3.1273 \(\int (a+b x)^9 (c+d x)^7 \, dx\)

Optimal. Leaf size=200 \[ \frac{7 d^6 (a+b x)^{16} (b c-a d)}{16 b^8}+\frac{7 d^5 (a+b x)^{15} (b c-a d)^2}{5 b^8}+\frac{5 d^4 (a+b x)^{14} (b c-a d)^3}{2 b^8}+\frac{35 d^3 (a+b x)^{13} (b c-a d)^4}{13 b^8}+\frac{7 d^2 (a+b x)^{12} (b c-a d)^5}{4 b^8}+\frac{7 d (a+b x)^{11} (b c-a d)^6}{11 b^8}+\frac{(a+b x)^{10} (b c-a d)^7}{10 b^8}+\frac{d^7 (a+b x)^{17}}{17 b^8} \]

[Out]

((b*c - a*d)^7*(a + b*x)^10)/(10*b^8) + (7*d*(b*c - a*d)^6*(a + b*x)^11)/(11*b^8) + (7*d^2*(b*c - a*d)^5*(a +
b*x)^12)/(4*b^8) + (35*d^3*(b*c - a*d)^4*(a + b*x)^13)/(13*b^8) + (5*d^4*(b*c - a*d)^3*(a + b*x)^14)/(2*b^8) +
(7*d^5*(b*c - a*d)^2*(a + b*x)^15)/(5*b^8) + (7*d^6*(b*c - a*d)*(a + b*x)^16)/(16*b^8) + (d^7*(a + b*x)^17)/(
17*b^8)

________________________________________________________________________________________

Rubi [A]  time = 0.676466, 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, Rules used = {43} \[ \frac{7 d^6 (a+b x)^{16} (b c-a d)}{16 b^8}+\frac{7 d^5 (a+b x)^{15} (b c-a d)^2}{5 b^8}+\frac{5 d^4 (a+b x)^{14} (b c-a d)^3}{2 b^8}+\frac{35 d^3 (a+b x)^{13} (b c-a d)^4}{13 b^8}+\frac{7 d^2 (a+b x)^{12} (b c-a d)^5}{4 b^8}+\frac{7 d (a+b x)^{11} (b c-a d)^6}{11 b^8}+\frac{(a+b x)^{10} (b c-a d)^7}{10 b^8}+\frac{d^7 (a+b x)^{17}}{17 b^8} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^9*(c + d*x)^7,x]

[Out]

((b*c - a*d)^7*(a + b*x)^10)/(10*b^8) + (7*d*(b*c - a*d)^6*(a + b*x)^11)/(11*b^8) + (7*d^2*(b*c - a*d)^5*(a +
b*x)^12)/(4*b^8) + (35*d^3*(b*c - a*d)^4*(a + b*x)^13)/(13*b^8) + (5*d^4*(b*c - a*d)^3*(a + b*x)^14)/(2*b^8) +
(7*d^5*(b*c - a*d)^2*(a + b*x)^15)/(5*b^8) + (7*d^6*(b*c - a*d)*(a + b*x)^16)/(16*b^8) + (d^7*(a + b*x)^17)/(
17*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)^9 (c+d x)^7 \, dx &=\int \left (\frac{(b c-a d)^7 (a+b x)^9}{b^7}+\frac{7 d (b c-a d)^6 (a+b x)^{10}}{b^7}+\frac{21 d^2 (b c-a d)^5 (a+b x)^{11}}{b^7}+\frac{35 d^3 (b c-a d)^4 (a+b x)^{12}}{b^7}+\frac{35 d^4 (b c-a d)^3 (a+b x)^{13}}{b^7}+\frac{21 d^5 (b c-a d)^2 (a+b x)^{14}}{b^7}+\frac{7 d^6 (b c-a d) (a+b x)^{15}}{b^7}+\frac{d^7 (a+b x)^{16}}{b^7}\right ) \, dx\\ &=\frac{(b c-a d)^7 (a+b x)^{10}}{10 b^8}+\frac{7 d (b c-a d)^6 (a+b x)^{11}}{11 b^8}+\frac{7 d^2 (b c-a d)^5 (a+b x)^{12}}{4 b^8}+\frac{35 d^3 (b c-a d)^4 (a+b x)^{13}}{13 b^8}+\frac{5 d^4 (b c-a d)^3 (a+b x)^{14}}{2 b^8}+\frac{7 d^5 (b c-a d)^2 (a+b x)^{15}}{5 b^8}+\frac{7 d^6 (b c-a d) (a+b x)^{16}}{16 b^8}+\frac{d^7 (a+b x)^{17}}{17 b^8}\\ \end{align*}

Mathematica [B]  time = 0.139842, size = 993, normalized size = 4.96 \[ \frac{1}{17} b^9 d^7 x^{17}+\frac{1}{16} b^8 d^6 (7 b c+9 a d) x^{16}+\frac{1}{5} b^7 d^5 \left (7 b^2 c^2+21 a b d c+12 a^2 d^2\right ) x^{15}+\frac{1}{2} b^6 d^4 \left (5 b^3 c^3+27 a b^2 d c^2+36 a^2 b d^2 c+12 a^3 d^3\right ) x^{14}+\frac{7}{13} b^5 d^3 \left (5 b^4 c^4+45 a b^3 d c^3+108 a^2 b^2 d^2 c^2+84 a^3 b d^3 c+18 a^4 d^4\right ) x^{13}+\frac{7}{4} b^4 d^2 \left (b^5 c^5+15 a b^4 d c^4+60 a^2 b^3 d^2 c^3+84 a^3 b^2 d^3 c^2+42 a^4 b d^4 c+6 a^5 d^5\right ) x^{12}+\frac{7}{11} b^3 d \left (b^6 c^6+27 a b^5 d c^5+180 a^2 b^4 d^2 c^4+420 a^3 b^3 d^3 c^3+378 a^4 b^2 d^4 c^2+126 a^5 b d^5 c+12 a^6 d^6\right ) x^{11}+\frac{1}{10} b^2 \left (b^7 c^7+63 a b^6 d c^6+756 a^2 b^5 d^2 c^5+2940 a^3 b^4 d^3 c^4+4410 a^4 b^3 d^4 c^3+2646 a^5 b^2 d^5 c^2+588 a^6 b d^6 c+36 a^7 d^7\right ) x^{10}+a b \left (b^7 c^7+28 a b^6 d c^6+196 a^2 b^5 d^2 c^5+490 a^3 b^4 d^3 c^4+490 a^4 b^3 d^4 c^3+196 a^5 b^2 d^5 c^2+28 a^6 b d^6 c+a^7 d^7\right ) x^9+\frac{1}{8} a^2 \left (36 b^7 c^7+588 a b^6 d c^6+2646 a^2 b^5 d^2 c^5+4410 a^3 b^4 d^3 c^4+2940 a^4 b^3 d^4 c^3+756 a^5 b^2 d^5 c^2+63 a^6 b d^6 c+a^7 d^7\right ) x^8+a^3 c \left (12 b^6 c^6+126 a b^5 d c^5+378 a^2 b^4 d^2 c^4+420 a^3 b^3 d^3 c^3+180 a^4 b^2 d^4 c^2+27 a^5 b d^5 c+a^6 d^6\right ) x^7+\frac{7}{2} a^4 c^2 \left (6 b^5 c^5+42 a b^4 d c^4+84 a^2 b^3 d^2 c^3+60 a^3 b^2 d^3 c^2+15 a^4 b d^4 c+a^5 d^5\right ) x^6+\frac{7}{5} a^5 c^3 \left (18 b^4 c^4+84 a b^3 d c^3+108 a^2 b^2 d^2 c^2+45 a^3 b d^3 c+5 a^4 d^4\right ) x^5+\frac{7}{4} a^6 c^4 \left (12 b^3 c^3+36 a b^2 d c^2+27 a^2 b d^2 c+5 a^3 d^3\right ) x^4+a^7 c^5 \left (12 b^2 c^2+21 a b d c+7 a^2 d^2\right ) x^3+\frac{1}{2} a^8 c^6 (9 b c+7 a d) x^2+a^9 c^7 x \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^9*(c + d*x)^7,x]

[Out]

a^9*c^7*x + (a^8*c^6*(9*b*c + 7*a*d)*x^2)/2 + a^7*c^5*(12*b^2*c^2 + 21*a*b*c*d + 7*a^2*d^2)*x^3 + (7*a^6*c^4*(
12*b^3*c^3 + 36*a*b^2*c^2*d + 27*a^2*b*c*d^2 + 5*a^3*d^3)*x^4)/4 + (7*a^5*c^3*(18*b^4*c^4 + 84*a*b^3*c^3*d + 1
08*a^2*b^2*c^2*d^2 + 45*a^3*b*c*d^3 + 5*a^4*d^4)*x^5)/5 + (7*a^4*c^2*(6*b^5*c^5 + 42*a*b^4*c^4*d + 84*a^2*b^3*
c^3*d^2 + 60*a^3*b^2*c^2*d^3 + 15*a^4*b*c*d^4 + a^5*d^5)*x^6)/2 + a^3*c*(12*b^6*c^6 + 126*a*b^5*c^5*d + 378*a^
2*b^4*c^4*d^2 + 420*a^3*b^3*c^3*d^3 + 180*a^4*b^2*c^2*d^4 + 27*a^5*b*c*d^5 + a^6*d^6)*x^7 + (a^2*(36*b^7*c^7 +
588*a*b^6*c^6*d + 2646*a^2*b^5*c^5*d^2 + 4410*a^3*b^4*c^4*d^3 + 2940*a^4*b^3*c^3*d^4 + 756*a^5*b^2*c^2*d^5 +
63*a^6*b*c*d^6 + a^7*d^7)*x^8)/8 + a*b*(b^7*c^7 + 28*a*b^6*c^6*d + 196*a^2*b^5*c^5*d^2 + 490*a^3*b^4*c^4*d^3 +
490*a^4*b^3*c^3*d^4 + 196*a^5*b^2*c^2*d^5 + 28*a^6*b*c*d^6 + a^7*d^7)*x^9 + (b^2*(b^7*c^7 + 63*a*b^6*c^6*d +
756*a^2*b^5*c^5*d^2 + 2940*a^3*b^4*c^4*d^3 + 4410*a^4*b^3*c^3*d^4 + 2646*a^5*b^2*c^2*d^5 + 588*a^6*b*c*d^6 + 3
6*a^7*d^7)*x^10)/10 + (7*b^3*d*(b^6*c^6 + 27*a*b^5*c^5*d + 180*a^2*b^4*c^4*d^2 + 420*a^3*b^3*c^3*d^3 + 378*a^4
*b^2*c^2*d^4 + 126*a^5*b*c*d^5 + 12*a^6*d^6)*x^11)/11 + (7*b^4*d^2*(b^5*c^5 + 15*a*b^4*c^4*d + 60*a^2*b^3*c^3*
d^2 + 84*a^3*b^2*c^2*d^3 + 42*a^4*b*c*d^4 + 6*a^5*d^5)*x^12)/4 + (7*b^5*d^3*(5*b^4*c^4 + 45*a*b^3*c^3*d + 108*
a^2*b^2*c^2*d^2 + 84*a^3*b*c*d^3 + 18*a^4*d^4)*x^13)/13 + (b^6*d^4*(5*b^3*c^3 + 27*a*b^2*c^2*d + 36*a^2*b*c*d^
2 + 12*a^3*d^3)*x^14)/2 + (b^7*d^5*(7*b^2*c^2 + 21*a*b*c*d + 12*a^2*d^2)*x^15)/5 + (b^8*d^6*(7*b*c + 9*a*d)*x^
16)/16 + (b^9*d^7*x^17)/17

________________________________________________________________________________________

Maple [B]  time = 0.003, size = 1033, normalized size = 5.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^9*(d*x+c)^7,x)

[Out]

1/17*b^9*d^7*x^17+1/16*(9*a*b^8*d^7+7*b^9*c*d^6)*x^16+1/15*(36*a^2*b^7*d^7+63*a*b^8*c*d^6+21*b^9*c^2*d^5)*x^15
+1/14*(84*a^3*b^6*d^7+252*a^2*b^7*c*d^6+189*a*b^8*c^2*d^5+35*b^9*c^3*d^4)*x^14+1/13*(126*a^4*b^5*d^7+588*a^3*b
^6*c*d^6+756*a^2*b^7*c^2*d^5+315*a*b^8*c^3*d^4+35*b^9*c^4*d^3)*x^13+1/12*(126*a^5*b^4*d^7+882*a^4*b^5*c*d^6+17
64*a^3*b^6*c^2*d^5+1260*a^2*b^7*c^3*d^4+315*a*b^8*c^4*d^3+21*b^9*c^5*d^2)*x^12+1/11*(84*a^6*b^3*d^7+882*a^5*b^
4*c*d^6+2646*a^4*b^5*c^2*d^5+2940*a^3*b^6*c^3*d^4+1260*a^2*b^7*c^4*d^3+189*a*b^8*c^5*d^2+7*b^9*c^6*d)*x^11+1/1
0*(36*a^7*b^2*d^7+588*a^6*b^3*c*d^6+2646*a^5*b^4*c^2*d^5+4410*a^4*b^5*c^3*d^4+2940*a^3*b^6*c^4*d^3+756*a^2*b^7
*c^5*d^2+63*a*b^8*c^6*d+b^9*c^7)*x^10+1/9*(9*a^8*b*d^7+252*a^7*b^2*c*d^6+1764*a^6*b^3*c^2*d^5+4410*a^5*b^4*c^3
*d^4+4410*a^4*b^5*c^4*d^3+1764*a^3*b^6*c^5*d^2+252*a^2*b^7*c^6*d+9*a*b^8*c^7)*x^9+1/8*(a^9*d^7+63*a^8*b*c*d^6+
756*a^7*b^2*c^2*d^5+2940*a^6*b^3*c^3*d^4+4410*a^5*b^4*c^4*d^3+2646*a^4*b^5*c^5*d^2+588*a^3*b^6*c^6*d+36*a^2*b^
7*c^7)*x^8+1/7*(7*a^9*c*d^6+189*a^8*b*c^2*d^5+1260*a^7*b^2*c^3*d^4+2940*a^6*b^3*c^4*d^3+2646*a^5*b^4*c^5*d^2+8
82*a^4*b^5*c^6*d+84*a^3*b^6*c^7)*x^7+1/6*(21*a^9*c^2*d^5+315*a^8*b*c^3*d^4+1260*a^7*b^2*c^4*d^3+1764*a^6*b^3*c
^5*d^2+882*a^5*b^4*c^6*d+126*a^4*b^5*c^7)*x^6+1/5*(35*a^9*c^3*d^4+315*a^8*b*c^4*d^3+756*a^7*b^2*c^5*d^2+588*a^
6*b^3*c^6*d+126*a^5*b^4*c^7)*x^5+1/4*(35*a^9*c^4*d^3+189*a^8*b*c^5*d^2+252*a^7*b^2*c^6*d+84*a^6*b^3*c^7)*x^4+1
/3*(21*a^9*c^5*d^2+63*a^8*b*c^6*d+36*a^7*b^2*c^7)*x^3+1/2*(7*a^9*c^6*d+9*a^8*b*c^7)*x^2+a^9*c^7*x

________________________________________________________________________________________

Maxima [B]  time = 0.980516, size = 1381, normalized size = 6.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^9*(d*x+c)^7,x, algorithm="maxima")

[Out]

1/17*b^9*d^7*x^17 + a^9*c^7*x + 1/16*(7*b^9*c*d^6 + 9*a*b^8*d^7)*x^16 + 1/5*(7*b^9*c^2*d^5 + 21*a*b^8*c*d^6 +
12*a^2*b^7*d^7)*x^15 + 1/2*(5*b^9*c^3*d^4 + 27*a*b^8*c^2*d^5 + 36*a^2*b^7*c*d^6 + 12*a^3*b^6*d^7)*x^14 + 7/13*
(5*b^9*c^4*d^3 + 45*a*b^8*c^3*d^4 + 108*a^2*b^7*c^2*d^5 + 84*a^3*b^6*c*d^6 + 18*a^4*b^5*d^7)*x^13 + 7/4*(b^9*c
^5*d^2 + 15*a*b^8*c^4*d^3 + 60*a^2*b^7*c^3*d^4 + 84*a^3*b^6*c^2*d^5 + 42*a^4*b^5*c*d^6 + 6*a^5*b^4*d^7)*x^12 +
7/11*(b^9*c^6*d + 27*a*b^8*c^5*d^2 + 180*a^2*b^7*c^4*d^3 + 420*a^3*b^6*c^3*d^4 + 378*a^4*b^5*c^2*d^5 + 126*a^
5*b^4*c*d^6 + 12*a^6*b^3*d^7)*x^11 + 1/10*(b^9*c^7 + 63*a*b^8*c^6*d + 756*a^2*b^7*c^5*d^2 + 2940*a^3*b^6*c^4*d
^3 + 4410*a^4*b^5*c^3*d^4 + 2646*a^5*b^4*c^2*d^5 + 588*a^6*b^3*c*d^6 + 36*a^7*b^2*d^7)*x^10 + (a*b^8*c^7 + 28*
a^2*b^7*c^6*d + 196*a^3*b^6*c^5*d^2 + 490*a^4*b^5*c^4*d^3 + 490*a^5*b^4*c^3*d^4 + 196*a^6*b^3*c^2*d^5 + 28*a^7
*b^2*c*d^6 + a^8*b*d^7)*x^9 + 1/8*(36*a^2*b^7*c^7 + 588*a^3*b^6*c^6*d + 2646*a^4*b^5*c^5*d^2 + 4410*a^5*b^4*c^
4*d^3 + 2940*a^6*b^3*c^3*d^4 + 756*a^7*b^2*c^2*d^5 + 63*a^8*b*c*d^6 + a^9*d^7)*x^8 + (12*a^3*b^6*c^7 + 126*a^4
*b^5*c^6*d + 378*a^5*b^4*c^5*d^2 + 420*a^6*b^3*c^4*d^3 + 180*a^7*b^2*c^3*d^4 + 27*a^8*b*c^2*d^5 + a^9*c*d^6)*x
^7 + 7/2*(6*a^4*b^5*c^7 + 42*a^5*b^4*c^6*d + 84*a^6*b^3*c^5*d^2 + 60*a^7*b^2*c^4*d^3 + 15*a^8*b*c^3*d^4 + a^9*
c^2*d^5)*x^6 + 7/5*(18*a^5*b^4*c^7 + 84*a^6*b^3*c^6*d + 108*a^7*b^2*c^5*d^2 + 45*a^8*b*c^4*d^3 + 5*a^9*c^3*d^4
)*x^5 + 7/4*(12*a^6*b^3*c^7 + 36*a^7*b^2*c^6*d + 27*a^8*b*c^5*d^2 + 5*a^9*c^4*d^3)*x^4 + (12*a^7*b^2*c^7 + 21*
a^8*b*c^6*d + 7*a^9*c^5*d^2)*x^3 + 1/2*(9*a^8*b*c^7 + 7*a^9*c^6*d)*x^2

________________________________________________________________________________________

Fricas [B]  time = 1.92556, size = 2626, normalized size = 13.13 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^9*(d*x+c)^7,x, algorithm="fricas")

[Out]

1/17*x^17*d^7*b^9 + 7/16*x^16*d^6*c*b^9 + 9/16*x^16*d^7*b^8*a + 7/5*x^15*d^5*c^2*b^9 + 21/5*x^15*d^6*c*b^8*a +
12/5*x^15*d^7*b^7*a^2 + 5/2*x^14*d^4*c^3*b^9 + 27/2*x^14*d^5*c^2*b^8*a + 18*x^14*d^6*c*b^7*a^2 + 6*x^14*d^7*b
^6*a^3 + 35/13*x^13*d^3*c^4*b^9 + 315/13*x^13*d^4*c^3*b^8*a + 756/13*x^13*d^5*c^2*b^7*a^2 + 588/13*x^13*d^6*c*
b^6*a^3 + 126/13*x^13*d^7*b^5*a^4 + 7/4*x^12*d^2*c^5*b^9 + 105/4*x^12*d^3*c^4*b^8*a + 105*x^12*d^4*c^3*b^7*a^2
+ 147*x^12*d^5*c^2*b^6*a^3 + 147/2*x^12*d^6*c*b^5*a^4 + 21/2*x^12*d^7*b^4*a^5 + 7/11*x^11*d*c^6*b^9 + 189/11*
x^11*d^2*c^5*b^8*a + 1260/11*x^11*d^3*c^4*b^7*a^2 + 2940/11*x^11*d^4*c^3*b^6*a^3 + 2646/11*x^11*d^5*c^2*b^5*a^
4 + 882/11*x^11*d^6*c*b^4*a^5 + 84/11*x^11*d^7*b^3*a^6 + 1/10*x^10*c^7*b^9 + 63/10*x^10*d*c^6*b^8*a + 378/5*x^
10*d^2*c^5*b^7*a^2 + 294*x^10*d^3*c^4*b^6*a^3 + 441*x^10*d^4*c^3*b^5*a^4 + 1323/5*x^10*d^5*c^2*b^4*a^5 + 294/5
*x^10*d^6*c*b^3*a^6 + 18/5*x^10*d^7*b^2*a^7 + x^9*c^7*b^8*a + 28*x^9*d*c^6*b^7*a^2 + 196*x^9*d^2*c^5*b^6*a^3 +
490*x^9*d^3*c^4*b^5*a^4 + 490*x^9*d^4*c^3*b^4*a^5 + 196*x^9*d^5*c^2*b^3*a^6 + 28*x^9*d^6*c*b^2*a^7 + x^9*d^7*
b*a^8 + 9/2*x^8*c^7*b^7*a^2 + 147/2*x^8*d*c^6*b^6*a^3 + 1323/4*x^8*d^2*c^5*b^5*a^4 + 2205/4*x^8*d^3*c^4*b^4*a^
5 + 735/2*x^8*d^4*c^3*b^3*a^6 + 189/2*x^8*d^5*c^2*b^2*a^7 + 63/8*x^8*d^6*c*b*a^8 + 1/8*x^8*d^7*a^9 + 12*x^7*c^
7*b^6*a^3 + 126*x^7*d*c^6*b^5*a^4 + 378*x^7*d^2*c^5*b^4*a^5 + 420*x^7*d^3*c^4*b^3*a^6 + 180*x^7*d^4*c^3*b^2*a^
7 + 27*x^7*d^5*c^2*b*a^8 + x^7*d^6*c*a^9 + 21*x^6*c^7*b^5*a^4 + 147*x^6*d*c^6*b^4*a^5 + 294*x^6*d^2*c^5*b^3*a^
6 + 210*x^6*d^3*c^4*b^2*a^7 + 105/2*x^6*d^4*c^3*b*a^8 + 7/2*x^6*d^5*c^2*a^9 + 126/5*x^5*c^7*b^4*a^5 + 588/5*x^
5*d*c^6*b^3*a^6 + 756/5*x^5*d^2*c^5*b^2*a^7 + 63*x^5*d^3*c^4*b*a^8 + 7*x^5*d^4*c^3*a^9 + 21*x^4*c^7*b^3*a^6 +
63*x^4*d*c^6*b^2*a^7 + 189/4*x^4*d^2*c^5*b*a^8 + 35/4*x^4*d^3*c^4*a^9 + 12*x^3*c^7*b^2*a^7 + 21*x^3*d*c^6*b*a^
8 + 7*x^3*d^2*c^5*a^9 + 9/2*x^2*c^7*b*a^8 + 7/2*x^2*d*c^6*a^9 + x*c^7*a^9

________________________________________________________________________________________

Sympy [B]  time = 0.204455, size = 1163, normalized size = 5.82 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**9*(d*x+c)**7,x)

[Out]

a**9*c**7*x + b**9*d**7*x**17/17 + x**16*(9*a*b**8*d**7/16 + 7*b**9*c*d**6/16) + x**15*(12*a**2*b**7*d**7/5 +
21*a*b**8*c*d**6/5 + 7*b**9*c**2*d**5/5) + x**14*(6*a**3*b**6*d**7 + 18*a**2*b**7*c*d**6 + 27*a*b**8*c**2*d**5
/2 + 5*b**9*c**3*d**4/2) + x**13*(126*a**4*b**5*d**7/13 + 588*a**3*b**6*c*d**6/13 + 756*a**2*b**7*c**2*d**5/13
+ 315*a*b**8*c**3*d**4/13 + 35*b**9*c**4*d**3/13) + x**12*(21*a**5*b**4*d**7/2 + 147*a**4*b**5*c*d**6/2 + 147
*a**3*b**6*c**2*d**5 + 105*a**2*b**7*c**3*d**4 + 105*a*b**8*c**4*d**3/4 + 7*b**9*c**5*d**2/4) + x**11*(84*a**6
*b**3*d**7/11 + 882*a**5*b**4*c*d**6/11 + 2646*a**4*b**5*c**2*d**5/11 + 2940*a**3*b**6*c**3*d**4/11 + 1260*a**
2*b**7*c**4*d**3/11 + 189*a*b**8*c**5*d**2/11 + 7*b**9*c**6*d/11) + x**10*(18*a**7*b**2*d**7/5 + 294*a**6*b**3
*c*d**6/5 + 1323*a**5*b**4*c**2*d**5/5 + 441*a**4*b**5*c**3*d**4 + 294*a**3*b**6*c**4*d**3 + 378*a**2*b**7*c**
5*d**2/5 + 63*a*b**8*c**6*d/10 + b**9*c**7/10) + x**9*(a**8*b*d**7 + 28*a**7*b**2*c*d**6 + 196*a**6*b**3*c**2*
d**5 + 490*a**5*b**4*c**3*d**4 + 490*a**4*b**5*c**4*d**3 + 196*a**3*b**6*c**5*d**2 + 28*a**2*b**7*c**6*d + a*b
**8*c**7) + x**8*(a**9*d**7/8 + 63*a**8*b*c*d**6/8 + 189*a**7*b**2*c**2*d**5/2 + 735*a**6*b**3*c**3*d**4/2 + 2
205*a**5*b**4*c**4*d**3/4 + 1323*a**4*b**5*c**5*d**2/4 + 147*a**3*b**6*c**6*d/2 + 9*a**2*b**7*c**7/2) + x**7*(
a**9*c*d**6 + 27*a**8*b*c**2*d**5 + 180*a**7*b**2*c**3*d**4 + 420*a**6*b**3*c**4*d**3 + 378*a**5*b**4*c**5*d**
2 + 126*a**4*b**5*c**6*d + 12*a**3*b**6*c**7) + x**6*(7*a**9*c**2*d**5/2 + 105*a**8*b*c**3*d**4/2 + 210*a**7*b
**2*c**4*d**3 + 294*a**6*b**3*c**5*d**2 + 147*a**5*b**4*c**6*d + 21*a**4*b**5*c**7) + x**5*(7*a**9*c**3*d**4 +
63*a**8*b*c**4*d**3 + 756*a**7*b**2*c**5*d**2/5 + 588*a**6*b**3*c**6*d/5 + 126*a**5*b**4*c**7/5) + x**4*(35*a
**9*c**4*d**3/4 + 189*a**8*b*c**5*d**2/4 + 63*a**7*b**2*c**6*d + 21*a**6*b**3*c**7) + x**3*(7*a**9*c**5*d**2 +
21*a**8*b*c**6*d + 12*a**7*b**2*c**7) + x**2*(7*a**9*c**6*d/2 + 9*a**8*b*c**7/2)

________________________________________________________________________________________

Giac [B]  time = 1.07534, size = 1586, normalized size = 7.93 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^9*(d*x+c)^7,x, algorithm="giac")

[Out]

1/17*b^9*d^7*x^17 + 7/16*b^9*c*d^6*x^16 + 9/16*a*b^8*d^7*x^16 + 7/5*b^9*c^2*d^5*x^15 + 21/5*a*b^8*c*d^6*x^15 +
12/5*a^2*b^7*d^7*x^15 + 5/2*b^9*c^3*d^4*x^14 + 27/2*a*b^8*c^2*d^5*x^14 + 18*a^2*b^7*c*d^6*x^14 + 6*a^3*b^6*d^
7*x^14 + 35/13*b^9*c^4*d^3*x^13 + 315/13*a*b^8*c^3*d^4*x^13 + 756/13*a^2*b^7*c^2*d^5*x^13 + 588/13*a^3*b^6*c*d
^6*x^13 + 126/13*a^4*b^5*d^7*x^13 + 7/4*b^9*c^5*d^2*x^12 + 105/4*a*b^8*c^4*d^3*x^12 + 105*a^2*b^7*c^3*d^4*x^12
+ 147*a^3*b^6*c^2*d^5*x^12 + 147/2*a^4*b^5*c*d^6*x^12 + 21/2*a^5*b^4*d^7*x^12 + 7/11*b^9*c^6*d*x^11 + 189/11*
a*b^8*c^5*d^2*x^11 + 1260/11*a^2*b^7*c^4*d^3*x^11 + 2940/11*a^3*b^6*c^3*d^4*x^11 + 2646/11*a^4*b^5*c^2*d^5*x^1
1 + 882/11*a^5*b^4*c*d^6*x^11 + 84/11*a^6*b^3*d^7*x^11 + 1/10*b^9*c^7*x^10 + 63/10*a*b^8*c^6*d*x^10 + 378/5*a^
2*b^7*c^5*d^2*x^10 + 294*a^3*b^6*c^4*d^3*x^10 + 441*a^4*b^5*c^3*d^4*x^10 + 1323/5*a^5*b^4*c^2*d^5*x^10 + 294/5
*a^6*b^3*c*d^6*x^10 + 18/5*a^7*b^2*d^7*x^10 + a*b^8*c^7*x^9 + 28*a^2*b^7*c^6*d*x^9 + 196*a^3*b^6*c^5*d^2*x^9 +
490*a^4*b^5*c^4*d^3*x^9 + 490*a^5*b^4*c^3*d^4*x^9 + 196*a^6*b^3*c^2*d^5*x^9 + 28*a^7*b^2*c*d^6*x^9 + a^8*b*d^
7*x^9 + 9/2*a^2*b^7*c^7*x^8 + 147/2*a^3*b^6*c^6*d*x^8 + 1323/4*a^4*b^5*c^5*d^2*x^8 + 2205/4*a^5*b^4*c^4*d^3*x^
8 + 735/2*a^6*b^3*c^3*d^4*x^8 + 189/2*a^7*b^2*c^2*d^5*x^8 + 63/8*a^8*b*c*d^6*x^8 + 1/8*a^9*d^7*x^8 + 12*a^3*b^
6*c^7*x^7 + 126*a^4*b^5*c^6*d*x^7 + 378*a^5*b^4*c^5*d^2*x^7 + 420*a^6*b^3*c^4*d^3*x^7 + 180*a^7*b^2*c^3*d^4*x^
7 + 27*a^8*b*c^2*d^5*x^7 + a^9*c*d^6*x^7 + 21*a^4*b^5*c^7*x^6 + 147*a^5*b^4*c^6*d*x^6 + 294*a^6*b^3*c^5*d^2*x^
6 + 210*a^7*b^2*c^4*d^3*x^6 + 105/2*a^8*b*c^3*d^4*x^6 + 7/2*a^9*c^2*d^5*x^6 + 126/5*a^5*b^4*c^7*x^5 + 588/5*a^
6*b^3*c^6*d*x^5 + 756/5*a^7*b^2*c^5*d^2*x^5 + 63*a^8*b*c^4*d^3*x^5 + 7*a^9*c^3*d^4*x^5 + 21*a^6*b^3*c^7*x^4 +
63*a^7*b^2*c^6*d*x^4 + 189/4*a^8*b*c^5*d^2*x^4 + 35/4*a^9*c^4*d^3*x^4 + 12*a^7*b^2*c^7*x^3 + 21*a^8*b*c^6*d*x^
3 + 7*a^9*c^5*d^2*x^3 + 9/2*a^8*b*c^7*x^2 + 7/2*a^9*c^6*d*x^2 + a^9*c^7*x