\(\int (a+b x)^5 (c+d x)^{10} \, dx\) [1306]

Optimal result

Integrand size = 15, antiderivative size = 146 \[ \int (a+b x)^5 (c+d x)^{10} \, dx=-\frac {(b c-a d)^5 (c+d x)^{11}}{11 d^6}+\frac {5 b (b c-a d)^4 (c+d x)^{12}}{12 d^6}-\frac {10 b^2 (b c-a d)^3 (c+d x)^{13}}{13 d^6}+\frac {5 b^3 (b c-a d)^2 (c+d x)^{14}}{7 d^6}-\frac {b^4 (b c-a d) (c+d x)^{15}}{3 d^6}+\frac {b^5 (c+d x)^{16}}{16 d^6} \]

[Out]

-1/11*(-a*d+b*c)^5*(d*x+c)^11/d^6+5/12*b*(-a*d+b*c)^4*(d*x+c)^12/d^6-10/13*b^2*(-a*d+b*c)^3*(d*x+c)^13/d^6+5/7
*b^3*(-a*d+b*c)^2*(d*x+c)^14/d^6-1/3*b^4*(-a*d+b*c)*(d*x+c)^15/d^6+1/16*b^5*(d*x+c)^16/d^6

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45} \[ \int (a+b x)^5 (c+d x)^{10} \, dx=-\frac {b^4 (c+d x)^{15} (b c-a d)}{3 d^6}+\frac {5 b^3 (c+d x)^{14} (b c-a d)^2}{7 d^6}-\frac {10 b^2 (c+d x)^{13} (b c-a d)^3}{13 d^6}+\frac {5 b (c+d x)^{12} (b c-a d)^4}{12 d^6}-\frac {(c+d x)^{11} (b c-a d)^5}{11 d^6}+\frac {b^5 (c+d x)^{16}}{16 d^6} \]

[In]

Int[(a + b*x)^5*(c + d*x)^10,x]

[Out]

-1/11*((b*c - a*d)^5*(c + d*x)^11)/d^6 + (5*b*(b*c - a*d)^4*(c + d*x)^12)/(12*d^6) - (10*b^2*(b*c - a*d)^3*(c
+ d*x)^13)/(13*d^6) + (5*b^3*(b*c - a*d)^2*(c + d*x)^14)/(7*d^6) - (b^4*(b*c - a*d)*(c + d*x)^15)/(3*d^6) + (b
^5*(c + d*x)^16)/(16*d^6)

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(811\) vs. \(2(146)=292\).

Time = 0.06 (sec) , antiderivative size = 811, normalized size of antiderivative = 5.55 \[ \int (a+b x)^5 (c+d x)^{10} \, dx=a^5 c^{10} x+\frac {5}{2} a^4 c^9 (b c+2 a d) x^2+\frac {5}{3} a^3 c^8 \left (2 b^2 c^2+10 a b c d+9 a^2 d^2\right ) x^3+\frac {5}{4} a^2 c^7 \left (2 b^3 c^3+20 a b^2 c^2 d+45 a^2 b c d^2+24 a^3 d^3\right ) x^4+a c^6 \left (b^4 c^4+20 a b^3 c^3 d+90 a^2 b^2 c^2 d^2+120 a^3 b c d^3+42 a^4 d^4\right ) x^5+\frac {1}{6} c^5 \left (b^5 c^5+50 a b^4 c^4 d+450 a^2 b^3 c^3 d^2+1200 a^3 b^2 c^2 d^3+1050 a^4 b c d^4+252 a^5 d^5\right ) x^6+\frac {5}{7} c^4 d \left (2 b^5 c^5+45 a b^4 c^4 d+240 a^2 b^3 c^3 d^2+420 a^3 b^2 c^2 d^3+252 a^4 b c d^4+42 a^5 d^5\right ) x^7+\frac {15}{8} c^3 d^2 \left (3 b^5 c^5+40 a b^4 c^4 d+140 a^2 b^3 c^3 d^2+168 a^3 b^2 c^2 d^3+70 a^4 b c d^4+8 a^5 d^5\right ) x^8+\frac {5}{3} c^2 d^3 \left (8 b^5 c^5+70 a b^4 c^4 d+168 a^2 b^3 c^3 d^2+140 a^3 b^2 c^2 d^3+40 a^4 b c d^4+3 a^5 d^5\right ) x^9+\frac {1}{2} c d^4 \left (42 b^5 c^5+252 a b^4 c^4 d+420 a^2 b^3 c^3 d^2+240 a^3 b^2 c^2 d^3+45 a^4 b c d^4+2 a^5 d^5\right ) x^{10}+\frac {1}{11} d^5 \left (252 b^5 c^5+1050 a b^4 c^4 d+1200 a^2 b^3 c^3 d^2+450 a^3 b^2 c^2 d^3+50 a^4 b c d^4+a^5 d^5\right ) x^{11}+\frac {5}{12} b d^6 \left (42 b^4 c^4+120 a b^3 c^3 d+90 a^2 b^2 c^2 d^2+20 a^3 b c d^3+a^4 d^4\right ) x^{12}+\frac {5}{13} b^2 d^7 \left (24 b^3 c^3+45 a b^2 c^2 d+20 a^2 b c d^2+2 a^3 d^3\right ) x^{13}+\frac {5}{14} b^3 d^8 \left (9 b^2 c^2+10 a b c d+2 a^2 d^2\right ) x^{14}+\frac {1}{3} b^4 d^9 (2 b c+a d) x^{15}+\frac {1}{16} b^5 d^{10} x^{16} \]

[In]

Integrate[(a + b*x)^5*(c + d*x)^10,x]

[Out]

a^5*c^10*x + (5*a^4*c^9*(b*c + 2*a*d)*x^2)/2 + (5*a^3*c^8*(2*b^2*c^2 + 10*a*b*c*d + 9*a^2*d^2)*x^3)/3 + (5*a^2
*c^7*(2*b^3*c^3 + 20*a*b^2*c^2*d + 45*a^2*b*c*d^2 + 24*a^3*d^3)*x^4)/4 + a*c^6*(b^4*c^4 + 20*a*b^3*c^3*d + 90*
a^2*b^2*c^2*d^2 + 120*a^3*b*c*d^3 + 42*a^4*d^4)*x^5 + (c^5*(b^5*c^5 + 50*a*b^4*c^4*d + 450*a^2*b^3*c^3*d^2 + 1
200*a^3*b^2*c^2*d^3 + 1050*a^4*b*c*d^4 + 252*a^5*d^5)*x^6)/6 + (5*c^4*d*(2*b^5*c^5 + 45*a*b^4*c^4*d + 240*a^2*
b^3*c^3*d^2 + 420*a^3*b^2*c^2*d^3 + 252*a^4*b*c*d^4 + 42*a^5*d^5)*x^7)/7 + (15*c^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^8)/8 + (5*c^2*d^3*(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^9)/3 + (c*d^4*(42
*b^5*c^5 + 252*a*b^4*c^4*d + 420*a^2*b^3*c^3*d^2 + 240*a^3*b^2*c^2*d^3 + 45*a^4*b*c*d^4 + 2*a^5*d^5)*x^10)/2 +
(d^5*(252*b^5*c^5 + 1050*a*b^4*c^4*d + 1200*a^2*b^3*c^3*d^2 + 450*a^3*b^2*c^2*d^3 + 50*a^4*b*c*d^4 + a^5*d^5)
*x^11)/11 + (5*b*d^6*(42*b^4*c^4 + 120*a*b^3*c^3*d + 90*a^2*b^2*c^2*d^2 + 20*a^3*b*c*d^3 + a^4*d^4)*x^12)/12 +
(5*b^2*d^7*(24*b^3*c^3 + 45*a*b^2*c^2*d + 20*a^2*b*c*d^2 + 2*a^3*d^3)*x^13)/13 + (5*b^3*d^8*(9*b^2*c^2 + 10*a
*b*c*d + 2*a^2*d^2)*x^14)/14 + (b^4*d^9*(2*b*c + a*d)*x^15)/3 + (b^5*d^10*x^16)/16

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(826\) vs. \(2(134)=268\).

Time = 0.20 (sec) , antiderivative size = 827, normalized size of antiderivative = 5.66

method	result	size
norman	\(a^{5} c^{10} x +\left (5 a^{5} c^{9} d +\frac {5}{2} a^{4} b \,c^{10}\right ) x^{2}+\left (15 a^{5} c^{8} d^{2}+\frac {50}{3} a^{4} b \,c^{9} d +\frac {10}{3} a^{3} b^{2} c^{10}\right ) x^{3}+\left (30 a^{5} c^{7} d^{3}+\frac {225}{4} a^{4} b \,c^{8} d^{2}+25 a^{3} b^{2} c^{9} d +\frac {5}{2} a^{2} b^{3} c^{10}\right ) x^{4}+\left (42 a^{5} c^{6} d^{4}+120 a^{4} b \,c^{7} d^{3}+90 a^{3} b^{2} c^{8} d^{2}+20 a^{2} b^{3} c^{9} d +a \,b^{4} c^{10}\right ) x^{5}+\left (42 a^{5} c^{5} d^{5}+175 a^{4} b \,c^{6} d^{4}+200 a^{3} b^{2} c^{7} d^{3}+75 a^{2} b^{3} c^{8} d^{2}+\frac {25}{3} a \,b^{4} c^{9} d +\frac {1}{6} b^{5} c^{10}\right ) x^{6}+\left (30 a^{5} c^{4} d^{6}+180 a^{4} b \,c^{5} d^{5}+300 a^{3} b^{2} c^{6} d^{4}+\frac {1200}{7} a^{2} b^{3} c^{7} d^{3}+\frac {225}{7} a \,b^{4} c^{8} d^{2}+\frac {10}{7} b^{5} c^{9} d \right ) x^{7}+\left (15 a^{5} c^{3} d^{7}+\frac {525}{4} a^{4} b \,c^{4} d^{6}+315 a^{3} b^{2} c^{5} d^{5}+\frac {525}{2} a^{2} b^{3} c^{6} d^{4}+75 a \,b^{4} c^{7} d^{3}+\frac {45}{8} b^{5} c^{8} d^{2}\right ) x^{8}+\left (5 a^{5} c^{2} d^{8}+\frac {200}{3} a^{4} b \,c^{3} d^{7}+\frac {700}{3} a^{3} b^{2} c^{4} d^{6}+280 a^{2} b^{3} c^{5} d^{5}+\frac {350}{3} a \,b^{4} c^{6} d^{4}+\frac {40}{3} b^{5} c^{7} d^{3}\right ) x^{9}+\left (a^{5} c \,d^{9}+\frac {45}{2} a^{4} b \,c^{2} d^{8}+120 a^{3} b^{2} c^{3} d^{7}+210 a^{2} b^{3} c^{4} d^{6}+126 a \,b^{4} c^{5} d^{5}+21 b^{5} c^{6} d^{4}\right ) x^{10}+\left (\frac {1}{11} a^{5} d^{10}+\frac {50}{11} a^{4} b c \,d^{9}+\frac {450}{11} a^{3} b^{2} c^{2} d^{8}+\frac {1200}{11} a^{2} b^{3} c^{3} d^{7}+\frac {1050}{11} a \,b^{4} c^{4} d^{6}+\frac {252}{11} b^{5} c^{5} d^{5}\right ) x^{11}+\left (\frac {5}{12} a^{4} b \,d^{10}+\frac {25}{3} a^{3} b^{2} c \,d^{9}+\frac {75}{2} a^{2} b^{3} c^{2} d^{8}+50 a \,b^{4} c^{3} d^{7}+\frac {35}{2} b^{5} c^{4} d^{6}\right ) x^{12}+\left (\frac {10}{13} a^{3} b^{2} d^{10}+\frac {100}{13} a^{2} b^{3} c \,d^{9}+\frac {225}{13} a \,b^{4} c^{2} d^{8}+\frac {120}{13} b^{5} c^{3} d^{7}\right ) x^{13}+\left (\frac {5}{7} a^{2} b^{3} d^{10}+\frac {25}{7} a \,b^{4} c \,d^{9}+\frac {45}{14} b^{5} c^{2} d^{8}\right ) x^{14}+\left (\frac {1}{3} a \,b^{4} d^{10}+\frac {2}{3} b^{5} c \,d^{9}\right ) x^{15}+\frac {b^{5} d^{10} x^{16}}{16}\)	\(827\)
default	\(\frac {b^{5} d^{10} x^{16}}{16}+\frac {\left (5 a \,b^{4} d^{10}+10 b^{5} c \,d^{9}\right ) x^{15}}{15}+\frac {\left (10 a^{2} b^{3} d^{10}+50 a \,b^{4} c \,d^{9}+45 b^{5} c^{2} d^{8}\right ) x^{14}}{14}+\frac {\left (10 a^{3} b^{2} d^{10}+100 a^{2} b^{3} c \,d^{9}+225 a \,b^{4} c^{2} d^{8}+120 b^{5} c^{3} d^{7}\right ) x^{13}}{13}+\frac {\left (5 a^{4} b \,d^{10}+100 a^{3} b^{2} c \,d^{9}+450 a^{2} b^{3} c^{2} d^{8}+600 a \,b^{4} c^{3} d^{7}+210 b^{5} c^{4} d^{6}\right ) x^{12}}{12}+\frac {\left (a^{5} d^{10}+50 a^{4} b c \,d^{9}+450 a^{3} b^{2} c^{2} d^{8}+1200 a^{2} b^{3} c^{3} d^{7}+1050 a \,b^{4} c^{4} d^{6}+252 b^{5} c^{5} d^{5}\right ) x^{11}}{11}+\frac {\left (10 a^{5} c \,d^{9}+225 a^{4} b \,c^{2} d^{8}+1200 a^{3} b^{2} c^{3} d^{7}+2100 a^{2} b^{3} c^{4} d^{6}+1260 a \,b^{4} c^{5} d^{5}+210 b^{5} c^{6} d^{4}\right ) x^{10}}{10}+\frac {\left (45 a^{5} c^{2} d^{8}+600 a^{4} b \,c^{3} d^{7}+2100 a^{3} b^{2} c^{4} d^{6}+2520 a^{2} b^{3} c^{5} d^{5}+1050 a \,b^{4} c^{6} d^{4}+120 b^{5} c^{7} d^{3}\right ) x^{9}}{9}+\frac {\left (120 a^{5} c^{3} d^{7}+1050 a^{4} b \,c^{4} d^{6}+2520 a^{3} b^{2} c^{5} d^{5}+2100 a^{2} b^{3} c^{6} d^{4}+600 a \,b^{4} c^{7} d^{3}+45 b^{5} c^{8} d^{2}\right ) x^{8}}{8}+\frac {\left (210 a^{5} c^{4} d^{6}+1260 a^{4} b \,c^{5} d^{5}+2100 a^{3} b^{2} c^{6} d^{4}+1200 a^{2} b^{3} c^{7} d^{3}+225 a \,b^{4} c^{8} d^{2}+10 b^{5} c^{9} d \right ) x^{7}}{7}+\frac {\left (252 a^{5} c^{5} d^{5}+1050 a^{4} b \,c^{6} d^{4}+1200 a^{3} b^{2} c^{7} d^{3}+450 a^{2} b^{3} c^{8} d^{2}+50 a \,b^{4} c^{9} d +b^{5} c^{10}\right ) x^{6}}{6}+\frac {\left (210 a^{5} c^{6} d^{4}+600 a^{4} b \,c^{7} d^{3}+450 a^{3} b^{2} c^{8} d^{2}+100 a^{2} b^{3} c^{9} d +5 a \,b^{4} c^{10}\right ) x^{5}}{5}+\frac {\left (120 a^{5} c^{7} d^{3}+225 a^{4} b \,c^{8} d^{2}+100 a^{3} b^{2} c^{9} d +10 a^{2} b^{3} c^{10}\right ) x^{4}}{4}+\frac {\left (45 a^{5} c^{8} d^{2}+50 a^{4} b \,c^{9} d +10 a^{3} b^{2} c^{10}\right ) x^{3}}{3}+\frac {\left (10 a^{5} c^{9} d +5 a^{4} b \,c^{10}\right ) x^{2}}{2}+a^{5} c^{10} x\)	\(841\)
gosper	\(\frac {1}{6} x^{6} b^{5} c^{10}+\frac {1}{11} x^{11} a^{5} d^{10}+a^{5} c^{10} x +\frac {1}{16} b^{5} d^{10} x^{16}+\frac {25}{3} x^{12} a^{3} b^{2} c \,d^{9}+\frac {1050}{11} x^{11} a \,b^{4} c^{4} d^{6}+\frac {50}{3} x^{3} a^{4} b \,c^{9} d +\frac {225}{4} x^{4} a^{4} b \,c^{8} d^{2}+25 x^{4} a^{3} b^{2} c^{9} d +175 x^{6} a^{4} b \,c^{6} d^{4}+200 x^{6} a^{3} b^{2} c^{7} d^{3}+75 x^{6} a^{2} b^{3} c^{8} d^{2}+\frac {25}{3} x^{6} a \,b^{4} c^{9} d +180 x^{7} a^{4} b \,c^{5} d^{5}+300 x^{7} a^{3} b^{2} c^{6} d^{4}+\frac {1200}{7} x^{7} a^{2} b^{3} c^{7} d^{3}+\frac {225}{7} x^{7} a \,b^{4} c^{8} d^{2}+\frac {525}{4} x^{8} a^{4} b \,c^{4} d^{6}+315 x^{8} a^{3} b^{2} c^{5} d^{5}+\frac {525}{2} x^{8} a^{2} b^{3} c^{6} d^{4}+75 x^{8} a \,b^{4} c^{7} d^{3}+\frac {200}{3} x^{9} a^{4} b \,c^{3} d^{7}+\frac {700}{3} x^{9} a^{3} b^{2} c^{4} d^{6}+280 x^{9} a^{2} b^{3} c^{5} d^{5}+\frac {350}{3} x^{9} a \,b^{4} c^{6} d^{4}+\frac {45}{2} x^{10} a^{4} b \,c^{2} d^{8}+120 x^{10} a^{3} b^{2} c^{3} d^{7}+210 x^{10} a^{2} b^{3} c^{4} d^{6}+126 x^{10} a \,b^{4} c^{5} d^{5}+\frac {50}{11} x^{11} a^{4} b c \,d^{9}+\frac {450}{11} x^{11} a^{3} b^{2} c^{2} d^{8}+\frac {1200}{11} x^{11} a^{2} b^{3} c^{3} d^{7}+\frac {25}{7} x^{14} a \,b^{4} c \,d^{9}+120 a^{4} b \,c^{7} d^{3} x^{5}+90 a^{3} b^{2} c^{8} d^{2} x^{5}+20 a^{2} b^{3} c^{9} d \,x^{5}+5 x^{2} a^{5} c^{9} d +\frac {5}{2} x^{2} a^{4} b \,c^{10}+15 x^{3} a^{5} c^{8} d^{2}+\frac {10}{3} x^{3} a^{3} b^{2} c^{10}+30 x^{4} a^{5} c^{7} d^{3}+\frac {5}{2} x^{4} a^{2} b^{3} c^{10}+42 x^{6} a^{5} c^{5} d^{5}+30 x^{7} a^{5} c^{4} d^{6}+\frac {10}{7} x^{7} b^{5} c^{9} d +15 x^{8} a^{5} c^{3} d^{7}+\frac {45}{8} x^{8} b^{5} c^{8} d^{2}+5 x^{9} a^{5} c^{2} d^{8}+\frac {40}{3} x^{9} b^{5} c^{7} d^{3}+x^{10} a^{5} c \,d^{9}+21 x^{10} b^{5} c^{6} d^{4}+\frac {252}{11} x^{11} b^{5} c^{5} d^{5}+\frac {5}{12} x^{12} a^{4} b \,d^{10}+\frac {35}{2} x^{12} b^{5} c^{4} d^{6}+\frac {10}{13} x^{13} a^{3} b^{2} d^{10}+\frac {120}{13} x^{13} b^{5} c^{3} d^{7}+\frac {75}{2} x^{12} a^{2} b^{3} c^{2} d^{8}+50 x^{12} a \,b^{4} c^{3} d^{7}+\frac {100}{13} x^{13} a^{2} b^{3} c \,d^{9}+\frac {225}{13} x^{13} a \,b^{4} c^{2} d^{8}+\frac {5}{7} x^{14} a^{2} b^{3} d^{10}+\frac {45}{14} x^{14} b^{5} c^{2} d^{8}+\frac {1}{3} x^{15} a \,b^{4} d^{10}+\frac {2}{3} x^{15} b^{5} c \,d^{9}+42 a^{5} c^{6} d^{4} x^{5}+a \,b^{4} c^{10} x^{5}\)	\(949\)
risch	\(\frac {1}{6} x^{6} b^{5} c^{10}+\frac {1}{11} x^{11} a^{5} d^{10}+a^{5} c^{10} x +\frac {1}{16} b^{5} d^{10} x^{16}+\frac {25}{3} x^{12} a^{3} b^{2} c \,d^{9}+\frac {1050}{11} x^{11} a \,b^{4} c^{4} d^{6}+\frac {50}{3} x^{3} a^{4} b \,c^{9} d +\frac {225}{4} x^{4} a^{4} b \,c^{8} d^{2}+25 x^{4} a^{3} b^{2} c^{9} d +175 x^{6} a^{4} b \,c^{6} d^{4}+200 x^{6} a^{3} b^{2} c^{7} d^{3}+75 x^{6} a^{2} b^{3} c^{8} d^{2}+\frac {25}{3} x^{6} a \,b^{4} c^{9} d +180 x^{7} a^{4} b \,c^{5} d^{5}+300 x^{7} a^{3} b^{2} c^{6} d^{4}+\frac {1200}{7} x^{7} a^{2} b^{3} c^{7} d^{3}+\frac {225}{7} x^{7} a \,b^{4} c^{8} d^{2}+\frac {525}{4} x^{8} a^{4} b \,c^{4} d^{6}+315 x^{8} a^{3} b^{2} c^{5} d^{5}+\frac {525}{2} x^{8} a^{2} b^{3} c^{6} d^{4}+75 x^{8} a \,b^{4} c^{7} d^{3}+\frac {200}{3} x^{9} a^{4} b \,c^{3} d^{7}+\frac {700}{3} x^{9} a^{3} b^{2} c^{4} d^{6}+280 x^{9} a^{2} b^{3} c^{5} d^{5}+\frac {350}{3} x^{9} a \,b^{4} c^{6} d^{4}+\frac {45}{2} x^{10} a^{4} b \,c^{2} d^{8}+120 x^{10} a^{3} b^{2} c^{3} d^{7}+210 x^{10} a^{2} b^{3} c^{4} d^{6}+126 x^{10} a \,b^{4} c^{5} d^{5}+\frac {50}{11} x^{11} a^{4} b c \,d^{9}+\frac {450}{11} x^{11} a^{3} b^{2} c^{2} d^{8}+\frac {1200}{11} x^{11} a^{2} b^{3} c^{3} d^{7}+\frac {25}{7} x^{14} a \,b^{4} c \,d^{9}+120 a^{4} b \,c^{7} d^{3} x^{5}+90 a^{3} b^{2} c^{8} d^{2} x^{5}+20 a^{2} b^{3} c^{9} d \,x^{5}+5 x^{2} a^{5} c^{9} d +\frac {5}{2} x^{2} a^{4} b \,c^{10}+15 x^{3} a^{5} c^{8} d^{2}+\frac {10}{3} x^{3} a^{3} b^{2} c^{10}+30 x^{4} a^{5} c^{7} d^{3}+\frac {5}{2} x^{4} a^{2} b^{3} c^{10}+42 x^{6} a^{5} c^{5} d^{5}+30 x^{7} a^{5} c^{4} d^{6}+\frac {10}{7} x^{7} b^{5} c^{9} d +15 x^{8} a^{5} c^{3} d^{7}+\frac {45}{8} x^{8} b^{5} c^{8} d^{2}+5 x^{9} a^{5} c^{2} d^{8}+\frac {40}{3} x^{9} b^{5} c^{7} d^{3}+x^{10} a^{5} c \,d^{9}+21 x^{10} b^{5} c^{6} d^{4}+\frac {252}{11} x^{11} b^{5} c^{5} d^{5}+\frac {5}{12} x^{12} a^{4} b \,d^{10}+\frac {35}{2} x^{12} b^{5} c^{4} d^{6}+\frac {10}{13} x^{13} a^{3} b^{2} d^{10}+\frac {120}{13} x^{13} b^{5} c^{3} d^{7}+\frac {75}{2} x^{12} a^{2} b^{3} c^{2} d^{8}+50 x^{12} a \,b^{4} c^{3} d^{7}+\frac {100}{13} x^{13} a^{2} b^{3} c \,d^{9}+\frac {225}{13} x^{13} a \,b^{4} c^{2} d^{8}+\frac {5}{7} x^{14} a^{2} b^{3} d^{10}+\frac {45}{14} x^{14} b^{5} c^{2} d^{8}+\frac {1}{3} x^{15} a \,b^{4} d^{10}+\frac {2}{3} x^{15} b^{5} c \,d^{9}+42 a^{5} c^{6} d^{4} x^{5}+a \,b^{4} c^{10} x^{5}\)	\(949\)
parallelrisch	\(\frac {1}{6} x^{6} b^{5} c^{10}+\frac {1}{11} x^{11} a^{5} d^{10}+a^{5} c^{10} x +\frac {1}{16} b^{5} d^{10} x^{16}+\frac {25}{3} x^{12} a^{3} b^{2} c \,d^{9}+\frac {1050}{11} x^{11} a \,b^{4} c^{4} d^{6}+\frac {50}{3} x^{3} a^{4} b \,c^{9} d +\frac {225}{4} x^{4} a^{4} b \,c^{8} d^{2}+25 x^{4} a^{3} b^{2} c^{9} d +175 x^{6} a^{4} b \,c^{6} d^{4}+200 x^{6} a^{3} b^{2} c^{7} d^{3}+75 x^{6} a^{2} b^{3} c^{8} d^{2}+\frac {25}{3} x^{6} a \,b^{4} c^{9} d +180 x^{7} a^{4} b \,c^{5} d^{5}+300 x^{7} a^{3} b^{2} c^{6} d^{4}+\frac {1200}{7} x^{7} a^{2} b^{3} c^{7} d^{3}+\frac {225}{7} x^{7} a \,b^{4} c^{8} d^{2}+\frac {525}{4} x^{8} a^{4} b \,c^{4} d^{6}+315 x^{8} a^{3} b^{2} c^{5} d^{5}+\frac {525}{2} x^{8} a^{2} b^{3} c^{6} d^{4}+75 x^{8} a \,b^{4} c^{7} d^{3}+\frac {200}{3} x^{9} a^{4} b \,c^{3} d^{7}+\frac {700}{3} x^{9} a^{3} b^{2} c^{4} d^{6}+280 x^{9} a^{2} b^{3} c^{5} d^{5}+\frac {350}{3} x^{9} a \,b^{4} c^{6} d^{4}+\frac {45}{2} x^{10} a^{4} b \,c^{2} d^{8}+120 x^{10} a^{3} b^{2} c^{3} d^{7}+210 x^{10} a^{2} b^{3} c^{4} d^{6}+126 x^{10} a \,b^{4} c^{5} d^{5}+\frac {50}{11} x^{11} a^{4} b c \,d^{9}+\frac {450}{11} x^{11} a^{3} b^{2} c^{2} d^{8}+\frac {1200}{11} x^{11} a^{2} b^{3} c^{3} d^{7}+\frac {25}{7} x^{14} a \,b^{4} c \,d^{9}+120 a^{4} b \,c^{7} d^{3} x^{5}+90 a^{3} b^{2} c^{8} d^{2} x^{5}+20 a^{2} b^{3} c^{9} d \,x^{5}+5 x^{2} a^{5} c^{9} d +\frac {5}{2} x^{2} a^{4} b \,c^{10}+15 x^{3} a^{5} c^{8} d^{2}+\frac {10}{3} x^{3} a^{3} b^{2} c^{10}+30 x^{4} a^{5} c^{7} d^{3}+\frac {5}{2} x^{4} a^{2} b^{3} c^{10}+42 x^{6} a^{5} c^{5} d^{5}+30 x^{7} a^{5} c^{4} d^{6}+\frac {10}{7} x^{7} b^{5} c^{9} d +15 x^{8} a^{5} c^{3} d^{7}+\frac {45}{8} x^{8} b^{5} c^{8} d^{2}+5 x^{9} a^{5} c^{2} d^{8}+\frac {40}{3} x^{9} b^{5} c^{7} d^{3}+x^{10} a^{5} c \,d^{9}+21 x^{10} b^{5} c^{6} d^{4}+\frac {252}{11} x^{11} b^{5} c^{5} d^{5}+\frac {5}{12} x^{12} a^{4} b \,d^{10}+\frac {35}{2} x^{12} b^{5} c^{4} d^{6}+\frac {10}{13} x^{13} a^{3} b^{2} d^{10}+\frac {120}{13} x^{13} b^{5} c^{3} d^{7}+\frac {75}{2} x^{12} a^{2} b^{3} c^{2} d^{8}+50 x^{12} a \,b^{4} c^{3} d^{7}+\frac {100}{13} x^{13} a^{2} b^{3} c \,d^{9}+\frac {225}{13} x^{13} a \,b^{4} c^{2} d^{8}+\frac {5}{7} x^{14} a^{2} b^{3} d^{10}+\frac {45}{14} x^{14} b^{5} c^{2} d^{8}+\frac {1}{3} x^{15} a \,b^{4} d^{10}+\frac {2}{3} x^{15} b^{5} c \,d^{9}+42 a^{5} c^{6} d^{4} x^{5}+a \,b^{4} c^{10} x^{5}\)	\(949\)

[In]

int((b*x+a)^5*(d*x+c)^10,x,method=_RETURNVERBOSE)

[Out]

a^5*c^10*x+(5*a^5*c^9*d+5/2*a^4*b*c^10)*x^2+(15*a^5*c^8*d^2+50/3*a^4*b*c^9*d+10/3*a^3*b^2*c^10)*x^3+(30*a^5*c^
7*d^3+225/4*a^4*b*c^8*d^2+25*a^3*b^2*c^9*d+5/2*a^2*b^3*c^10)*x^4+(42*a^5*c^6*d^4+120*a^4*b*c^7*d^3+90*a^3*b^2*
c^8*d^2+20*a^2*b^3*c^9*d+a*b^4*c^10)*x^5+(42*a^5*c^5*d^5+175*a^4*b*c^6*d^4+200*a^3*b^2*c^7*d^3+75*a^2*b^3*c^8*
d^2+25/3*a*b^4*c^9*d+1/6*b^5*c^10)*x^6+(30*a^5*c^4*d^6+180*a^4*b*c^5*d^5+300*a^3*b^2*c^6*d^4+1200/7*a^2*b^3*c^
7*d^3+225/7*a*b^4*c^8*d^2+10/7*b^5*c^9*d)*x^7+(15*a^5*c^3*d^7+525/4*a^4*b*c^4*d^6+315*a^3*b^2*c^5*d^5+525/2*a^
2*b^3*c^6*d^4+75*a*b^4*c^7*d^3+45/8*b^5*c^8*d^2)*x^8+(5*a^5*c^2*d^8+200/3*a^4*b*c^3*d^7+700/3*a^3*b^2*c^4*d^6+
280*a^2*b^3*c^5*d^5+350/3*a*b^4*c^6*d^4+40/3*b^5*c^7*d^3)*x^9+(a^5*c*d^9+45/2*a^4*b*c^2*d^8+120*a^3*b^2*c^3*d^
7+210*a^2*b^3*c^4*d^6+126*a*b^4*c^5*d^5+21*b^5*c^6*d^4)*x^10+(1/11*a^5*d^10+50/11*a^4*b*c*d^9+450/11*a^3*b^2*c
^2*d^8+1200/11*a^2*b^3*c^3*d^7+1050/11*a*b^4*c^4*d^6+252/11*b^5*c^5*d^5)*x^11+(5/12*a^4*b*d^10+25/3*a^3*b^2*c*
d^9+75/2*a^2*b^3*c^2*d^8+50*a*b^4*c^3*d^7+35/2*b^5*c^4*d^6)*x^12+(10/13*a^3*b^2*d^10+100/13*a^2*b^3*c*d^9+225/
13*a*b^4*c^2*d^8+120/13*b^5*c^3*d^7)*x^13+(5/7*a^2*b^3*d^10+25/7*a*b^4*c*d^9+45/14*b^5*c^2*d^8)*x^14+(1/3*a*b^
4*d^10+2/3*b^5*c*d^9)*x^15+1/16*b^5*d^10*x^16

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 835 vs. \(2 (134) = 268\).

Time = 0.22 (sec) , antiderivative size = 835, normalized size of antiderivative = 5.72 \[ \int (a+b x)^5 (c+d x)^{10} \, dx=\frac {1}{16} \, b^{5} d^{10} x^{16} + a^{5} c^{10} x + \frac {1}{3} \, {\left (2 \, b^{5} c d^{9} + a b^{4} d^{10}\right )} x^{15} + \frac {5}{14} \, {\left (9 \, b^{5} c^{2} d^{8} + 10 \, a b^{4} c d^{9} + 2 \, a^{2} b^{3} d^{10}\right )} x^{14} + \frac {5}{13} \, {\left (24 \, b^{5} c^{3} d^{7} + 45 \, a b^{4} c^{2} d^{8} + 20 \, a^{2} b^{3} c d^{9} + 2 \, a^{3} b^{2} d^{10}\right )} x^{13} + \frac {5}{12} \, {\left (42 \, b^{5} c^{4} d^{6} + 120 \, a b^{4} c^{3} d^{7} + 90 \, a^{2} b^{3} c^{2} d^{8} + 20 \, a^{3} b^{2} c d^{9} + a^{4} b d^{10}\right )} x^{12} + \frac {1}{11} \, {\left (252 \, b^{5} c^{5} d^{5} + 1050 \, a b^{4} c^{4} d^{6} + 1200 \, a^{2} b^{3} c^{3} d^{7} + 450 \, a^{3} b^{2} c^{2} d^{8} + 50 \, a^{4} b c d^{9} + a^{5} d^{10}\right )} x^{11} + \frac {1}{2} \, {\left (42 \, b^{5} c^{6} d^{4} + 252 \, a b^{4} c^{5} d^{5} + 420 \, a^{2} b^{3} c^{4} d^{6} + 240 \, a^{3} b^{2} c^{3} d^{7} + 45 \, a^{4} b c^{2} d^{8} + 2 \, a^{5} c d^{9}\right )} x^{10} + \frac {5}{3} \, {\left (8 \, b^{5} c^{7} d^{3} + 70 \, a b^{4} c^{6} d^{4} + 168 \, a^{2} b^{3} c^{5} d^{5} + 140 \, a^{3} b^{2} c^{4} d^{6} + 40 \, a^{4} b c^{3} d^{7} + 3 \, a^{5} c^{2} d^{8}\right )} x^{9} + \frac {15}{8} \, {\left (3 \, b^{5} c^{8} d^{2} + 40 \, a b^{4} c^{7} d^{3} + 140 \, a^{2} b^{3} c^{6} d^{4} + 168 \, a^{3} b^{2} c^{5} d^{5} + 70 \, a^{4} b c^{4} d^{6} + 8 \, a^{5} c^{3} d^{7}\right )} x^{8} + \frac {5}{7} \, {\left (2 \, b^{5} c^{9} d + 45 \, a b^{4} c^{8} d^{2} + 240 \, a^{2} b^{3} c^{7} d^{3} + 420 \, a^{3} b^{2} c^{6} d^{4} + 252 \, a^{4} b c^{5} d^{5} + 42 \, a^{5} c^{4} d^{6}\right )} x^{7} + \frac {1}{6} \, {\left (b^{5} c^{10} + 50 \, a b^{4} c^{9} d + 450 \, a^{2} b^{3} c^{8} d^{2} + 1200 \, a^{3} b^{2} c^{7} d^{3} + 1050 \, a^{4} b c^{6} d^{4} + 252 \, a^{5} c^{5} d^{5}\right )} x^{6} + {\left (a b^{4} c^{10} + 20 \, a^{2} b^{3} c^{9} d + 90 \, a^{3} b^{2} c^{8} d^{2} + 120 \, a^{4} b c^{7} d^{3} + 42 \, a^{5} c^{6} d^{4}\right )} x^{5} + \frac {5}{4} \, {\left (2 \, a^{2} b^{3} c^{10} + 20 \, a^{3} b^{2} c^{9} d + 45 \, a^{4} b c^{8} d^{2} + 24 \, a^{5} c^{7} d^{3}\right )} x^{4} + \frac {5}{3} \, {\left (2 \, a^{3} b^{2} c^{10} + 10 \, a^{4} b c^{9} d + 9 \, a^{5} c^{8} d^{2}\right )} x^{3} + \frac {5}{2} \, {\left (a^{4} b c^{10} + 2 \, a^{5} c^{9} d\right )} x^{2} \]

[In]

integrate((b*x+a)^5*(d*x+c)^10,x, algorithm="fricas")

[Out]

1/16*b^5*d^10*x^16 + a^5*c^10*x + 1/3*(2*b^5*c*d^9 + a*b^4*d^10)*x^15 + 5/14*(9*b^5*c^2*d^8 + 10*a*b^4*c*d^9 +
2*a^2*b^3*d^10)*x^14 + 5/13*(24*b^5*c^3*d^7 + 45*a*b^4*c^2*d^8 + 20*a^2*b^3*c*d^9 + 2*a^3*b^2*d^10)*x^13 + 5/
12*(42*b^5*c^4*d^6 + 120*a*b^4*c^3*d^7 + 90*a^2*b^3*c^2*d^8 + 20*a^3*b^2*c*d^9 + a^4*b*d^10)*x^12 + 1/11*(252*
b^5*c^5*d^5 + 1050*a*b^4*c^4*d^6 + 1200*a^2*b^3*c^3*d^7 + 450*a^3*b^2*c^2*d^8 + 50*a^4*b*c*d^9 + a^5*d^10)*x^1
1 + 1/2*(42*b^5*c^6*d^4 + 252*a*b^4*c^5*d^5 + 420*a^2*b^3*c^4*d^6 + 240*a^3*b^2*c^3*d^7 + 45*a^4*b*c^2*d^8 + 2
*a^5*c*d^9)*x^10 + 5/3*(8*b^5*c^7*d^3 + 70*a*b^4*c^6*d^4 + 168*a^2*b^3*c^5*d^5 + 140*a^3*b^2*c^4*d^6 + 40*a^4*
b*c^3*d^7 + 3*a^5*c^2*d^8)*x^9 + 15/8*(3*b^5*c^8*d^2 + 40*a*b^4*c^7*d^3 + 140*a^2*b^3*c^6*d^4 + 168*a^3*b^2*c^
5*d^5 + 70*a^4*b*c^4*d^6 + 8*a^5*c^3*d^7)*x^8 + 5/7*(2*b^5*c^9*d + 45*a*b^4*c^8*d^2 + 240*a^2*b^3*c^7*d^3 + 42
0*a^3*b^2*c^6*d^4 + 252*a^4*b*c^5*d^5 + 42*a^5*c^4*d^6)*x^7 + 1/6*(b^5*c^10 + 50*a*b^4*c^9*d + 450*a^2*b^3*c^8
*d^2 + 1200*a^3*b^2*c^7*d^3 + 1050*a^4*b*c^6*d^4 + 252*a^5*c^5*d^5)*x^6 + (a*b^4*c^10 + 20*a^2*b^3*c^9*d + 90*
a^3*b^2*c^8*d^2 + 120*a^4*b*c^7*d^3 + 42*a^5*c^6*d^4)*x^5 + 5/4*(2*a^2*b^3*c^10 + 20*a^3*b^2*c^9*d + 45*a^4*b*
c^8*d^2 + 24*a^5*c^7*d^3)*x^4 + 5/3*(2*a^3*b^2*c^10 + 10*a^4*b*c^9*d + 9*a^5*c^8*d^2)*x^3 + 5/2*(a^4*b*c^10 +
2*a^5*c^9*d)*x^2

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 940 vs. \(2 (131) = 262\).

Time = 0.08 (sec) , antiderivative size = 940, normalized size of antiderivative = 6.44 \[ \int (a+b x)^5 (c+d x)^{10} \, dx=a^{5} c^{10} x + \frac {b^{5} d^{10} x^{16}}{16} + x^{15} \left (\frac {a b^{4} d^{10}}{3} + \frac {2 b^{5} c d^{9}}{3}\right ) + x^{14} \cdot \left (\frac {5 a^{2} b^{3} d^{10}}{7} + \frac {25 a b^{4} c d^{9}}{7} + \frac {45 b^{5} c^{2} d^{8}}{14}\right ) + x^{13} \cdot \left (\frac {10 a^{3} b^{2} d^{10}}{13} + \frac {100 a^{2} b^{3} c d^{9}}{13} + \frac {225 a b^{4} c^{2} d^{8}}{13} + \frac {120 b^{5} c^{3} d^{7}}{13}\right ) + x^{12} \cdot \left (\frac {5 a^{4} b d^{10}}{12} + \frac {25 a^{3} b^{2} c d^{9}}{3} + \frac {75 a^{2} b^{3} c^{2} d^{8}}{2} + 50 a b^{4} c^{3} d^{7} + \frac {35 b^{5} c^{4} d^{6}}{2}\right ) + x^{11} \left (\frac {a^{5} d^{10}}{11} + \frac {50 a^{4} b c d^{9}}{11} + \frac {450 a^{3} b^{2} c^{2} d^{8}}{11} + \frac {1200 a^{2} b^{3} c^{3} d^{7}}{11} + \frac {1050 a b^{4} c^{4} d^{6}}{11} + \frac {252 b^{5} c^{5} d^{5}}{11}\right ) + x^{10} \left (a^{5} c d^{9} + \frac {45 a^{4} b c^{2} d^{8}}{2} + 120 a^{3} b^{2} c^{3} d^{7} + 210 a^{2} b^{3} c^{4} d^{6} + 126 a b^{4} c^{5} d^{5} + 21 b^{5} c^{6} d^{4}\right ) + x^{9} \cdot \left (5 a^{5} c^{2} d^{8} + \frac {200 a^{4} b c^{3} d^{7}}{3} + \frac {700 a^{3} b^{2} c^{4} d^{6}}{3} + 280 a^{2} b^{3} c^{5} d^{5} + \frac {350 a b^{4} c^{6} d^{4}}{3} + \frac {40 b^{5} c^{7} d^{3}}{3}\right ) + x^{8} \cdot \left (15 a^{5} c^{3} d^{7} + \frac {525 a^{4} b c^{4} d^{6}}{4} + 315 a^{3} b^{2} c^{5} d^{5} + \frac {525 a^{2} b^{3} c^{6} d^{4}}{2} + 75 a b^{4} c^{7} d^{3} + \frac {45 b^{5} c^{8} d^{2}}{8}\right ) + x^{7} \cdot \left (30 a^{5} c^{4} d^{6} + 180 a^{4} b c^{5} d^{5} + 300 a^{3} b^{2} c^{6} d^{4} + \frac {1200 a^{2} b^{3} c^{7} d^{3}}{7} + \frac {225 a b^{4} c^{8} d^{2}}{7} + \frac {10 b^{5} c^{9} d}{7}\right ) + x^{6} \cdot \left (42 a^{5} c^{5} d^{5} + 175 a^{4} b c^{6} d^{4} + 200 a^{3} b^{2} c^{7} d^{3} + 75 a^{2} b^{3} c^{8} d^{2} + \frac {25 a b^{4} c^{9} d}{3} + \frac {b^{5} c^{10}}{6}\right ) + x^{5} \cdot \left (42 a^{5} c^{6} d^{4} + 120 a^{4} b c^{7} d^{3} + 90 a^{3} b^{2} c^{8} d^{2} + 20 a^{2} b^{3} c^{9} d + a b^{4} c^{10}\right ) + x^{4} \cdot \left (30 a^{5} c^{7} d^{3} + \frac {225 a^{4} b c^{8} d^{2}}{4} + 25 a^{3} b^{2} c^{9} d + \frac {5 a^{2} b^{3} c^{10}}{2}\right ) + x^{3} \cdot \left (15 a^{5} c^{8} d^{2} + \frac {50 a^{4} b c^{9} d}{3} + \frac {10 a^{3} b^{2} c^{10}}{3}\right ) + x^{2} \cdot \left (5 a^{5} c^{9} d + \frac {5 a^{4} b c^{10}}{2}\right ) \]

[In]

integrate((b*x+a)**5*(d*x+c)**10,x)

[Out]

a**5*c**10*x + b**5*d**10*x**16/16 + x**15*(a*b**4*d**10/3 + 2*b**5*c*d**9/3) + x**14*(5*a**2*b**3*d**10/7 + 2
5*a*b**4*c*d**9/7 + 45*b**5*c**2*d**8/14) + x**13*(10*a**3*b**2*d**10/13 + 100*a**2*b**3*c*d**9/13 + 225*a*b**
4*c**2*d**8/13 + 120*b**5*c**3*d**7/13) + x**12*(5*a**4*b*d**10/12 + 25*a**3*b**2*c*d**9/3 + 75*a**2*b**3*c**2
*d**8/2 + 50*a*b**4*c**3*d**7 + 35*b**5*c**4*d**6/2) + x**11*(a**5*d**10/11 + 50*a**4*b*c*d**9/11 + 450*a**3*b
**2*c**2*d**8/11 + 1200*a**2*b**3*c**3*d**7/11 + 1050*a*b**4*c**4*d**6/11 + 252*b**5*c**5*d**5/11) + x**10*(a*
*5*c*d**9 + 45*a**4*b*c**2*d**8/2 + 120*a**3*b**2*c**3*d**7 + 210*a**2*b**3*c**4*d**6 + 126*a*b**4*c**5*d**5 +
21*b**5*c**6*d**4) + x**9*(5*a**5*c**2*d**8 + 200*a**4*b*c**3*d**7/3 + 700*a**3*b**2*c**4*d**6/3 + 280*a**2*b
**3*c**5*d**5 + 350*a*b**4*c**6*d**4/3 + 40*b**5*c**7*d**3/3) + x**8*(15*a**5*c**3*d**7 + 525*a**4*b*c**4*d**6
/4 + 315*a**3*b**2*c**5*d**5 + 525*a**2*b**3*c**6*d**4/2 + 75*a*b**4*c**7*d**3 + 45*b**5*c**8*d**2/8) + x**7*(
30*a**5*c**4*d**6 + 180*a**4*b*c**5*d**5 + 300*a**3*b**2*c**6*d**4 + 1200*a**2*b**3*c**7*d**3/7 + 225*a*b**4*c
**8*d**2/7 + 10*b**5*c**9*d/7) + x**6*(42*a**5*c**5*d**5 + 175*a**4*b*c**6*d**4 + 200*a**3*b**2*c**7*d**3 + 75
*a**2*b**3*c**8*d**2 + 25*a*b**4*c**9*d/3 + b**5*c**10/6) + x**5*(42*a**5*c**6*d**4 + 120*a**4*b*c**7*d**3 + 9
0*a**3*b**2*c**8*d**2 + 20*a**2*b**3*c**9*d + a*b**4*c**10) + x**4*(30*a**5*c**7*d**3 + 225*a**4*b*c**8*d**2/4
+ 25*a**3*b**2*c**9*d + 5*a**2*b**3*c**10/2) + x**3*(15*a**5*c**8*d**2 + 50*a**4*b*c**9*d/3 + 10*a**3*b**2*c*
*10/3) + x**2*(5*a**5*c**9*d + 5*a**4*b*c**10/2)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 835 vs. \(2 (134) = 268\).

Time = 0.23 (sec) , antiderivative size = 835, normalized size of antiderivative = 5.72 \[ \int (a+b x)^5 (c+d x)^{10} \, dx=\frac {1}{16} \, b^{5} d^{10} x^{16} + a^{5} c^{10} x + \frac {1}{3} \, {\left (2 \, b^{5} c d^{9} + a b^{4} d^{10}\right )} x^{15} + \frac {5}{14} \, {\left (9 \, b^{5} c^{2} d^{8} + 10 \, a b^{4} c d^{9} + 2 \, a^{2} b^{3} d^{10}\right )} x^{14} + \frac {5}{13} \, {\left (24 \, b^{5} c^{3} d^{7} + 45 \, a b^{4} c^{2} d^{8} + 20 \, a^{2} b^{3} c d^{9} + 2 \, a^{3} b^{2} d^{10}\right )} x^{13} + \frac {5}{12} \, {\left (42 \, b^{5} c^{4} d^{6} + 120 \, a b^{4} c^{3} d^{7} + 90 \, a^{2} b^{3} c^{2} d^{8} + 20 \, a^{3} b^{2} c d^{9} + a^{4} b d^{10}\right )} x^{12} + \frac {1}{11} \, {\left (252 \, b^{5} c^{5} d^{5} + 1050 \, a b^{4} c^{4} d^{6} + 1200 \, a^{2} b^{3} c^{3} d^{7} + 450 \, a^{3} b^{2} c^{2} d^{8} + 50 \, a^{4} b c d^{9} + a^{5} d^{10}\right )} x^{11} + \frac {1}{2} \, {\left (42 \, b^{5} c^{6} d^{4} + 252 \, a b^{4} c^{5} d^{5} + 420 \, a^{2} b^{3} c^{4} d^{6} + 240 \, a^{3} b^{2} c^{3} d^{7} + 45 \, a^{4} b c^{2} d^{8} + 2 \, a^{5} c d^{9}\right )} x^{10} + \frac {5}{3} \, {\left (8 \, b^{5} c^{7} d^{3} + 70 \, a b^{4} c^{6} d^{4} + 168 \, a^{2} b^{3} c^{5} d^{5} + 140 \, a^{3} b^{2} c^{4} d^{6} + 40 \, a^{4} b c^{3} d^{7} + 3 \, a^{5} c^{2} d^{8}\right )} x^{9} + \frac {15}{8} \, {\left (3 \, b^{5} c^{8} d^{2} + 40 \, a b^{4} c^{7} d^{3} + 140 \, a^{2} b^{3} c^{6} d^{4} + 168 \, a^{3} b^{2} c^{5} d^{5} + 70 \, a^{4} b c^{4} d^{6} + 8 \, a^{5} c^{3} d^{7}\right )} x^{8} + \frac {5}{7} \, {\left (2 \, b^{5} c^{9} d + 45 \, a b^{4} c^{8} d^{2} + 240 \, a^{2} b^{3} c^{7} d^{3} + 420 \, a^{3} b^{2} c^{6} d^{4} + 252 \, a^{4} b c^{5} d^{5} + 42 \, a^{5} c^{4} d^{6}\right )} x^{7} + \frac {1}{6} \, {\left (b^{5} c^{10} + 50 \, a b^{4} c^{9} d + 450 \, a^{2} b^{3} c^{8} d^{2} + 1200 \, a^{3} b^{2} c^{7} d^{3} + 1050 \, a^{4} b c^{6} d^{4} + 252 \, a^{5} c^{5} d^{5}\right )} x^{6} + {\left (a b^{4} c^{10} + 20 \, a^{2} b^{3} c^{9} d + 90 \, a^{3} b^{2} c^{8} d^{2} + 120 \, a^{4} b c^{7} d^{3} + 42 \, a^{5} c^{6} d^{4}\right )} x^{5} + \frac {5}{4} \, {\left (2 \, a^{2} b^{3} c^{10} + 20 \, a^{3} b^{2} c^{9} d + 45 \, a^{4} b c^{8} d^{2} + 24 \, a^{5} c^{7} d^{3}\right )} x^{4} + \frac {5}{3} \, {\left (2 \, a^{3} b^{2} c^{10} + 10 \, a^{4} b c^{9} d + 9 \, a^{5} c^{8} d^{2}\right )} x^{3} + \frac {5}{2} \, {\left (a^{4} b c^{10} + 2 \, a^{5} c^{9} d\right )} x^{2} \]

[In]

integrate((b*x+a)^5*(d*x+c)^10,x, algorithm="maxima")

[Out]

1/16*b^5*d^10*x^16 + a^5*c^10*x + 1/3*(2*b^5*c*d^9 + a*b^4*d^10)*x^15 + 5/14*(9*b^5*c^2*d^8 + 10*a*b^4*c*d^9 +
2*a^2*b^3*d^10)*x^14 + 5/13*(24*b^5*c^3*d^7 + 45*a*b^4*c^2*d^8 + 20*a^2*b^3*c*d^9 + 2*a^3*b^2*d^10)*x^13 + 5/
12*(42*b^5*c^4*d^6 + 120*a*b^4*c^3*d^7 + 90*a^2*b^3*c^2*d^8 + 20*a^3*b^2*c*d^9 + a^4*b*d^10)*x^12 + 1/11*(252*
b^5*c^5*d^5 + 1050*a*b^4*c^4*d^6 + 1200*a^2*b^3*c^3*d^7 + 450*a^3*b^2*c^2*d^8 + 50*a^4*b*c*d^9 + a^5*d^10)*x^1
1 + 1/2*(42*b^5*c^6*d^4 + 252*a*b^4*c^5*d^5 + 420*a^2*b^3*c^4*d^6 + 240*a^3*b^2*c^3*d^7 + 45*a^4*b*c^2*d^8 + 2
*a^5*c*d^9)*x^10 + 5/3*(8*b^5*c^7*d^3 + 70*a*b^4*c^6*d^4 + 168*a^2*b^3*c^5*d^5 + 140*a^3*b^2*c^4*d^6 + 40*a^4*
b*c^3*d^7 + 3*a^5*c^2*d^8)*x^9 + 15/8*(3*b^5*c^8*d^2 + 40*a*b^4*c^7*d^3 + 140*a^2*b^3*c^6*d^4 + 168*a^3*b^2*c^
5*d^5 + 70*a^4*b*c^4*d^6 + 8*a^5*c^3*d^7)*x^8 + 5/7*(2*b^5*c^9*d + 45*a*b^4*c^8*d^2 + 240*a^2*b^3*c^7*d^3 + 42
0*a^3*b^2*c^6*d^4 + 252*a^4*b*c^5*d^5 + 42*a^5*c^4*d^6)*x^7 + 1/6*(b^5*c^10 + 50*a*b^4*c^9*d + 450*a^2*b^3*c^8
*d^2 + 1200*a^3*b^2*c^7*d^3 + 1050*a^4*b*c^6*d^4 + 252*a^5*c^5*d^5)*x^6 + (a*b^4*c^10 + 20*a^2*b^3*c^9*d + 90*
a^3*b^2*c^8*d^2 + 120*a^4*b*c^7*d^3 + 42*a^5*c^6*d^4)*x^5 + 5/4*(2*a^2*b^3*c^10 + 20*a^3*b^2*c^9*d + 45*a^4*b*
c^8*d^2 + 24*a^5*c^7*d^3)*x^4 + 5/3*(2*a^3*b^2*c^10 + 10*a^4*b*c^9*d + 9*a^5*c^8*d^2)*x^3 + 5/2*(a^4*b*c^10 +
2*a^5*c^9*d)*x^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 948 vs. \(2 (134) = 268\).

Time = 0.30 (sec) , antiderivative size = 948, normalized size of antiderivative = 6.49 \[ \int (a+b x)^5 (c+d x)^{10} \, dx=\frac {1}{16} \, b^{5} d^{10} x^{16} + \frac {2}{3} \, b^{5} c d^{9} x^{15} + \frac {1}{3} \, a b^{4} d^{10} x^{15} + \frac {45}{14} \, b^{5} c^{2} d^{8} x^{14} + \frac {25}{7} \, a b^{4} c d^{9} x^{14} + \frac {5}{7} \, a^{2} b^{3} d^{10} x^{14} + \frac {120}{13} \, b^{5} c^{3} d^{7} x^{13} + \frac {225}{13} \, a b^{4} c^{2} d^{8} x^{13} + \frac {100}{13} \, a^{2} b^{3} c d^{9} x^{13} + \frac {10}{13} \, a^{3} b^{2} d^{10} x^{13} + \frac {35}{2} \, b^{5} c^{4} d^{6} x^{12} + 50 \, a b^{4} c^{3} d^{7} x^{12} + \frac {75}{2} \, a^{2} b^{3} c^{2} d^{8} x^{12} + \frac {25}{3} \, a^{3} b^{2} c d^{9} x^{12} + \frac {5}{12} \, a^{4} b d^{10} x^{12} + \frac {252}{11} \, b^{5} c^{5} d^{5} x^{11} + \frac {1050}{11} \, a b^{4} c^{4} d^{6} x^{11} + \frac {1200}{11} \, a^{2} b^{3} c^{3} d^{7} x^{11} + \frac {450}{11} \, a^{3} b^{2} c^{2} d^{8} x^{11} + \frac {50}{11} \, a^{4} b c d^{9} x^{11} + \frac {1}{11} \, a^{5} d^{10} x^{11} + 21 \, b^{5} c^{6} d^{4} x^{10} + 126 \, a b^{4} c^{5} d^{5} x^{10} + 210 \, a^{2} b^{3} c^{4} d^{6} x^{10} + 120 \, a^{3} b^{2} c^{3} d^{7} x^{10} + \frac {45}{2} \, a^{4} b c^{2} d^{8} x^{10} + a^{5} c d^{9} x^{10} + \frac {40}{3} \, b^{5} c^{7} d^{3} x^{9} + \frac {350}{3} \, a b^{4} c^{6} d^{4} x^{9} + 280 \, a^{2} b^{3} c^{5} d^{5} x^{9} + \frac {700}{3} \, a^{3} b^{2} c^{4} d^{6} x^{9} + \frac {200}{3} \, a^{4} b c^{3} d^{7} x^{9} + 5 \, a^{5} c^{2} d^{8} x^{9} + \frac {45}{8} \, b^{5} c^{8} d^{2} x^{8} + 75 \, a b^{4} c^{7} d^{3} x^{8} + \frac {525}{2} \, a^{2} b^{3} c^{6} d^{4} x^{8} + 315 \, a^{3} b^{2} c^{5} d^{5} x^{8} + \frac {525}{4} \, a^{4} b c^{4} d^{6} x^{8} + 15 \, a^{5} c^{3} d^{7} x^{8} + \frac {10}{7} \, b^{5} c^{9} d x^{7} + \frac {225}{7} \, a b^{4} c^{8} d^{2} x^{7} + \frac {1200}{7} \, a^{2} b^{3} c^{7} d^{3} x^{7} + 300 \, a^{3} b^{2} c^{6} d^{4} x^{7} + 180 \, a^{4} b c^{5} d^{5} x^{7} + 30 \, a^{5} c^{4} d^{6} x^{7} + \frac {1}{6} \, b^{5} c^{10} x^{6} + \frac {25}{3} \, a b^{4} c^{9} d x^{6} + 75 \, a^{2} b^{3} c^{8} d^{2} x^{6} + 200 \, a^{3} b^{2} c^{7} d^{3} x^{6} + 175 \, a^{4} b c^{6} d^{4} x^{6} + 42 \, a^{5} c^{5} d^{5} x^{6} + a b^{4} c^{10} x^{5} + 20 \, a^{2} b^{3} c^{9} d x^{5} + 90 \, a^{3} b^{2} c^{8} d^{2} x^{5} + 120 \, a^{4} b c^{7} d^{3} x^{5} + 42 \, a^{5} c^{6} d^{4} x^{5} + \frac {5}{2} \, a^{2} b^{3} c^{10} x^{4} + 25 \, a^{3} b^{2} c^{9} d x^{4} + \frac {225}{4} \, a^{4} b c^{8} d^{2} x^{4} + 30 \, a^{5} c^{7} d^{3} x^{4} + \frac {10}{3} \, a^{3} b^{2} c^{10} x^{3} + \frac {50}{3} \, a^{4} b c^{9} d x^{3} + 15 \, a^{5} c^{8} d^{2} x^{3} + \frac {5}{2} \, a^{4} b c^{10} x^{2} + 5 \, a^{5} c^{9} d x^{2} + a^{5} c^{10} x \]

[In]

integrate((b*x+a)^5*(d*x+c)^10,x, algorithm="giac")

[Out]

1/16*b^5*d^10*x^16 + 2/3*b^5*c*d^9*x^15 + 1/3*a*b^4*d^10*x^15 + 45/14*b^5*c^2*d^8*x^14 + 25/7*a*b^4*c*d^9*x^14
+ 5/7*a^2*b^3*d^10*x^14 + 120/13*b^5*c^3*d^7*x^13 + 225/13*a*b^4*c^2*d^8*x^13 + 100/13*a^2*b^3*c*d^9*x^13 + 1
0/13*a^3*b^2*d^10*x^13 + 35/2*b^5*c^4*d^6*x^12 + 50*a*b^4*c^3*d^7*x^12 + 75/2*a^2*b^3*c^2*d^8*x^12 + 25/3*a^3*
b^2*c*d^9*x^12 + 5/12*a^4*b*d^10*x^12 + 252/11*b^5*c^5*d^5*x^11 + 1050/11*a*b^4*c^4*d^6*x^11 + 1200/11*a^2*b^3
*c^3*d^7*x^11 + 450/11*a^3*b^2*c^2*d^8*x^11 + 50/11*a^4*b*c*d^9*x^11 + 1/11*a^5*d^10*x^11 + 21*b^5*c^6*d^4*x^1
0 + 126*a*b^4*c^5*d^5*x^10 + 210*a^2*b^3*c^4*d^6*x^10 + 120*a^3*b^2*c^3*d^7*x^10 + 45/2*a^4*b*c^2*d^8*x^10 + a
^5*c*d^9*x^10 + 40/3*b^5*c^7*d^3*x^9 + 350/3*a*b^4*c^6*d^4*x^9 + 280*a^2*b^3*c^5*d^5*x^9 + 700/3*a^3*b^2*c^4*d
^6*x^9 + 200/3*a^4*b*c^3*d^7*x^9 + 5*a^5*c^2*d^8*x^9 + 45/8*b^5*c^8*d^2*x^8 + 75*a*b^4*c^7*d^3*x^8 + 525/2*a^2
*b^3*c^6*d^4*x^8 + 315*a^3*b^2*c^5*d^5*x^8 + 525/4*a^4*b*c^4*d^6*x^8 + 15*a^5*c^3*d^7*x^8 + 10/7*b^5*c^9*d*x^7
+ 225/7*a*b^4*c^8*d^2*x^7 + 1200/7*a^2*b^3*c^7*d^3*x^7 + 300*a^3*b^2*c^6*d^4*x^7 + 180*a^4*b*c^5*d^5*x^7 + 30
*a^5*c^4*d^6*x^7 + 1/6*b^5*c^10*x^6 + 25/3*a*b^4*c^9*d*x^6 + 75*a^2*b^3*c^8*d^2*x^6 + 200*a^3*b^2*c^7*d^3*x^6
+ 175*a^4*b*c^6*d^4*x^6 + 42*a^5*c^5*d^5*x^6 + a*b^4*c^10*x^5 + 20*a^2*b^3*c^9*d*x^5 + 90*a^3*b^2*c^8*d^2*x^5
+ 120*a^4*b*c^7*d^3*x^5 + 42*a^5*c^6*d^4*x^5 + 5/2*a^2*b^3*c^10*x^4 + 25*a^3*b^2*c^9*d*x^4 + 225/4*a^4*b*c^8*d
^2*x^4 + 30*a^5*c^7*d^3*x^4 + 10/3*a^3*b^2*c^10*x^3 + 50/3*a^4*b*c^9*d*x^3 + 15*a^5*c^8*d^2*x^3 + 5/2*a^4*b*c^
10*x^2 + 5*a^5*c^9*d*x^2 + a^5*c^10*x

Mupad [B] (verification not implemented)

Time = 0.52 (sec) , antiderivative size = 806, normalized size of antiderivative = 5.52 \[ \int (a+b x)^5 (c+d x)^{10} \, dx=x^{10}\,\left (a^5\,c\,d^9+\frac {45\,a^4\,b\,c^2\,d^8}{2}+120\,a^3\,b^2\,c^3\,d^7+210\,a^2\,b^3\,c^4\,d^6+126\,a\,b^4\,c^5\,d^5+21\,b^5\,c^6\,d^4\right )+x^7\,\left (30\,a^5\,c^4\,d^6+180\,a^4\,b\,c^5\,d^5+300\,a^3\,b^2\,c^6\,d^4+\frac {1200\,a^2\,b^3\,c^7\,d^3}{7}+\frac {225\,a\,b^4\,c^8\,d^2}{7}+\frac {10\,b^5\,c^9\,d}{7}\right )+x^6\,\left (42\,a^5\,c^5\,d^5+175\,a^4\,b\,c^6\,d^4+200\,a^3\,b^2\,c^7\,d^3+75\,a^2\,b^3\,c^8\,d^2+\frac {25\,a\,b^4\,c^9\,d}{3}+\frac {b^5\,c^{10}}{6}\right )+x^{11}\,\left (\frac {a^5\,d^{10}}{11}+\frac {50\,a^4\,b\,c\,d^9}{11}+\frac {450\,a^3\,b^2\,c^2\,d^8}{11}+\frac {1200\,a^2\,b^3\,c^3\,d^7}{11}+\frac {1050\,a\,b^4\,c^4\,d^6}{11}+\frac {252\,b^5\,c^5\,d^5}{11}\right )+x^8\,\left (15\,a^5\,c^3\,d^7+\frac {525\,a^4\,b\,c^4\,d^6}{4}+315\,a^3\,b^2\,c^5\,d^5+\frac {525\,a^2\,b^3\,c^6\,d^4}{2}+75\,a\,b^4\,c^7\,d^3+\frac {45\,b^5\,c^8\,d^2}{8}\right )+x^9\,\left (5\,a^5\,c^2\,d^8+\frac {200\,a^4\,b\,c^3\,d^7}{3}+\frac {700\,a^3\,b^2\,c^4\,d^6}{3}+280\,a^2\,b^3\,c^5\,d^5+\frac {350\,a\,b^4\,c^6\,d^4}{3}+\frac {40\,b^5\,c^7\,d^3}{3}\right )+x^5\,\left (42\,a^5\,c^6\,d^4+120\,a^4\,b\,c^7\,d^3+90\,a^3\,b^2\,c^8\,d^2+20\,a^2\,b^3\,c^9\,d+a\,b^4\,c^{10}\right )+x^{12}\,\left (\frac {5\,a^4\,b\,d^{10}}{12}+\frac {25\,a^3\,b^2\,c\,d^9}{3}+\frac {75\,a^2\,b^3\,c^2\,d^8}{2}+50\,a\,b^4\,c^3\,d^7+\frac {35\,b^5\,c^4\,d^6}{2}\right )+a^5\,c^{10}\,x+\frac {b^5\,d^{10}\,x^{16}}{16}+\frac {5\,a^2\,c^7\,x^4\,\left (24\,a^3\,d^3+45\,a^2\,b\,c\,d^2+20\,a\,b^2\,c^2\,d+2\,b^3\,c^3\right )}{4}+\frac {5\,b^2\,d^7\,x^{13}\,\left (2\,a^3\,d^3+20\,a^2\,b\,c\,d^2+45\,a\,b^2\,c^2\,d+24\,b^3\,c^3\right )}{13}+\frac {5\,a^4\,c^9\,x^2\,\left (2\,a\,d+b\,c\right )}{2}+\frac {b^4\,d^9\,x^{15}\,\left (a\,d+2\,b\,c\right )}{3}+\frac {5\,a^3\,c^8\,x^3\,\left (9\,a^2\,d^2+10\,a\,b\,c\,d+2\,b^2\,c^2\right )}{3}+\frac {5\,b^3\,d^8\,x^{14}\,\left (2\,a^2\,d^2+10\,a\,b\,c\,d+9\,b^2\,c^2\right )}{14} \]

[In]

int((a + b*x)^5*(c + d*x)^10,x)

[Out]

x^10*(a^5*c*d^9 + 21*b^5*c^6*d^4 + 126*a*b^4*c^5*d^5 + (45*a^4*b*c^2*d^8)/2 + 210*a^2*b^3*c^4*d^6 + 120*a^3*b^
2*c^3*d^7) + x^7*((10*b^5*c^9*d)/7 + 30*a^5*c^4*d^6 + (225*a*b^4*c^8*d^2)/7 + 180*a^4*b*c^5*d^5 + (1200*a^2*b^
3*c^7*d^3)/7 + 300*a^3*b^2*c^6*d^4) + x^6*((b^5*c^10)/6 + 42*a^5*c^5*d^5 + 175*a^4*b*c^6*d^4 + 75*a^2*b^3*c^8*
d^2 + 200*a^3*b^2*c^7*d^3 + (25*a*b^4*c^9*d)/3) + x^11*((a^5*d^10)/11 + (252*b^5*c^5*d^5)/11 + (1050*a*b^4*c^4
*d^6)/11 + (1200*a^2*b^3*c^3*d^7)/11 + (450*a^3*b^2*c^2*d^8)/11 + (50*a^4*b*c*d^9)/11) + x^8*(15*a^5*c^3*d^7 +
(45*b^5*c^8*d^2)/8 + 75*a*b^4*c^7*d^3 + (525*a^4*b*c^4*d^6)/4 + (525*a^2*b^3*c^6*d^4)/2 + 315*a^3*b^2*c^5*d^5
) + x^9*(5*a^5*c^2*d^8 + (40*b^5*c^7*d^3)/3 + (350*a*b^4*c^6*d^4)/3 + (200*a^4*b*c^3*d^7)/3 + 280*a^2*b^3*c^5*
d^5 + (700*a^3*b^2*c^4*d^6)/3) + x^5*(a*b^4*c^10 + 42*a^5*c^6*d^4 + 20*a^2*b^3*c^9*d + 120*a^4*b*c^7*d^3 + 90*
a^3*b^2*c^8*d^2) + x^12*((5*a^4*b*d^10)/12 + (35*b^5*c^4*d^6)/2 + 50*a*b^4*c^3*d^7 + (25*a^3*b^2*c*d^9)/3 + (7
5*a^2*b^3*c^2*d^8)/2) + a^5*c^10*x + (b^5*d^10*x^16)/16 + (5*a^2*c^7*x^4*(24*a^3*d^3 + 2*b^3*c^3 + 20*a*b^2*c^
2*d + 45*a^2*b*c*d^2))/4 + (5*b^2*d^7*x^13*(2*a^3*d^3 + 24*b^3*c^3 + 45*a*b^2*c^2*d + 20*a^2*b*c*d^2))/13 + (5
*a^4*c^9*x^2*(2*a*d + b*c))/2 + (b^4*d^9*x^15*(a*d + 2*b*c))/3 + (5*a^3*c^8*x^3*(9*a^2*d^2 + 2*b^2*c^2 + 10*a*
b*c*d))/3 + (5*b^3*d^8*x^14*(2*a^2*d^2 + 9*b^2*c^2 + 10*a*b*c*d))/14