\(\int \frac {(c+d x)^{10}}{(a+b x)^5} \, dx\) [1316]

Optimal result

Integrand size = 15, antiderivative size = 262 \[ \int \frac {(c+d x)^{10}}{(a+b x)^5} \, dx=\frac {252 d^5 (b c-a d)^5 x}{b^{10}}-\frac {(b c-a d)^{10}}{4 b^{11} (a+b x)^4}-\frac {10 d (b c-a d)^9}{3 b^{11} (a+b x)^3}-\frac {45 d^2 (b c-a d)^8}{2 b^{11} (a+b x)^2}-\frac {120 d^3 (b c-a d)^7}{b^{11} (a+b x)}+\frac {105 d^6 (b c-a d)^4 (a+b x)^2}{b^{11}}+\frac {40 d^7 (b c-a d)^3 (a+b x)^3}{b^{11}}+\frac {45 d^8 (b c-a d)^2 (a+b x)^4}{4 b^{11}}+\frac {2 d^9 (b c-a d) (a+b x)^5}{b^{11}}+\frac {d^{10} (a+b x)^6}{6 b^{11}}+\frac {210 d^4 (b c-a d)^6 \log (a+b x)}{b^{11}} \]

[Out]

252*d^5*(-a*d+b*c)^5*x/b^10-1/4*(-a*d+b*c)^10/b^11/(b*x+a)^4-10/3*d*(-a*d+b*c)^9/b^11/(b*x+a)^3-45/2*d^2*(-a*d
+b*c)^8/b^11/(b*x+a)^2-120*d^3*(-a*d+b*c)^7/b^11/(b*x+a)+105*d^6*(-a*d+b*c)^4*(b*x+a)^2/b^11+40*d^7*(-a*d+b*c)
^3*(b*x+a)^3/b^11+45/4*d^8*(-a*d+b*c)^2*(b*x+a)^4/b^11+2*d^9*(-a*d+b*c)*(b*x+a)^5/b^11+1/6*d^10*(b*x+a)^6/b^11
+210*d^4*(-a*d+b*c)^6*ln(b*x+a)/b^11

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 262, 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} \[ \int \frac {(c+d x)^{10}}{(a+b x)^5} \, dx=\frac {2 d^9 (a+b x)^5 (b c-a d)}{b^{11}}+\frac {45 d^8 (a+b x)^4 (b c-a d)^2}{4 b^{11}}+\frac {40 d^7 (a+b x)^3 (b c-a d)^3}{b^{11}}+\frac {105 d^6 (a+b x)^2 (b c-a d)^4}{b^{11}}+\frac {210 d^4 (b c-a d)^6 \log (a+b x)}{b^{11}}-\frac {120 d^3 (b c-a d)^7}{b^{11} (a+b x)}-\frac {45 d^2 (b c-a d)^8}{2 b^{11} (a+b x)^2}-\frac {10 d (b c-a d)^9}{3 b^{11} (a+b x)^3}-\frac {(b c-a d)^{10}}{4 b^{11} (a+b x)^4}+\frac {d^{10} (a+b x)^6}{6 b^{11}}+\frac {252 d^5 x (b c-a d)^5}{b^{10}} \]

[In]

Int[(c + d*x)^10/(a + b*x)^5,x]

[Out]

(252*d^5*(b*c - a*d)^5*x)/b^10 - (b*c - a*d)^10/(4*b^11*(a + b*x)^4) - (10*d*(b*c - a*d)^9)/(3*b^11*(a + b*x)^
3) - (45*d^2*(b*c - a*d)^8)/(2*b^11*(a + b*x)^2) - (120*d^3*(b*c - a*d)^7)/(b^11*(a + b*x)) + (105*d^6*(b*c -
a*d)^4*(a + b*x)^2)/b^11 + (40*d^7*(b*c - a*d)^3*(a + b*x)^3)/b^11 + (45*d^8*(b*c - a*d)^2*(a + b*x)^4)/(4*b^1
1) + (2*d^9*(b*c - a*d)*(a + b*x)^5)/b^11 + (d^10*(a + b*x)^6)/(6*b^11) + (210*d^4*(b*c - a*d)^6*Log[a + b*x])
/b^11

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 {252 d^5 (b c-a d)^5}{b^{10}}+\frac {(b c-a d)^{10}}{b^{10} (a+b x)^5}+\frac {10 d (b c-a d)^9}{b^{10} (a+b x)^4}+\frac {45 d^2 (b c-a d)^8}{b^{10} (a+b x)^3}+\frac {120 d^3 (b c-a d)^7}{b^{10} (a+b x)^2}+\frac {210 d^4 (b c-a d)^6}{b^{10} (a+b x)}+\frac {210 d^6 (b c-a d)^4 (a+b x)}{b^{10}}+\frac {120 d^7 (b c-a d)^3 (a+b x)^2}{b^{10}}+\frac {45 d^8 (b c-a d)^2 (a+b x)^3}{b^{10}}+\frac {10 d^9 (b c-a d) (a+b x)^4}{b^{10}}+\frac {d^{10} (a+b x)^5}{b^{10}}\right ) \, dx \\ & = \frac {252 d^5 (b c-a d)^5 x}{b^{10}}-\frac {(b c-a d)^{10}}{4 b^{11} (a+b x)^4}-\frac {10 d (b c-a d)^9}{3 b^{11} (a+b x)^3}-\frac {45 d^2 (b c-a d)^8}{2 b^{11} (a+b x)^2}-\frac {120 d^3 (b c-a d)^7}{b^{11} (a+b x)}+\frac {105 d^6 (b c-a d)^4 (a+b x)^2}{b^{11}}+\frac {40 d^7 (b c-a d)^3 (a+b x)^3}{b^{11}}+\frac {45 d^8 (b c-a d)^2 (a+b x)^4}{4 b^{11}}+\frac {2 d^9 (b c-a d) (a+b x)^5}{b^{11}}+\frac {d^{10} (a+b x)^6}{6 b^{11}}+\frac {210 d^4 (b c-a d)^6 \log (a+b x)}{b^{11}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.37 \[ \int \frac {(c+d x)^{10}}{(a+b x)^5} \, dx=\frac {12 b d^5 \left (252 b^5 c^5-1050 a b^4 c^4 d+1800 a^2 b^3 c^3 d^2-1575 a^3 b^2 c^2 d^3+700 a^4 b c d^4-126 a^5 d^5\right ) x+30 b^2 d^6 \left (42 b^4 c^4-120 a b^3 c^3 d+135 a^2 b^2 c^2 d^2-70 a^3 b c d^3+14 a^4 d^4\right ) x^2+20 b^3 d^7 \left (24 b^3 c^3-45 a b^2 c^2 d+30 a^2 b c d^2-7 a^3 d^3\right ) x^3+15 b^4 d^8 \left (9 b^2 c^2-10 a b c d+3 a^2 d^2\right ) x^4+12 b^5 d^9 (2 b c-a d) x^5+2 b^6 d^{10} x^6-\frac {3 (b c-a d)^{10}}{(a+b x)^4}+\frac {40 d (-b c+a d)^9}{(a+b x)^3}-\frac {270 d^2 (b c-a d)^8}{(a+b x)^2}+\frac {1440 d^3 (-b c+a d)^7}{a+b x}+2520 d^4 (b c-a d)^6 \log (a+b x)}{12 b^{11}} \]

[In]

Integrate[(c + d*x)^10/(a + b*x)^5,x]

[Out]

(12*b*d^5*(252*b^5*c^5 - 1050*a*b^4*c^4*d + 1800*a^2*b^3*c^3*d^2 - 1575*a^3*b^2*c^2*d^3 + 700*a^4*b*c*d^4 - 12
6*a^5*d^5)*x + 30*b^2*d^6*(42*b^4*c^4 - 120*a*b^3*c^3*d + 135*a^2*b^2*c^2*d^2 - 70*a^3*b*c*d^3 + 14*a^4*d^4)*x
^2 + 20*b^3*d^7*(24*b^3*c^3 - 45*a*b^2*c^2*d + 30*a^2*b*c*d^2 - 7*a^3*d^3)*x^3 + 15*b^4*d^8*(9*b^2*c^2 - 10*a*
b*c*d + 3*a^2*d^2)*x^4 + 12*b^5*d^9*(2*b*c - a*d)*x^5 + 2*b^6*d^10*x^6 - (3*(b*c - a*d)^10)/(a + b*x)^4 + (40*
d*(-(b*c) + a*d)^9)/(a + b*x)^3 - (270*d^2*(b*c - a*d)^8)/(a + b*x)^2 + (1440*d^3*(-(b*c) + a*d)^7)/(a + b*x)
+ 2520*d^4*(b*c - a*d)^6*Log[a + b*x])/(12*b^11)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(843\) vs. \(2(252)=504\).

Time = 0.23 (sec) , antiderivative size = 844, normalized size of antiderivative = 3.22

method	result	size
norman	\(\frac {\frac {5250 a^{10} d^{10}-31500 a^{9} b c \,d^{9}+78750 a^{8} b^{2} c^{2} d^{8}-105000 a^{7} b^{3} c^{3} d^{7}+78750 a^{6} b^{4} c^{4} d^{6}-31500 a^{5} b^{5} c^{5} d^{5}+5250 a^{4} b^{6} c^{6} d^{4}-360 a^{3} b^{7} c^{7} d^{3}-45 a^{2} b^{8} c^{8} d^{2}-10 a \,b^{9} c^{9} d -3 b^{10} c^{10}}{12 b^{11}}+\frac {d^{10} x^{10}}{6 b}+\frac {4 \left (210 a^{7} d^{10}-1260 a^{6} b c \,d^{9}+3150 a^{5} b^{2} c^{2} d^{8}-4200 a^{4} b^{3} c^{3} d^{7}+3150 a^{3} b^{4} c^{4} d^{6}-1260 a^{2} b^{5} c^{5} d^{5}+210 a \,b^{6} c^{6} d^{4}-30 b^{7} c^{7} d^{3}\right ) x^{3}}{b^{8}}+\frac {3 \left (1260 a^{8} d^{10}-7560 a^{7} b c \,d^{9}+18900 a^{6} b^{2} c^{2} d^{8}-25200 a^{5} b^{3} c^{3} d^{7}+18900 a^{4} b^{4} c^{4} d^{6}-7560 a^{3} b^{5} c^{5} d^{5}+1260 a^{2} b^{6} c^{6} d^{4}-120 a \,b^{7} c^{7} d^{3}-15 b^{8} c^{8} d^{2}\right ) x^{2}}{2 b^{9}}+\frac {\left (4620 a^{9} d^{10}-27720 a^{8} b c \,d^{9}+69300 a^{7} b^{2} c^{2} d^{8}-92400 a^{6} b^{3} c^{3} d^{7}+69300 a^{5} b^{4} c^{4} d^{6}-27720 a^{4} b^{5} c^{5} d^{5}+4620 a^{3} b^{6} c^{6} d^{4}-360 a^{2} b^{7} c^{7} d^{3}-45 a \,b^{8} c^{8} d^{2}-10 b^{9} c^{9} d \right ) x}{3 b^{10}}-\frac {42 d^{5} \left (a^{5} d^{5}-6 a^{4} b c \,d^{4}+15 a^{3} b^{2} c^{2} d^{3}-20 a^{2} b^{3} c^{3} d^{2}+15 a \,b^{4} c^{4} d -6 b^{5} c^{5}\right ) x^{5}}{b^{6}}+\frac {7 d^{6} \left (a^{4} d^{4}-6 a^{3} b c \,d^{3}+15 a^{2} b^{2} c^{2} d^{2}-20 a \,b^{3} c^{3} d +15 b^{4} c^{4}\right ) x^{6}}{b^{5}}-\frac {2 d^{7} \left (a^{3} d^{3}-6 a^{2} b c \,d^{2}+15 a \,b^{2} c^{2} d -20 b^{3} c^{3}\right ) x^{7}}{b^{4}}+\frac {3 d^{8} \left (a^{2} d^{2}-6 a b c d +15 b^{2} c^{2}\right ) x^{8}}{4 b^{3}}-\frac {d^{9} \left (a d -6 b c \right ) x^{9}}{3 b^{2}}}{\left (b x +a \right )^{4}}+\frac {210 d^{4} \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right ) \ln \left (b x +a \right )}{b^{11}}\)	\(844\)
default	\(-\frac {d^{5} \left (-\frac {1}{6} d^{5} x^{6} b^{5}+a \,b^{4} d^{5} x^{5}-2 b^{5} c \,d^{4} x^{5}-\frac {15}{4} a^{2} b^{3} d^{5} x^{4}+\frac {25}{2} a \,b^{4} c \,d^{4} x^{4}-\frac {45}{4} b^{5} c^{2} d^{3} x^{4}+\frac {35}{3} a^{3} b^{2} d^{5} x^{3}-50 a^{2} b^{3} c \,d^{4} x^{3}+75 a \,b^{4} c^{2} d^{3} x^{3}-40 b^{5} c^{3} d^{2} x^{3}-35 a^{4} b \,d^{5} x^{2}+175 a^{3} b^{2} c \,d^{4} x^{2}-\frac {675}{2} a^{2} b^{3} c^{2} d^{3} x^{2}+300 a \,b^{4} c^{3} d^{2} x^{2}-105 b^{5} c^{4} d \,x^{2}+126 a^{5} d^{5} x -700 a^{4} b c \,d^{4} x +1575 a^{3} b^{2} c^{2} d^{3} x -1800 a^{2} b^{3} c^{3} d^{2} x +1050 a \,b^{4} c^{4} d x -252 b^{5} c^{5} x \right )}{b^{10}}+\frac {10 d \left (a^{9} d^{9}-9 a^{8} b c \,d^{8}+36 a^{7} b^{2} c^{2} d^{7}-84 a^{6} b^{3} c^{3} d^{6}+126 a^{5} b^{4} c^{4} d^{5}-126 a^{4} b^{5} c^{5} d^{4}+84 a^{3} b^{6} c^{6} d^{3}-36 a^{2} b^{7} c^{7} d^{2}+9 a \,b^{8} c^{8} d -b^{9} c^{9}\right )}{3 b^{11} \left (b x +a \right )^{3}}+\frac {210 d^{4} \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right ) \ln \left (b x +a \right )}{b^{11}}-\frac {a^{10} d^{10}-10 a^{9} b c \,d^{9}+45 a^{8} b^{2} c^{2} d^{8}-120 a^{7} b^{3} c^{3} d^{7}+210 a^{6} b^{4} c^{4} d^{6}-252 a^{5} b^{5} c^{5} d^{5}+210 a^{4} b^{6} c^{6} d^{4}-120 a^{3} b^{7} c^{7} d^{3}+45 a^{2} b^{8} c^{8} d^{2}-10 a \,b^{9} c^{9} d +b^{10} c^{10}}{4 b^{11} \left (b x +a \right )^{4}}-\frac {45 d^{2} \left (a^{8} d^{8}-8 a^{7} b c \,d^{7}+28 a^{6} b^{2} c^{2} d^{6}-56 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} c^{4} d^{4}-56 a^{3} b^{5} c^{5} d^{3}+28 a^{2} b^{6} c^{6} d^{2}-8 a \,b^{7} c^{7} d +b^{8} c^{8}\right )}{2 b^{11} \left (b x +a \right )^{2}}+\frac {120 d^{3} \left (a^{7} d^{7}-7 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}-35 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}-21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d -b^{7} c^{7}\right )}{b^{11} \left (b x +a \right )}\)	\(881\)
risch	\(-\frac {25 d^{9} a c \,x^{4}}{2 b^{6}}+\frac {50 d^{9} a^{2} c \,x^{3}}{b^{7}}-\frac {75 d^{8} a \,c^{2} x^{3}}{b^{6}}-\frac {175 d^{9} a^{3} c \,x^{2}}{b^{8}}+\frac {675 d^{8} a^{2} c^{2} x^{2}}{2 b^{7}}-\frac {300 d^{7} a \,c^{3} x^{2}}{b^{6}}+\frac {700 d^{9} a^{4} c x}{b^{9}}-\frac {1575 d^{8} a^{3} c^{2} x}{b^{8}}+\frac {1800 d^{7} a^{2} c^{3} x}{b^{7}}-\frac {1050 d^{6} a \,c^{4} x}{b^{6}}-\frac {1260 d^{9} \ln \left (b x +a \right ) a^{5} c}{b^{10}}+\frac {3150 d^{8} \ln \left (b x +a \right ) a^{4} c^{2}}{b^{9}}-\frac {4200 d^{7} \ln \left (b x +a \right ) a^{3} c^{3}}{b^{8}}+\frac {3150 d^{6} \ln \left (b x +a \right ) a^{2} c^{4}}{b^{7}}-\frac {1260 d^{5} \ln \left (b x +a \right ) a \,c^{5}}{b^{6}}+\frac {d^{10} x^{6}}{6 b^{5}}-\frac {d^{10} a \,x^{5}}{b^{6}}+\frac {2 d^{9} c \,x^{5}}{b^{5}}+\frac {15 d^{10} a^{2} x^{4}}{4 b^{7}}+\frac {45 d^{8} c^{2} x^{4}}{4 b^{5}}-\frac {35 d^{10} a^{3} x^{3}}{3 b^{8}}+\frac {40 d^{7} c^{3} x^{3}}{b^{5}}+\frac {35 d^{10} a^{4} x^{2}}{b^{9}}+\frac {105 d^{6} c^{4} x^{2}}{b^{5}}-\frac {126 d^{10} a^{5} x}{b^{10}}+\frac {252 d^{5} c^{5} x}{b^{5}}+\frac {210 d^{10} \ln \left (b x +a \right ) a^{6}}{b^{11}}+\frac {210 d^{4} \ln \left (b x +a \right ) c^{6}}{b^{5}}+\frac {\left (120 a^{7} b^{2} d^{10}-840 a^{6} b^{3} c \,d^{9}+2520 a^{5} b^{4} c^{2} d^{8}-4200 a^{4} b^{5} c^{3} d^{7}+4200 a^{3} b^{6} c^{4} d^{6}-2520 a^{2} b^{7} c^{5} d^{5}+840 a \,b^{8} c^{6} d^{4}-120 b^{9} c^{7} d^{3}\right ) x^{3}+\frac {45 b \,d^{2} \left (15 a^{8} d^{8}-104 a^{7} b c \,d^{7}+308 a^{6} b^{2} c^{2} d^{6}-504 a^{5} b^{3} c^{3} d^{5}+490 a^{4} b^{4} c^{4} d^{4}-280 a^{3} b^{5} c^{5} d^{3}+84 a^{2} b^{6} c^{6} d^{2}-8 a \,b^{7} c^{7} d -b^{8} c^{8}\right ) x^{2}}{2}+\frac {5 d \left (191 a^{9} d^{9}-1314 a^{8} b c \,d^{8}+3852 a^{7} b^{2} c^{2} d^{7}-6216 a^{6} b^{3} c^{3} d^{6}+5922 a^{5} b^{4} c^{4} d^{5}-3276 a^{4} b^{5} c^{5} d^{4}+924 a^{3} b^{6} c^{6} d^{3}-72 a^{2} b^{7} c^{7} d^{2}-9 a \,b^{8} c^{8} d -2 b^{9} c^{9}\right ) x}{3}+\frac {1207 a^{10} d^{10}-8250 a^{9} b c \,d^{9}+23985 a^{8} b^{2} c^{2} d^{8}-38280 a^{7} b^{3} c^{3} d^{7}+35910 a^{6} b^{4} c^{4} d^{6}-19404 a^{5} b^{5} c^{5} d^{5}+5250 a^{4} b^{6} c^{6} d^{4}-360 a^{3} b^{7} c^{7} d^{3}-45 a^{2} b^{8} c^{8} d^{2}-10 a \,b^{9} c^{9} d -3 b^{10} c^{10}}{12 b}}{b^{10} \left (b x +a \right )^{4}}\)	\(921\)
parallelrisch	\(\text {Expression too large to display}\)	\(1585\)

[In]

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

[Out]

(1/12*(5250*a^10*d^10-31500*a^9*b*c*d^9+78750*a^8*b^2*c^2*d^8-105000*a^7*b^3*c^3*d^7+78750*a^6*b^4*c^4*d^6-315
00*a^5*b^5*c^5*d^5+5250*a^4*b^6*c^6*d^4-360*a^3*b^7*c^7*d^3-45*a^2*b^8*c^8*d^2-10*a*b^9*c^9*d-3*b^10*c^10)/b^1
1+1/6/b*d^10*x^10+4*(210*a^7*d^10-1260*a^6*b*c*d^9+3150*a^5*b^2*c^2*d^8-4200*a^4*b^3*c^3*d^7+3150*a^3*b^4*c^4*
d^6-1260*a^2*b^5*c^5*d^5+210*a*b^6*c^6*d^4-30*b^7*c^7*d^3)/b^8*x^3+3/2*(1260*a^8*d^10-7560*a^7*b*c*d^9+18900*a
^6*b^2*c^2*d^8-25200*a^5*b^3*c^3*d^7+18900*a^4*b^4*c^4*d^6-7560*a^3*b^5*c^5*d^5+1260*a^2*b^6*c^6*d^4-120*a*b^7
*c^7*d^3-15*b^8*c^8*d^2)/b^9*x^2+1/3*(4620*a^9*d^10-27720*a^8*b*c*d^9+69300*a^7*b^2*c^2*d^8-92400*a^6*b^3*c^3*
d^7+69300*a^5*b^4*c^4*d^6-27720*a^4*b^5*c^5*d^5+4620*a^3*b^6*c^6*d^4-360*a^2*b^7*c^7*d^3-45*a*b^8*c^8*d^2-10*b
^9*c^9*d)/b^10*x-42*d^5*(a^5*d^5-6*a^4*b*c*d^4+15*a^3*b^2*c^2*d^3-20*a^2*b^3*c^3*d^2+15*a*b^4*c^4*d-6*b^5*c^5)
/b^6*x^5+7*d^6*(a^4*d^4-6*a^3*b*c*d^3+15*a^2*b^2*c^2*d^2-20*a*b^3*c^3*d+15*b^4*c^4)/b^5*x^6-2*d^7*(a^3*d^3-6*a
^2*b*c*d^2+15*a*b^2*c^2*d-20*b^3*c^3)/b^4*x^7+3/4*d^8*(a^2*d^2-6*a*b*c*d+15*b^2*c^2)/b^3*x^8-1/3*d^9*(a*d-6*b*
c)/b^2*x^9)/(b*x+a)^4+210/b^11*d^4*(a^6*d^6-6*a^5*b*c*d^5+15*a^4*b^2*c^2*d^4-20*a^3*b^3*c^3*d^3+15*a^2*b^4*c^4
*d^2-6*a*b^5*c^5*d+b^6*c^6)*ln(b*x+a)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1365 vs. \(2 (252) = 504\).

Time = 0.23 (sec) , antiderivative size = 1365, normalized size of antiderivative = 5.21 \[ \int \frac {(c+d x)^{10}}{(a+b x)^5} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/12*(2*b^10*d^10*x^10 - 3*b^10*c^10 - 10*a*b^9*c^9*d - 45*a^2*b^8*c^8*d^2 - 360*a^3*b^7*c^7*d^3 + 5250*a^4*b^
6*c^6*d^4 - 19404*a^5*b^5*c^5*d^5 + 35910*a^6*b^4*c^4*d^6 - 38280*a^7*b^3*c^3*d^7 + 23985*a^8*b^2*c^2*d^8 - 82
50*a^9*b*c*d^9 + 1207*a^10*d^10 + 4*(6*b^10*c*d^9 - a*b^9*d^10)*x^9 + 9*(15*b^10*c^2*d^8 - 6*a*b^9*c*d^9 + a^2
*b^8*d^10)*x^8 + 24*(20*b^10*c^3*d^7 - 15*a*b^9*c^2*d^8 + 6*a^2*b^8*c*d^9 - a^3*b^7*d^10)*x^7 + 84*(15*b^10*c^
4*d^6 - 20*a*b^9*c^3*d^7 + 15*a^2*b^8*c^2*d^8 - 6*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 504*(6*b^10*c^5*d^5 - 15
*a*b^9*c^4*d^6 + 20*a^2*b^8*c^3*d^7 - 15*a^3*b^7*c^2*d^8 + 6*a^4*b^6*c*d^9 - a^5*b^5*d^10)*x^5 + (12096*a*b^9*
c^5*d^5 - 42840*a^2*b^8*c^4*d^6 + 66720*a^3*b^7*c^3*d^7 - 54765*a^4*b^6*c^2*d^8 + 23250*a^5*b^5*c*d^9 - 4043*a
^6*b^4*d^10)*x^4 - 4*(360*b^10*c^7*d^3 - 2520*a*b^9*c^6*d^4 + 3024*a^2*b^8*c^5*d^5 + 5040*a^3*b^7*c^4*d^6 - 16
320*a^4*b^6*c^3*d^7 + 16965*a^5*b^5*c^2*d^8 - 8130*a^6*b^4*c*d^9 + 1523*a^7*b^3*d^10)*x^3 - 6*(45*b^10*c^8*d^2
+ 360*a*b^9*c^7*d^3 - 3780*a^2*b^8*c^6*d^4 + 10584*a^3*b^7*c^5*d^5 - 13860*a^4*b^6*c^4*d^6 + 8880*a^5*b^5*c^3
*d^7 - 1935*a^6*b^4*c^2*d^8 - 570*a^7*b^3*c*d^9 + 263*a^8*b^2*d^10)*x^2 - 4*(10*b^10*c^9*d + 45*a*b^9*c^8*d^2
+ 360*a^2*b^8*c^7*d^3 - 4620*a^3*b^7*c^6*d^4 + 15624*a^4*b^6*c^5*d^5 - 26460*a^5*b^5*c^4*d^6 + 25680*a^6*b^4*c
^3*d^7 - 14535*a^7*b^3*c^2*d^8 + 4470*a^8*b^2*c*d^9 - 577*a^9*b*d^10)*x + 2520*(a^4*b^6*c^6*d^4 - 6*a^5*b^5*c^
5*d^5 + 15*a^6*b^4*c^4*d^6 - 20*a^7*b^3*c^3*d^7 + 15*a^8*b^2*c^2*d^8 - 6*a^9*b*c*d^9 + a^10*d^10 + (b^10*c^6*d
^4 - 6*a*b^9*c^5*d^5 + 15*a^2*b^8*c^4*d^6 - 20*a^3*b^7*c^3*d^7 + 15*a^4*b^6*c^2*d^8 - 6*a^5*b^5*c*d^9 + a^6*b^
4*d^10)*x^4 + 4*(a*b^9*c^6*d^4 - 6*a^2*b^8*c^5*d^5 + 15*a^3*b^7*c^4*d^6 - 20*a^4*b^6*c^3*d^7 + 15*a^5*b^5*c^2*
d^8 - 6*a^6*b^4*c*d^9 + a^7*b^3*d^10)*x^3 + 6*(a^2*b^8*c^6*d^4 - 6*a^3*b^7*c^5*d^5 + 15*a^4*b^6*c^4*d^6 - 20*a
^5*b^5*c^3*d^7 + 15*a^6*b^4*c^2*d^8 - 6*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 4*(a^3*b^7*c^6*d^4 - 6*a^4*b^6*c^5
*d^5 + 15*a^5*b^5*c^4*d^6 - 20*a^6*b^4*c^3*d^7 + 15*a^7*b^3*c^2*d^8 - 6*a^8*b^2*c*d^9 + a^9*b*d^10)*x)*log(b*x
+ a))/(b^15*x^4 + 4*a*b^14*x^3 + 6*a^2*b^13*x^2 + 4*a^3*b^12*x + a^4*b^11)

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{10}}{(a+b x)^5} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 903 vs. \(2 (252) = 504\).

Time = 0.24 (sec) , antiderivative size = 903, normalized size of antiderivative = 3.45 \[ \int \frac {(c+d x)^{10}}{(a+b x)^5} \, dx=-\frac {3 \, b^{10} c^{10} + 10 \, a b^{9} c^{9} d + 45 \, a^{2} b^{8} c^{8} d^{2} + 360 \, a^{3} b^{7} c^{7} d^{3} - 5250 \, a^{4} b^{6} c^{6} d^{4} + 19404 \, a^{5} b^{5} c^{5} d^{5} - 35910 \, a^{6} b^{4} c^{4} d^{6} + 38280 \, a^{7} b^{3} c^{3} d^{7} - 23985 \, a^{8} b^{2} c^{2} d^{8} + 8250 \, a^{9} b c d^{9} - 1207 \, a^{10} d^{10} + 1440 \, {\left (b^{10} c^{7} d^{3} - 7 \, a b^{9} c^{6} d^{4} + 21 \, a^{2} b^{8} c^{5} d^{5} - 35 \, a^{3} b^{7} c^{4} d^{6} + 35 \, a^{4} b^{6} c^{3} d^{7} - 21 \, a^{5} b^{5} c^{2} d^{8} + 7 \, a^{6} b^{4} c d^{9} - a^{7} b^{3} d^{10}\right )} x^{3} + 270 \, {\left (b^{10} c^{8} d^{2} + 8 \, a b^{9} c^{7} d^{3} - 84 \, a^{2} b^{8} c^{6} d^{4} + 280 \, a^{3} b^{7} c^{5} d^{5} - 490 \, a^{4} b^{6} c^{4} d^{6} + 504 \, a^{5} b^{5} c^{3} d^{7} - 308 \, a^{6} b^{4} c^{2} d^{8} + 104 \, a^{7} b^{3} c d^{9} - 15 \, a^{8} b^{2} d^{10}\right )} x^{2} + 20 \, {\left (2 \, b^{10} c^{9} d + 9 \, a b^{9} c^{8} d^{2} + 72 \, a^{2} b^{8} c^{7} d^{3} - 924 \, a^{3} b^{7} c^{6} d^{4} + 3276 \, a^{4} b^{6} c^{5} d^{5} - 5922 \, a^{5} b^{5} c^{4} d^{6} + 6216 \, a^{6} b^{4} c^{3} d^{7} - 3852 \, a^{7} b^{3} c^{2} d^{8} + 1314 \, a^{8} b^{2} c d^{9} - 191 \, a^{9} b d^{10}\right )} x}{12 \, {\left (b^{15} x^{4} + 4 \, a b^{14} x^{3} + 6 \, a^{2} b^{13} x^{2} + 4 \, a^{3} b^{12} x + a^{4} b^{11}\right )}} + \frac {2 \, b^{5} d^{10} x^{6} + 12 \, {\left (2 \, b^{5} c d^{9} - a b^{4} d^{10}\right )} x^{5} + 15 \, {\left (9 \, b^{5} c^{2} d^{8} - 10 \, a b^{4} c d^{9} + 3 \, a^{2} b^{3} d^{10}\right )} x^{4} + 20 \, {\left (24 \, b^{5} c^{3} d^{7} - 45 \, a b^{4} c^{2} d^{8} + 30 \, a^{2} b^{3} c d^{9} - 7 \, a^{3} b^{2} d^{10}\right )} x^{3} + 30 \, {\left (42 \, b^{5} c^{4} d^{6} - 120 \, a b^{4} c^{3} d^{7} + 135 \, a^{2} b^{3} c^{2} d^{8} - 70 \, a^{3} b^{2} c d^{9} + 14 \, a^{4} b d^{10}\right )} x^{2} + 12 \, {\left (252 \, b^{5} c^{5} d^{5} - 1050 \, a b^{4} c^{4} d^{6} + 1800 \, a^{2} b^{3} c^{3} d^{7} - 1575 \, a^{3} b^{2} c^{2} d^{8} + 700 \, a^{4} b c d^{9} - 126 \, a^{5} d^{10}\right )} x}{12 \, b^{10}} + \frac {210 \, {\left (b^{6} c^{6} d^{4} - 6 \, a b^{5} c^{5} d^{5} + 15 \, a^{2} b^{4} c^{4} d^{6} - 20 \, a^{3} b^{3} c^{3} d^{7} + 15 \, a^{4} b^{2} c^{2} d^{8} - 6 \, a^{5} b c d^{9} + a^{6} d^{10}\right )} \log \left (b x + a\right )}{b^{11}} \]

[In]

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

[Out]

-1/12*(3*b^10*c^10 + 10*a*b^9*c^9*d + 45*a^2*b^8*c^8*d^2 + 360*a^3*b^7*c^7*d^3 - 5250*a^4*b^6*c^6*d^4 + 19404*
a^5*b^5*c^5*d^5 - 35910*a^6*b^4*c^4*d^6 + 38280*a^7*b^3*c^3*d^7 - 23985*a^8*b^2*c^2*d^8 + 8250*a^9*b*c*d^9 - 1
207*a^10*d^10 + 1440*(b^10*c^7*d^3 - 7*a*b^9*c^6*d^4 + 21*a^2*b^8*c^5*d^5 - 35*a^3*b^7*c^4*d^6 + 35*a^4*b^6*c^
3*d^7 - 21*a^5*b^5*c^2*d^8 + 7*a^6*b^4*c*d^9 - a^7*b^3*d^10)*x^3 + 270*(b^10*c^8*d^2 + 8*a*b^9*c^7*d^3 - 84*a^
2*b^8*c^6*d^4 + 280*a^3*b^7*c^5*d^5 - 490*a^4*b^6*c^4*d^6 + 504*a^5*b^5*c^3*d^7 - 308*a^6*b^4*c^2*d^8 + 104*a^
7*b^3*c*d^9 - 15*a^8*b^2*d^10)*x^2 + 20*(2*b^10*c^9*d + 9*a*b^9*c^8*d^2 + 72*a^2*b^8*c^7*d^3 - 924*a^3*b^7*c^6
*d^4 + 3276*a^4*b^6*c^5*d^5 - 5922*a^5*b^5*c^4*d^6 + 6216*a^6*b^4*c^3*d^7 - 3852*a^7*b^3*c^2*d^8 + 1314*a^8*b^
2*c*d^9 - 191*a^9*b*d^10)*x)/(b^15*x^4 + 4*a*b^14*x^3 + 6*a^2*b^13*x^2 + 4*a^3*b^12*x + a^4*b^11) + 1/12*(2*b^
5*d^10*x^6 + 12*(2*b^5*c*d^9 - a*b^4*d^10)*x^5 + 15*(9*b^5*c^2*d^8 - 10*a*b^4*c*d^9 + 3*a^2*b^3*d^10)*x^4 + 20
*(24*b^5*c^3*d^7 - 45*a*b^4*c^2*d^8 + 30*a^2*b^3*c*d^9 - 7*a^3*b^2*d^10)*x^3 + 30*(42*b^5*c^4*d^6 - 120*a*b^4*
c^3*d^7 + 135*a^2*b^3*c^2*d^8 - 70*a^3*b^2*c*d^9 + 14*a^4*b*d^10)*x^2 + 12*(252*b^5*c^5*d^5 - 1050*a*b^4*c^4*d
^6 + 1800*a^2*b^3*c^3*d^7 - 1575*a^3*b^2*c^2*d^8 + 700*a^4*b*c*d^9 - 126*a^5*d^10)*x)/b^10 + 210*(b^6*c^6*d^4
- 6*a*b^5*c^5*d^5 + 15*a^2*b^4*c^4*d^6 - 20*a^3*b^3*c^3*d^7 + 15*a^4*b^2*c^2*d^8 - 6*a^5*b*c*d^9 + a^6*d^10)*l
og(b*x + a)/b^11

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1168 vs. \(2 (252) = 504\).

Time = 0.34 (sec) , antiderivative size = 1168, normalized size of antiderivative = 4.46 \[ \int \frac {(c+d x)^{10}}{(a+b x)^5} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/12*(2*d^10 + 24*(b^2*c*d^9 - a*b*d^10)/((b*x + a)*b) + 135*(b^4*c^2*d^8 - 2*a*b^3*c*d^9 + a^2*b^2*d^10)/((b*
x + a)^2*b^2) + 480*(b^6*c^3*d^7 - 3*a*b^5*c^2*d^8 + 3*a^2*b^4*c*d^9 - a^3*b^3*d^10)/((b*x + a)^3*b^3) + 1260*
(b^8*c^4*d^6 - 4*a*b^7*c^3*d^7 + 6*a^2*b^6*c^2*d^8 - 4*a^3*b^5*c*d^9 + a^4*b^4*d^10)/((b*x + a)^4*b^4) + 3024*
(b^10*c^5*d^5 - 5*a*b^9*c^4*d^6 + 10*a^2*b^8*c^3*d^7 - 10*a^3*b^7*c^2*d^8 + 5*a^4*b^6*c*d^9 - a^5*b^5*d^10)/((
b*x + a)^5*b^5))*(b*x + a)^6/b^11 - 210*(b^6*c^6*d^4 - 6*a*b^5*c^5*d^5 + 15*a^2*b^4*c^4*d^6 - 20*a^3*b^3*c^3*d
^7 + 15*a^4*b^2*c^2*d^8 - 6*a^5*b*c*d^9 + a^6*d^10)*log(abs(b*x + a)/((b*x + a)^2*abs(b)))/b^11 - 1/12*(3*b^67
*c^10/(b*x + a)^4 + 40*b^66*c^9*d/(b*x + a)^3 - 30*a*b^66*c^9*d/(b*x + a)^4 + 270*b^65*c^8*d^2/(b*x + a)^2 - 3
60*a*b^65*c^8*d^2/(b*x + a)^3 + 135*a^2*b^65*c^8*d^2/(b*x + a)^4 + 1440*b^64*c^7*d^3/(b*x + a) - 2160*a*b^64*c
^7*d^3/(b*x + a)^2 + 1440*a^2*b^64*c^7*d^3/(b*x + a)^3 - 360*a^3*b^64*c^7*d^3/(b*x + a)^4 - 10080*a*b^63*c^6*d
^4/(b*x + a) + 7560*a^2*b^63*c^6*d^4/(b*x + a)^2 - 3360*a^3*b^63*c^6*d^4/(b*x + a)^3 + 630*a^4*b^63*c^6*d^4/(b
*x + a)^4 + 30240*a^2*b^62*c^5*d^5/(b*x + a) - 15120*a^3*b^62*c^5*d^5/(b*x + a)^2 + 5040*a^4*b^62*c^5*d^5/(b*x
+ a)^3 - 756*a^5*b^62*c^5*d^5/(b*x + a)^4 - 50400*a^3*b^61*c^4*d^6/(b*x + a) + 18900*a^4*b^61*c^4*d^6/(b*x +
a)^2 - 5040*a^5*b^61*c^4*d^6/(b*x + a)^3 + 630*a^6*b^61*c^4*d^6/(b*x + a)^4 + 50400*a^4*b^60*c^3*d^7/(b*x + a)
- 15120*a^5*b^60*c^3*d^7/(b*x + a)^2 + 3360*a^6*b^60*c^3*d^7/(b*x + a)^3 - 360*a^7*b^60*c^3*d^7/(b*x + a)^4 -
30240*a^5*b^59*c^2*d^8/(b*x + a) + 7560*a^6*b^59*c^2*d^8/(b*x + a)^2 - 1440*a^7*b^59*c^2*d^8/(b*x + a)^3 + 13
5*a^8*b^59*c^2*d^8/(b*x + a)^4 + 10080*a^6*b^58*c*d^9/(b*x + a) - 2160*a^7*b^58*c*d^9/(b*x + a)^2 + 360*a^8*b^
58*c*d^9/(b*x + a)^3 - 30*a^9*b^58*c*d^9/(b*x + a)^4 - 1440*a^7*b^57*d^10/(b*x + a) + 270*a^8*b^57*d^10/(b*x +
a)^2 - 40*a^9*b^57*d^10/(b*x + a)^3 + 3*a^10*b^57*d^10/(b*x + a)^4)/b^68

Mupad [B] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 1494, normalized size of antiderivative = 5.70 \[ \int \frac {(c+d x)^{10}}{(a+b x)^5} \, dx=\text {Too large to display} \]

[In]

int((c + d*x)^10/(a + b*x)^5,x)

[Out]

x^2*((5*a*((5*a*((5*a*((5*a*d^10)/b^6 - (10*c*d^9)/b^5))/b - (10*a^2*d^10)/b^7 + (45*c^2*d^8)/b^5))/b + (10*a^
3*d^10)/b^8 - (120*c^3*d^7)/b^5 - (10*a^2*((5*a*d^10)/b^6 - (10*c*d^9)/b^5))/b^2))/(2*b) - (5*a^4*d^10)/(2*b^9
) + (105*c^4*d^6)/b^5 + (5*a^3*((5*a*d^10)/b^6 - (10*c*d^9)/b^5))/b^3 - (5*a^2*((5*a*((5*a*d^10)/b^6 - (10*c*d
^9)/b^5))/b - (10*a^2*d^10)/b^7 + (45*c^2*d^8)/b^5))/b^2) - x^5*((a*d^10)/b^6 - (2*c*d^9)/b^5) - x^3*((5*a*((5
*a*((5*a*d^10)/b^6 - (10*c*d^9)/b^5))/b - (10*a^2*d^10)/b^7 + (45*c^2*d^8)/b^5))/(3*b) + (10*a^3*d^10)/(3*b^8)
- (40*c^3*d^7)/b^5 - (10*a^2*((5*a*d^10)/b^6 - (10*c*d^9)/b^5))/(3*b^2)) + x^4*((5*a*((5*a*d^10)/b^6 - (10*c*
d^9)/b^5))/(4*b) - (5*a^2*d^10)/(2*b^7) + (45*c^2*d^8)/(4*b^5)) - x*((5*a*((5*a*((5*a*((5*a*((5*a*d^10)/b^6 -
(10*c*d^9)/b^5))/b - (10*a^2*d^10)/b^7 + (45*c^2*d^8)/b^5))/b + (10*a^3*d^10)/b^8 - (120*c^3*d^7)/b^5 - (10*a^
2*((5*a*d^10)/b^6 - (10*c*d^9)/b^5))/b^2))/b - (5*a^4*d^10)/b^9 + (210*c^4*d^6)/b^5 + (10*a^3*((5*a*d^10)/b^6
- (10*c*d^9)/b^5))/b^3 - (10*a^2*((5*a*((5*a*d^10)/b^6 - (10*c*d^9)/b^5))/b - (10*a^2*d^10)/b^7 + (45*c^2*d^8)
/b^5))/b^2))/b + (a^5*d^10)/b^10 - (252*c^5*d^5)/b^5 - (5*a^4*((5*a*d^10)/b^6 - (10*c*d^9)/b^5))/b^4 - (10*a^2
*((5*a*((5*a*((5*a*d^10)/b^6 - (10*c*d^9)/b^5))/b - (10*a^2*d^10)/b^7 + (45*c^2*d^8)/b^5))/b + (10*a^3*d^10)/b
^8 - (120*c^3*d^7)/b^5 - (10*a^2*((5*a*d^10)/b^6 - (10*c*d^9)/b^5))/b^2))/b^2 + (10*a^3*((5*a*((5*a*d^10)/b^6
- (10*c*d^9)/b^5))/b - (10*a^2*d^10)/b^7 + (45*c^2*d^8)/b^5))/b^3) - ((3*b^10*c^10 - 1207*a^10*d^10 + 45*a^2*b
^8*c^8*d^2 + 360*a^3*b^7*c^7*d^3 - 5250*a^4*b^6*c^6*d^4 + 19404*a^5*b^5*c^5*d^5 - 35910*a^6*b^4*c^4*d^6 + 3828
0*a^7*b^3*c^3*d^7 - 23985*a^8*b^2*c^2*d^8 + 10*a*b^9*c^9*d + 8250*a^9*b*c*d^9)/(12*b) + x*((10*b^9*c^9*d)/3 -
(955*a^9*d^10)/3 + 15*a*b^8*c^8*d^2 + 120*a^2*b^7*c^7*d^3 - 1540*a^3*b^6*c^6*d^4 + 5460*a^4*b^5*c^5*d^5 - 9870
*a^5*b^4*c^4*d^6 + 10360*a^6*b^3*c^3*d^7 - 6420*a^7*b^2*c^2*d^8 + 2190*a^8*b*c*d^9) - x^3*(120*a^7*b^2*d^10 -
120*b^9*c^7*d^3 + 840*a*b^8*c^6*d^4 - 840*a^6*b^3*c*d^9 - 2520*a^2*b^7*c^5*d^5 + 4200*a^3*b^6*c^4*d^6 - 4200*a
^4*b^5*c^3*d^7 + 2520*a^5*b^4*c^2*d^8) + x^2*((45*b^9*c^8*d^2)/2 - (675*a^8*b*d^10)/2 + 180*a*b^8*c^7*d^3 + 23
40*a^7*b^2*c*d^9 - 1890*a^2*b^7*c^6*d^4 + 6300*a^3*b^6*c^5*d^5 - 11025*a^4*b^5*c^4*d^6 + 11340*a^5*b^4*c^3*d^7
- 6930*a^6*b^3*c^2*d^8))/(a^4*b^10 + b^14*x^4 + 4*a^3*b^11*x + 4*a*b^13*x^3 + 6*a^2*b^12*x^2) + (log(a + b*x)
*(210*a^6*d^10 + 210*b^6*c^6*d^4 - 1260*a*b^5*c^5*d^5 + 3150*a^2*b^4*c^4*d^6 - 4200*a^3*b^3*c^3*d^7 + 3150*a^4
*b^2*c^2*d^8 - 1260*a^5*b*c*d^9))/b^11 + (d^10*x^6)/(6*b^5)