3.1275 \(\int (a+b x)^7 (c+d x)^7 \, dx\)

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

[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)^1
1)/(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)/(15*b^8)

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Rubi [A]  time = 0.893588, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{d^6 (a+b x)^{14} (b c-a d)}{2 b^8}+\frac{21 d^5 (a+b x)^{13} (b c-a d)^2}{13 b^8}+\frac{35 d^4 (a+b x)^{12} (b c-a d)^3}{12 b^8}+\frac{35 d^3 (a+b x)^{11} (b c-a d)^4}{11 b^8}+\frac{21 d^2 (a+b x)^{10} (b c-a d)^5}{10 b^8}+\frac{7 d (a+b x)^9 (b c-a d)^6}{9 b^8}+\frac{(a+b x)^8 (b c-a d)^7}{8 b^8}+\frac{d^7 (a+b x)^{15}}{15 b^8} \]

Antiderivative was successfully verified.

[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)^1
1)/(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)/(15*b^8)

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Rubi in Sympy [A]  time = 107.709, size = 182, normalized size = 0.91 \[ \frac{d^{7} \left (a + b x\right )^{15}}{15 b^{8}} - \frac{d^{6} \left (a + b x\right )^{14} \left (a d - b c\right )}{2 b^{8}} + \frac{21 d^{5} \left (a + b x\right )^{13} \left (a d - b c\right )^{2}}{13 b^{8}} - \frac{35 d^{4} \left (a + b x\right )^{12} \left (a d - b c\right )^{3}}{12 b^{8}} + \frac{35 d^{3} \left (a + b x\right )^{11} \left (a d - b c\right )^{4}}{11 b^{8}} - \frac{21 d^{2} \left (a + b x\right )^{10} \left (a d - b c\right )^{5}}{10 b^{8}} + \frac{7 d \left (a + b x\right )^{9} \left (a d - b c\right )^{6}}{9 b^{8}} - \frac{\left (a + b x\right )^{8} \left (a d - b c\right )^{7}}{8 b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**7*(d*x+c)**7,x)

[Out]

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

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Mathematica [B]  time = 0.156137, size = 785, normalized size = 3.92 \[ a^7 c^7 x+\frac{7}{2} a^6 c^6 x^2 (a d+b c)+\frac{7}{13} b^5 d^5 x^{13} \left (3 a^2 d^2+7 a b c d+3 b^2 c^2\right )+\frac{7}{3} a^5 c^5 x^3 \left (3 a^2 d^2+7 a b c d+3 b^2 c^2\right )+\frac{7}{12} b^4 d^4 x^{12} \left (5 a^3 d^3+21 a^2 b c d^2+21 a b^2 c^2 d+5 b^3 c^3\right )+\frac{7}{4} a^4 c^4 x^4 \left (5 a^3 d^3+21 a^2 b c d^2+21 a b^2 c^2 d+5 b^3 c^3\right )+\frac{7}{11} b^3 d^3 x^{11} \left (5 a^4 d^4+35 a^3 b c d^3+63 a^2 b^2 c^2 d^2+35 a b^3 c^3 d+5 b^4 c^4\right )+\frac{7}{5} a^3 c^3 x^5 \left (5 a^4 d^4+35 a^3 b c d^3+63 a^2 b^2 c^2 d^2+35 a b^3 c^3 d+5 b^4 c^4\right )+\frac{7}{10} b^2 d^2 x^{10} \left (3 a^5 d^5+35 a^4 b c d^4+105 a^3 b^2 c^2 d^3+105 a^2 b^3 c^3 d^2+35 a b^4 c^4 d+3 b^5 c^5\right )+\frac{7}{6} a^2 c^2 x^6 \left (3 a^5 d^5+35 a^4 b c d^4+105 a^3 b^2 c^2 d^3+105 a^2 b^3 c^3 d^2+35 a b^4 c^4 d+3 b^5 c^5\right )+\frac{7}{9} b d x^9 \left (a^6 d^6+21 a^5 b c d^5+105 a^4 b^2 c^2 d^4+175 a^3 b^3 c^3 d^3+105 a^2 b^4 c^4 d^2+21 a b^5 c^5 d+b^6 c^6\right )+a c x^7 \left (a^6 d^6+21 a^5 b c d^5+105 a^4 b^2 c^2 d^4+175 a^3 b^3 c^3 d^3+105 a^2 b^4 c^4 d^2+21 a b^5 c^5 d+b^6 c^6\right )+\frac{1}{8} x^8 \left (a^7 d^7+49 a^6 b c d^6+441 a^5 b^2 c^2 d^5+1225 a^4 b^3 c^3 d^4+1225 a^3 b^4 c^4 d^3+441 a^2 b^5 c^5 d^2+49 a b^6 c^6 d+b^7 c^7\right )+\frac{1}{2} b^6 d^6 x^{14} (a d+b c)+\frac{1}{15} b^7 d^7 x^{15} \]

Antiderivative was successfully verified.

[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 + 63*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 + 10
5*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 + 44
1*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 + 10
5*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)/1
1 + (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)/1
2 + (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

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Maple [B]  time = 0.004, size = 817, normalized size = 4.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^7*(d*x+c)^7,x)

[Out]

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

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Maxima [A]  time = 1.34894, size = 1089, normalized size = 5.44 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 0.189082, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.450143, size = 935, normalized size = 4.68 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.2215, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done