3.12.68 \(\int (a+b x)^8 (c+d x)^7 \, dx\)

Optimal. Leaf size=200 \[ \frac {7 d^6 (a+b x)^{15} (b c-a d)}{15 b^8}+\frac {3 d^5 (a+b x)^{14} (b c-a d)^2}{2 b^8}+\frac {35 d^4 (a+b x)^{13} (b c-a d)^3}{13 b^8}+\frac {35 d^3 (a+b x)^{12} (b c-a d)^4}{12 b^8}+\frac {21 d^2 (a+b x)^{11} (b c-a d)^5}{11 b^8}+\frac {7 d (a+b x)^{10} (b c-a d)^6}{10 b^8}+\frac {(a+b x)^9 (b c-a d)^7}{9 b^8}+\frac {d^7 (a+b x)^{16}}{16 b^8} \]

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Rubi [A]  time = 0.57, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {43} \begin {gather*} \frac {7 d^6 (a+b x)^{15} (b c-a d)}{15 b^8}+\frac {3 d^5 (a+b x)^{14} (b c-a d)^2}{2 b^8}+\frac {35 d^4 (a+b x)^{13} (b c-a d)^3}{13 b^8}+\frac {35 d^3 (a+b x)^{12} (b c-a d)^4}{12 b^8}+\frac {21 d^2 (a+b x)^{11} (b c-a d)^5}{11 b^8}+\frac {7 d (a+b x)^{10} (b c-a d)^6}{10 b^8}+\frac {(a+b x)^9 (b c-a d)^7}{9 b^8}+\frac {d^7 (a+b x)^{16}}{16 b^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^8*(c + d*x)^7,x]

[Out]

((b*c - a*d)^7*(a + b*x)^9)/(9*b^8) + (7*d*(b*c - a*d)^6*(a + b*x)^10)/(10*b^8) + (21*d^2*(b*c - a*d)^5*(a + b
*x)^11)/(11*b^8) + (35*d^3*(b*c - a*d)^4*(a + b*x)^12)/(12*b^8) + (35*d^4*(b*c - a*d)^3*(a + b*x)^13)/(13*b^8)
+ (3*d^5*(b*c - a*d)^2*(a + b*x)^14)/(2*b^8) + (7*d^6*(b*c - a*d)*(a + b*x)^15)/(15*b^8) + (d^7*(a + b*x)^16)
/(16*b^8)

Rule 43

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*} \int (a+b x)^8 (c+d x)^7 \, dx &=\int \left (\frac {(b c-a d)^7 (a+b x)^8}{b^7}+\frac {7 d (b c-a d)^6 (a+b x)^9}{b^7}+\frac {21 d^2 (b c-a d)^5 (a+b x)^{10}}{b^7}+\frac {35 d^3 (b c-a d)^4 (a+b x)^{11}}{b^7}+\frac {35 d^4 (b c-a d)^3 (a+b x)^{12}}{b^7}+\frac {21 d^5 (b c-a d)^2 (a+b x)^{13}}{b^7}+\frac {7 d^6 (b c-a d) (a+b x)^{14}}{b^7}+\frac {d^7 (a+b x)^{15}}{b^7}\right ) \, dx\\ &=\frac {(b c-a d)^7 (a+b x)^9}{9 b^8}+\frac {7 d (b c-a d)^6 (a+b x)^{10}}{10 b^8}+\frac {21 d^2 (b c-a d)^5 (a+b x)^{11}}{11 b^8}+\frac {35 d^3 (b c-a d)^4 (a+b x)^{12}}{12 b^8}+\frac {35 d^4 (b c-a d)^3 (a+b x)^{13}}{13 b^8}+\frac {3 d^5 (b c-a d)^2 (a+b x)^{14}}{2 b^8}+\frac {7 d^6 (b c-a d) (a+b x)^{15}}{15 b^8}+\frac {d^7 (a+b x)^{16}}{16 b^8}\\ \end {align*}

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Mathematica [B]  time = 0.11, size = 897, normalized size = 4.48 \begin {gather*} \frac {1}{16} b^8 d^7 x^{16}+\frac {1}{15} b^7 d^6 (7 b c+8 a d) x^{15}+\frac {1}{2} b^6 d^5 \left (3 b^2 c^2+8 a b d c+4 a^2 d^2\right ) x^{14}+\frac {7}{13} b^5 d^4 \left (5 b^3 c^3+24 a b^2 d c^2+28 a^2 b d^2 c+8 a^3 d^3\right ) x^{13}+\frac {7}{12} b^4 d^3 \left (5 b^4 c^4+40 a b^3 d c^3+84 a^2 b^2 d^2 c^2+56 a^3 b d^3 c+10 a^4 d^4\right ) x^{12}+\frac {7}{11} b^3 d^2 \left (3 b^5 c^5+40 a b^4 d c^4+140 a^2 b^3 d^2 c^3+168 a^3 b^2 d^3 c^2+70 a^4 b d^4 c+8 a^5 d^5\right ) x^{11}+\frac {7}{10} b^2 d \left (b^6 c^6+24 a b^5 d c^5+140 a^2 b^4 d^2 c^4+280 a^3 b^3 d^3 c^3+210 a^4 b^2 d^4 c^2+56 a^5 b d^5 c+4 a^6 d^6\right ) x^{10}+\frac {1}{9} b \left (b^7 c^7+56 a b^6 d c^6+588 a^2 b^5 d^2 c^5+1960 a^3 b^4 d^3 c^4+2450 a^4 b^3 d^4 c^3+1176 a^5 b^2 d^5 c^2+196 a^6 b d^6 c+8 a^7 d^7\right ) x^9+\frac {1}{8} a \left (8 b^7 c^7+196 a b^6 d c^6+1176 a^2 b^5 d^2 c^5+2450 a^3 b^4 d^3 c^4+1960 a^4 b^3 d^4 c^3+588 a^5 b^2 d^5 c^2+56 a^6 b d^6 c+a^7 d^7\right ) x^8+a^2 c \left (4 b^6 c^6+56 a b^5 d c^5+210 a^2 b^4 d^2 c^4+280 a^3 b^3 d^3 c^3+140 a^4 b^2 d^4 c^2+24 a^5 b d^5 c+a^6 d^6\right ) x^7+\frac {7}{6} a^3 c^2 \left (8 b^5 c^5+70 a b^4 d c^4+168 a^2 b^3 d^2 c^3+140 a^3 b^2 d^3 c^2+40 a^4 b d^4 c+3 a^5 d^5\right ) x^6+\frac {7}{5} a^4 c^3 \left (10 b^4 c^4+56 a b^3 d c^3+84 a^2 b^2 d^2 c^2+40 a^3 b d^3 c+5 a^4 d^4\right ) x^5+\frac {7}{4} a^5 c^4 \left (8 b^3 c^3+28 a b^2 d c^2+24 a^2 b d^2 c+5 a^3 d^3\right ) x^4+\frac {7}{3} a^6 c^5 \left (4 b^2 c^2+8 a b d c+3 a^2 d^2\right ) x^3+\frac {1}{2} a^7 c^6 (8 b c+7 a d) x^2+a^8 c^7 x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^8*(c + d*x)^7,x]

[Out]

a^8*c^7*x + (a^7*c^6*(8*b*c + 7*a*d)*x^2)/2 + (7*a^6*c^5*(4*b^2*c^2 + 8*a*b*c*d + 3*a^2*d^2)*x^3)/3 + (7*a^5*c
^4*(8*b^3*c^3 + 28*a*b^2*c^2*d + 24*a^2*b*c*d^2 + 5*a^3*d^3)*x^4)/4 + (7*a^4*c^3*(10*b^4*c^4 + 56*a*b^3*c^3*d
+ 84*a^2*b^2*c^2*d^2 + 40*a^3*b*c*d^3 + 5*a^4*d^4)*x^5)/5 + (7*a^3*c^2*(8*b^5*c^5 + 70*a*b^4*c^4*d + 168*a^2*b
^3*c^3*d^2 + 140*a^3*b^2*c^2*d^3 + 40*a^4*b*c*d^4 + 3*a^5*d^5)*x^6)/6 + a^2*c*(4*b^6*c^6 + 56*a*b^5*c^5*d + 21
0*a^2*b^4*c^4*d^2 + 280*a^3*b^3*c^3*d^3 + 140*a^4*b^2*c^2*d^4 + 24*a^5*b*c*d^5 + a^6*d^6)*x^7 + (a*(8*b^7*c^7
+ 196*a*b^6*c^6*d + 1176*a^2*b^5*c^5*d^2 + 2450*a^3*b^4*c^4*d^3 + 1960*a^4*b^3*c^3*d^4 + 588*a^5*b^2*c^2*d^5 +
56*a^6*b*c*d^6 + a^7*d^7)*x^8)/8 + (b*(b^7*c^7 + 56*a*b^6*c^6*d + 588*a^2*b^5*c^5*d^2 + 1960*a^3*b^4*c^4*d^3
+ 2450*a^4*b^3*c^3*d^4 + 1176*a^5*b^2*c^2*d^5 + 196*a^6*b*c*d^6 + 8*a^7*d^7)*x^9)/9 + (7*b^2*d*(b^6*c^6 + 24*a
*b^5*c^5*d + 140*a^2*b^4*c^4*d^2 + 280*a^3*b^3*c^3*d^3 + 210*a^4*b^2*c^2*d^4 + 56*a^5*b*c*d^5 + 4*a^6*d^6)*x^1
0)/10 + (7*b^3*d^2*(3*b^5*c^5 + 40*a*b^4*c^4*d + 140*a^2*b^3*c^3*d^2 + 168*a^3*b^2*c^2*d^3 + 70*a^4*b*c*d^4 +
8*a^5*d^5)*x^11)/11 + (7*b^4*d^3*(5*b^4*c^4 + 40*a*b^3*c^3*d + 84*a^2*b^2*c^2*d^2 + 56*a^3*b*c*d^3 + 10*a^4*d^
4)*x^12)/12 + (7*b^5*d^4*(5*b^3*c^3 + 24*a*b^2*c^2*d + 28*a^2*b*c*d^2 + 8*a^3*d^3)*x^13)/13 + (b^6*d^5*(3*b^2*
c^2 + 8*a*b*c*d + 4*a^2*d^2)*x^14)/2 + (b^7*d^6*(7*b*c + 8*a*d)*x^15)/15 + (b^8*d^7*x^16)/16

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x)^8 (c+d x)^7 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)^8*(c + d*x)^7,x]

[Out]

IntegrateAlgebraic[(a + b*x)^8*(c + d*x)^7, x]

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fricas [B]  time = 1.28, size = 1050, normalized size = 5.25 \begin {gather*} \frac {1}{16} x^{16} d^{7} b^{8} + \frac {7}{15} x^{15} d^{6} c b^{8} + \frac {8}{15} x^{15} d^{7} b^{7} a + \frac {3}{2} x^{14} d^{5} c^{2} b^{8} + 4 x^{14} d^{6} c b^{7} a + 2 x^{14} d^{7} b^{6} a^{2} + \frac {35}{13} x^{13} d^{4} c^{3} b^{8} + \frac {168}{13} x^{13} d^{5} c^{2} b^{7} a + \frac {196}{13} x^{13} d^{6} c b^{6} a^{2} + \frac {56}{13} x^{13} d^{7} b^{5} a^{3} + \frac {35}{12} x^{12} d^{3} c^{4} b^{8} + \frac {70}{3} x^{12} d^{4} c^{3} b^{7} a + 49 x^{12} d^{5} c^{2} b^{6} a^{2} + \frac {98}{3} x^{12} d^{6} c b^{5} a^{3} + \frac {35}{6} x^{12} d^{7} b^{4} a^{4} + \frac {21}{11} x^{11} d^{2} c^{5} b^{8} + \frac {280}{11} x^{11} d^{3} c^{4} b^{7} a + \frac {980}{11} x^{11} d^{4} c^{3} b^{6} a^{2} + \frac {1176}{11} x^{11} d^{5} c^{2} b^{5} a^{3} + \frac {490}{11} x^{11} d^{6} c b^{4} a^{4} + \frac {56}{11} x^{11} d^{7} b^{3} a^{5} + \frac {7}{10} x^{10} d c^{6} b^{8} + \frac {84}{5} x^{10} d^{2} c^{5} b^{7} a + 98 x^{10} d^{3} c^{4} b^{6} a^{2} + 196 x^{10} d^{4} c^{3} b^{5} a^{3} + 147 x^{10} d^{5} c^{2} b^{4} a^{4} + \frac {196}{5} x^{10} d^{6} c b^{3} a^{5} + \frac {14}{5} x^{10} d^{7} b^{2} a^{6} + \frac {1}{9} x^{9} c^{7} b^{8} + \frac {56}{9} x^{9} d c^{6} b^{7} a + \frac {196}{3} x^{9} d^{2} c^{5} b^{6} a^{2} + \frac {1960}{9} x^{9} d^{3} c^{4} b^{5} a^{3} + \frac {2450}{9} x^{9} d^{4} c^{3} b^{4} a^{4} + \frac {392}{3} x^{9} d^{5} c^{2} b^{3} a^{5} + \frac {196}{9} x^{9} d^{6} c b^{2} a^{6} + \frac {8}{9} x^{9} d^{7} b a^{7} + x^{8} c^{7} b^{7} a + \frac {49}{2} x^{8} d c^{6} b^{6} a^{2} + 147 x^{8} d^{2} c^{5} b^{5} a^{3} + \frac {1225}{4} x^{8} d^{3} c^{4} b^{4} a^{4} + 245 x^{8} d^{4} c^{3} b^{3} a^{5} + \frac {147}{2} x^{8} d^{5} c^{2} b^{2} a^{6} + 7 x^{8} d^{6} c b a^{7} + \frac {1}{8} x^{8} d^{7} a^{8} + 4 x^{7} c^{7} b^{6} a^{2} + 56 x^{7} d c^{6} b^{5} a^{3} + 210 x^{7} d^{2} c^{5} b^{4} a^{4} + 280 x^{7} d^{3} c^{4} b^{3} a^{5} + 140 x^{7} d^{4} c^{3} b^{2} a^{6} + 24 x^{7} d^{5} c^{2} b a^{7} + x^{7} d^{6} c a^{8} + \frac {28}{3} x^{6} c^{7} b^{5} a^{3} + \frac {245}{3} x^{6} d c^{6} b^{4} a^{4} + 196 x^{6} d^{2} c^{5} b^{3} a^{5} + \frac {490}{3} x^{6} d^{3} c^{4} b^{2} a^{6} + \frac {140}{3} x^{6} d^{4} c^{3} b a^{7} + \frac {7}{2} x^{6} d^{5} c^{2} a^{8} + 14 x^{5} c^{7} b^{4} a^{4} + \frac {392}{5} x^{5} d c^{6} b^{3} a^{5} + \frac {588}{5} x^{5} d^{2} c^{5} b^{2} a^{6} + 56 x^{5} d^{3} c^{4} b a^{7} + 7 x^{5} d^{4} c^{3} a^{8} + 14 x^{4} c^{7} b^{3} a^{5} + 49 x^{4} d c^{6} b^{2} a^{6} + 42 x^{4} d^{2} c^{5} b a^{7} + \frac {35}{4} x^{4} d^{3} c^{4} a^{8} + \frac {28}{3} x^{3} c^{7} b^{2} a^{6} + \frac {56}{3} x^{3} d c^{6} b a^{7} + 7 x^{3} d^{2} c^{5} a^{8} + 4 x^{2} c^{7} b a^{7} + \frac {7}{2} x^{2} d c^{6} a^{8} + x c^{7} a^{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^8*(d*x+c)^7,x, algorithm="fricas")

[Out]

1/16*x^16*d^7*b^8 + 7/15*x^15*d^6*c*b^8 + 8/15*x^15*d^7*b^7*a + 3/2*x^14*d^5*c^2*b^8 + 4*x^14*d^6*c*b^7*a + 2*
x^14*d^7*b^6*a^2 + 35/13*x^13*d^4*c^3*b^8 + 168/13*x^13*d^5*c^2*b^7*a + 196/13*x^13*d^6*c*b^6*a^2 + 56/13*x^13
*d^7*b^5*a^3 + 35/12*x^12*d^3*c^4*b^8 + 70/3*x^12*d^4*c^3*b^7*a + 49*x^12*d^5*c^2*b^6*a^2 + 98/3*x^12*d^6*c*b^
5*a^3 + 35/6*x^12*d^7*b^4*a^4 + 21/11*x^11*d^2*c^5*b^8 + 280/11*x^11*d^3*c^4*b^7*a + 980/11*x^11*d^4*c^3*b^6*a
^2 + 1176/11*x^11*d^5*c^2*b^5*a^3 + 490/11*x^11*d^6*c*b^4*a^4 + 56/11*x^11*d^7*b^3*a^5 + 7/10*x^10*d*c^6*b^8 +
84/5*x^10*d^2*c^5*b^7*a + 98*x^10*d^3*c^4*b^6*a^2 + 196*x^10*d^4*c^3*b^5*a^3 + 147*x^10*d^5*c^2*b^4*a^4 + 196
/5*x^10*d^6*c*b^3*a^5 + 14/5*x^10*d^7*b^2*a^6 + 1/9*x^9*c^7*b^8 + 56/9*x^9*d*c^6*b^7*a + 196/3*x^9*d^2*c^5*b^6
*a^2 + 1960/9*x^9*d^3*c^4*b^5*a^3 + 2450/9*x^9*d^4*c^3*b^4*a^4 + 392/3*x^9*d^5*c^2*b^3*a^5 + 196/9*x^9*d^6*c*b
^2*a^6 + 8/9*x^9*d^7*b*a^7 + x^8*c^7*b^7*a + 49/2*x^8*d*c^6*b^6*a^2 + 147*x^8*d^2*c^5*b^5*a^3 + 1225/4*x^8*d^3
*c^4*b^4*a^4 + 245*x^8*d^4*c^3*b^3*a^5 + 147/2*x^8*d^5*c^2*b^2*a^6 + 7*x^8*d^6*c*b*a^7 + 1/8*x^8*d^7*a^8 + 4*x
^7*c^7*b^6*a^2 + 56*x^7*d*c^6*b^5*a^3 + 210*x^7*d^2*c^5*b^4*a^4 + 280*x^7*d^3*c^4*b^3*a^5 + 140*x^7*d^4*c^3*b^
2*a^6 + 24*x^7*d^5*c^2*b*a^7 + x^7*d^6*c*a^8 + 28/3*x^6*c^7*b^5*a^3 + 245/3*x^6*d*c^6*b^4*a^4 + 196*x^6*d^2*c^
5*b^3*a^5 + 490/3*x^6*d^3*c^4*b^2*a^6 + 140/3*x^6*d^4*c^3*b*a^7 + 7/2*x^6*d^5*c^2*a^8 + 14*x^5*c^7*b^4*a^4 + 3
92/5*x^5*d*c^6*b^3*a^5 + 588/5*x^5*d^2*c^5*b^2*a^6 + 56*x^5*d^3*c^4*b*a^7 + 7*x^5*d^4*c^3*a^8 + 14*x^4*c^7*b^3
*a^5 + 49*x^4*d*c^6*b^2*a^6 + 42*x^4*d^2*c^5*b*a^7 + 35/4*x^4*d^3*c^4*a^8 + 28/3*x^3*c^7*b^2*a^6 + 56/3*x^3*d*
c^6*b*a^7 + 7*x^3*d^2*c^5*a^8 + 4*x^2*c^7*b*a^7 + 7/2*x^2*d*c^6*a^8 + x*c^7*a^8

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giac [B]  time = 1.01, size = 1050, normalized size = 5.25 \begin {gather*} \frac {1}{16} \, b^{8} d^{7} x^{16} + \frac {7}{15} \, b^{8} c d^{6} x^{15} + \frac {8}{15} \, a b^{7} d^{7} x^{15} + \frac {3}{2} \, b^{8} c^{2} d^{5} x^{14} + 4 \, a b^{7} c d^{6} x^{14} + 2 \, a^{2} b^{6} d^{7} x^{14} + \frac {35}{13} \, b^{8} c^{3} d^{4} x^{13} + \frac {168}{13} \, a b^{7} c^{2} d^{5} x^{13} + \frac {196}{13} \, a^{2} b^{6} c d^{6} x^{13} + \frac {56}{13} \, a^{3} b^{5} d^{7} x^{13} + \frac {35}{12} \, b^{8} c^{4} d^{3} x^{12} + \frac {70}{3} \, a b^{7} c^{3} d^{4} x^{12} + 49 \, a^{2} b^{6} c^{2} d^{5} x^{12} + \frac {98}{3} \, a^{3} b^{5} c d^{6} x^{12} + \frac {35}{6} \, a^{4} b^{4} d^{7} x^{12} + \frac {21}{11} \, b^{8} c^{5} d^{2} x^{11} + \frac {280}{11} \, a b^{7} c^{4} d^{3} x^{11} + \frac {980}{11} \, a^{2} b^{6} c^{3} d^{4} x^{11} + \frac {1176}{11} \, a^{3} b^{5} c^{2} d^{5} x^{11} + \frac {490}{11} \, a^{4} b^{4} c d^{6} x^{11} + \frac {56}{11} \, a^{5} b^{3} d^{7} x^{11} + \frac {7}{10} \, b^{8} c^{6} d x^{10} + \frac {84}{5} \, a b^{7} c^{5} d^{2} x^{10} + 98 \, a^{2} b^{6} c^{4} d^{3} x^{10} + 196 \, a^{3} b^{5} c^{3} d^{4} x^{10} + 147 \, a^{4} b^{4} c^{2} d^{5} x^{10} + \frac {196}{5} \, a^{5} b^{3} c d^{6} x^{10} + \frac {14}{5} \, a^{6} b^{2} d^{7} x^{10} + \frac {1}{9} \, b^{8} c^{7} x^{9} + \frac {56}{9} \, a b^{7} c^{6} d x^{9} + \frac {196}{3} \, a^{2} b^{6} c^{5} d^{2} x^{9} + \frac {1960}{9} \, a^{3} b^{5} c^{4} d^{3} x^{9} + \frac {2450}{9} \, a^{4} b^{4} c^{3} d^{4} x^{9} + \frac {392}{3} \, a^{5} b^{3} c^{2} d^{5} x^{9} + \frac {196}{9} \, a^{6} b^{2} c d^{6} x^{9} + \frac {8}{9} \, a^{7} b d^{7} x^{9} + a b^{7} c^{7} x^{8} + \frac {49}{2} \, a^{2} b^{6} c^{6} d x^{8} + 147 \, a^{3} b^{5} c^{5} d^{2} x^{8} + \frac {1225}{4} \, a^{4} b^{4} c^{4} d^{3} x^{8} + 245 \, a^{5} b^{3} c^{3} d^{4} x^{8} + \frac {147}{2} \, a^{6} b^{2} c^{2} d^{5} x^{8} + 7 \, a^{7} b c d^{6} x^{8} + \frac {1}{8} \, a^{8} d^{7} x^{8} + 4 \, a^{2} b^{6} c^{7} x^{7} + 56 \, a^{3} b^{5} c^{6} d x^{7} + 210 \, a^{4} b^{4} c^{5} d^{2} x^{7} + 280 \, a^{5} b^{3} c^{4} d^{3} x^{7} + 140 \, a^{6} b^{2} c^{3} d^{4} x^{7} + 24 \, a^{7} b c^{2} d^{5} x^{7} + a^{8} c d^{6} x^{7} + \frac {28}{3} \, a^{3} b^{5} c^{7} x^{6} + \frac {245}{3} \, a^{4} b^{4} c^{6} d x^{6} + 196 \, a^{5} b^{3} c^{5} d^{2} x^{6} + \frac {490}{3} \, a^{6} b^{2} c^{4} d^{3} x^{6} + \frac {140}{3} \, a^{7} b c^{3} d^{4} x^{6} + \frac {7}{2} \, a^{8} c^{2} d^{5} x^{6} + 14 \, a^{4} b^{4} c^{7} x^{5} + \frac {392}{5} \, a^{5} b^{3} c^{6} d x^{5} + \frac {588}{5} \, a^{6} b^{2} c^{5} d^{2} x^{5} + 56 \, a^{7} b c^{4} d^{3} x^{5} + 7 \, a^{8} c^{3} d^{4} x^{5} + 14 \, a^{5} b^{3} c^{7} x^{4} + 49 \, a^{6} b^{2} c^{6} d x^{4} + 42 \, a^{7} b c^{5} d^{2} x^{4} + \frac {35}{4} \, a^{8} c^{4} d^{3} x^{4} + \frac {28}{3} \, a^{6} b^{2} c^{7} x^{3} + \frac {56}{3} \, a^{7} b c^{6} d x^{3} + 7 \, a^{8} c^{5} d^{2} x^{3} + 4 \, a^{7} b c^{7} x^{2} + \frac {7}{2} \, a^{8} c^{6} d x^{2} + a^{8} c^{7} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^8*(d*x+c)^7,x, algorithm="giac")

[Out]

1/16*b^8*d^7*x^16 + 7/15*b^8*c*d^6*x^15 + 8/15*a*b^7*d^7*x^15 + 3/2*b^8*c^2*d^5*x^14 + 4*a*b^7*c*d^6*x^14 + 2*
a^2*b^6*d^7*x^14 + 35/13*b^8*c^3*d^4*x^13 + 168/13*a*b^7*c^2*d^5*x^13 + 196/13*a^2*b^6*c*d^6*x^13 + 56/13*a^3*
b^5*d^7*x^13 + 35/12*b^8*c^4*d^3*x^12 + 70/3*a*b^7*c^3*d^4*x^12 + 49*a^2*b^6*c^2*d^5*x^12 + 98/3*a^3*b^5*c*d^6
*x^12 + 35/6*a^4*b^4*d^7*x^12 + 21/11*b^8*c^5*d^2*x^11 + 280/11*a*b^7*c^4*d^3*x^11 + 980/11*a^2*b^6*c^3*d^4*x^
11 + 1176/11*a^3*b^5*c^2*d^5*x^11 + 490/11*a^4*b^4*c*d^6*x^11 + 56/11*a^5*b^3*d^7*x^11 + 7/10*b^8*c^6*d*x^10 +
84/5*a*b^7*c^5*d^2*x^10 + 98*a^2*b^6*c^4*d^3*x^10 + 196*a^3*b^5*c^3*d^4*x^10 + 147*a^4*b^4*c^2*d^5*x^10 + 196
/5*a^5*b^3*c*d^6*x^10 + 14/5*a^6*b^2*d^7*x^10 + 1/9*b^8*c^7*x^9 + 56/9*a*b^7*c^6*d*x^9 + 196/3*a^2*b^6*c^5*d^2
*x^9 + 1960/9*a^3*b^5*c^4*d^3*x^9 + 2450/9*a^4*b^4*c^3*d^4*x^9 + 392/3*a^5*b^3*c^2*d^5*x^9 + 196/9*a^6*b^2*c*d
^6*x^9 + 8/9*a^7*b*d^7*x^9 + a*b^7*c^7*x^8 + 49/2*a^2*b^6*c^6*d*x^8 + 147*a^3*b^5*c^5*d^2*x^8 + 1225/4*a^4*b^4
*c^4*d^3*x^8 + 245*a^5*b^3*c^3*d^4*x^8 + 147/2*a^6*b^2*c^2*d^5*x^8 + 7*a^7*b*c*d^6*x^8 + 1/8*a^8*d^7*x^8 + 4*a
^2*b^6*c^7*x^7 + 56*a^3*b^5*c^6*d*x^7 + 210*a^4*b^4*c^5*d^2*x^7 + 280*a^5*b^3*c^4*d^3*x^7 + 140*a^6*b^2*c^3*d^
4*x^7 + 24*a^7*b*c^2*d^5*x^7 + a^8*c*d^6*x^7 + 28/3*a^3*b^5*c^7*x^6 + 245/3*a^4*b^4*c^6*d*x^6 + 196*a^5*b^3*c^
5*d^2*x^6 + 490/3*a^6*b^2*c^4*d^3*x^6 + 140/3*a^7*b*c^3*d^4*x^6 + 7/2*a^8*c^2*d^5*x^6 + 14*a^4*b^4*c^7*x^5 + 3
92/5*a^5*b^3*c^6*d*x^5 + 588/5*a^6*b^2*c^5*d^2*x^5 + 56*a^7*b*c^4*d^3*x^5 + 7*a^8*c^3*d^4*x^5 + 14*a^5*b^3*c^7
*x^4 + 49*a^6*b^2*c^6*d*x^4 + 42*a^7*b*c^5*d^2*x^4 + 35/4*a^8*c^4*d^3*x^4 + 28/3*a^6*b^2*c^7*x^3 + 56/3*a^7*b*
c^6*d*x^3 + 7*a^8*c^5*d^2*x^3 + 4*a^7*b*c^7*x^2 + 7/2*a^8*c^6*d*x^2 + a^8*c^7*x

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maple [B]  time = 0.00, size = 925, normalized size = 4.62 \begin {gather*} \frac {b^{8} d^{7} x^{16}}{16}+a^{8} c^{7} x +\frac {\left (8 a \,b^{7} d^{7}+7 b^{8} c \,d^{6}\right ) x^{15}}{15}+\frac {\left (28 a^{2} b^{6} d^{7}+56 a \,b^{7} c \,d^{6}+21 b^{8} c^{2} d^{5}\right ) x^{14}}{14}+\frac {\left (56 a^{3} b^{5} d^{7}+196 a^{2} b^{6} c \,d^{6}+168 a \,b^{7} c^{2} d^{5}+35 b^{8} c^{3} d^{4}\right ) x^{13}}{13}+\frac {\left (70 a^{4} b^{4} d^{7}+392 a^{3} b^{5} c \,d^{6}+588 a^{2} b^{6} c^{2} d^{5}+280 a \,b^{7} c^{3} d^{4}+35 b^{8} c^{4} d^{3}\right ) x^{12}}{12}+\frac {\left (56 a^{5} b^{3} d^{7}+490 a^{4} b^{4} c \,d^{6}+1176 a^{3} b^{5} c^{2} d^{5}+980 a^{2} b^{6} c^{3} d^{4}+280 a \,b^{7} c^{4} d^{3}+21 b^{8} c^{5} d^{2}\right ) x^{11}}{11}+\frac {\left (28 a^{6} b^{2} d^{7}+392 a^{5} b^{3} c \,d^{6}+1470 a^{4} b^{4} c^{2} d^{5}+1960 a^{3} b^{5} c^{3} d^{4}+980 a^{2} b^{6} c^{4} d^{3}+168 a \,b^{7} c^{5} d^{2}+7 b^{8} c^{6} d \right ) x^{10}}{10}+\frac {\left (8 a^{7} b \,d^{7}+196 a^{6} b^{2} c \,d^{6}+1176 a^{5} b^{3} c^{2} d^{5}+2450 a^{4} b^{4} c^{3} d^{4}+1960 a^{3} b^{5} c^{4} d^{3}+588 a^{2} b^{6} c^{5} d^{2}+56 a \,b^{7} c^{6} d +b^{8} c^{7}\right ) x^{9}}{9}+\frac {\left (a^{8} d^{7}+56 a^{7} b c \,d^{6}+588 a^{6} b^{2} c^{2} d^{5}+1960 a^{5} b^{3} c^{3} d^{4}+2450 a^{4} b^{4} c^{4} d^{3}+1176 a^{3} b^{5} c^{5} d^{2}+196 a^{2} b^{6} c^{6} d +8 a \,b^{7} c^{7}\right ) x^{8}}{8}+\frac {\left (7 a^{8} c \,d^{6}+168 a^{7} b \,c^{2} d^{5}+980 a^{6} b^{2} c^{3} d^{4}+1960 a^{5} b^{3} c^{4} d^{3}+1470 a^{4} b^{4} c^{5} d^{2}+392 a^{3} b^{5} c^{6} d +28 a^{2} b^{6} c^{7}\right ) x^{7}}{7}+\frac {\left (21 a^{8} c^{2} d^{5}+280 a^{7} b \,c^{3} d^{4}+980 a^{6} b^{2} c^{4} d^{3}+1176 a^{5} b^{3} c^{5} d^{2}+490 a^{4} b^{4} c^{6} d +56 a^{3} b^{5} c^{7}\right ) x^{6}}{6}+\frac {\left (35 a^{8} c^{3} d^{4}+280 a^{7} b \,c^{4} d^{3}+588 a^{6} b^{2} c^{5} d^{2}+392 a^{5} b^{3} c^{6} d +70 a^{4} b^{4} c^{7}\right ) x^{5}}{5}+\frac {\left (35 a^{8} c^{4} d^{3}+168 a^{7} b \,c^{5} d^{2}+196 a^{6} b^{2} c^{6} d +56 a^{5} b^{3} c^{7}\right ) x^{4}}{4}+\frac {\left (21 a^{8} c^{5} d^{2}+56 a^{7} b \,c^{6} d +28 a^{6} b^{2} c^{7}\right ) x^{3}}{3}+\frac {\left (7 a^{8} c^{6} d +8 a^{7} b \,c^{7}\right ) x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^8*(d*x+c)^7,x)

[Out]

1/16*b^8*d^7*x^16+1/15*(8*a*b^7*d^7+7*b^8*c*d^6)*x^15+1/14*(28*a^2*b^6*d^7+56*a*b^7*c*d^6+21*b^8*c^2*d^5)*x^14
+1/13*(56*a^3*b^5*d^7+196*a^2*b^6*c*d^6+168*a*b^7*c^2*d^5+35*b^8*c^3*d^4)*x^13+1/12*(70*a^4*b^4*d^7+392*a^3*b^
5*c*d^6+588*a^2*b^6*c^2*d^5+280*a*b^7*c^3*d^4+35*b^8*c^4*d^3)*x^12+1/11*(56*a^5*b^3*d^7+490*a^4*b^4*c*d^6+1176
*a^3*b^5*c^2*d^5+980*a^2*b^6*c^3*d^4+280*a*b^7*c^4*d^3+21*b^8*c^5*d^2)*x^11+1/10*(28*a^6*b^2*d^7+392*a^5*b^3*c
*d^6+1470*a^4*b^4*c^2*d^5+1960*a^3*b^5*c^3*d^4+980*a^2*b^6*c^4*d^3+168*a*b^7*c^5*d^2+7*b^8*c^6*d)*x^10+1/9*(8*
a^7*b*d^7+196*a^6*b^2*c*d^6+1176*a^5*b^3*c^2*d^5+2450*a^4*b^4*c^3*d^4+1960*a^3*b^5*c^4*d^3+588*a^2*b^6*c^5*d^2
+56*a*b^7*c^6*d+b^8*c^7)*x^9+1/8*(a^8*d^7+56*a^7*b*c*d^6+588*a^6*b^2*c^2*d^5+1960*a^5*b^3*c^3*d^4+2450*a^4*b^4
*c^4*d^3+1176*a^3*b^5*c^5*d^2+196*a^2*b^6*c^6*d+8*a*b^7*c^7)*x^8+1/7*(7*a^8*c*d^6+168*a^7*b*c^2*d^5+980*a^6*b^
2*c^3*d^4+1960*a^5*b^3*c^4*d^3+1470*a^4*b^4*c^5*d^2+392*a^3*b^5*c^6*d+28*a^2*b^6*c^7)*x^7+1/6*(21*a^8*c^2*d^5+
280*a^7*b*c^3*d^4+980*a^6*b^2*c^4*d^3+1176*a^5*b^3*c^5*d^2+490*a^4*b^4*c^6*d+56*a^3*b^5*c^7)*x^6+1/5*(35*a^8*c
^3*d^4+280*a^7*b*c^4*d^3+588*a^6*b^2*c^5*d^2+392*a^5*b^3*c^6*d+70*a^4*b^4*c^7)*x^5+1/4*(35*a^8*c^4*d^3+168*a^7
*b*c^5*d^2+196*a^6*b^2*c^6*d+56*a^5*b^3*c^7)*x^4+1/3*(21*a^8*c^5*d^2+56*a^7*b*c^6*d+28*a^6*b^2*c^7)*x^3+1/2*(7
*a^8*c^6*d+8*a^7*b*c^7)*x^2+a^8*c^7*x

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maxima [B]  time = 1.40, size = 921, normalized size = 4.60 \begin {gather*} \frac {1}{16} \, b^{8} d^{7} x^{16} + a^{8} c^{7} x + \frac {1}{15} \, {\left (7 \, b^{8} c d^{6} + 8 \, a b^{7} d^{7}\right )} x^{15} + \frac {1}{2} \, {\left (3 \, b^{8} c^{2} d^{5} + 8 \, a b^{7} c d^{6} + 4 \, a^{2} b^{6} d^{7}\right )} x^{14} + \frac {7}{13} \, {\left (5 \, b^{8} c^{3} d^{4} + 24 \, a b^{7} c^{2} d^{5} + 28 \, a^{2} b^{6} c d^{6} + 8 \, a^{3} b^{5} d^{7}\right )} x^{13} + \frac {7}{12} \, {\left (5 \, b^{8} c^{4} d^{3} + 40 \, a b^{7} c^{3} d^{4} + 84 \, a^{2} b^{6} c^{2} d^{5} + 56 \, a^{3} b^{5} c d^{6} + 10 \, a^{4} b^{4} d^{7}\right )} x^{12} + \frac {7}{11} \, {\left (3 \, b^{8} c^{5} d^{2} + 40 \, a b^{7} c^{4} d^{3} + 140 \, a^{2} b^{6} c^{3} d^{4} + 168 \, a^{3} b^{5} c^{2} d^{5} + 70 \, a^{4} b^{4} c d^{6} + 8 \, a^{5} b^{3} d^{7}\right )} x^{11} + \frac {7}{10} \, {\left (b^{8} c^{6} d + 24 \, a b^{7} c^{5} d^{2} + 140 \, a^{2} b^{6} c^{4} d^{3} + 280 \, a^{3} b^{5} c^{3} d^{4} + 210 \, a^{4} b^{4} c^{2} d^{5} + 56 \, a^{5} b^{3} c d^{6} + 4 \, a^{6} b^{2} d^{7}\right )} x^{10} + \frac {1}{9} \, {\left (b^{8} c^{7} + 56 \, a b^{7} c^{6} d + 588 \, a^{2} b^{6} c^{5} d^{2} + 1960 \, a^{3} b^{5} c^{4} d^{3} + 2450 \, a^{4} b^{4} c^{3} d^{4} + 1176 \, a^{5} b^{3} c^{2} d^{5} + 196 \, a^{6} b^{2} c d^{6} + 8 \, a^{7} b d^{7}\right )} x^{9} + \frac {1}{8} \, {\left (8 \, a b^{7} c^{7} + 196 \, a^{2} b^{6} c^{6} d + 1176 \, a^{3} b^{5} c^{5} d^{2} + 2450 \, a^{4} b^{4} c^{4} d^{3} + 1960 \, a^{5} b^{3} c^{3} d^{4} + 588 \, a^{6} b^{2} c^{2} d^{5} + 56 \, a^{7} b c d^{6} + a^{8} d^{7}\right )} x^{8} + {\left (4 \, a^{2} b^{6} c^{7} + 56 \, a^{3} b^{5} c^{6} d + 210 \, a^{4} b^{4} c^{5} d^{2} + 280 \, a^{5} b^{3} c^{4} d^{3} + 140 \, a^{6} b^{2} c^{3} d^{4} + 24 \, a^{7} b c^{2} d^{5} + a^{8} c d^{6}\right )} x^{7} + \frac {7}{6} \, {\left (8 \, a^{3} b^{5} c^{7} + 70 \, a^{4} b^{4} c^{6} d + 168 \, a^{5} b^{3} c^{5} d^{2} + 140 \, a^{6} b^{2} c^{4} d^{3} + 40 \, a^{7} b c^{3} d^{4} + 3 \, a^{8} c^{2} d^{5}\right )} x^{6} + \frac {7}{5} \, {\left (10 \, a^{4} b^{4} c^{7} + 56 \, a^{5} b^{3} c^{6} d + 84 \, a^{6} b^{2} c^{5} d^{2} + 40 \, a^{7} b c^{4} d^{3} + 5 \, a^{8} c^{3} d^{4}\right )} x^{5} + \frac {7}{4} \, {\left (8 \, a^{5} b^{3} c^{7} + 28 \, a^{6} b^{2} c^{6} d + 24 \, a^{7} b c^{5} d^{2} + 5 \, a^{8} c^{4} d^{3}\right )} x^{4} + \frac {7}{3} \, {\left (4 \, a^{6} b^{2} c^{7} + 8 \, a^{7} b c^{6} d + 3 \, a^{8} c^{5} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (8 \, a^{7} b c^{7} + 7 \, a^{8} c^{6} d\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^8*(d*x+c)^7,x, algorithm="maxima")

[Out]

1/16*b^8*d^7*x^16 + a^8*c^7*x + 1/15*(7*b^8*c*d^6 + 8*a*b^7*d^7)*x^15 + 1/2*(3*b^8*c^2*d^5 + 8*a*b^7*c*d^6 + 4
*a^2*b^6*d^7)*x^14 + 7/13*(5*b^8*c^3*d^4 + 24*a*b^7*c^2*d^5 + 28*a^2*b^6*c*d^6 + 8*a^3*b^5*d^7)*x^13 + 7/12*(5
*b^8*c^4*d^3 + 40*a*b^7*c^3*d^4 + 84*a^2*b^6*c^2*d^5 + 56*a^3*b^5*c*d^6 + 10*a^4*b^4*d^7)*x^12 + 7/11*(3*b^8*c
^5*d^2 + 40*a*b^7*c^4*d^3 + 140*a^2*b^6*c^3*d^4 + 168*a^3*b^5*c^2*d^5 + 70*a^4*b^4*c*d^6 + 8*a^5*b^3*d^7)*x^11
+ 7/10*(b^8*c^6*d + 24*a*b^7*c^5*d^2 + 140*a^2*b^6*c^4*d^3 + 280*a^3*b^5*c^3*d^4 + 210*a^4*b^4*c^2*d^5 + 56*a
^5*b^3*c*d^6 + 4*a^6*b^2*d^7)*x^10 + 1/9*(b^8*c^7 + 56*a*b^7*c^6*d + 588*a^2*b^6*c^5*d^2 + 1960*a^3*b^5*c^4*d^
3 + 2450*a^4*b^4*c^3*d^4 + 1176*a^5*b^3*c^2*d^5 + 196*a^6*b^2*c*d^6 + 8*a^7*b*d^7)*x^9 + 1/8*(8*a*b^7*c^7 + 19
6*a^2*b^6*c^6*d + 1176*a^3*b^5*c^5*d^2 + 2450*a^4*b^4*c^4*d^3 + 1960*a^5*b^3*c^3*d^4 + 588*a^6*b^2*c^2*d^5 + 5
6*a^7*b*c*d^6 + a^8*d^7)*x^8 + (4*a^2*b^6*c^7 + 56*a^3*b^5*c^6*d + 210*a^4*b^4*c^5*d^2 + 280*a^5*b^3*c^4*d^3 +
140*a^6*b^2*c^3*d^4 + 24*a^7*b*c^2*d^5 + a^8*c*d^6)*x^7 + 7/6*(8*a^3*b^5*c^7 + 70*a^4*b^4*c^6*d + 168*a^5*b^3
*c^5*d^2 + 140*a^6*b^2*c^4*d^3 + 40*a^7*b*c^3*d^4 + 3*a^8*c^2*d^5)*x^6 + 7/5*(10*a^4*b^4*c^7 + 56*a^5*b^3*c^6*
d + 84*a^6*b^2*c^5*d^2 + 40*a^7*b*c^4*d^3 + 5*a^8*c^3*d^4)*x^5 + 7/4*(8*a^5*b^3*c^7 + 28*a^6*b^2*c^6*d + 24*a^
7*b*c^5*d^2 + 5*a^8*c^4*d^3)*x^4 + 7/3*(4*a^6*b^2*c^7 + 8*a^7*b*c^6*d + 3*a^8*c^5*d^2)*x^3 + 1/2*(8*a^7*b*c^7
+ 7*a^8*c^6*d)*x^2

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mupad [B]  time = 0.36, size = 892, normalized size = 4.46 \begin {gather*} x^8\,\left (\frac {a^8\,d^7}{8}+7\,a^7\,b\,c\,d^6+\frac {147\,a^6\,b^2\,c^2\,d^5}{2}+245\,a^5\,b^3\,c^3\,d^4+\frac {1225\,a^4\,b^4\,c^4\,d^3}{4}+147\,a^3\,b^5\,c^5\,d^2+\frac {49\,a^2\,b^6\,c^6\,d}{2}+a\,b^7\,c^7\right )+x^9\,\left (\frac {8\,a^7\,b\,d^7}{9}+\frac {196\,a^6\,b^2\,c\,d^6}{9}+\frac {392\,a^5\,b^3\,c^2\,d^5}{3}+\frac {2450\,a^4\,b^4\,c^3\,d^4}{9}+\frac {1960\,a^3\,b^5\,c^4\,d^3}{9}+\frac {196\,a^2\,b^6\,c^5\,d^2}{3}+\frac {56\,a\,b^7\,c^6\,d}{9}+\frac {b^8\,c^7}{9}\right )+x^5\,\left (7\,a^8\,c^3\,d^4+56\,a^7\,b\,c^4\,d^3+\frac {588\,a^6\,b^2\,c^5\,d^2}{5}+\frac {392\,a^5\,b^3\,c^6\,d}{5}+14\,a^4\,b^4\,c^7\right )+x^{12}\,\left (\frac {35\,a^4\,b^4\,d^7}{6}+\frac {98\,a^3\,b^5\,c\,d^6}{3}+49\,a^2\,b^6\,c^2\,d^5+\frac {70\,a\,b^7\,c^3\,d^4}{3}+\frac {35\,b^8\,c^4\,d^3}{12}\right )+x^6\,\left (\frac {7\,a^8\,c^2\,d^5}{2}+\frac {140\,a^7\,b\,c^3\,d^4}{3}+\frac {490\,a^6\,b^2\,c^4\,d^3}{3}+196\,a^5\,b^3\,c^5\,d^2+\frac {245\,a^4\,b^4\,c^6\,d}{3}+\frac {28\,a^3\,b^5\,c^7}{3}\right )+x^{11}\,\left (\frac {56\,a^5\,b^3\,d^7}{11}+\frac {490\,a^4\,b^4\,c\,d^6}{11}+\frac {1176\,a^3\,b^5\,c^2\,d^5}{11}+\frac {980\,a^2\,b^6\,c^3\,d^4}{11}+\frac {280\,a\,b^7\,c^4\,d^3}{11}+\frac {21\,b^8\,c^5\,d^2}{11}\right )+x^7\,\left (a^8\,c\,d^6+24\,a^7\,b\,c^2\,d^5+140\,a^6\,b^2\,c^3\,d^4+280\,a^5\,b^3\,c^4\,d^3+210\,a^4\,b^4\,c^5\,d^2+56\,a^3\,b^5\,c^6\,d+4\,a^2\,b^6\,c^7\right )+x^{10}\,\left (\frac {14\,a^6\,b^2\,d^7}{5}+\frac {196\,a^5\,b^3\,c\,d^6}{5}+147\,a^4\,b^4\,c^2\,d^5+196\,a^3\,b^5\,c^3\,d^4+98\,a^2\,b^6\,c^4\,d^3+\frac {84\,a\,b^7\,c^5\,d^2}{5}+\frac {7\,b^8\,c^6\,d}{10}\right )+a^8\,c^7\,x+\frac {b^8\,d^7\,x^{16}}{16}+\frac {7\,a^5\,c^4\,x^4\,\left (5\,a^3\,d^3+24\,a^2\,b\,c\,d^2+28\,a\,b^2\,c^2\,d+8\,b^3\,c^3\right )}{4}+\frac {7\,b^5\,d^4\,x^{13}\,\left (8\,a^3\,d^3+28\,a^2\,b\,c\,d^2+24\,a\,b^2\,c^2\,d+5\,b^3\,c^3\right )}{13}+\frac {a^7\,c^6\,x^2\,\left (7\,a\,d+8\,b\,c\right )}{2}+\frac {b^7\,d^6\,x^{15}\,\left (8\,a\,d+7\,b\,c\right )}{15}+\frac {7\,a^6\,c^5\,x^3\,\left (3\,a^2\,d^2+8\,a\,b\,c\,d+4\,b^2\,c^2\right )}{3}+\frac {b^6\,d^5\,x^{14}\,\left (4\,a^2\,d^2+8\,a\,b\,c\,d+3\,b^2\,c^2\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^8*(c + d*x)^7,x)

[Out]

x^8*((a^8*d^7)/8 + a*b^7*c^7 + (49*a^2*b^6*c^6*d)/2 + 147*a^3*b^5*c^5*d^2 + (1225*a^4*b^4*c^4*d^3)/4 + 245*a^5
*b^3*c^3*d^4 + (147*a^6*b^2*c^2*d^5)/2 + 7*a^7*b*c*d^6) + x^9*((b^8*c^7)/9 + (8*a^7*b*d^7)/9 + (196*a^6*b^2*c*
d^6)/9 + (196*a^2*b^6*c^5*d^2)/3 + (1960*a^3*b^5*c^4*d^3)/9 + (2450*a^4*b^4*c^3*d^4)/9 + (392*a^5*b^3*c^2*d^5)
/3 + (56*a*b^7*c^6*d)/9) + x^5*(14*a^4*b^4*c^7 + 7*a^8*c^3*d^4 + (392*a^5*b^3*c^6*d)/5 + 56*a^7*b*c^4*d^3 + (5
88*a^6*b^2*c^5*d^2)/5) + x^12*((35*a^4*b^4*d^7)/6 + (35*b^8*c^4*d^3)/12 + (70*a*b^7*c^3*d^4)/3 + (98*a^3*b^5*c
*d^6)/3 + 49*a^2*b^6*c^2*d^5) + x^6*((28*a^3*b^5*c^7)/3 + (7*a^8*c^2*d^5)/2 + (245*a^4*b^4*c^6*d)/3 + (140*a^7
*b*c^3*d^4)/3 + 196*a^5*b^3*c^5*d^2 + (490*a^6*b^2*c^4*d^3)/3) + x^11*((56*a^5*b^3*d^7)/11 + (21*b^8*c^5*d^2)/
11 + (280*a*b^7*c^4*d^3)/11 + (490*a^4*b^4*c*d^6)/11 + (980*a^2*b^6*c^3*d^4)/11 + (1176*a^3*b^5*c^2*d^5)/11) +
x^7*(a^8*c*d^6 + 4*a^2*b^6*c^7 + 56*a^3*b^5*c^6*d + 24*a^7*b*c^2*d^5 + 210*a^4*b^4*c^5*d^2 + 280*a^5*b^3*c^4*
d^3 + 140*a^6*b^2*c^3*d^4) + x^10*((7*b^8*c^6*d)/10 + (14*a^6*b^2*d^7)/5 + (84*a*b^7*c^5*d^2)/5 + (196*a^5*b^3
*c*d^6)/5 + 98*a^2*b^6*c^4*d^3 + 196*a^3*b^5*c^3*d^4 + 147*a^4*b^4*c^2*d^5) + a^8*c^7*x + (b^8*d^7*x^16)/16 +
(7*a^5*c^4*x^4*(5*a^3*d^3 + 8*b^3*c^3 + 28*a*b^2*c^2*d + 24*a^2*b*c*d^2))/4 + (7*b^5*d^4*x^13*(8*a^3*d^3 + 5*b
^3*c^3 + 24*a*b^2*c^2*d + 28*a^2*b*c*d^2))/13 + (a^7*c^6*x^2*(7*a*d + 8*b*c))/2 + (b^7*d^6*x^15*(8*a*d + 7*b*c
))/15 + (7*a^6*c^5*x^3*(3*a^2*d^2 + 4*b^2*c^2 + 8*a*b*c*d))/3 + (b^6*d^5*x^14*(4*a^2*d^2 + 3*b^2*c^2 + 8*a*b*c
*d))/2

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sympy [B]  time = 0.21, size = 1046, normalized size = 5.23 \begin {gather*} a^{8} c^{7} x + \frac {b^{8} d^{7} x^{16}}{16} + x^{15} \left (\frac {8 a b^{7} d^{7}}{15} + \frac {7 b^{8} c d^{6}}{15}\right ) + x^{14} \left (2 a^{2} b^{6} d^{7} + 4 a b^{7} c d^{6} + \frac {3 b^{8} c^{2} d^{5}}{2}\right ) + x^{13} \left (\frac {56 a^{3} b^{5} d^{7}}{13} + \frac {196 a^{2} b^{6} c d^{6}}{13} + \frac {168 a b^{7} c^{2} d^{5}}{13} + \frac {35 b^{8} c^{3} d^{4}}{13}\right ) + x^{12} \left (\frac {35 a^{4} b^{4} d^{7}}{6} + \frac {98 a^{3} b^{5} c d^{6}}{3} + 49 a^{2} b^{6} c^{2} d^{5} + \frac {70 a b^{7} c^{3} d^{4}}{3} + \frac {35 b^{8} c^{4} d^{3}}{12}\right ) + x^{11} \left (\frac {56 a^{5} b^{3} d^{7}}{11} + \frac {490 a^{4} b^{4} c d^{6}}{11} + \frac {1176 a^{3} b^{5} c^{2} d^{5}}{11} + \frac {980 a^{2} b^{6} c^{3} d^{4}}{11} + \frac {280 a b^{7} c^{4} d^{3}}{11} + \frac {21 b^{8} c^{5} d^{2}}{11}\right ) + x^{10} \left (\frac {14 a^{6} b^{2} d^{7}}{5} + \frac {196 a^{5} b^{3} c d^{6}}{5} + 147 a^{4} b^{4} c^{2} d^{5} + 196 a^{3} b^{5} c^{3} d^{4} + 98 a^{2} b^{6} c^{4} d^{3} + \frac {84 a b^{7} c^{5} d^{2}}{5} + \frac {7 b^{8} c^{6} d}{10}\right ) + x^{9} \left (\frac {8 a^{7} b d^{7}}{9} + \frac {196 a^{6} b^{2} c d^{6}}{9} + \frac {392 a^{5} b^{3} c^{2} d^{5}}{3} + \frac {2450 a^{4} b^{4} c^{3} d^{4}}{9} + \frac {1960 a^{3} b^{5} c^{4} d^{3}}{9} + \frac {196 a^{2} b^{6} c^{5} d^{2}}{3} + \frac {56 a b^{7} c^{6} d}{9} + \frac {b^{8} c^{7}}{9}\right ) + x^{8} \left (\frac {a^{8} d^{7}}{8} + 7 a^{7} b c d^{6} + \frac {147 a^{6} b^{2} c^{2} d^{5}}{2} + 245 a^{5} b^{3} c^{3} d^{4} + \frac {1225 a^{4} b^{4} c^{4} d^{3}}{4} + 147 a^{3} b^{5} c^{5} d^{2} + \frac {49 a^{2} b^{6} c^{6} d}{2} + a b^{7} c^{7}\right ) + x^{7} \left (a^{8} c d^{6} + 24 a^{7} b c^{2} d^{5} + 140 a^{6} b^{2} c^{3} d^{4} + 280 a^{5} b^{3} c^{4} d^{3} + 210 a^{4} b^{4} c^{5} d^{2} + 56 a^{3} b^{5} c^{6} d + 4 a^{2} b^{6} c^{7}\right ) + x^{6} \left (\frac {7 a^{8} c^{2} d^{5}}{2} + \frac {140 a^{7} b c^{3} d^{4}}{3} + \frac {490 a^{6} b^{2} c^{4} d^{3}}{3} + 196 a^{5} b^{3} c^{5} d^{2} + \frac {245 a^{4} b^{4} c^{6} d}{3} + \frac {28 a^{3} b^{5} c^{7}}{3}\right ) + x^{5} \left (7 a^{8} c^{3} d^{4} + 56 a^{7} b c^{4} d^{3} + \frac {588 a^{6} b^{2} c^{5} d^{2}}{5} + \frac {392 a^{5} b^{3} c^{6} d}{5} + 14 a^{4} b^{4} c^{7}\right ) + x^{4} \left (\frac {35 a^{8} c^{4} d^{3}}{4} + 42 a^{7} b c^{5} d^{2} + 49 a^{6} b^{2} c^{6} d + 14 a^{5} b^{3} c^{7}\right ) + x^{3} \left (7 a^{8} c^{5} d^{2} + \frac {56 a^{7} b c^{6} d}{3} + \frac {28 a^{6} b^{2} c^{7}}{3}\right ) + x^{2} \left (\frac {7 a^{8} c^{6} d}{2} + 4 a^{7} b c^{7}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**8*(d*x+c)**7,x)

[Out]

a**8*c**7*x + b**8*d**7*x**16/16 + x**15*(8*a*b**7*d**7/15 + 7*b**8*c*d**6/15) + x**14*(2*a**2*b**6*d**7 + 4*a
*b**7*c*d**6 + 3*b**8*c**2*d**5/2) + x**13*(56*a**3*b**5*d**7/13 + 196*a**2*b**6*c*d**6/13 + 168*a*b**7*c**2*d
**5/13 + 35*b**8*c**3*d**4/13) + x**12*(35*a**4*b**4*d**7/6 + 98*a**3*b**5*c*d**6/3 + 49*a**2*b**6*c**2*d**5 +
70*a*b**7*c**3*d**4/3 + 35*b**8*c**4*d**3/12) + x**11*(56*a**5*b**3*d**7/11 + 490*a**4*b**4*c*d**6/11 + 1176*
a**3*b**5*c**2*d**5/11 + 980*a**2*b**6*c**3*d**4/11 + 280*a*b**7*c**4*d**3/11 + 21*b**8*c**5*d**2/11) + x**10*
(14*a**6*b**2*d**7/5 + 196*a**5*b**3*c*d**6/5 + 147*a**4*b**4*c**2*d**5 + 196*a**3*b**5*c**3*d**4 + 98*a**2*b*
*6*c**4*d**3 + 84*a*b**7*c**5*d**2/5 + 7*b**8*c**6*d/10) + x**9*(8*a**7*b*d**7/9 + 196*a**6*b**2*c*d**6/9 + 39
2*a**5*b**3*c**2*d**5/3 + 2450*a**4*b**4*c**3*d**4/9 + 1960*a**3*b**5*c**4*d**3/9 + 196*a**2*b**6*c**5*d**2/3
+ 56*a*b**7*c**6*d/9 + b**8*c**7/9) + x**8*(a**8*d**7/8 + 7*a**7*b*c*d**6 + 147*a**6*b**2*c**2*d**5/2 + 245*a*
*5*b**3*c**3*d**4 + 1225*a**4*b**4*c**4*d**3/4 + 147*a**3*b**5*c**5*d**2 + 49*a**2*b**6*c**6*d/2 + a*b**7*c**7
) + x**7*(a**8*c*d**6 + 24*a**7*b*c**2*d**5 + 140*a**6*b**2*c**3*d**4 + 280*a**5*b**3*c**4*d**3 + 210*a**4*b**
4*c**5*d**2 + 56*a**3*b**5*c**6*d + 4*a**2*b**6*c**7) + x**6*(7*a**8*c**2*d**5/2 + 140*a**7*b*c**3*d**4/3 + 49
0*a**6*b**2*c**4*d**3/3 + 196*a**5*b**3*c**5*d**2 + 245*a**4*b**4*c**6*d/3 + 28*a**3*b**5*c**7/3) + x**5*(7*a*
*8*c**3*d**4 + 56*a**7*b*c**4*d**3 + 588*a**6*b**2*c**5*d**2/5 + 392*a**5*b**3*c**6*d/5 + 14*a**4*b**4*c**7) +
x**4*(35*a**8*c**4*d**3/4 + 42*a**7*b*c**5*d**2 + 49*a**6*b**2*c**6*d + 14*a**5*b**3*c**7) + x**3*(7*a**8*c**
5*d**2 + 56*a**7*b*c**6*d/3 + 28*a**6*b**2*c**7/3) + x**2*(7*a**8*c**6*d/2 + 4*a**7*b*c**7)

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