d) x^2+\frac {7}{3} a^5 c^5 \left (3 b^2 c^2+7 a b c d+3 a^2 d^2\right ) x^3+\frac {7}{4} a^4 c^4 \left (5 b^3 c^3+21 a b^2 c^2 d+21 a^2 b c d^2+5 a^3 d^3\right ) x^4+\frac {7}{5} a^3 c^3 \left (5 b^4 c^4+35 a b^3 c^3 d+63 a^2 b^2 c^2 d^2+35 a^3 b c d^3+5 a^4 d^4\right ) x^5+\frac {7}{6} a^2 c^2 \left (3 b^5 c^5+35 a b^4 c^4 d+105 a^2 b^3 c^3 d^2+105 a^3 b^2 c^2 d^3+35 a^4 b c d^4+3 a^5 d^5\right ) x^6+a c \left (b^6 c^6+21 a b^5 c^5 d+105 a^2 b^4 c^4 d^2+175 a^3 b^3 c^3 d^3+105 a^4 b^2 c^2 d^4+21 a^5 b c d^5+a^6 d^6\right ) x^7+\frac {1}{8} \left (b^7 c^7+49 a b^6 c^6 d+441 a^2 b^5 c^5 d^2+1225 a^3 b^4 c^4 d^3+1225 a^4 b^3 c^3 d^4+441 a^5 b^2 c^2 d^5+49 a^6 b c d^6+a^7 d^7\right ) x^8+\frac {7}{9} b d \left (b^6 c^6+21 a b^5 c^5 d+105 a^2 b^4 c^4 d^2+175 a^3 b^3 c^3 d^3+105 a^4 b^2 c^2 d^4+21 a^5 b c d^5+a^6 d^6\right ) x^9+\frac {7}{10} b^2 d^2 \left (3 b^5 c^5+35 a b^4 c^4 d+105 a^2 b^3 c^3 d^2+105 a^3 b^2 c^2 d^3+35 a^4 b c d^4+3 a^5 d^5\right ) x^{10}+\frac {7}{11} b^3 d^3 \left (5 b^4 c^4+35 a b^3 c^3 d+63 a^2 b^2 c^2 d^2+35 a^3 b c d^3+5 a^4 d^4\right ) x^{11}+\frac {7}{12} b^4 d^4 \left (5 b^3 c^3+21 a b^2 c^2 d+21 a^2 b c d^2+5 a^3 d^3\right ) x^{12}+\frac {7}{13} b^5 d^5 \left (3 b^2 c^2+7 a b c d+3 a^2 d^2\right ) x^{13}+\frac {1}{2} b^6 d^6 (b c+a d) x^{14}+\frac {1}{15} b^7 d^7 x^{15} \]

[In]

Integrate[(a + b*x)^7*(c + d*x)^7,x]

[Out]

a^7*c^7*x + (7*a^6*c^6*(b*c + a*d)*x^2)/2 + (7*a^5*c^5*(3*b^2*c^2 + 7*a*b*c*d + 3*a^2*d^2)*x^3)/3 + (7*a^4*c^4
*(5*b^3*c^3 + 21*a*b^2*c^2*d + 21*a^2*b*c*d^2 + 5*a^3*d^3)*x^4)/4 + (7*a^3*c^3*(5*b^4*c^4 + 35*a*b^3*c^3*d + 6
3*a^2*b^2*c^2*d^2 + 35*a^3*b*c*d^3 + 5*a^4*d^4)*x^5)/5 + (7*a^2*c^2*(3*b^5*c^5 + 35*a*b^4*c^4*d + 105*a^2*b^3*
c^3*d^2 + 105*a^3*b^2*c^2*d^3 + 35*a^4*b*c*d^4 + 3*a^5*d^5)*x^6)/6 + a*c*(b^6*c^6 + 21*a*b^5*c^5*d + 105*a^2*b
^4*c^4*d^2 + 175*a^3*b^3*c^3*d^3 + 105*a^4*b^2*c^2*d^4 + 21*a^5*b*c*d^5 + a^6*d^6)*x^7 + ((b^7*c^7 + 49*a*b^6*
c^6*d + 441*a^2*b^5*c^5*d^2 + 1225*a^3*b^4*c^4*d^3 + 1225*a^4*b^3*c^3*d^4 + 441*a^5*b^2*c^2*d^5 + 49*a^6*b*c*d
^6 + a^7*d^7)*x^8)/8 + (7*b*d*(b^6*c^6 + 21*a*b^5*c^5*d + 105*a^2*b^4*c^4*d^2 + 175*a^3*b^3*c^3*d^3 + 105*a^4*
b^2*c^2*d^4 + 21*a^5*b*c*d^5 + a^6*d^6)*x^9)/9 + (7*b^2*d^2*(3*b^5*c^5 + 35*a*b^4*c^4*d + 105*a^2*b^3*c^3*d^2
+ 105*a^3*b^2*c^2*d^3 + 35*a^4*b*c*d^4 + 3*a^5*d^5)*x^10)/10 + (7*b^3*d^3*(5*b^4*c^4 + 35*a*b^3*c^3*d + 63*a^2
*b^2*c^2*d^2 + 35*a^3*b*c*d^3 + 5*a^4*d^4)*x^11)/11 + (7*b^4*d^4*(5*b^3*c^3 + 21*a*b^2*c^2*d + 21*a^2*b*c*d^2
+ 5*a^3*d^3)*x^12)/12 + (7*b^5*d^5*(3*b^2*c^2 + 7*a*b*c*d + 3*a^2*d^2)*x^13)/13 + (b^6*d^6*(b*c + a*d)*x^14)/2
+ (b^7*d^7*x^15)/15

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(803\) vs. \(2(184)=368\).

Time = 0.20 (sec) , antiderivative size = 804, normalized size of antiderivative = 4.02

method	result	size
norman	\(a^{7} c^{7} x +\left (\frac {7}{2} a^{7} c^{6} d +\frac {7}{2} a^{6} b \,c^{7}\right ) x^{2}+\left (7 a^{7} c^{5} d^{2}+\frac {49}{3} a^{6} b \,c^{6} d +7 a^{5} b^{2} c^{7}\right ) x^{3}+\left (\frac {35}{4} a^{7} c^{4} d^{3}+\frac {147}{4} a^{6} b \,c^{5} d^{2}+\frac {147}{4} a^{5} b^{2} c^{6} d +\frac {35}{4} a^{4} b^{3} c^{7}\right ) x^{4}+\left (7 a^{7} c^{3} d^{4}+49 a^{6} b \,c^{4} d^{3}+\frac {441}{5} a^{5} b^{2} c^{5} d^{2}+49 a^{4} b^{3} c^{6} d +7 a^{3} b^{4} c^{7}\right ) x^{5}+\left (\frac {7}{2} a^{7} c^{2} d^{5}+\frac {245}{6} a^{6} b \,c^{3} d^{4}+\frac {245}{2} a^{5} b^{2} c^{4} d^{3}+\frac {245}{2} a^{4} b^{3} c^{5} d^{2}+\frac {245}{6} a^{3} b^{4} c^{6} d +\frac {7}{2} a^{2} b^{5} c^{7}\right ) x^{6}+\left (a^{7} c \,d^{6}+21 a^{6} b \,c^{2} d^{5}+105 a^{5} b^{2} c^{3} d^{4}+175 a^{4} b^{3} c^{4} d^{3}+105 a^{3} b^{4} c^{5} d^{2}+21 a^{2} b^{5} c^{6} d +a \,b^{6} c^{7}\right ) x^{7}+\left (\frac {1}{8} a^{7} d^{7}+\frac {49}{8} a^{6} b c \,d^{6}+\frac {441}{8} a^{5} b^{2} c^{2} d^{5}+\frac {1225}{8} a^{4} b^{3} c^{3} d^{4}+\frac {1225}{8} a^{3} b^{4} c^{4} d^{3}+\frac {441}{8} a^{2} b^{5} c^{5} d^{2}+\frac {49}{8} a \,b^{6} c^{6} d +\frac {1}{8} b^{7} c^{7}\right ) x^{8}+\left (\frac {7}{9} a^{6} b \,d^{7}+\frac {49}{3} a^{5} b^{2} c \,d^{6}+\frac {245}{3} a^{4} b^{3} c^{2} d^{5}+\frac {1225}{9} a^{3} b^{4} c^{3} d^{4}+\frac {245}{3} a^{2} b^{5} c^{4} d^{3}+\frac {49}{3} a \,b^{6} c^{5} d^{2}+\frac {7}{9} b^{7} c^{6} d \right ) x^{9}+\left (\frac {21}{10} a^{5} b^{2} d^{7}+\frac {49}{2} a^{4} b^{3} c \,d^{6}+\frac {147}{2} a^{3} b^{4} c^{2} d^{5}+\frac {147}{2} a^{2} b^{5} c^{3} d^{4}+\frac {49}{2} a \,b^{6} c^{4} d^{3}+\frac {21}{10} b^{7} c^{5} d^{2}\right ) x^{10}+\left (\frac {35}{11} a^{4} b^{3} d^{7}+\frac {245}{11} a^{3} b^{4} c \,d^{6}+\frac {441}{11} a^{2} b^{5} c^{2} d^{5}+\frac {245}{11} a \,b^{6} c^{3} d^{4}+\frac {35}{11} b^{7} c^{4} d^{3}\right ) x^{11}+\left (\frac {35}{12} a^{3} b^{4} d^{7}+\frac {49}{4} a^{2} b^{5} c \,d^{6}+\frac {49}{4} a \,b^{6} c^{2} d^{5}+\frac {35}{12} b^{7} c^{3} d^{4}\right ) x^{12}+\left (\frac {21}{13} a^{2} b^{5} d^{7}+\frac {49}{13} a \,b^{6} c \,d^{6}+\frac {21}{13} b^{7} c^{2} d^{5}\right ) x^{13}+\left (\frac {1}{2} a \,b^{6} d^{7}+\frac {1}{2} b^{7} c \,d^{6}\right ) x^{14}+\frac {b^{7} d^{7} x^{15}}{15}\)	\(804\)
default	\(\frac {b^{7} d^{7} x^{15}}{15}+\frac {\left (7 a \,b^{6} d^{7}+7 b^{7} c \,d^{6}\right ) x^{14}}{14}+\frac {\left (21 a^{2} b^{5} d^{7}+49 a \,b^{6} c \,d^{6}+21 b^{7} c^{2} d^{5}\right ) x^{13}}{13}+\frac {\left (35 a^{3} b^{4} d^{7}+147 a^{2} b^{5} c \,d^{6}+147 a \,b^{6} c^{2} d^{5}+35 b^{7} c^{3} d^{4}\right ) x^{12}}{12}+\frac {\left (35 a^{4} b^{3} d^{7}+245 a^{3} b^{4} c \,d^{6}+441 a^{2} b^{5} c^{2} d^{5}+245 a \,b^{6} c^{3} d^{4}+35 b^{7} c^{4} d^{3}\right ) x^{11}}{11}+\frac {\left (21 a^{5} b^{2} d^{7}+245 a^{4} b^{3} c \,d^{6}+735 a^{3} b^{4} c^{2} d^{5}+735 a^{2} b^{5} c^{3} d^{4}+245 a \,b^{6} c^{4} d^{3}+21 b^{7} c^{5} d^{2}\right ) x^{10}}{10}+\frac {\left (7 a^{6} b \,d^{7}+147 a^{5} b^{2} c \,d^{6}+735 a^{4} b^{3} c^{2} d^{5}+1225 a^{3} b^{4} c^{3} d^{4}+735 a^{2} b^{5} c^{4} d^{3}+147 a \,b^{6} c^{5} d^{2}+7 b^{7} c^{6} d \right ) x^{9}}{9}+\frac {\left (a^{7} d^{7}+49 a^{6} b c \,d^{6}+441 a^{5} b^{2} c^{2} d^{5}+1225 a^{4} b^{3} c^{3} d^{4}+1225 a^{3} b^{4} c^{4} d^{3}+441 a^{2} b^{5} c^{5} d^{2}+49 a \,b^{6} c^{6} d +b^{7} c^{7}\right ) x^{8}}{8}+\frac {\left (7 a^{7} c \,d^{6}+147 a^{6} b \,c^{2} d^{5}+735 a^{5} b^{2} c^{3} d^{4}+1225 a^{4} b^{3} c^{4} d^{3}+735 a^{3} b^{4} c^{5} d^{2}+147 a^{2} b^{5} c^{6} d +7 a \,b^{6} c^{7}\right ) x^{7}}{7}+\frac {\left (21 a^{7} c^{2} d^{5}+245 a^{6} b \,c^{3} d^{4}+735 a^{5} b^{2} c^{4} d^{3}+735 a^{4} b^{3} c^{5} d^{2}+245 a^{3} b^{4} c^{6} d +21 a^{2} b^{5} c^{7}\right ) x^{6}}{6}+\frac {\left (35 a^{7} c^{3} d^{4}+245 a^{6} b \,c^{4} d^{3}+441 a^{5} b^{2} c^{5} d^{2}+245 a^{4} b^{3} c^{6} d +35 a^{3} b^{4} c^{7}\right ) x^{5}}{5}+\frac {\left (35 a^{7} c^{4} d^{3}+147 a^{6} b \,c^{5} d^{2}+147 a^{5} b^{2} c^{6} d +35 a^{4} b^{3} c^{7}\right ) x^{4}}{4}+\frac {\left (21 a^{7} c^{5} d^{2}+49 a^{6} b \,c^{6} d +21 a^{5} b^{2} c^{7}\right ) x^{3}}{3}+\frac {\left (7 a^{7} c^{6} d +7 a^{6} b \,c^{7}\right ) x^{2}}{2}+a^{7} c^{7} x\)	\(817\)
gosper	\(\frac {1}{2} x^{14} b^{7} c \,d^{6}+a^{7} c \,d^{6} x^{7}+a \,b^{6} c^{7} x^{7}+\frac {35}{12} x^{12} a^{3} b^{4} d^{7}+\frac {35}{12} x^{12} b^{7} c^{3} d^{4}+\frac {21}{13} x^{13} a^{2} b^{5} d^{7}+\frac {21}{13} x^{13} b^{7} c^{2} d^{5}+\frac {1}{2} x^{14} a \,b^{6} d^{7}+\frac {7}{2} x^{2} a^{7} c^{6} d +\frac {7}{2} x^{2} a^{6} b \,c^{7}+7 x^{3} a^{7} c^{5} d^{2}+7 x^{3} a^{5} b^{2} c^{7}+\frac {35}{4} x^{4} a^{7} c^{4} d^{3}+\frac {35}{4} x^{4} a^{4} b^{3} c^{7}+7 x^{5} a^{7} c^{3} d^{4}+7 x^{5} a^{3} b^{4} c^{7}+\frac {7}{2} x^{6} a^{7} c^{2} d^{5}+\frac {7}{2} x^{6} a^{2} b^{5} c^{7}+\frac {7}{9} x^{9} a^{6} b \,d^{7}+\frac {7}{9} x^{9} b^{7} c^{6} d +\frac {21}{10} x^{10} a^{5} b^{2} d^{7}+\frac {21}{10} x^{10} b^{7} c^{5} d^{2}+\frac {35}{11} x^{11} a^{4} b^{3} d^{7}+\frac {35}{11} x^{11} b^{7} c^{4} d^{3}+\frac {49}{8} x^{8} a \,b^{6} c^{6} d +\frac {49}{3} x^{9} a^{5} b^{2} c \,d^{6}+\frac {245}{3} x^{9} a^{4} b^{3} c^{2} d^{5}+\frac {1225}{9} x^{9} a^{3} b^{4} c^{3} d^{4}+\frac {245}{3} x^{9} a^{2} b^{5} c^{4} d^{3}+\frac {49}{3} x^{9} a \,b^{6} c^{5} d^{2}+\frac {49}{2} x^{10} a^{4} b^{3} c \,d^{6}+\frac {147}{2} x^{10} a^{3} b^{4} c^{2} d^{5}+\frac {147}{2} x^{10} a^{2} b^{5} c^{3} d^{4}+\frac {49}{2} x^{10} a \,b^{6} c^{4} d^{3}+\frac {245}{11} x^{11} a^{3} b^{4} c \,d^{6}+\frac {441}{11} x^{11} a^{2} b^{5} c^{2} d^{5}+\frac {245}{11} x^{11} a \,b^{6} c^{3} d^{4}+\frac {49}{4} x^{12} a^{2} b^{5} c \,d^{6}+a^{7} c^{7} x +\frac {1}{15} b^{7} d^{7} x^{15}+\frac {1}{8} x^{8} a^{7} d^{7}+\frac {1}{8} x^{8} b^{7} c^{7}+\frac {49}{4} x^{12} a \,b^{6} c^{2} d^{5}+\frac {49}{13} x^{13} a \,b^{6} c \,d^{6}+\frac {245}{2} x^{6} a^{5} b^{2} c^{4} d^{3}+\frac {245}{2} x^{6} a^{4} b^{3} c^{5} d^{2}+\frac {245}{6} x^{6} a^{3} b^{4} c^{6} d +\frac {49}{8} x^{8} a^{6} b c \,d^{6}+\frac {441}{8} x^{8} a^{5} b^{2} c^{2} d^{5}+\frac {1225}{8} x^{8} a^{4} b^{3} c^{3} d^{4}+\frac {1225}{8} x^{8} a^{3} b^{4} c^{4} d^{3}+\frac {441}{8} x^{8} a^{2} b^{5} c^{5} d^{2}+21 a^{6} b \,c^{2} d^{5} x^{7}+105 a^{5} b^{2} c^{3} d^{4} x^{7}+175 a^{4} b^{3} c^{4} d^{3} x^{7}+105 a^{3} b^{4} c^{5} d^{2} x^{7}+21 a^{2} b^{5} c^{6} d \,x^{7}+\frac {49}{3} x^{3} a^{6} b \,c^{6} d +\frac {147}{4} x^{4} a^{6} b \,c^{5} d^{2}+\frac {147}{4} x^{4} a^{5} b^{2} c^{6} d +49 x^{5} a^{6} b \,c^{4} d^{3}+\frac {441}{5} x^{5} a^{5} b^{2} c^{5} d^{2}+49 x^{5} a^{4} b^{3} c^{6} d +\frac {245}{6} x^{6} a^{6} b \,c^{3} d^{4}\)	\(925\)
risch	\(\frac {1}{2} x^{14} b^{7} c \,d^{6}+a^{7} c \,d^{6} x^{7}+a \,b^{6} c^{7} x^{7}+\frac {35}{12} x^{12} a^{3} b^{4} d^{7}+\frac {35}{12} x^{12} b^{7} c^{3} d^{4}+\frac {21}{13} x^{13} a^{2} b^{5} d^{7}+\frac {21}{13} x^{13} b^{7} c^{2} d^{5}+\frac {1}{2} x^{14} a \,b^{6} d^{7}+\frac {7}{2} x^{2} a^{7} c^{6} d +\frac {7}{2} x^{2} a^{6} b \,c^{7}+7 x^{3} a^{7} c^{5} d^{2}+7 x^{3} a^{5} b^{2} c^{7}+\frac {35}{4} x^{4} a^{7} c^{4} d^{3}+\frac {35}{4} x^{4} a^{4} b^{3} c^{7}+7 x^{5} a^{7} c^{3} d^{4}+7 x^{5} a^{3} b^{4} c^{7}+\frac {7}{2} x^{6} a^{7} c^{2} d^{5}+\frac {7}{2} x^{6} a^{2} b^{5} c^{7}+\frac {7}{9} x^{9} a^{6} b \,d^{7}+\frac {7}{9} x^{9} b^{7} c^{6} d +\frac {21}{10} x^{10} a^{5} b^{2} d^{7}+\frac {21}{10} x^{10} b^{7} c^{5} d^{2}+\frac {35}{11} x^{11} a^{4} b^{3} d^{7}+\frac {35}{11} x^{11} b^{7} c^{4} d^{3}+\frac {49}{8} x^{8} a \,b^{6} c^{6} d +\frac {49}{3} x^{9} a^{5} b^{2} c \,d^{6}+\frac {245}{3} x^{9} a^{4} b^{3} c^{2} d^{5}+\frac {1225}{9} x^{9} a^{3} b^{4} c^{3} d^{4}+\frac {245}{3} x^{9} a^{2} b^{5} c^{4} d^{3}+\frac {49}{3} x^{9} a \,b^{6} c^{5} d^{2}+\frac {49}{2} x^{10} a^{4} b^{3} c \,d^{6}+\frac {147}{2} x^{10} a^{3} b^{4} c^{2} d^{5}+\frac {147}{2} x^{10} a^{2} b^{5} c^{3} d^{4}+\frac {49}{2} x^{10} a \,b^{6} c^{4} d^{3}+\frac {245}{11} x^{11} a^{3} b^{4} c \,d^{6}+\frac {441}{11} x^{11} a^{2} b^{5} c^{2} d^{5}+\frac {245}{11} x^{11} a \,b^{6} c^{3} d^{4}+\frac {49}{4} x^{12} a^{2} b^{5} c \,d^{6}+a^{7} c^{7} x +\frac {1}{15} b^{7} d^{7} x^{15}+\frac {1}{8} x^{8} a^{7} d^{7}+\frac {1}{8} x^{8} b^{7} c^{7}+\frac {49}{4} x^{12} a \,b^{6} c^{2} d^{5}+\frac {49}{13} x^{13} a \,b^{6} c \,d^{6}+\frac {245}{2} x^{6} a^{5} b^{2} c^{4} d^{3}+\frac {245}{2} x^{6} a^{4} b^{3} c^{5} d^{2}+\frac {245}{6} x^{6} a^{3} b^{4} c^{6} d +\frac {49}{8} x^{8} a^{6} b c \,d^{6}+\frac {441}{8} x^{8} a^{5} b^{2} c^{2} d^{5}+\frac {1225}{8} x^{8} a^{4} b^{3} c^{3} d^{4}+\frac {1225}{8} x^{8} a^{3} b^{4} c^{4} d^{3}+\frac {441}{8} x^{8} a^{2} b^{5} c^{5} d^{2}+21 a^{6} b \,c^{2} d^{5} x^{7}+105 a^{5} b^{2} c^{3} d^{4} x^{7}+175 a^{4} b^{3} c^{4} d^{3} x^{7}+105 a^{3} b^{4} c^{5} d^{2} x^{7}+21 a^{2} b^{5} c^{6} d \,x^{7}+\frac {49}{3} x^{3} a^{6} b \,c^{6} d +\frac {147}{4} x^{4} a^{6} b \,c^{5} d^{2}+\frac {147}{4} x^{4} a^{5} b^{2} c^{6} d +49 x^{5} a^{6} b \,c^{4} d^{3}+\frac {441}{5} x^{5} a^{5} b^{2} c^{5} d^{2}+49 x^{5} a^{4} b^{3} c^{6} d +\frac {245}{6} x^{6} a^{6} b \,c^{3} d^{4}\)	\(925\)
parallelrisch	\(\frac {1}{2} x^{14} b^{7} c \,d^{6}+a^{7} c \,d^{6} x^{7}+a \,b^{6} c^{7} x^{7}+\frac {35}{12} x^{12} a^{3} b^{4} d^{7}+\frac {35}{12} x^{12} b^{7} c^{3} d^{4}+\frac {21}{13} x^{13} a^{2} b^{5} d^{7}+\frac {21}{13} x^{13} b^{7} c^{2} d^{5}+\frac {1}{2} x^{14} a \,b^{6} d^{7}+\frac {7}{2} x^{2} a^{7} c^{6} d +\frac {7}{2} x^{2} a^{6} b \,c^{7}+7 x^{3} a^{7} c^{5} d^{2}+7 x^{3} a^{5} b^{2} c^{7}+\frac {35}{4} x^{4} a^{7} c^{4} d^{3}+\frac {35}{4} x^{4} a^{4} b^{3} c^{7}+7 x^{5} a^{7} c^{3} d^{4}+7 x^{5} a^{3} b^{4} c^{7}+\frac {7}{2} x^{6} a^{7} c^{2} d^{5}+\frac {7}{2} x^{6} a^{2} b^{5} c^{7}+\frac {7}{9} x^{9} a^{6} b \,d^{7}+\frac {7}{9} x^{9} b^{7} c^{6} d +\frac {21}{10} x^{10} a^{5} b^{2} d^{7}+\frac {21}{10} x^{10} b^{7} c^{5} d^{2}+\frac {35}{11} x^{11} a^{4} b^{3} d^{7}+\frac {35}{11} x^{11} b^{7} c^{4} d^{3}+\frac {49}{8} x^{8} a \,b^{6} c^{6} d +\frac {49}{3} x^{9} a^{5} b^{2} c \,d^{6}+\frac {245}{3} x^{9} a^{4} b^{3} c^{2} d^{5}+\frac {1225}{9} x^{9} a^{3} b^{4} c^{3} d^{4}+\frac {245}{3} x^{9} a^{2} b^{5} c^{4} d^{3}+\frac {49}{3} x^{9} a \,b^{6} c^{5} d^{2}+\frac {49}{2} x^{10} a^{4} b^{3} c \,d^{6}+\frac {147}{2} x^{10} a^{3} b^{4} c^{2} d^{5}+\frac {147}{2} x^{10} a^{2} b^{5} c^{3} d^{4}+\frac {49}{2} x^{10} a \,b^{6} c^{4} d^{3}+\frac {245}{11} x^{11} a^{3} b^{4} c \,d^{6}+\frac {441}{11} x^{11} a^{2} b^{5} c^{2} d^{5}+\frac {245}{11} x^{11} a \,b^{6} c^{3} d^{4}+\frac {49}{4} x^{12} a^{2} b^{5} c \,d^{6}+a^{7} c^{7} x +\frac {1}{15} b^{7} d^{7} x^{15}+\frac {1}{8} x^{8} a^{7} d^{7}+\frac {1}{8} x^{8} b^{7} c^{7}+\frac {49}{4} x^{12} a \,b^{6} c^{2} d^{5}+\frac {49}{13} x^{13} a \,b^{6} c \,d^{6}+\frac {245}{2} x^{6} a^{5} b^{2} c^{4} d^{3}+\frac {245}{2} x^{6} a^{4} b^{3} c^{5} d^{2}+\frac {245}{6} x^{6} a^{3} b^{4} c^{6} d +\frac {49}{8} x^{8} a^{6} b c \,d^{6}+\frac {441}{8} x^{8} a^{5} b^{2} c^{2} d^{5}+\frac {1225}{8} x^{8} a^{4} b^{3} c^{3} d^{4}+\frac {1225}{8} x^{8} a^{3} b^{4} c^{4} d^{3}+\frac {441}{8} x^{8} a^{2} b^{5} c^{5} d^{2}+21 a^{6} b \,c^{2} d^{5} x^{7}+105 a^{5} b^{2} c^{3} d^{4} x^{7}+175 a^{4} b^{3} c^{4} d^{3} x^{7}+105 a^{3} b^{4} c^{5} d^{2} x^{7}+21 a^{2} b^{5} c^{6} d \,x^{7}+\frac {49}{3} x^{3} a^{6} b \,c^{6} d +\frac {147}{4} x^{4} a^{6} b \,c^{5} d^{2}+\frac {147}{4} x^{4} a^{5} b^{2} c^{6} d +49 x^{5} a^{6} b \,c^{4} d^{3}+\frac {441}{5} x^{5} a^{5} b^{2} c^{5} d^{2}+49 x^{5} a^{4} b^{3} c^{6} d +\frac {245}{6} x^{6} a^{6} b \,c^{3} d^{4}\)	\(925\)

[In]

int((b*x+a)^7*(d*x+c)^7,x,method=_RETURNVERBOSE)

[Out]

a^7*c^7*x+(7/2*a^7*c^6*d+7/2*a^6*b*c^7)*x^2+(7*a^7*c^5*d^2+49/3*a^6*b*c^6*d+7*a^5*b^2*c^7)*x^3+(35/4*a^7*c^4*d
^3+147/4*a^6*b*c^5*d^2+147/4*a^5*b^2*c^6*d+35/4*a^4*b^3*c^7)*x^4+(7*a^7*c^3*d^4+49*a^6*b*c^4*d^3+441/5*a^5*b^2
*c^5*d^2+49*a^4*b^3*c^6*d+7*a^3*b^4*c^7)*x^5+(7/2*a^7*c^2*d^5+245/6*a^6*b*c^3*d^4+245/2*a^5*b^2*c^4*d^3+245/2*
a^4*b^3*c^5*d^2+245/6*a^3*b^4*c^6*d+7/2*a^2*b^5*c^7)*x^6+(a^7*c*d^6+21*a^6*b*c^2*d^5+105*a^5*b^2*c^3*d^4+175*a
^4*b^3*c^4*d^3+105*a^3*b^4*c^5*d^2+21*a^2*b^5*c^6*d+a*b^6*c^7)*x^7+(1/8*a^7*d^7+49/8*a^6*b*c*d^6+441/8*a^5*b^2
*c^2*d^5+1225/8*a^4*b^3*c^3*d^4+1225/8*a^3*b^4*c^4*d^3+441/8*a^2*b^5*c^5*d^2+49/8*a*b^6*c^6*d+1/8*b^7*c^7)*x^8
+(7/9*a^6*b*d^7+49/3*a^5*b^2*c*d^6+245/3*a^4*b^3*c^2*d^5+1225/9*a^3*b^4*c^3*d^4+245/3*a^2*b^5*c^4*d^3+49/3*a*b
^6*c^5*d^2+7/9*b^7*c^6*d)*x^9+(21/10*a^5*b^2*d^7+49/2*a^4*b^3*c*d^6+147/2*a^3*b^4*c^2*d^5+147/2*a^2*b^5*c^3*d^
4+49/2*a*b^6*c^4*d^3+21/10*b^7*c^5*d^2)*x^10+(35/11*a^4*b^3*d^7+245/11*a^3*b^4*c*d^6+441/11*a^2*b^5*c^2*d^5+24
5/11*a*b^6*c^3*d^4+35/11*b^7*c^4*d^3)*x^11+(35/12*a^3*b^4*d^7+49/4*a^2*b^5*c*d^6+49/4*a*b^6*c^2*d^5+35/12*b^7*
c^3*d^4)*x^12+(21/13*a^2*b^5*d^7+49/13*a*b^6*c*d^6+21/13*b^7*c^2*d^5)*x^13+(1/2*a*b^6*d^7+1/2*b^7*c*d^6)*x^14+
1/15*b^7*d^7*x^15

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 807 vs. \(2 (184) = 368\).

Time = 0.23 (sec) , antiderivative size = 807, normalized size of antiderivative = 4.04 \[ \int (a+b x)^7 (c+d x)^7 \, dx=\frac {1}{15} \, b^{7} d^{7} x^{15} + a^{7} c^{7} x + \frac {1}{2} \, {\left (b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{14} + \frac {7}{13} \, {\left (3 \, b^{7} c^{2} d^{5} + 7 \, a b^{6} c d^{6} + 3 \, a^{2} b^{5} d^{7}\right )} x^{13} + \frac {7}{12} \, {\left (5 \, b^{7} c^{3} d^{4} + 21 \, a b^{6} c^{2} d^{5} + 21 \, a^{2} b^{5} c d^{6} + 5 \, a^{3} b^{4} d^{7}\right )} x^{12} + \frac {7}{11} \, {\left (5 \, b^{7} c^{4} d^{3} + 35 \, a b^{6} c^{3} d^{4} + 63 \, a^{2} b^{5} c^{2} d^{5} + 35 \, a^{3} b^{4} c d^{6} + 5 \, a^{4} b^{3} d^{7}\right )} x^{11} + \frac {7}{10} \, {\left (3 \, b^{7} c^{5} d^{2} + 35 \, a b^{6} c^{4} d^{3} + 105 \, a^{2} b^{5} c^{3} d^{4} + 105 \, a^{3} b^{4} c^{2} d^{5} + 35 \, a^{4} b^{3} c d^{6} + 3 \, a^{5} b^{2} d^{7}\right )} x^{10} + \frac {7}{9} \, {\left (b^{7} c^{6} d + 21 \, a b^{6} c^{5} d^{2} + 105 \, a^{2} b^{5} c^{4} d^{3} + 175 \, a^{3} b^{4} c^{3} d^{4} + 105 \, a^{4} b^{3} c^{2} d^{5} + 21 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x^{9} + \frac {1}{8} \, {\left (b^{7} c^{7} + 49 \, a b^{6} c^{6} d + 441 \, a^{2} b^{5} c^{5} d^{2} + 1225 \, a^{3} b^{4} c^{4} d^{3} + 1225 \, a^{4} b^{3} c^{3} d^{4} + 441 \, a^{5} b^{2} c^{2} d^{5} + 49 \, a^{6} b c d^{6} + a^{7} d^{7}\right )} x^{8} + {\left (a b^{6} c^{7} + 21 \, a^{2} b^{5} c^{6} d + 105 \, a^{3} b^{4} c^{5} d^{2} + 175 \, a^{4} b^{3} c^{4} d^{3} + 105 \, a^{5} b^{2} c^{3} d^{4} + 21 \, a^{6} b c^{2} d^{5} + a^{7} c d^{6}\right )} x^{7} + \frac {7}{6} \, {\left (3 \, a^{2} b^{5} c^{7} + 35 \, a^{3} b^{4} c^{6} d + 105 \, a^{4} b^{3} c^{5} d^{2} + 105 \, a^{5} b^{2} c^{4} d^{3} + 35 \, a^{6} b c^{3} d^{4} + 3 \, a^{7} c^{2} d^{5}\right )} x^{6} + \frac {7}{5} \, {\left (5 \, a^{3} b^{4} c^{7} + 35 \, a^{4} b^{3} c^{6} d + 63 \, a^{5} b^{2} c^{5} d^{2} + 35 \, a^{6} b c^{4} d^{3} + 5 \, a^{7} c^{3} d^{4}\right )} x^{5} + \frac {7}{4} \, {\left (5 \, a^{4} b^{3} c^{7} + 21 \, a^{5} b^{2} c^{6} d + 21 \, a^{6} b c^{5} d^{2} + 5 \, a^{7} c^{4} d^{3}\right )} x^{4} + \frac {7}{3} \, {\left (3 \, a^{5} b^{2} c^{7} + 7 \, a^{6} b c^{6} d + 3 \, a^{7} c^{5} d^{2}\right )} x^{3} + \frac {7}{2} \, {\left (a^{6} b c^{7} + a^{7} c^{6} d\right )} x^{2} \]

[In]

integrate((b*x+a)^7*(d*x+c)^7,x, algorithm="fricas")

[Out]

1/15*b^7*d^7*x^15 + a^7*c^7*x + 1/2*(b^7*c*d^6 + a*b^6*d^7)*x^14 + 7/13*(3*b^7*c^2*d^5 + 7*a*b^6*c*d^6 + 3*a^2
*b^5*d^7)*x^13 + 7/12*(5*b^7*c^3*d^4 + 21*a*b^6*c^2*d^5 + 21*a^2*b^5*c*d^6 + 5*a^3*b^4*d^7)*x^12 + 7/11*(5*b^7
*c^4*d^3 + 35*a*b^6*c^3*d^4 + 63*a^2*b^5*c^2*d^5 + 35*a^3*b^4*c*d^6 + 5*a^4*b^3*d^7)*x^11 + 7/10*(3*b^7*c^5*d^
2 + 35*a*b^6*c^4*d^3 + 105*a^2*b^5*c^3*d^4 + 105*a^3*b^4*c^2*d^5 + 35*a^4*b^3*c*d^6 + 3*a^5*b^2*d^7)*x^10 + 7/
9*(b^7*c^6*d + 21*a*b^6*c^5*d^2 + 105*a^2*b^5*c^4*d^3 + 175*a^3*b^4*c^3*d^4 + 105*a^4*b^3*c^2*d^5 + 21*a^5*b^2
*c*d^6 + a^6*b*d^7)*x^9 + 1/8*(b^7*c^7 + 49*a*b^6*c^6*d + 441*a^2*b^5*c^5*d^2 + 1225*a^3*b^4*c^4*d^3 + 1225*a^
4*b^3*c^3*d^4 + 441*a^5*b^2*c^2*d^5 + 49*a^6*b*c*d^6 + a^7*d^7)*x^8 + (a*b^6*c^7 + 21*a^2*b^5*c^6*d + 105*a^3*
b^4*c^5*d^2 + 175*a^4*b^3*c^4*d^3 + 105*a^5*b^2*c^3*d^4 + 21*a^6*b*c^2*d^5 + a^7*c*d^6)*x^7 + 7/6*(3*a^2*b^5*c
^7 + 35*a^3*b^4*c^6*d + 105*a^4*b^3*c^5*d^2 + 105*a^5*b^2*c^4*d^3 + 35*a^6*b*c^3*d^4 + 3*a^7*c^2*d^5)*x^6 + 7/
5*(5*a^3*b^4*c^7 + 35*a^4*b^3*c^6*d + 63*a^5*b^2*c^5*d^2 + 35*a^6*b*c^4*d^3 + 5*a^7*c^3*d^4)*x^5 + 7/4*(5*a^4*
b^3*c^7 + 21*a^5*b^2*c^6*d + 21*a^6*b*c^5*d^2 + 5*a^7*c^4*d^3)*x^4 + 7/3*(3*a^5*b^2*c^7 + 7*a^6*b*c^6*d + 3*a^
7*c^5*d^2)*x^3 + 7/2*(a^6*b*c^7 + a^7*c^6*d)*x^2

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 935 vs. \(2 (182) = 364\).

Time = 0.07 (sec) , antiderivative size = 935, normalized size of antiderivative = 4.68 \[ \int (a+b x)^7 (c+d x)^7 \, dx=a^{7} c^{7} x + \frac {b^{7} d^{7} x^{15}}{15} + x^{14} \left (\frac {a b^{6} d^{7}}{2} + \frac {b^{7} c d^{6}}{2}\right ) + x^{13} \cdot \left (\frac {21 a^{2} b^{5} d^{7}}{13} + \frac {49 a b^{6} c d^{6}}{13} + \frac {21 b^{7} c^{2} d^{5}}{13}\right ) + x^{12} \cdot \left (\frac {35 a^{3} b^{4} d^{7}}{12} + \frac {49 a^{2} b^{5} c d^{6}}{4} + \frac {49 a b^{6} c^{2} d^{5}}{4} + \frac {35 b^{7} c^{3} d^{4}}{12}\right ) + x^{11} \cdot \left (\frac {35 a^{4} b^{3} d^{7}}{11} + \frac {245 a^{3} b^{4} c d^{6}}{11} + \frac {441 a^{2} b^{5} c^{2} d^{5}}{11} + \frac {245 a b^{6} c^{3} d^{4}}{11} + \frac {35 b^{7} c^{4} d^{3}}{11}\right ) + x^{10} \cdot \left (\frac {21 a^{5} b^{2} d^{7}}{10} + \frac {49 a^{4} b^{3} c d^{6}}{2} + \frac {147 a^{3} b^{4} c^{2} d^{5}}{2} + \frac {147 a^{2} b^{5} c^{3} d^{4}}{2} + \frac {49 a b^{6} c^{4} d^{3}}{2} + \frac {21 b^{7} c^{5} d^{2}}{10}\right ) + x^{9} \cdot \left (\frac {7 a^{6} b d^{7}}{9} + \frac {49 a^{5} b^{2} c d^{6}}{3} + \frac {245 a^{4} b^{3} c^{2} d^{5}}{3} + \frac {1225 a^{3} b^{4} c^{3} d^{4}}{9} + \frac {245 a^{2} b^{5} c^{4} d^{3}}{3} + \frac {49 a b^{6} c^{5} d^{2}}{3} + \frac {7 b^{7} c^{6} d}{9}\right ) + x^{8} \left (\frac {a^{7} d^{7}}{8} + \frac {49 a^{6} b c d^{6}}{8} + \frac {441 a^{5} b^{2} c^{2} d^{5}}{8} + \frac {1225 a^{4} b^{3} c^{3} d^{4}}{8} + \frac {1225 a^{3} b^{4} c^{4} d^{3}}{8} + \frac {441 a^{2} b^{5} c^{5} d^{2}}{8} + \frac {49 a b^{6} c^{6} d}{8} + \frac {b^{7} c^{7}}{8}\right ) + x^{7} \left (a^{7} c d^{6} + 21 a^{6} b c^{2} d^{5} + 105 a^{5} b^{2} c^{3} d^{4} + 175 a^{4} b^{3} c^{4} d^{3} + 105 a^{3} b^{4} c^{5} d^{2} + 21 a^{2} b^{5} c^{6} d + a b^{6} c^{7}\right ) + x^{6} \cdot \left (\frac {7 a^{7} c^{2} d^{5}}{2} + \frac {245 a^{6} b c^{3} d^{4}}{6} + \frac {245 a^{5} b^{2} c^{4} d^{3}}{2} + \frac {245 a^{4} b^{3} c^{5} d^{2}}{2} + \frac {245 a^{3} b^{4} c^{6} d}{6} + \frac {7 a^{2} b^{5} c^{7}}{2}\right ) + x^{5} \cdot \left (7 a^{7} c^{3} d^{4} + 49 a^{6} b c^{4} d^{3} + \frac {441 a^{5} b^{2} c^{5} d^{2}}{5} + 49 a^{4} b^{3} c^{6} d + 7 a^{3} b^{4} c^{7}\right ) + x^{4} \cdot \left (\frac {35 a^{7} c^{4} d^{3}}{4} + \frac {147 a^{6} b c^{5} d^{2}}{4} + \frac {147 a^{5} b^{2} c^{6} d}{4} + \frac {35 a^{4} b^{3} c^{7}}{4}\right ) + x^{3} \cdot \left (7 a^{7} c^{5} d^{2} + \frac {49 a^{6} b c^{6} d}{3} + 7 a^{5} b^{2} c^{7}\right ) + x^{2} \cdot \left (\frac {7 a^{7} c^{6} d}{2} + \frac {7 a^{6} b c^{7}}{2}\right ) \]

[In]

integrate((b*x+a)**7*(d*x+c)**7,x)

[Out]

a**7*c**7*x + b**7*d**7*x**15/15 + x**14*(a*b**6*d**7/2 + b**7*c*d**6/2) + x**13*(21*a**2*b**5*d**7/13 + 49*a*
b**6*c*d**6/13 + 21*b**7*c**2*d**5/13) + x**12*(35*a**3*b**4*d**7/12 + 49*a**2*b**5*c*d**6/4 + 49*a*b**6*c**2*
d**5/4 + 35*b**7*c**3*d**4/12) + x**11*(35*a**4*b**3*d**7/11 + 245*a**3*b**4*c*d**6/11 + 441*a**2*b**5*c**2*d*
*5/11 + 245*a*b**6*c**3*d**4/11 + 35*b**7*c**4*d**3/11) + x**10*(21*a**5*b**2*d**7/10 + 49*a**4*b**3*c*d**6/2
+ 147*a**3*b**4*c**2*d**5/2 + 147*a**2*b**5*c**3*d**4/2 + 49*a*b**6*c**4*d**3/2 + 21*b**7*c**5*d**2/10) + x**9
*(7*a**6*b*d**7/9 + 49*a**5*b**2*c*d**6/3 + 245*a**4*b**3*c**2*d**5/3 + 1225*a**3*b**4*c**3*d**4/9 + 245*a**2*
b**5*c**4*d**3/3 + 49*a*b**6*c**5*d**2/3 + 7*b**7*c**6*d/9) + x**8*(a**7*d**7/8 + 49*a**6*b*c*d**6/8 + 441*a**
5*b**2*c**2*d**5/8 + 1225*a**4*b**3*c**3*d**4/8 + 1225*a**3*b**4*c**4*d**3/8 + 441*a**2*b**5*c**5*d**2/8 + 49*
a*b**6*c**6*d/8 + b**7*c**7/8) + x**7*(a**7*c*d**6 + 21*a**6*b*c**2*d**5 + 105*a**5*b**2*c**3*d**4 + 175*a**4*
b**3*c**4*d**3 + 105*a**3*b**4*c**5*d**2 + 21*a**2*b**5*c**6*d + a*b**6*c**7) + x**6*(7*a**7*c**2*d**5/2 + 245
*a**6*b*c**3*d**4/6 + 245*a**5*b**2*c**4*d**3/2 + 245*a**4*b**3*c**5*d**2/2 + 245*a**3*b**4*c**6*d/6 + 7*a**2*
b**5*c**7/2) + x**5*(7*a**7*c**3*d**4 + 49*a**6*b*c**4*d**3 + 441*a**5*b**2*c**5*d**2/5 + 49*a**4*b**3*c**6*d
+ 7*a**3*b**4*c**7) + x**4*(35*a**7*c**4*d**3/4 + 147*a**6*b*c**5*d**2/4 + 147*a**5*b**2*c**6*d/4 + 35*a**4*b*
*3*c**7/4) + x**3*(7*a**7*c**5*d**2 + 49*a**6*b*c**6*d/3 + 7*a**5*b**2*c**7) + x**2*(7*a**7*c**6*d/2 + 7*a**6*
b*c**7/2)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 807 vs. \(2 (184) = 368\).

Time = 0.22 (sec) , antiderivative size = 807, normalized size of antiderivative = 4.04 \[ \int (a+b x)^7 (c+d x)^7 \, dx=\frac {1}{15} \, b^{7} d^{7} x^{15} + a^{7} c^{7} x + \frac {1}{2} \, {\left (b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{14} + \frac {7}{13} \, {\left (3 \, b^{7} c^{2} d^{5} + 7 \, a b^{6} c d^{6} + 3 \, a^{2} b^{5} d^{7}\right )} x^{13} + \frac {7}{12} \, {\left (5 \, b^{7} c^{3} d^{4} + 21 \, a b^{6} c^{2} d^{5} + 21 \, a^{2} b^{5} c d^{6} + 5 \, a^{3} b^{4} d^{7}\right )} x^{12} + \frac {7}{11} \, {\left (5 \, b^{7} c^{4} d^{3} + 35 \, a b^{6} c^{3} d^{4} + 63 \, a^{2} b^{5} c^{2} d^{5} + 35 \, a^{3} b^{4} c d^{6} + 5 \, a^{4} b^{3} d^{7}\right )} x^{11} + \frac {7}{10} \, {\left (3 \, b^{7} c^{5} d^{2} + 35 \, a b^{6} c^{4} d^{3} + 105 \, a^{2} b^{5} c^{3} d^{4} + 105 \, a^{3} b^{4} c^{2} d^{5} + 35 \, a^{4} b^{3} c d^{6} + 3 \, a^{5} b^{2} d^{7}\right )} x^{10} + \frac {7}{9} \, {\left (b^{7} c^{6} d + 21 \, a b^{6} c^{5} d^{2} + 105 \, a^{2} b^{5} c^{4} d^{3} + 175 \, a^{3} b^{4} c^{3} d^{4} + 105 \, a^{4} b^{3} c^{2} d^{5} + 21 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x^{9} + \frac {1}{8} \, {\left (b^{7} c^{7} + 49 \, a b^{6} c^{6} d + 441 \, a^{2} b^{5} c^{5} d^{2} + 1225 \, a^{3} b^{4} c^{4} d^{3} + 1225 \, a^{4} b^{3} c^{3} d^{4} + 441 \, a^{5} b^{2} c^{2} d^{5} + 49 \, a^{6} b c d^{6} + a^{7} d^{7}\right )} x^{8} + {\left (a b^{6} c^{7} + 21 \, a^{2} b^{5} c^{6} d + 105 \, a^{3} b^{4} c^{5} d^{2} + 175 \, a^{4} b^{3} c^{4} d^{3} + 105 \, a^{5} b^{2} c^{3} d^{4} + 21 \, a^{6} b c^{2} d^{5} + a^{7} c d^{6}\right )} x^{7} + \frac {7}{6} \, {\left (3 \, a^{2} b^{5} c^{7} + 35 \, a^{3} b^{4} c^{6} d + 105 \, a^{4} b^{3} c^{5} d^{2} + 105 \, a^{5} b^{2} c^{4} d^{3} + 35 \, a^{6} b c^{3} d^{4} + 3 \, a^{7} c^{2} d^{5}\right )} x^{6} + \frac {7}{5} \, {\left (5 \, a^{3} b^{4} c^{7} + 35 \, a^{4} b^{3} c^{6} d + 63 \, a^{5} b^{2} c^{5} d^{2} + 35 \, a^{6} b c^{4} d^{3} + 5 \, a^{7} c^{3} d^{4}\right )} x^{5} + \frac {7}{4} \, {\left (5 \, a^{4} b^{3} c^{7} + 21 \, a^{5} b^{2} c^{6} d + 21 \, a^{6} b c^{5} d^{2} + 5 \, a^{7} c^{4} d^{3}\right )} x^{4} + \frac {7}{3} \, {\left (3 \, a^{5} b^{2} c^{7} + 7 \, a^{6} b c^{6} d + 3 \, a^{7} c^{5} d^{2}\right )} x^{3} + \frac {7}{2} \, {\left (a^{6} b c^{7} + a^{7} c^{6} d\right )} x^{2} \]

[In]

integrate((b*x+a)^7*(d*x+c)^7,x, algorithm="maxima")

[Out]

1/15*b^7*d^7*x^15 + a^7*c^7*x + 1/2*(b^7*c*d^6 + a*b^6*d^7)*x^14 + 7/13*(3*b^7*c^2*d^5 + 7*a*b^6*c*d^6 + 3*a^2
*b^5*d^7)*x^13 + 7/12*(5*b^7*c^3*d^4 + 21*a*b^6*c^2*d^5 + 21*a^2*b^5*c*d^6 + 5*a^3*b^4*d^7)*x^12 + 7/11*(5*b^7
*c^4*d^3 + 35*a*b^6*c^3*d^4 + 63*a^2*b^5*c^2*d^5 + 35*a^3*b^4*c*d^6 + 5*a^4*b^3*d^7)*x^11 + 7/10*(3*b^7*c^5*d^
2 + 35*a*b^6*c^4*d^3 + 105*a^2*b^5*c^3*d^4 + 105*a^3*b^4*c^2*d^5 + 35*a^4*b^3*c*d^6 + 3*a^5*b^2*d^7)*x^10 + 7/
9*(b^7*c^6*d + 21*a*b^6*c^5*d^2 + 105*a^2*b^5*c^4*d^3 + 175*a^3*b^4*c^3*d^4 + 105*a^4*b^3*c^2*d^5 + 21*a^5*b^2
*c*d^6 + a^6*b*d^7)*x^9 + 1/8*(b^7*c^7 + 49*a*b^6*c^6*d + 441*a^2*b^5*c^5*d^2 + 1225*a^3*b^4*c^4*d^3 + 1225*a^
4*b^3*c^3*d^4 + 441*a^5*b^2*c^2*d^5 + 49*a^6*b*c*d^6 + a^7*d^7)*x^8 + (a*b^6*c^7 + 21*a^2*b^5*c^6*d + 105*a^3*
b^4*c^5*d^2 + 175*a^4*b^3*c^4*d^3 + 105*a^5*b^2*c^3*d^4 + 21*a^6*b*c^2*d^5 + a^7*c*d^6)*x^7 + 7/6*(3*a^2*b^5*c
^7 + 35*a^3*b^4*c^6*d + 105*a^4*b^3*c^5*d^2 + 105*a^5*b^2*c^4*d^3 + 35*a^6*b*c^3*d^4 + 3*a^7*c^2*d^5)*x^6 + 7/
5*(5*a^3*b^4*c^7 + 35*a^4*b^3*c^6*d + 63*a^5*b^2*c^5*d^2 + 35*a^6*b*c^4*d^3 + 5*a^7*c^3*d^4)*x^5 + 7/4*(5*a^4*
b^3*c^7 + 21*a^5*b^2*c^6*d + 21*a^6*b*c^5*d^2 + 5*a^7*c^4*d^3)*x^4 + 7/3*(3*a^5*b^2*c^7 + 7*a^6*b*c^6*d + 3*a^
7*c^5*d^2)*x^3 + 7/2*(a^6*b*c^7 + a^7*c^6*d)*x^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 924 vs. \(2 (184) = 368\).

Time = 0.31 (sec) , antiderivative size = 924, normalized size of antiderivative = 4.62 \[ \int (a+b x)^7 (c+d x)^7 \, dx=\frac {1}{15} \, b^{7} d^{7} x^{15} + \frac {1}{2} \, b^{7} c d^{6} x^{14} + \frac {1}{2} \, a b^{6} d^{7} x^{14} + \frac {21}{13} \, b^{7} c^{2} d^{5} x^{13} + \frac {49}{13} \, a b^{6} c d^{6} x^{13} + \frac {21}{13} \, a^{2} b^{5} d^{7} x^{13} + \frac {35}{12} \, b^{7} c^{3} d^{4} x^{12} + \frac {49}{4} \, a b^{6} c^{2} d^{5} x^{12} + \frac {49}{4} \, a^{2} b^{5} c d^{6} x^{12} + \frac {35}{12} \, a^{3} b^{4} d^{7} x^{12} + \frac {35}{11} \, b^{7} c^{4} d^{3} x^{11} + \frac {245}{11} \, a b^{6} c^{3} d^{4} x^{11} + \frac {441}{11} \, a^{2} b^{5} c^{2} d^{5} x^{11} + \frac {245}{11} \, a^{3} b^{4} c d^{6} x^{11} + \frac {35}{11} \, a^{4} b^{3} d^{7} x^{11} + \frac {21}{10} \, b^{7} c^{5} d^{2} x^{10} + \frac {49}{2} \, a b^{6} c^{4} d^{3} x^{10} + \frac {147}{2} \, a^{2} b^{5} c^{3} d^{4} x^{10} + \frac {147}{2} \, a^{3} b^{4} c^{2} d^{5} x^{10} + \frac {49}{2} \, a^{4} b^{3} c d^{6} x^{10} + \frac {21}{10} \, a^{5} b^{2} d^{7} x^{10} + \frac {7}{9} \, b^{7} c^{6} d x^{9} + \frac {49}{3} \, a b^{6} c^{5} d^{2} x^{9} + \frac {245}{3} \, a^{2} b^{5} c^{4} d^{3} x^{9} + \frac {1225}{9} \, a^{3} b^{4} c^{3} d^{4} x^{9} + \frac {245}{3} \, a^{4} b^{3} c^{2} d^{5} x^{9} + \frac {49}{3} \, a^{5} b^{2} c d^{6} x^{9} + \frac {7}{9} \, a^{6} b d^{7} x^{9} + \frac {1}{8} \, b^{7} c^{7} x^{8} + \frac {49}{8} \, a b^{6} c^{6} d x^{8} + \frac {441}{8} \, a^{2} b^{5} c^{5} d^{2} x^{8} + \frac {1225}{8} \, a^{3} b^{4} c^{4} d^{3} x^{8} + \frac {1225}{8} \, a^{4} b^{3} c^{3} d^{4} x^{8} + \frac {441}{8} \, a^{5} b^{2} c^{2} d^{5} x^{8} + \frac {49}{8} \, a^{6} b c d^{6} x^{8} + \frac {1}{8} \, a^{7} d^{7} x^{8} + a b^{6} c^{7} x^{7} + 21 \, a^{2} b^{5} c^{6} d x^{7} + 105 \, a^{3} b^{4} c^{5} d^{2} x^{7} + 175 \, a^{4} b^{3} c^{4} d^{3} x^{7} + 105 \, a^{5} b^{2} c^{3} d^{4} x^{7} + 21 \, a^{6} b c^{2} d^{5} x^{7} + a^{7} c d^{6} x^{7} + \frac {7}{2} \, a^{2} b^{5} c^{7} x^{6} + \frac {245}{6} \, a^{3} b^{4} c^{6} d x^{6} + \frac {245}{2} \, a^{4} b^{3} c^{5} d^{2} x^{6} + \frac {245}{2} \, a^{5} b^{2} c^{4} d^{3} x^{6} + \frac {245}{6} \, a^{6} b c^{3} d^{4} x^{6} + \frac {7}{2} \, a^{7} c^{2} d^{5} x^{6} + 7 \, a^{3} b^{4} c^{7} x^{5} + 49 \, a^{4} b^{3} c^{6} d x^{5} + \frac {441}{5} \, a^{5} b^{2} c^{5} d^{2} x^{5} + 49 \, a^{6} b c^{4} d^{3} x^{5} + 7 \, a^{7} c^{3} d^{4} x^{5} + \frac {35}{4} \, a^{4} b^{3} c^{7} x^{4} + \frac {147}{4} \, a^{5} b^{2} c^{6} d x^{4} + \frac {147}{4} \, a^{6} b c^{5} d^{2} x^{4} + \frac {35}{4} \, a^{7} c^{4} d^{3} x^{4} + 7 \, a^{5} b^{2} c^{7} x^{3} + \frac {49}{3} \, a^{6} b c^{6} d x^{3} + 7 \, a^{7} c^{5} d^{2} x^{3} + \frac {7}{2} \, a^{6} b c^{7} x^{2} + \frac {7}{2} \, a^{7} c^{6} d x^{2} + a^{7} c^{7} x \]

[In]

integrate((b*x+a)^7*(d*x+c)^7,x, algorithm="giac")

[Out]

1/15*b^7*d^7*x^15 + 1/2*b^7*c*d^6*x^14 + 1/2*a*b^6*d^7*x^14 + 21/13*b^7*c^2*d^5*x^13 + 49/13*a*b^6*c*d^6*x^13
+ 21/13*a^2*b^5*d^7*x^13 + 35/12*b^7*c^3*d^4*x^12 + 49/4*a*b^6*c^2*d^5*x^12 + 49/4*a^2*b^5*c*d^6*x^12 + 35/12*
a^3*b^4*d^7*x^12 + 35/11*b^7*c^4*d^3*x^11 + 245/11*a*b^6*c^3*d^4*x^11 + 441/11*a^2*b^5*c^2*d^5*x^11 + 245/11*a
^3*b^4*c*d^6*x^11 + 35/11*a^4*b^3*d^7*x^11 + 21/10*b^7*c^5*d^2*x^10 + 49/2*a*b^6*c^4*d^3*x^10 + 147/2*a^2*b^5*
c^3*d^4*x^10 + 147/2*a^3*b^4*c^2*d^5*x^10 + 49/2*a^4*b^3*c*d^6*x^10 + 21/10*a^5*b^2*d^7*x^10 + 7/9*b^7*c^6*d*x
^9 + 49/3*a*b^6*c^5*d^2*x^9 + 245/3*a^2*b^5*c^4*d^3*x^9 + 1225/9*a^3*b^4*c^3*d^4*x^9 + 245/3*a^4*b^3*c^2*d^5*x
^9 + 49/3*a^5*b^2*c*d^6*x^9 + 7/9*a^6*b*d^7*x^9 + 1/8*b^7*c^7*x^8 + 49/8*a*b^6*c^6*d*x^8 + 441/8*a^2*b^5*c^5*d
^2*x^8 + 1225/8*a^3*b^4*c^4*d^3*x^8 + 1225/8*a^4*b^3*c^3*d^4*x^8 + 441/8*a^5*b^2*c^2*d^5*x^8 + 49/8*a^6*b*c*d^
6*x^8 + 1/8*a^7*d^7*x^8 + a*b^6*c^7*x^7 + 21*a^2*b^5*c^6*d*x^7 + 105*a^3*b^4*c^5*d^2*x^7 + 175*a^4*b^3*c^4*d^3
*x^7 + 105*a^5*b^2*c^3*d^4*x^7 + 21*a^6*b*c^2*d^5*x^7 + a^7*c*d^6*x^7 + 7/2*a^2*b^5*c^7*x^6 + 245/6*a^3*b^4*c^
6*d*x^6 + 245/2*a^4*b^3*c^5*d^2*x^6 + 245/2*a^5*b^2*c^4*d^3*x^6 + 245/6*a^6*b*c^3*d^4*x^6 + 7/2*a^7*c^2*d^5*x^
6 + 7*a^3*b^4*c^7*x^5 + 49*a^4*b^3*c^6*d*x^5 + 441/5*a^5*b^2*c^5*d^2*x^5 + 49*a^6*b*c^4*d^3*x^5 + 7*a^7*c^3*d^
4*x^5 + 35/4*a^4*b^3*c^7*x^4 + 147/4*a^5*b^2*c^6*d*x^4 + 147/4*a^6*b*c^5*d^2*x^4 + 35/4*a^7*c^4*d^3*x^4 + 7*a^
5*b^2*c^7*x^3 + 49/3*a^6*b*c^6*d*x^3 + 7*a^7*c^5*d^2*x^3 + 7/2*a^6*b*c^7*x^2 + 7/2*a^7*c^6*d*x^2 + a^7*c^7*x

Mupad [B] (verification not implemented)

Time = 0.62 (sec) , antiderivative size = 781, normalized size of antiderivative = 3.90 \[ \int (a+b x)^7 (c+d x)^7 \, dx=x^8\,\left (\frac {a^7\,d^7}{8}+\frac {49\,a^6\,b\,c\,d^6}{8}+\frac {441\,a^5\,b^2\,c^2\,d^5}{8}+\frac {1225\,a^4\,b^3\,c^3\,d^4}{8}+\frac {1225\,a^3\,b^4\,c^4\,d^3}{8}+\frac {441\,a^2\,b^5\,c^5\,d^2}{8}+\frac {49\,a\,b^6\,c^6\,d}{8}+\frac {b^7\,c^7}{8}\right )+x^5\,\left (7\,a^7\,c^3\,d^4+49\,a^6\,b\,c^4\,d^3+\frac {441\,a^5\,b^2\,c^5\,d^2}{5}+49\,a^4\,b^3\,c^6\,d+7\,a^3\,b^4\,c^7\right )+x^{11}\,\left (\frac {35\,a^4\,b^3\,d^7}{11}+\frac {245\,a^3\,b^4\,c\,d^6}{11}+\frac {441\,a^2\,b^5\,c^2\,d^5}{11}+\frac {245\,a\,b^6\,c^3\,d^4}{11}+\frac {35\,b^7\,c^4\,d^3}{11}\right )+x^7\,\left (a^7\,c\,d^6+21\,a^6\,b\,c^2\,d^5+105\,a^5\,b^2\,c^3\,d^4+175\,a^4\,b^3\,c^4\,d^3+105\,a^3\,b^4\,c^5\,d^2+21\,a^2\,b^5\,c^6\,d+a\,b^6\,c^7\right )+x^9\,\left (\frac {7\,a^6\,b\,d^7}{9}+\frac {49\,a^5\,b^2\,c\,d^6}{3}+\frac {245\,a^4\,b^3\,c^2\,d^5}{3}+\frac {1225\,a^3\,b^4\,c^3\,d^4}{9}+\frac {245\,a^2\,b^5\,c^4\,d^3}{3}+\frac {49\,a\,b^6\,c^5\,d^2}{3}+\frac {7\,b^7\,c^6\,d}{9}\right )+x^6\,\left (\frac {7\,a^7\,c^2\,d^5}{2}+\frac {245\,a^6\,b\,c^3\,d^4}{6}+\frac {245\,a^5\,b^2\,c^4\,d^3}{2}+\frac {245\,a^4\,b^3\,c^5\,d^2}{2}+\frac {245\,a^3\,b^4\,c^6\,d}{6}+\frac {7\,a^2\,b^5\,c^7}{2}\right )+x^{10}\,\left (\frac {21\,a^5\,b^2\,d^7}{10}+\frac {49\,a^4\,b^3\,c\,d^6}{2}+\frac {147\,a^3\,b^4\,c^2\,d^5}{2}+\frac {147\,a^2\,b^5\,c^3\,d^4}{2}+\frac {49\,a\,b^6\,c^4\,d^3}{2}+\frac {21\,b^7\,c^5\,d^2}{10}\right )+a^7\,c^7\,x+\frac {b^7\,d^7\,x^{15}}{15}+\frac {7\,a^4\,c^4\,x^4\,\left (5\,a^3\,d^3+21\,a^2\,b\,c\,d^2+21\,a\,b^2\,c^2\,d+5\,b^3\,c^3\right )}{4}+\frac {7\,b^4\,d^4\,x^{12}\,\left (5\,a^3\,d^3+21\,a^2\,b\,c\,d^2+21\,a\,b^2\,c^2\,d+5\,b^3\,c^3\right )}{12}+\frac {7\,a^6\,c^6\,x^2\,\left (a\,d+b\,c\right )}{2}+\frac {b^6\,d^6\,x^{14}\,\left (a\,d+b\,c\right )}{2}+\frac {7\,a^5\,c^5\,x^3\,\left (3\,a^2\,d^2+7\,a\,b\,c\,d+3\,b^2\,c^2\right )}{3}+\frac {7\,b^5\,d^5\,x^{13}\,\left (3\,a^2\,d^2+7\,a\,b\,c\,d+3\,b^2\,c^2\right )}{13} \]

[In]

int((a + b*x)^7*(c + d*x)^7,x)

[Out]

x^8*((a^7*d^7)/8 + (b^7*c^7)/8 + (441*a^2*b^5*c^5*d^2)/8 + (1225*a^3*b^4*c^4*d^3)/8 + (1225*a^4*b^3*c^3*d^4)/8
+ (441*a^5*b^2*c^2*d^5)/8 + (49*a*b^6*c^6*d)/8 + (49*a^6*b*c*d^6)/8) + x^5*(7*a^3*b^4*c^7 + 7*a^7*c^3*d^4 + 4
9*a^4*b^3*c^6*d + 49*a^6*b*c^4*d^3 + (441*a^5*b^2*c^5*d^2)/5) + x^11*((35*a^4*b^3*d^7)/11 + (35*b^7*c^4*d^3)/1
1 + (245*a*b^6*c^3*d^4)/11 + (245*a^3*b^4*c*d^6)/11 + (441*a^2*b^5*c^2*d^5)/11) + x^7*(a*b^6*c^7 + a^7*c*d^6 +
21*a^2*b^5*c^6*d + 21*a^6*b*c^2*d^5 + 105*a^3*b^4*c^5*d^2 + 175*a^4*b^3*c^4*d^3 + 105*a^5*b^2*c^3*d^4) + x^9*
((7*a^6*b*d^7)/9 + (7*b^7*c^6*d)/9 + (49*a*b^6*c^5*d^2)/3 + (49*a^5*b^2*c*d^6)/3 + (245*a^2*b^5*c^4*d^3)/3 + (
1225*a^3*b^4*c^3*d^4)/9 + (245*a^4*b^3*c^2*d^5)/3) + x^6*((7*a^2*b^5*c^7)/2 + (7*a^7*c^2*d^5)/2 + (245*a^3*b^4
*c^6*d)/6 + (245*a^6*b*c^3*d^4)/6 + (245*a^4*b^3*c^5*d^2)/2 + (245*a^5*b^2*c^4*d^3)/2) + x^10*((21*a^5*b^2*d^7
)/10 + (21*b^7*c^5*d^2)/10 + (49*a*b^6*c^4*d^3)/2 + (49*a^4*b^3*c*d^6)/2 + (147*a^2*b^5*c^3*d^4)/2 + (147*a^3*
b^4*c^2*d^5)/2) + a^7*c^7*x + (b^7*d^7*x^15)/15 + (7*a^4*c^4*x^4*(5*a^3*d^3 + 5*b^3*c^3 + 21*a*b^2*c^2*d + 21*
a^2*b*c*d^2))/4 + (7*b^4*d^4*x^12*(5*a^3*d^3 + 5*b^3*c^3 + 21*a*b^2*c^2*d + 21*a^2*b*c*d^2))/12 + (7*a^6*c^6*x
^2*(a*d + b*c))/2 + (b^6*d^6*x^14*(a*d + b*c))/2 + (7*a^5*c^5*x^3*(3*a^2*d^2 + 3*b^2*c^2 + 7*a*b*c*d))/3 + (7*
b^5*d^5*x^13*(3*a^2*d^2 + 3*b^2*c^2 + 7*a*b*c*d))/13