3.1304 \(\int (a+b x)^7 (c+d x)^{10} \, dx\)
Optimal. Leaf size=200 \[ -\frac{7 b^6 (c+d x)^{17} (b c-a d)}{17 d^8}+\frac{21 b^5 (c+d x)^{16} (b c-a d)^2}{16 d^8}-\frac{7 b^4 (c+d x)^{15} (b c-a d)^3}{3 d^8}+\frac{5 b^3 (c+d x)^{14} (b c-a d)^4}{2 d^8}-\frac{21 b^2 (c+d x)^{13} (b c-a d)^5}{13 d^8}+\frac{7 b (c+d x)^{12} (b c-a d)^6}{12 d^8}-\frac{(c+d x)^{11} (b c-a d)^7}{11 d^8}+\frac{b^7 (c+d x)^{18}}{18 d^8} \]
[Out]
-((b*c - a*d)^7*(c + d*x)^11)/(11*d^8) + (7*b*(b*c - a*d)^6*(c + d*x)^12)/(12*d^8) - (21*b^2*(b*c - a*d)^5*(c
+ d*x)^13)/(13*d^8) + (5*b^3*(b*c - a*d)^4*(c + d*x)^14)/(2*d^8) - (7*b^4*(b*c - a*d)^3*(c + d*x)^15)/(3*d^8)
+ (21*b^5*(b*c - a*d)^2*(c + d*x)^16)/(16*d^8) - (7*b^6*(b*c - a*d)*(c + d*x)^17)/(17*d^8) + (b^7*(c + d*x)^18
)/(18*d^8)
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Rubi [A] time = 0.76581, 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} \[ -\frac{7 b^6 (c+d x)^{17} (b c-a d)}{17 d^8}+\frac{21 b^5 (c+d x)^{16} (b c-a d)^2}{16 d^8}-\frac{7 b^4 (c+d x)^{15} (b c-a d)^3}{3 d^8}+\frac{5 b^3 (c+d x)^{14} (b c-a d)^4}{2 d^8}-\frac{21 b^2 (c+d x)^{13} (b c-a d)^5}{13 d^8}+\frac{7 b (c+d x)^{12} (b c-a d)^6}{12 d^8}-\frac{(c+d x)^{11} (b c-a d)^7}{11 d^8}+\frac{b^7 (c+d x)^{18}}{18 d^8} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*x)^7*(c + d*x)^10,x]
[Out]
-((b*c - a*d)^7*(c + d*x)^11)/(11*d^8) + (7*b*(b*c - a*d)^6*(c + d*x)^12)/(12*d^8) - (21*b^2*(b*c - a*d)^5*(c
+ d*x)^13)/(13*d^8) + (5*b^3*(b*c - a*d)^4*(c + d*x)^14)/(2*d^8) - (7*b^4*(b*c - a*d)^3*(c + d*x)^15)/(3*d^8)
+ (21*b^5*(b*c - a*d)^2*(c + d*x)^16)/(16*d^8) - (7*b^6*(b*c - a*d)*(c + d*x)^17)/(17*d^8) + (b^7*(c + d*x)^18
)/(18*d^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)^7 (c+d x)^{10} \, dx &=\int \left (\frac{(-b c+a d)^7 (c+d x)^{10}}{d^7}+\frac{7 b (b c-a d)^6 (c+d x)^{11}}{d^7}-\frac{21 b^2 (b c-a d)^5 (c+d x)^{12}}{d^7}+\frac{35 b^3 (b c-a d)^4 (c+d x)^{13}}{d^7}-\frac{35 b^4 (b c-a d)^3 (c+d x)^{14}}{d^7}+\frac{21 b^5 (b c-a d)^2 (c+d x)^{15}}{d^7}-\frac{7 b^6 (b c-a d) (c+d x)^{16}}{d^7}+\frac{b^7 (c+d x)^{17}}{d^7}\right ) \, dx\\ &=-\frac{(b c-a d)^7 (c+d x)^{11}}{11 d^8}+\frac{7 b (b c-a d)^6 (c+d x)^{12}}{12 d^8}-\frac{21 b^2 (b c-a d)^5 (c+d x)^{13}}{13 d^8}+\frac{5 b^3 (b c-a d)^4 (c+d x)^{14}}{2 d^8}-\frac{7 b^4 (b c-a d)^3 (c+d x)^{15}}{3 d^8}+\frac{21 b^5 (b c-a d)^2 (c+d x)^{16}}{16 d^8}-\frac{7 b^6 (b c-a d) (c+d x)^{17}}{17 d^8}+\frac{b^7 (c+d x)^{18}}{18 d^8}\\ \end{align*}
Mathematica [B] time = 0.124828, size = 1105, normalized size = 5.52 \[ \frac{1}{18} b^7 d^{10} x^{18}+\frac{1}{17} b^6 d^9 (10 b c+7 a d) x^{17}+\frac{1}{16} b^5 d^8 \left (45 b^2 c^2+70 a b d c+21 a^2 d^2\right ) x^{16}+\frac{1}{3} b^4 d^7 \left (24 b^3 c^3+63 a b^2 d c^2+42 a^2 b d^2 c+7 a^3 d^3\right ) x^{15}+\frac{5}{2} b^3 d^6 \left (6 b^4 c^4+24 a b^3 d c^3+27 a^2 b^2 d^2 c^2+10 a^3 b d^3 c+a^4 d^4\right ) x^{14}+\frac{7}{13} b^2 d^5 \left (36 b^5 c^5+210 a b^4 d c^4+360 a^2 b^3 d^2 c^3+225 a^3 b^2 d^3 c^2+50 a^4 b d^4 c+3 a^5 d^5\right ) x^{13}+\frac{7}{12} b d^4 \left (30 b^6 c^6+252 a b^5 d c^5+630 a^2 b^4 d^2 c^4+600 a^3 b^3 d^3 c^3+225 a^4 b^2 d^4 c^2+30 a^5 b d^5 c+a^6 d^6\right ) x^{12}+\frac{1}{11} d^3 \left (120 b^7 c^7+1470 a b^6 d c^6+5292 a^2 b^5 d^2 c^5+7350 a^3 b^4 d^3 c^4+4200 a^4 b^3 d^4 c^3+945 a^5 b^2 d^5 c^2+70 a^6 b d^6 c+a^7 d^7\right ) x^{11}+\frac{1}{2} c d^2 \left (9 b^7 c^7+168 a b^6 d c^6+882 a^2 b^5 d^2 c^5+1764 a^3 b^4 d^3 c^4+1470 a^4 b^3 d^4 c^3+504 a^5 b^2 d^5 c^2+63 a^6 b d^6 c+2 a^7 d^7\right ) x^{10}+\frac{5}{9} c^2 d \left (2 b^7 c^7+63 a b^6 d c^6+504 a^2 b^5 d^2 c^5+1470 a^3 b^4 d^3 c^4+1764 a^4 b^3 d^4 c^3+882 a^5 b^2 d^5 c^2+168 a^6 b d^6 c+9 a^7 d^7\right ) x^9+\frac{1}{8} c^3 \left (b^7 c^7+70 a b^6 d c^6+945 a^2 b^5 d^2 c^5+4200 a^3 b^4 d^3 c^4+7350 a^4 b^3 d^4 c^3+5292 a^5 b^2 d^5 c^2+1470 a^6 b d^6 c+120 a^7 d^7\right ) x^8+a c^4 \left (b^6 c^6+30 a b^5 d c^5+225 a^2 b^4 d^2 c^4+600 a^3 b^3 d^3 c^3+630 a^4 b^2 d^4 c^2+252 a^5 b d^5 c+30 a^6 d^6\right ) x^7+\frac{7}{6} a^2 c^5 \left (3 b^5 c^5+50 a b^4 d c^4+225 a^2 b^3 d^2 c^3+360 a^3 b^2 d^3 c^2+210 a^4 b d^4 c+36 a^5 d^5\right ) x^6+7 a^3 c^6 \left (b^4 c^4+10 a b^3 d c^3+27 a^2 b^2 d^2 c^2+24 a^3 b d^3 c+6 a^4 d^4\right ) x^5+\frac{5}{4} a^4 c^7 \left (7 b^3 c^3+42 a b^2 d c^2+63 a^2 b d^2 c+24 a^3 d^3\right ) x^4+\frac{1}{3} a^5 c^8 \left (21 b^2 c^2+70 a b d c+45 a^2 d^2\right ) x^3+\frac{1}{2} a^6 c^9 (7 b c+10 a d) x^2+a^7 c^{10} x \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x)^7*(c + d*x)^10,x]
[Out]
a^7*c^10*x + (a^6*c^9*(7*b*c + 10*a*d)*x^2)/2 + (a^5*c^8*(21*b^2*c^2 + 70*a*b*c*d + 45*a^2*d^2)*x^3)/3 + (5*a^
4*c^7*(7*b^3*c^3 + 42*a*b^2*c^2*d + 63*a^2*b*c*d^2 + 24*a^3*d^3)*x^4)/4 + 7*a^3*c^6*(b^4*c^4 + 10*a*b^3*c^3*d
+ 27*a^2*b^2*c^2*d^2 + 24*a^3*b*c*d^3 + 6*a^4*d^4)*x^5 + (7*a^2*c^5*(3*b^5*c^5 + 50*a*b^4*c^4*d + 225*a^2*b^3*
c^3*d^2 + 360*a^3*b^2*c^2*d^3 + 210*a^4*b*c*d^4 + 36*a^5*d^5)*x^6)/6 + a*c^4*(b^6*c^6 + 30*a*b^5*c^5*d + 225*a
^2*b^4*c^4*d^2 + 600*a^3*b^3*c^3*d^3 + 630*a^4*b^2*c^2*d^4 + 252*a^5*b*c*d^5 + 30*a^6*d^6)*x^7 + (c^3*(b^7*c^7
+ 70*a*b^6*c^6*d + 945*a^2*b^5*c^5*d^2 + 4200*a^3*b^4*c^4*d^3 + 7350*a^4*b^3*c^3*d^4 + 5292*a^5*b^2*c^2*d^5 +
1470*a^6*b*c*d^6 + 120*a^7*d^7)*x^8)/8 + (5*c^2*d*(2*b^7*c^7 + 63*a*b^6*c^6*d + 504*a^2*b^5*c^5*d^2 + 1470*a^
3*b^4*c^4*d^3 + 1764*a^4*b^3*c^3*d^4 + 882*a^5*b^2*c^2*d^5 + 168*a^6*b*c*d^6 + 9*a^7*d^7)*x^9)/9 + (c*d^2*(9*b
^7*c^7 + 168*a*b^6*c^6*d + 882*a^2*b^5*c^5*d^2 + 1764*a^3*b^4*c^4*d^3 + 1470*a^4*b^3*c^3*d^4 + 504*a^5*b^2*c^2
*d^5 + 63*a^6*b*c*d^6 + 2*a^7*d^7)*x^10)/2 + (d^3*(120*b^7*c^7 + 1470*a*b^6*c^6*d + 5292*a^2*b^5*c^5*d^2 + 735
0*a^3*b^4*c^4*d^3 + 4200*a^4*b^3*c^3*d^4 + 945*a^5*b^2*c^2*d^5 + 70*a^6*b*c*d^6 + a^7*d^7)*x^11)/11 + (7*b*d^4
*(30*b^6*c^6 + 252*a*b^5*c^5*d + 630*a^2*b^4*c^4*d^2 + 600*a^3*b^3*c^3*d^3 + 225*a^4*b^2*c^2*d^4 + 30*a^5*b*c*
d^5 + a^6*d^6)*x^12)/12 + (7*b^2*d^5*(36*b^5*c^5 + 210*a*b^4*c^4*d + 360*a^2*b^3*c^3*d^2 + 225*a^3*b^2*c^2*d^3
+ 50*a^4*b*c*d^4 + 3*a^5*d^5)*x^13)/13 + (5*b^3*d^6*(6*b^4*c^4 + 24*a*b^3*c^3*d + 27*a^2*b^2*c^2*d^2 + 10*a^3
*b*c*d^3 + a^4*d^4)*x^14)/2 + (b^4*d^7*(24*b^3*c^3 + 63*a*b^2*c^2*d + 42*a^2*b*c*d^2 + 7*a^3*d^3)*x^15)/3 + (b
^5*d^8*(45*b^2*c^2 + 70*a*b*c*d + 21*a^2*d^2)*x^16)/16 + (b^6*d^9*(10*b*c + 7*a*d)*x^17)/17 + (b^7*d^10*x^18)/
18
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Maple [B] time = 0.003, size = 1141, normalized size = 5.7 \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)^7*(d*x+c)^10,x)
[Out]
1/18*b^7*d^10*x^18+1/17*(7*a*b^6*d^10+10*b^7*c*d^9)*x^17+1/16*(21*a^2*b^5*d^10+70*a*b^6*c*d^9+45*b^7*c^2*d^8)*
x^16+1/15*(35*a^3*b^4*d^10+210*a^2*b^5*c*d^9+315*a*b^6*c^2*d^8+120*b^7*c^3*d^7)*x^15+1/14*(35*a^4*b^3*d^10+350
*a^3*b^4*c*d^9+945*a^2*b^5*c^2*d^8+840*a*b^6*c^3*d^7+210*b^7*c^4*d^6)*x^14+1/13*(21*a^5*b^2*d^10+350*a^4*b^3*c
*d^9+1575*a^3*b^4*c^2*d^8+2520*a^2*b^5*c^3*d^7+1470*a*b^6*c^4*d^6+252*b^7*c^5*d^5)*x^13+1/12*(7*a^6*b*d^10+210
*a^5*b^2*c*d^9+1575*a^4*b^3*c^2*d^8+4200*a^3*b^4*c^3*d^7+4410*a^2*b^5*c^4*d^6+1764*a*b^6*c^5*d^5+210*b^7*c^6*d
^4)*x^12+1/11*(a^7*d^10+70*a^6*b*c*d^9+945*a^5*b^2*c^2*d^8+4200*a^4*b^3*c^3*d^7+7350*a^3*b^4*c^4*d^6+5292*a^2*
b^5*c^5*d^5+1470*a*b^6*c^6*d^4+120*b^7*c^7*d^3)*x^11+1/10*(10*a^7*c*d^9+315*a^6*b*c^2*d^8+2520*a^5*b^2*c^3*d^7
+7350*a^4*b^3*c^4*d^6+8820*a^3*b^4*c^5*d^5+4410*a^2*b^5*c^6*d^4+840*a*b^6*c^7*d^3+45*b^7*c^8*d^2)*x^10+1/9*(45
*a^7*c^2*d^8+840*a^6*b*c^3*d^7+4410*a^5*b^2*c^4*d^6+8820*a^4*b^3*c^5*d^5+7350*a^3*b^4*c^6*d^4+2520*a^2*b^5*c^7
*d^3+315*a*b^6*c^8*d^2+10*b^7*c^9*d)*x^9+1/8*(120*a^7*c^3*d^7+1470*a^6*b*c^4*d^6+5292*a^5*b^2*c^5*d^5+7350*a^4
*b^3*c^6*d^4+4200*a^3*b^4*c^7*d^3+945*a^2*b^5*c^8*d^2+70*a*b^6*c^9*d+b^7*c^10)*x^8+1/7*(210*a^7*c^4*d^6+1764*a
^6*b*c^5*d^5+4410*a^5*b^2*c^6*d^4+4200*a^4*b^3*c^7*d^3+1575*a^3*b^4*c^8*d^2+210*a^2*b^5*c^9*d+7*a*b^6*c^10)*x^
7+1/6*(252*a^7*c^5*d^5+1470*a^6*b*c^6*d^4+2520*a^5*b^2*c^7*d^3+1575*a^4*b^3*c^8*d^2+350*a^3*b^4*c^9*d+21*a^2*b
^5*c^10)*x^6+1/5*(210*a^7*c^6*d^4+840*a^6*b*c^7*d^3+945*a^5*b^2*c^8*d^2+350*a^4*b^3*c^9*d+35*a^3*b^4*c^10)*x^5
+1/4*(120*a^7*c^7*d^3+315*a^6*b*c^8*d^2+210*a^5*b^2*c^9*d+35*a^4*b^3*c^10)*x^4+1/3*(45*a^7*c^8*d^2+70*a^6*b*c^
9*d+21*a^5*b^2*c^10)*x^3+1/2*(10*a^7*c^9*d+7*a^6*b*c^10)*x^2+a^7*c^10*x
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Maxima [B] time = 0.980027, size = 1532, normalized size = 7.66 \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)^7*(d*x+c)^10,x, algorithm="maxima")
[Out]
1/18*b^7*d^10*x^18 + a^7*c^10*x + 1/17*(10*b^7*c*d^9 + 7*a*b^6*d^10)*x^17 + 1/16*(45*b^7*c^2*d^8 + 70*a*b^6*c*
d^9 + 21*a^2*b^5*d^10)*x^16 + 1/3*(24*b^7*c^3*d^7 + 63*a*b^6*c^2*d^8 + 42*a^2*b^5*c*d^9 + 7*a^3*b^4*d^10)*x^15
+ 5/2*(6*b^7*c^4*d^6 + 24*a*b^6*c^3*d^7 + 27*a^2*b^5*c^2*d^8 + 10*a^3*b^4*c*d^9 + a^4*b^3*d^10)*x^14 + 7/13*(
36*b^7*c^5*d^5 + 210*a*b^6*c^4*d^6 + 360*a^2*b^5*c^3*d^7 + 225*a^3*b^4*c^2*d^8 + 50*a^4*b^3*c*d^9 + 3*a^5*b^2*
d^10)*x^13 + 7/12*(30*b^7*c^6*d^4 + 252*a*b^6*c^5*d^5 + 630*a^2*b^5*c^4*d^6 + 600*a^3*b^4*c^3*d^7 + 225*a^4*b^
3*c^2*d^8 + 30*a^5*b^2*c*d^9 + a^6*b*d^10)*x^12 + 1/11*(120*b^7*c^7*d^3 + 1470*a*b^6*c^6*d^4 + 5292*a^2*b^5*c^
5*d^5 + 7350*a^3*b^4*c^4*d^6 + 4200*a^4*b^3*c^3*d^7 + 945*a^5*b^2*c^2*d^8 + 70*a^6*b*c*d^9 + a^7*d^10)*x^11 +
1/2*(9*b^7*c^8*d^2 + 168*a*b^6*c^7*d^3 + 882*a^2*b^5*c^6*d^4 + 1764*a^3*b^4*c^5*d^5 + 1470*a^4*b^3*c^4*d^6 + 5
04*a^5*b^2*c^3*d^7 + 63*a^6*b*c^2*d^8 + 2*a^7*c*d^9)*x^10 + 5/9*(2*b^7*c^9*d + 63*a*b^6*c^8*d^2 + 504*a^2*b^5*
c^7*d^3 + 1470*a^3*b^4*c^6*d^4 + 1764*a^4*b^3*c^5*d^5 + 882*a^5*b^2*c^4*d^6 + 168*a^6*b*c^3*d^7 + 9*a^7*c^2*d^
8)*x^9 + 1/8*(b^7*c^10 + 70*a*b^6*c^9*d + 945*a^2*b^5*c^8*d^2 + 4200*a^3*b^4*c^7*d^3 + 7350*a^4*b^3*c^6*d^4 +
5292*a^5*b^2*c^5*d^5 + 1470*a^6*b*c^4*d^6 + 120*a^7*c^3*d^7)*x^8 + (a*b^6*c^10 + 30*a^2*b^5*c^9*d + 225*a^3*b^
4*c^8*d^2 + 600*a^4*b^3*c^7*d^3 + 630*a^5*b^2*c^6*d^4 + 252*a^6*b*c^5*d^5 + 30*a^7*c^4*d^6)*x^7 + 7/6*(3*a^2*b
^5*c^10 + 50*a^3*b^4*c^9*d + 225*a^4*b^3*c^8*d^2 + 360*a^5*b^2*c^7*d^3 + 210*a^6*b*c^6*d^4 + 36*a^7*c^5*d^5)*x
^6 + 7*(a^3*b^4*c^10 + 10*a^4*b^3*c^9*d + 27*a^5*b^2*c^8*d^2 + 24*a^6*b*c^7*d^3 + 6*a^7*c^6*d^4)*x^5 + 5/4*(7*
a^4*b^3*c^10 + 42*a^5*b^2*c^9*d + 63*a^6*b*c^8*d^2 + 24*a^7*c^7*d^3)*x^4 + 1/3*(21*a^5*b^2*c^10 + 70*a^6*b*c^9
*d + 45*a^7*c^8*d^2)*x^3 + 1/2*(7*a^6*b*c^10 + 10*a^7*c^9*d)*x^2
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Fricas [B] time = 1.61419, size = 2931, normalized size = 14.66 \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)^7*(d*x+c)^10,x, algorithm="fricas")
[Out]
1/18*x^18*d^10*b^7 + 10/17*x^17*d^9*c*b^7 + 7/17*x^17*d^10*b^6*a + 45/16*x^16*d^8*c^2*b^7 + 35/8*x^16*d^9*c*b^
6*a + 21/16*x^16*d^10*b^5*a^2 + 8*x^15*d^7*c^3*b^7 + 21*x^15*d^8*c^2*b^6*a + 14*x^15*d^9*c*b^5*a^2 + 7/3*x^15*
d^10*b^4*a^3 + 15*x^14*d^6*c^4*b^7 + 60*x^14*d^7*c^3*b^6*a + 135/2*x^14*d^8*c^2*b^5*a^2 + 25*x^14*d^9*c*b^4*a^
3 + 5/2*x^14*d^10*b^3*a^4 + 252/13*x^13*d^5*c^5*b^7 + 1470/13*x^13*d^6*c^4*b^6*a + 2520/13*x^13*d^7*c^3*b^5*a^
2 + 1575/13*x^13*d^8*c^2*b^4*a^3 + 350/13*x^13*d^9*c*b^3*a^4 + 21/13*x^13*d^10*b^2*a^5 + 35/2*x^12*d^4*c^6*b^7
+ 147*x^12*d^5*c^5*b^6*a + 735/2*x^12*d^6*c^4*b^5*a^2 + 350*x^12*d^7*c^3*b^4*a^3 + 525/4*x^12*d^8*c^2*b^3*a^4
+ 35/2*x^12*d^9*c*b^2*a^5 + 7/12*x^12*d^10*b*a^6 + 120/11*x^11*d^3*c^7*b^7 + 1470/11*x^11*d^4*c^6*b^6*a + 529
2/11*x^11*d^5*c^5*b^5*a^2 + 7350/11*x^11*d^6*c^4*b^4*a^3 + 4200/11*x^11*d^7*c^3*b^3*a^4 + 945/11*x^11*d^8*c^2*
b^2*a^5 + 70/11*x^11*d^9*c*b*a^6 + 1/11*x^11*d^10*a^7 + 9/2*x^10*d^2*c^8*b^7 + 84*x^10*d^3*c^7*b^6*a + 441*x^1
0*d^4*c^6*b^5*a^2 + 882*x^10*d^5*c^5*b^4*a^3 + 735*x^10*d^6*c^4*b^3*a^4 + 252*x^10*d^7*c^3*b^2*a^5 + 63/2*x^10
*d^8*c^2*b*a^6 + x^10*d^9*c*a^7 + 10/9*x^9*d*c^9*b^7 + 35*x^9*d^2*c^8*b^6*a + 280*x^9*d^3*c^7*b^5*a^2 + 2450/3
*x^9*d^4*c^6*b^4*a^3 + 980*x^9*d^5*c^5*b^3*a^4 + 490*x^9*d^6*c^4*b^2*a^5 + 280/3*x^9*d^7*c^3*b*a^6 + 5*x^9*d^8
*c^2*a^7 + 1/8*x^8*c^10*b^7 + 35/4*x^8*d*c^9*b^6*a + 945/8*x^8*d^2*c^8*b^5*a^2 + 525*x^8*d^3*c^7*b^4*a^3 + 367
5/4*x^8*d^4*c^6*b^3*a^4 + 1323/2*x^8*d^5*c^5*b^2*a^5 + 735/4*x^8*d^6*c^4*b*a^6 + 15*x^8*d^7*c^3*a^7 + x^7*c^10
*b^6*a + 30*x^7*d*c^9*b^5*a^2 + 225*x^7*d^2*c^8*b^4*a^3 + 600*x^7*d^3*c^7*b^3*a^4 + 630*x^7*d^4*c^6*b^2*a^5 +
252*x^7*d^5*c^5*b*a^6 + 30*x^7*d^6*c^4*a^7 + 7/2*x^6*c^10*b^5*a^2 + 175/3*x^6*d*c^9*b^4*a^3 + 525/2*x^6*d^2*c^
8*b^3*a^4 + 420*x^6*d^3*c^7*b^2*a^5 + 245*x^6*d^4*c^6*b*a^6 + 42*x^6*d^5*c^5*a^7 + 7*x^5*c^10*b^4*a^3 + 70*x^5
*d*c^9*b^3*a^4 + 189*x^5*d^2*c^8*b^2*a^5 + 168*x^5*d^3*c^7*b*a^6 + 42*x^5*d^4*c^6*a^7 + 35/4*x^4*c^10*b^3*a^4
+ 105/2*x^4*d*c^9*b^2*a^5 + 315/4*x^4*d^2*c^8*b*a^6 + 30*x^4*d^3*c^7*a^7 + 7*x^3*c^10*b^2*a^5 + 70/3*x^3*d*c^9
*b*a^6 + 15*x^3*d^2*c^8*a^7 + 7/2*x^2*c^10*b*a^6 + 5*x^2*d*c^9*a^7 + x*c^10*a^7
________________________________________________________________________________________
Sympy [B] time = 0.233145, size = 1280, normalized size = 6.4 \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)**7*(d*x+c)**10,x)
[Out]
a**7*c**10*x + b**7*d**10*x**18/18 + x**17*(7*a*b**6*d**10/17 + 10*b**7*c*d**9/17) + x**16*(21*a**2*b**5*d**10
/16 + 35*a*b**6*c*d**9/8 + 45*b**7*c**2*d**8/16) + x**15*(7*a**3*b**4*d**10/3 + 14*a**2*b**5*c*d**9 + 21*a*b**
6*c**2*d**8 + 8*b**7*c**3*d**7) + x**14*(5*a**4*b**3*d**10/2 + 25*a**3*b**4*c*d**9 + 135*a**2*b**5*c**2*d**8/2
+ 60*a*b**6*c**3*d**7 + 15*b**7*c**4*d**6) + x**13*(21*a**5*b**2*d**10/13 + 350*a**4*b**3*c*d**9/13 + 1575*a*
*3*b**4*c**2*d**8/13 + 2520*a**2*b**5*c**3*d**7/13 + 1470*a*b**6*c**4*d**6/13 + 252*b**7*c**5*d**5/13) + x**12
*(7*a**6*b*d**10/12 + 35*a**5*b**2*c*d**9/2 + 525*a**4*b**3*c**2*d**8/4 + 350*a**3*b**4*c**3*d**7 + 735*a**2*b
**5*c**4*d**6/2 + 147*a*b**6*c**5*d**5 + 35*b**7*c**6*d**4/2) + x**11*(a**7*d**10/11 + 70*a**6*b*c*d**9/11 + 9
45*a**5*b**2*c**2*d**8/11 + 4200*a**4*b**3*c**3*d**7/11 + 7350*a**3*b**4*c**4*d**6/11 + 5292*a**2*b**5*c**5*d*
*5/11 + 1470*a*b**6*c**6*d**4/11 + 120*b**7*c**7*d**3/11) + x**10*(a**7*c*d**9 + 63*a**6*b*c**2*d**8/2 + 252*a
**5*b**2*c**3*d**7 + 735*a**4*b**3*c**4*d**6 + 882*a**3*b**4*c**5*d**5 + 441*a**2*b**5*c**6*d**4 + 84*a*b**6*c
**7*d**3 + 9*b**7*c**8*d**2/2) + x**9*(5*a**7*c**2*d**8 + 280*a**6*b*c**3*d**7/3 + 490*a**5*b**2*c**4*d**6 + 9
80*a**4*b**3*c**5*d**5 + 2450*a**3*b**4*c**6*d**4/3 + 280*a**2*b**5*c**7*d**3 + 35*a*b**6*c**8*d**2 + 10*b**7*
c**9*d/9) + x**8*(15*a**7*c**3*d**7 + 735*a**6*b*c**4*d**6/4 + 1323*a**5*b**2*c**5*d**5/2 + 3675*a**4*b**3*c**
6*d**4/4 + 525*a**3*b**4*c**7*d**3 + 945*a**2*b**5*c**8*d**2/8 + 35*a*b**6*c**9*d/4 + b**7*c**10/8) + x**7*(30
*a**7*c**4*d**6 + 252*a**6*b*c**5*d**5 + 630*a**5*b**2*c**6*d**4 + 600*a**4*b**3*c**7*d**3 + 225*a**3*b**4*c**
8*d**2 + 30*a**2*b**5*c**9*d + a*b**6*c**10) + x**6*(42*a**7*c**5*d**5 + 245*a**6*b*c**6*d**4 + 420*a**5*b**2*
c**7*d**3 + 525*a**4*b**3*c**8*d**2/2 + 175*a**3*b**4*c**9*d/3 + 7*a**2*b**5*c**10/2) + x**5*(42*a**7*c**6*d**
4 + 168*a**6*b*c**7*d**3 + 189*a**5*b**2*c**8*d**2 + 70*a**4*b**3*c**9*d + 7*a**3*b**4*c**10) + x**4*(30*a**7*
c**7*d**3 + 315*a**6*b*c**8*d**2/4 + 105*a**5*b**2*c**9*d/2 + 35*a**4*b**3*c**10/4) + x**3*(15*a**7*c**8*d**2
+ 70*a**6*b*c**9*d/3 + 7*a**5*b**2*c**10) + x**2*(5*a**7*c**9*d + 7*a**6*b*c**10/2)
________________________________________________________________________________________
Giac [B] time = 1.06255, size = 1758, normalized size = 8.79 \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)^7*(d*x+c)^10,x, algorithm="giac")
[Out]
1/18*b^7*d^10*x^18 + 10/17*b^7*c*d^9*x^17 + 7/17*a*b^6*d^10*x^17 + 45/16*b^7*c^2*d^8*x^16 + 35/8*a*b^6*c*d^9*x
^16 + 21/16*a^2*b^5*d^10*x^16 + 8*b^7*c^3*d^7*x^15 + 21*a*b^6*c^2*d^8*x^15 + 14*a^2*b^5*c*d^9*x^15 + 7/3*a^3*b
^4*d^10*x^15 + 15*b^7*c^4*d^6*x^14 + 60*a*b^6*c^3*d^7*x^14 + 135/2*a^2*b^5*c^2*d^8*x^14 + 25*a^3*b^4*c*d^9*x^1
4 + 5/2*a^4*b^3*d^10*x^14 + 252/13*b^7*c^5*d^5*x^13 + 1470/13*a*b^6*c^4*d^6*x^13 + 2520/13*a^2*b^5*c^3*d^7*x^1
3 + 1575/13*a^3*b^4*c^2*d^8*x^13 + 350/13*a^4*b^3*c*d^9*x^13 + 21/13*a^5*b^2*d^10*x^13 + 35/2*b^7*c^6*d^4*x^12
+ 147*a*b^6*c^5*d^5*x^12 + 735/2*a^2*b^5*c^4*d^6*x^12 + 350*a^3*b^4*c^3*d^7*x^12 + 525/4*a^4*b^3*c^2*d^8*x^12
+ 35/2*a^5*b^2*c*d^9*x^12 + 7/12*a^6*b*d^10*x^12 + 120/11*b^7*c^7*d^3*x^11 + 1470/11*a*b^6*c^6*d^4*x^11 + 529
2/11*a^2*b^5*c^5*d^5*x^11 + 7350/11*a^3*b^4*c^4*d^6*x^11 + 4200/11*a^4*b^3*c^3*d^7*x^11 + 945/11*a^5*b^2*c^2*d
^8*x^11 + 70/11*a^6*b*c*d^9*x^11 + 1/11*a^7*d^10*x^11 + 9/2*b^7*c^8*d^2*x^10 + 84*a*b^6*c^7*d^3*x^10 + 441*a^2
*b^5*c^6*d^4*x^10 + 882*a^3*b^4*c^5*d^5*x^10 + 735*a^4*b^3*c^4*d^6*x^10 + 252*a^5*b^2*c^3*d^7*x^10 + 63/2*a^6*
b*c^2*d^8*x^10 + a^7*c*d^9*x^10 + 10/9*b^7*c^9*d*x^9 + 35*a*b^6*c^8*d^2*x^9 + 280*a^2*b^5*c^7*d^3*x^9 + 2450/3
*a^3*b^4*c^6*d^4*x^9 + 980*a^4*b^3*c^5*d^5*x^9 + 490*a^5*b^2*c^4*d^6*x^9 + 280/3*a^6*b*c^3*d^7*x^9 + 5*a^7*c^2
*d^8*x^9 + 1/8*b^7*c^10*x^8 + 35/4*a*b^6*c^9*d*x^8 + 945/8*a^2*b^5*c^8*d^2*x^8 + 525*a^3*b^4*c^7*d^3*x^8 + 367
5/4*a^4*b^3*c^6*d^4*x^8 + 1323/2*a^5*b^2*c^5*d^5*x^8 + 735/4*a^6*b*c^4*d^6*x^8 + 15*a^7*c^3*d^7*x^8 + a*b^6*c^
10*x^7 + 30*a^2*b^5*c^9*d*x^7 + 225*a^3*b^4*c^8*d^2*x^7 + 600*a^4*b^3*c^7*d^3*x^7 + 630*a^5*b^2*c^6*d^4*x^7 +
252*a^6*b*c^5*d^5*x^7 + 30*a^7*c^4*d^6*x^7 + 7/2*a^2*b^5*c^10*x^6 + 175/3*a^3*b^4*c^9*d*x^6 + 525/2*a^4*b^3*c^
8*d^2*x^6 + 420*a^5*b^2*c^7*d^3*x^6 + 245*a^6*b*c^6*d^4*x^6 + 42*a^7*c^5*d^5*x^6 + 7*a^3*b^4*c^10*x^5 + 70*a^4
*b^3*c^9*d*x^5 + 189*a^5*b^2*c^8*d^2*x^5 + 168*a^6*b*c^7*d^3*x^5 + 42*a^7*c^6*d^4*x^5 + 35/4*a^4*b^3*c^10*x^4
+ 105/2*a^5*b^2*c^9*d*x^4 + 315/4*a^6*b*c^8*d^2*x^4 + 30*a^7*c^7*d^3*x^4 + 7*a^5*b^2*c^10*x^3 + 70/3*a^6*b*c^9
*d*x^3 + 15*a^7*c^8*d^2*x^3 + 7/2*a^6*b*c^10*x^2 + 5*a^7*c^9*d*x^2 + a^7*c^10*x