3.14.4 \(\int (a+b x)^7 (c+d x)^{10} \, dx\) [1304]

Optimal. Leaf size=200 \[ -\frac {(b c-a d)^7 (c+d x)^{11}}{11 d^8}+\frac {7 b (b c-a d)^6 (c+d x)^{12}}{12 d^8}-\frac {21 b^2 (b c-a d)^5 (c+d x)^{13}}{13 d^8}+\frac {5 b^3 (b c-a d)^4 (c+d x)^{14}}{2 d^8}-\frac {7 b^4 (b c-a d)^3 (c+d x)^{15}}{3 d^8}+\frac {21 b^5 (b c-a d)^2 (c+d x)^{16}}{16 d^8}-\frac {7 b^6 (b c-a d) (c+d x)^{17}}{17 d^8}+\frac {b^7 (c+d x)^{18}}{18 d^8} \]

[Out]

-1/11*(-a*d+b*c)^7*(d*x+c)^11/d^8+7/12*b*(-a*d+b*c)^6*(d*x+c)^12/d^8-21/13*b^2*(-a*d+b*c)^5*(d*x+c)^13/d^8+5/2
*b^3*(-a*d+b*c)^4*(d*x+c)^14/d^8-7/3*b^4*(-a*d+b*c)^3*(d*x+c)^15/d^8+21/16*b^5*(-a*d+b*c)^2*(d*x+c)^16/d^8-7/1
7*b^6*(-a*d+b*c)*(d*x+c)^17/d^8+1/18*b^7*(d*x+c)^18/d^8

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Rubi [A]
time = 0.54, 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 = {45} \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^7*(c + d*x)^10,x]

[Out]

-1/11*((b*c - a*d)^7*(c + d*x)^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 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*} \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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1105\) vs. \(2(200)=400\).
time = 0.09, size = 1105, normalized size = 5.52 \begin {gather*} a^7 c^{10} x+\frac {1}{2} a^6 c^9 (7 b c+10 a d) x^2+\frac {1}{3} a^5 c^8 \left (21 b^2 c^2+70 a b c d+45 a^2 d^2\right ) x^3+\frac {5}{4} a^4 c^7 \left (7 b^3 c^3+42 a b^2 c^2 d+63 a^2 b c d^2+24 a^3 d^3\right ) x^4+7 a^3 c^6 \left (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\right ) x^5+\frac {7}{6} a^2 c^5 \left (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\right ) x^6+a c^4 \left (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\right ) x^7+\frac {1}{8} c^3 \left (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\right ) x^8+\frac {5}{9} c^2 d \left (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\right ) x^9+\frac {1}{2} c d^2 \left (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\right ) x^{10}+\frac {1}{11} d^3 \left (120 b^7 c^7+1470 a b^6 c^6 d+5292 a^2 b^5 c^5 d^2+7350 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\right ) x^{11}+\frac {7}{12} b d^4 \left (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\right ) x^{12}+\frac {7}{13} b^2 d^5 \left (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\right ) x^{13}+\frac {5}{2} b^3 d^6 \left (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\right ) x^{14}+\frac {1}{3} b^4 d^7 \left (24 b^3 c^3+63 a b^2 c^2 d+42 a^2 b c d^2+7 a^3 d^3\right ) x^{15}+\frac {1}{16} b^5 d^8 \left (45 b^2 c^2+70 a b c d+21 a^2 d^2\right ) x^{16}+\frac {1}{17} b^6 d^9 (10 b c+7 a d) x^{17}+\frac {1}{18} b^7 d^{10} x^{18} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(1140\) vs. \(2(184)=368\).
time = 0.13, size = 1141, normalized size = 5.70 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^7*(d*x+c)^10,x,method=_RETURNVERBOSE)

[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] Leaf count of result is larger than twice the leaf count of optimal. 1135 vs. \(2 (184) = 368\).
time = 0.30, size = 1135, normalized size = 5.68 \begin {gather*} \frac {1}{18} \, b^{7} d^{10} x^{18} + a^{7} c^{10} x + \frac {1}{17} \, {\left (10 \, b^{7} c d^{9} + 7 \, a b^{6} d^{10}\right )} x^{17} + \frac {1}{16} \, {\left (45 \, b^{7} c^{2} d^{8} + 70 \, a b^{6} c d^{9} + 21 \, a^{2} b^{5} d^{10}\right )} x^{16} + \frac {1}{3} \, {\left (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}\right )} x^{15} + \frac {5}{2} \, {\left (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}\right )} x^{14} + \frac {7}{13} \, {\left (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}\right )} x^{13} + \frac {7}{12} \, {\left (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}\right )} x^{12} + \frac {1}{11} \, {\left (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}\right )} x^{11} + \frac {1}{2} \, {\left (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} + 504 \, a^{5} b^{2} c^{3} d^{7} + 63 \, a^{6} b c^{2} d^{8} + 2 \, a^{7} c d^{9}\right )} x^{10} + \frac {5}{9} \, {\left (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}\right )} x^{9} + \frac {1}{8} \, {\left (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}\right )} x^{8} + {\left (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}\right )} x^{7} + \frac {7}{6} \, {\left (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}\right )} x^{6} + 7 \, {\left (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}\right )} x^{5} + \frac {5}{4} \, {\left (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}\right )} x^{4} + \frac {1}{3} \, {\left (21 \, a^{5} b^{2} c^{10} + 70 \, a^{6} b c^{9} d + 45 \, a^{7} c^{8} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (7 \, a^{6} b c^{10} + 10 \, a^{7} c^{9} d\right )} x^{2} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 1135 vs. \(2 (184) = 368\).
time = 0.50, size = 1135, normalized size = 5.68 \begin {gather*} \frac {1}{18} \, b^{7} d^{10} x^{18} + a^{7} c^{10} x + \frac {1}{17} \, {\left (10 \, b^{7} c d^{9} + 7 \, a b^{6} d^{10}\right )} x^{17} + \frac {1}{16} \, {\left (45 \, b^{7} c^{2} d^{8} + 70 \, a b^{6} c d^{9} + 21 \, a^{2} b^{5} d^{10}\right )} x^{16} + \frac {1}{3} \, {\left (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}\right )} x^{15} + \frac {5}{2} \, {\left (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}\right )} x^{14} + \frac {7}{13} \, {\left (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}\right )} x^{13} + \frac {7}{12} \, {\left (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}\right )} x^{12} + \frac {1}{11} \, {\left (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}\right )} x^{11} + \frac {1}{2} \, {\left (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} + 504 \, a^{5} b^{2} c^{3} d^{7} + 63 \, a^{6} b c^{2} d^{8} + 2 \, a^{7} c d^{9}\right )} x^{10} + \frac {5}{9} \, {\left (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}\right )} x^{9} + \frac {1}{8} \, {\left (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}\right )} x^{8} + {\left (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}\right )} x^{7} + \frac {7}{6} \, {\left (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}\right )} x^{6} + 7 \, {\left (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}\right )} x^{5} + \frac {5}{4} \, {\left (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}\right )} x^{4} + \frac {1}{3} \, {\left (21 \, a^{5} b^{2} c^{10} + 70 \, a^{6} b c^{9} d + 45 \, a^{7} c^{8} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (7 \, a^{6} b c^{10} + 10 \, a^{7} c^{9} d\right )} x^{2} \end {gather*}

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*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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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1280 vs. \(2 (184) = 368\).
time = 0.09, size = 1280, normalized size = 6.40 \begin {gather*} a^{7} c^{10} x + \frac {b^{7} d^{10} x^{18}}{18} + x^{17} \cdot \left (\frac {7 a b^{6} d^{10}}{17} + \frac {10 b^{7} c d^{9}}{17}\right ) + x^{16} \cdot \left (\frac {21 a^{2} b^{5} d^{10}}{16} + \frac {35 a b^{6} c d^{9}}{8} + \frac {45 b^{7} c^{2} d^{8}}{16}\right ) + x^{15} \cdot \left (\frac {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}\right ) + x^{14} \cdot \left (\frac {5 a^{4} b^{3} d^{10}}{2} + 25 a^{3} b^{4} c d^{9} + \frac {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}\right ) + x^{13} \cdot \left (\frac {21 a^{5} b^{2} d^{10}}{13} + \frac {350 a^{4} b^{3} c d^{9}}{13} + \frac {1575 a^{3} b^{4} c^{2} d^{8}}{13} + \frac {2520 a^{2} b^{5} c^{3} d^{7}}{13} + \frac {1470 a b^{6} c^{4} d^{6}}{13} + \frac {252 b^{7} c^{5} d^{5}}{13}\right ) + x^{12} \cdot \left (\frac {7 a^{6} b d^{10}}{12} + \frac {35 a^{5} b^{2} c d^{9}}{2} + \frac {525 a^{4} b^{3} c^{2} d^{8}}{4} + 350 a^{3} b^{4} c^{3} d^{7} + \frac {735 a^{2} b^{5} c^{4} d^{6}}{2} + 147 a b^{6} c^{5} d^{5} + \frac {35 b^{7} c^{6} d^{4}}{2}\right ) + x^{11} \left (\frac {a^{7} d^{10}}{11} + \frac {70 a^{6} b c d^{9}}{11} + \frac {945 a^{5} b^{2} c^{2} d^{8}}{11} + \frac {4200 a^{4} b^{3} c^{3} d^{7}}{11} + \frac {7350 a^{3} b^{4} c^{4} d^{6}}{11} + \frac {5292 a^{2} b^{5} c^{5} d^{5}}{11} + \frac {1470 a b^{6} c^{6} d^{4}}{11} + \frac {120 b^{7} c^{7} d^{3}}{11}\right ) + x^{10} \left (a^{7} c d^{9} + \frac {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} + \frac {9 b^{7} c^{8} d^{2}}{2}\right ) + x^{9} \cdot \left (5 a^{7} c^{2} d^{8} + \frac {280 a^{6} b c^{3} d^{7}}{3} + 490 a^{5} b^{2} c^{4} d^{6} + 980 a^{4} b^{3} c^{5} d^{5} + \frac {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} + \frac {10 b^{7} c^{9} d}{9}\right ) + x^{8} \cdot \left (15 a^{7} c^{3} d^{7} + \frac {735 a^{6} b c^{4} d^{6}}{4} + \frac {1323 a^{5} b^{2} c^{5} d^{5}}{2} + \frac {3675 a^{4} b^{3} c^{6} d^{4}}{4} + 525 a^{3} b^{4} c^{7} d^{3} + \frac {945 a^{2} b^{5} c^{8} d^{2}}{8} + \frac {35 a b^{6} c^{9} d}{4} + \frac {b^{7} c^{10}}{8}\right ) + x^{7} \cdot \left (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}\right ) + x^{6} \cdot \left (42 a^{7} c^{5} d^{5} + 245 a^{6} b c^{6} d^{4} + 420 a^{5} b^{2} c^{7} d^{3} + \frac {525 a^{4} b^{3} c^{8} d^{2}}{2} + \frac {175 a^{3} b^{4} c^{9} d}{3} + \frac {7 a^{2} b^{5} c^{10}}{2}\right ) + x^{5} \cdot \left (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}\right ) + x^{4} \cdot \left (30 a^{7} c^{7} d^{3} + \frac {315 a^{6} b c^{8} d^{2}}{4} + \frac {105 a^{5} b^{2} c^{9} d}{2} + \frac {35 a^{4} b^{3} c^{10}}{4}\right ) + x^{3} \cdot \left (15 a^{7} c^{8} d^{2} + \frac {70 a^{6} b c^{9} d}{3} + 7 a^{5} b^{2} c^{10}\right ) + x^{2} \cdot \left (5 a^{7} c^{9} d + \frac {7 a^{6} b c^{10}}{2}\right ) \end {gather*}

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)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1302 vs. \(2 (184) = 368\).
time = 0.81, size = 1302, normalized size = 6.51 \begin {gather*} \frac {1}{18} \, b^{7} d^{10} x^{18} + \frac {10}{17} \, b^{7} c d^{9} x^{17} + \frac {7}{17} \, a b^{6} d^{10} x^{17} + \frac {45}{16} \, b^{7} c^{2} d^{8} x^{16} + \frac {35}{8} \, a b^{6} c d^{9} x^{16} + \frac {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} + \frac {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} + \frac {135}{2} \, a^{2} b^{5} c^{2} d^{8} x^{14} + 25 \, a^{3} b^{4} c d^{9} x^{14} + \frac {5}{2} \, a^{4} b^{3} d^{10} x^{14} + \frac {252}{13} \, b^{7} c^{5} d^{5} x^{13} + \frac {1470}{13} \, a b^{6} c^{4} d^{6} x^{13} + \frac {2520}{13} \, a^{2} b^{5} c^{3} d^{7} x^{13} + \frac {1575}{13} \, a^{3} b^{4} c^{2} d^{8} x^{13} + \frac {350}{13} \, a^{4} b^{3} c d^{9} x^{13} + \frac {21}{13} \, a^{5} b^{2} d^{10} x^{13} + \frac {35}{2} \, b^{7} c^{6} d^{4} x^{12} + 147 \, a b^{6} c^{5} d^{5} x^{12} + \frac {735}{2} \, a^{2} b^{5} c^{4} d^{6} x^{12} + 350 \, a^{3} b^{4} c^{3} d^{7} x^{12} + \frac {525}{4} \, a^{4} b^{3} c^{2} d^{8} x^{12} + \frac {35}{2} \, a^{5} b^{2} c d^{9} x^{12} + \frac {7}{12} \, a^{6} b d^{10} x^{12} + \frac {120}{11} \, b^{7} c^{7} d^{3} x^{11} + \frac {1470}{11} \, a b^{6} c^{6} d^{4} x^{11} + \frac {5292}{11} \, a^{2} b^{5} c^{5} d^{5} x^{11} + \frac {7350}{11} \, a^{3} b^{4} c^{4} d^{6} x^{11} + \frac {4200}{11} \, a^{4} b^{3} c^{3} d^{7} x^{11} + \frac {945}{11} \, a^{5} b^{2} c^{2} d^{8} x^{11} + \frac {70}{11} \, a^{6} b c d^{9} x^{11} + \frac {1}{11} \, a^{7} d^{10} x^{11} + \frac {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} + \frac {63}{2} \, a^{6} b c^{2} d^{8} x^{10} + a^{7} c d^{9} x^{10} + \frac {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} + \frac {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} + \frac {280}{3} \, a^{6} b c^{3} d^{7} x^{9} + 5 \, a^{7} c^{2} d^{8} x^{9} + \frac {1}{8} \, b^{7} c^{10} x^{8} + \frac {35}{4} \, a b^{6} c^{9} d x^{8} + \frac {945}{8} \, a^{2} b^{5} c^{8} d^{2} x^{8} + 525 \, a^{3} b^{4} c^{7} d^{3} x^{8} + \frac {3675}{4} \, a^{4} b^{3} c^{6} d^{4} x^{8} + \frac {1323}{2} \, a^{5} b^{2} c^{5} d^{5} x^{8} + \frac {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} + \frac {7}{2} \, a^{2} b^{5} c^{10} x^{6} + \frac {175}{3} \, a^{3} b^{4} c^{9} d x^{6} + \frac {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} + \frac {35}{4} \, a^{4} b^{3} c^{10} x^{4} + \frac {105}{2} \, a^{5} b^{2} c^{9} d x^{4} + \frac {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} + \frac {70}{3} \, a^{6} b c^{9} d x^{3} + 15 \, a^{7} c^{8} d^{2} x^{3} + \frac {7}{2} \, a^{6} b c^{10} x^{2} + 5 \, a^{7} c^{9} d x^{2} + a^{7} c^{10} x \end {gather*}

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

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Mupad [B]
time = 0.61, size = 1106, normalized size = 5.53 \begin {gather*} x^{10}\,\left (a^7\,c\,d^9+\frac {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+\frac {9\,b^7\,c^8\,d^2}{2}\right )+x^9\,\left (5\,a^7\,c^2\,d^8+\frac {280\,a^6\,b\,c^3\,d^7}{3}+490\,a^5\,b^2\,c^4\,d^6+980\,a^4\,b^3\,c^5\,d^5+\frac {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+\frac {10\,b^7\,c^9\,d}{9}\right )+x^5\,\left (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}\right )+x^{14}\,\left (\frac {5\,a^4\,b^3\,d^{10}}{2}+25\,a^3\,b^4\,c\,d^9+\frac {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\right )+x^8\,\left (15\,a^7\,c^3\,d^7+\frac {735\,a^6\,b\,c^4\,d^6}{4}+\frac {1323\,a^5\,b^2\,c^5\,d^5}{2}+\frac {3675\,a^4\,b^3\,c^6\,d^4}{4}+525\,a^3\,b^4\,c^7\,d^3+\frac {945\,a^2\,b^5\,c^8\,d^2}{8}+\frac {35\,a\,b^6\,c^9\,d}{4}+\frac {b^7\,c^{10}}{8}\right )+x^{11}\,\left (\frac {a^7\,d^{10}}{11}+\frac {70\,a^6\,b\,c\,d^9}{11}+\frac {945\,a^5\,b^2\,c^2\,d^8}{11}+\frac {4200\,a^4\,b^3\,c^3\,d^7}{11}+\frac {7350\,a^3\,b^4\,c^4\,d^6}{11}+\frac {5292\,a^2\,b^5\,c^5\,d^5}{11}+\frac {1470\,a\,b^6\,c^6\,d^4}{11}+\frac {120\,b^7\,c^7\,d^3}{11}\right )+x^6\,\left (42\,a^7\,c^5\,d^5+245\,a^6\,b\,c^6\,d^4+420\,a^5\,b^2\,c^7\,d^3+\frac {525\,a^4\,b^3\,c^8\,d^2}{2}+\frac {175\,a^3\,b^4\,c^9\,d}{3}+\frac {7\,a^2\,b^5\,c^{10}}{2}\right )+x^{13}\,\left (\frac {21\,a^5\,b^2\,d^{10}}{13}+\frac {350\,a^4\,b^3\,c\,d^9}{13}+\frac {1575\,a^3\,b^4\,c^2\,d^8}{13}+\frac {2520\,a^2\,b^5\,c^3\,d^7}{13}+\frac {1470\,a\,b^6\,c^4\,d^6}{13}+\frac {252\,b^7\,c^5\,d^5}{13}\right )+x^7\,\left (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}\right )+x^{12}\,\left (\frac {7\,a^6\,b\,d^{10}}{12}+\frac {35\,a^5\,b^2\,c\,d^9}{2}+\frac {525\,a^4\,b^3\,c^2\,d^8}{4}+350\,a^3\,b^4\,c^3\,d^7+\frac {735\,a^2\,b^5\,c^4\,d^6}{2}+147\,a\,b^6\,c^5\,d^5+\frac {35\,b^7\,c^6\,d^4}{2}\right )+a^7\,c^{10}\,x+\frac {b^7\,d^{10}\,x^{18}}{18}+\frac {5\,a^4\,c^7\,x^4\,\left (24\,a^3\,d^3+63\,a^2\,b\,c\,d^2+42\,a\,b^2\,c^2\,d+7\,b^3\,c^3\right )}{4}+\frac {b^4\,d^7\,x^{15}\,\left (7\,a^3\,d^3+42\,a^2\,b\,c\,d^2+63\,a\,b^2\,c^2\,d+24\,b^3\,c^3\right )}{3}+\frac {a^6\,c^9\,x^2\,\left (10\,a\,d+7\,b\,c\right )}{2}+\frac {b^6\,d^9\,x^{17}\,\left (7\,a\,d+10\,b\,c\right )}{17}+\frac {a^5\,c^8\,x^3\,\left (45\,a^2\,d^2+70\,a\,b\,c\,d+21\,b^2\,c^2\right )}{3}+\frac {b^5\,d^8\,x^{16}\,\left (21\,a^2\,d^2+70\,a\,b\,c\,d+45\,b^2\,c^2\right )}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^7*(c + d*x)^10,x)

[Out]

x^10*(a^7*c*d^9 + (9*b^7*c^8*d^2)/2 + 84*a*b^6*c^7*d^3 + (63*a^6*b*c^2*d^8)/2 + 441*a^2*b^5*c^6*d^4 + 882*a^3*
b^4*c^5*d^5 + 735*a^4*b^3*c^4*d^6 + 252*a^5*b^2*c^3*d^7) + x^9*((10*b^7*c^9*d)/9 + 5*a^7*c^2*d^8 + 35*a*b^6*c^
8*d^2 + (280*a^6*b*c^3*d^7)/3 + 280*a^2*b^5*c^7*d^3 + (2450*a^3*b^4*c^6*d^4)/3 + 980*a^4*b^3*c^5*d^5 + 490*a^5
*b^2*c^4*d^6) + x^5*(7*a^3*b^4*c^10 + 42*a^7*c^6*d^4 + 70*a^4*b^3*c^9*d + 168*a^6*b*c^7*d^3 + 189*a^5*b^2*c^8*
d^2) + x^14*((5*a^4*b^3*d^10)/2 + 15*b^7*c^4*d^6 + 60*a*b^6*c^3*d^7 + 25*a^3*b^4*c*d^9 + (135*a^2*b^5*c^2*d^8)
/2) + x^8*((b^7*c^10)/8 + 15*a^7*c^3*d^7 + (735*a^6*b*c^4*d^6)/4 + (945*a^2*b^5*c^8*d^2)/8 + 525*a^3*b^4*c^7*d
^3 + (3675*a^4*b^3*c^6*d^4)/4 + (1323*a^5*b^2*c^5*d^5)/2 + (35*a*b^6*c^9*d)/4) + x^11*((a^7*d^10)/11 + (120*b^
7*c^7*d^3)/11 + (1470*a*b^6*c^6*d^4)/11 + (5292*a^2*b^5*c^5*d^5)/11 + (7350*a^3*b^4*c^4*d^6)/11 + (4200*a^4*b^
3*c^3*d^7)/11 + (945*a^5*b^2*c^2*d^8)/11 + (70*a^6*b*c*d^9)/11) + x^6*((7*a^2*b^5*c^10)/2 + 42*a^7*c^5*d^5 + (
175*a^3*b^4*c^9*d)/3 + 245*a^6*b*c^6*d^4 + (525*a^4*b^3*c^8*d^2)/2 + 420*a^5*b^2*c^7*d^3) + x^13*((21*a^5*b^2*
d^10)/13 + (252*b^7*c^5*d^5)/13 + (1470*a*b^6*c^4*d^6)/13 + (350*a^4*b^3*c*d^9)/13 + (2520*a^2*b^5*c^3*d^7)/13
+ (1575*a^3*b^4*c^2*d^8)/13) + x^7*(a*b^6*c^10 + 30*a^7*c^4*d^6 + 30*a^2*b^5*c^9*d + 252*a^6*b*c^5*d^5 + 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) + x^12*((7*a^6*b*d^10)/12 + (35*b^7*c^6*d^4)/2 +
147*a*b^6*c^5*d^5 + (35*a^5*b^2*c*d^9)/2 + (735*a^2*b^5*c^4*d^6)/2 + 350*a^3*b^4*c^3*d^7 + (525*a^4*b^3*c^2*d^
8)/4) + a^7*c^10*x + (b^7*d^10*x^18)/18 + (5*a^4*c^7*x^4*(24*a^3*d^3 + 7*b^3*c^3 + 42*a*b^2*c^2*d + 63*a^2*b*c
*d^2))/4 + (b^4*d^7*x^15*(7*a^3*d^3 + 24*b^3*c^3 + 63*a*b^2*c^2*d + 42*a^2*b*c*d^2))/3 + (a^6*c^9*x^2*(10*a*d
+ 7*b*c))/2 + (b^6*d^9*x^17*(7*a*d + 10*b*c))/17 + (a^5*c^8*x^3*(45*a^2*d^2 + 21*b^2*c^2 + 70*a*b*c*d))/3 + (b
^5*d^8*x^16*(21*a^2*d^2 + 45*b^2*c^2 + 70*a*b*c*d))/16

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