\(\int \frac {(c+d x)^7}{(a+b x)^{16}} \, dx\) [1298]

Optimal result

Integrand size = 15, antiderivative size = 200 \[ \int \frac {(c+d x)^7}{(a+b x)^{16}} \, dx=-\frac {(b c-a d)^7}{15 b^8 (a+b x)^{15}}-\frac {d (b c-a d)^6}{2 b^8 (a+b x)^{14}}-\frac {21 d^2 (b c-a d)^5}{13 b^8 (a+b x)^{13}}-\frac {35 d^3 (b c-a d)^4}{12 b^8 (a+b x)^{12}}-\frac {35 d^4 (b c-a d)^3}{11 b^8 (a+b x)^{11}}-\frac {21 d^5 (b c-a d)^2}{10 b^8 (a+b x)^{10}}-\frac {7 d^6 (b c-a d)}{9 b^8 (a+b x)^9}-\frac {d^7}{8 b^8 (a+b x)^8} \]

[Out]

-1/15*(-a*d+b*c)^7/b^8/(b*x+a)^15-1/2*d*(-a*d+b*c)^6/b^8/(b*x+a)^14-21/13*d^2*(-a*d+b*c)^5/b^8/(b*x+a)^13-35/1
2*d^3*(-a*d+b*c)^4/b^8/(b*x+a)^12-35/11*d^4*(-a*d+b*c)^3/b^8/(b*x+a)^11-21/10*d^5*(-a*d+b*c)^2/b^8/(b*x+a)^10-
7/9*d^6*(-a*d+b*c)/b^8/(b*x+a)^9-1/8*d^7/b^8/(b*x+a)^8

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 200, 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} \[ \int \frac {(c+d x)^7}{(a+b x)^{16}} \, dx=-\frac {7 d^6 (b c-a d)}{9 b^8 (a+b x)^9}-\frac {21 d^5 (b c-a d)^2}{10 b^8 (a+b x)^{10}}-\frac {35 d^4 (b c-a d)^3}{11 b^8 (a+b x)^{11}}-\frac {35 d^3 (b c-a d)^4}{12 b^8 (a+b x)^{12}}-\frac {21 d^2 (b c-a d)^5}{13 b^8 (a+b x)^{13}}-\frac {d (b c-a d)^6}{2 b^8 (a+b x)^{14}}-\frac {(b c-a d)^7}{15 b^8 (a+b x)^{15}}-\frac {d^7}{8 b^8 (a+b x)^8} \]

[In]

Int[(c + d*x)^7/(a + b*x)^16,x]

[Out]

-1/15*(b*c - a*d)^7/(b^8*(a + b*x)^15) - (d*(b*c - a*d)^6)/(2*b^8*(a + b*x)^14) - (21*d^2*(b*c - a*d)^5)/(13*b
^8*(a + b*x)^13) - (35*d^3*(b*c - a*d)^4)/(12*b^8*(a + b*x)^12) - (35*d^4*(b*c - a*d)^3)/(11*b^8*(a + b*x)^11)
- (21*d^5*(b*c - a*d)^2)/(10*b^8*(a + b*x)^10) - (7*d^6*(b*c - a*d))/(9*b^8*(a + b*x)^9) - d^7/(8*b^8*(a + b*
x)^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*} \text {integral}& = \int \left (\frac {(b c-a d)^7}{b^7 (a+b x)^{16}}+\frac {7 d (b c-a d)^6}{b^7 (a+b x)^{15}}+\frac {21 d^2 (b c-a d)^5}{b^7 (a+b x)^{14}}+\frac {35 d^3 (b c-a d)^4}{b^7 (a+b x)^{13}}+\frac {35 d^4 (b c-a d)^3}{b^7 (a+b x)^{12}}+\frac {21 d^5 (b c-a d)^2}{b^7 (a+b x)^{11}}+\frac {7 d^6 (b c-a d)}{b^7 (a+b x)^{10}}+\frac {d^7}{b^7 (a+b x)^9}\right ) \, dx \\ & = -\frac {(b c-a d)^7}{15 b^8 (a+b x)^{15}}-\frac {d (b c-a d)^6}{2 b^8 (a+b x)^{14}}-\frac {21 d^2 (b c-a d)^5}{13 b^8 (a+b x)^{13}}-\frac {35 d^3 (b c-a d)^4}{12 b^8 (a+b x)^{12}}-\frac {35 d^4 (b c-a d)^3}{11 b^8 (a+b x)^{11}}-\frac {21 d^5 (b c-a d)^2}{10 b^8 (a+b x)^{10}}-\frac {7 d^6 (b c-a d)}{9 b^8 (a+b x)^9}-\frac {d^7}{8 b^8 (a+b x)^8} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.86 \[ \int \frac {(c+d x)^7}{(a+b x)^{16}} \, dx=-\frac {a^7 d^7+a^6 b d^6 (8 c+15 d x)+3 a^5 b^2 d^5 \left (12 c^2+40 c d x+35 d^2 x^2\right )+5 a^4 b^3 d^4 \left (24 c^3+108 c^2 d x+168 c d^2 x^2+91 d^3 x^3\right )+5 a^3 b^4 d^3 \left (66 c^4+360 c^3 d x+756 c^2 d^2 x^2+728 c d^3 x^3+273 d^4 x^4\right )+3 a^2 b^5 d^2 \left (264 c^5+1650 c^4 d x+4200 c^3 d^2 x^2+5460 c^2 d^3 x^3+3640 c d^4 x^4+1001 d^5 x^5\right )+a b^6 d \left (1716 c^6+11880 c^5 d x+34650 c^4 d^2 x^2+54600 c^3 d^3 x^3+49140 c^2 d^4 x^4+24024 c d^5 x^5+5005 d^6 x^6\right )+b^7 \left (3432 c^7+25740 c^6 d x+83160 c^5 d^2 x^2+150150 c^4 d^3 x^3+163800 c^3 d^4 x^4+108108 c^2 d^5 x^5+40040 c d^6 x^6+6435 d^7 x^7\right )}{51480 b^8 (a+b x)^{15}} \]

[In]

Integrate[(c + d*x)^7/(a + b*x)^16,x]

[Out]

-1/51480*(a^7*d^7 + a^6*b*d^6*(8*c + 15*d*x) + 3*a^5*b^2*d^5*(12*c^2 + 40*c*d*x + 35*d^2*x^2) + 5*a^4*b^3*d^4*
(24*c^3 + 108*c^2*d*x + 168*c*d^2*x^2 + 91*d^3*x^3) + 5*a^3*b^4*d^3*(66*c^4 + 360*c^3*d*x + 756*c^2*d^2*x^2 +
728*c*d^3*x^3 + 273*d^4*x^4) + 3*a^2*b^5*d^2*(264*c^5 + 1650*c^4*d*x + 4200*c^3*d^2*x^2 + 5460*c^2*d^3*x^3 + 3
640*c*d^4*x^4 + 1001*d^5*x^5) + a*b^6*d*(1716*c^6 + 11880*c^5*d*x + 34650*c^4*d^2*x^2 + 54600*c^3*d^3*x^3 + 49
140*c^2*d^4*x^4 + 24024*c*d^5*x^5 + 5005*d^6*x^6) + b^7*(3432*c^7 + 25740*c^6*d*x + 83160*c^5*d^2*x^2 + 150150
*c^4*d^3*x^3 + 163800*c^3*d^4*x^4 + 108108*c^2*d^5*x^5 + 40040*c*d^6*x^6 + 6435*d^7*x^7))/(b^8*(a + b*x)^15)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(437\) vs. \(2(184)=368\).

Time = 0.24 (sec) , antiderivative size = 438, normalized size of antiderivative = 2.19

method	result	size
risch	\(\frac {-\frac {a^{7} d^{7}+8 a^{6} b c \,d^{6}+36 a^{5} b^{2} c^{2} d^{5}+120 a^{4} b^{3} c^{3} d^{4}+330 a^{3} b^{4} c^{4} d^{3}+792 a^{2} b^{5} c^{5} d^{2}+1716 a \,b^{6} c^{6} d +3432 b^{7} c^{7}}{51480 b^{8}}-\frac {d \left (a^{6} d^{6}+8 a^{5} b c \,d^{5}+36 a^{4} b^{2} c^{2} d^{4}+120 a^{3} b^{3} c^{3} d^{3}+330 a^{2} b^{4} c^{4} d^{2}+792 a \,b^{5} c^{5} d +1716 b^{6} c^{6}\right ) x}{3432 b^{7}}-\frac {7 d^{2} \left (a^{5} d^{5}+8 a^{4} b c \,d^{4}+36 a^{3} b^{2} c^{2} d^{3}+120 a^{2} b^{3} c^{3} d^{2}+330 a \,b^{4} c^{4} d +792 b^{5} c^{5}\right ) x^{2}}{3432 b^{6}}-\frac {7 d^{3} \left (a^{4} d^{4}+8 a^{3} b c \,d^{3}+36 a^{2} b^{2} c^{2} d^{2}+120 a \,b^{3} c^{3} d +330 b^{4} c^{4}\right ) x^{3}}{792 b^{5}}-\frac {7 d^{4} \left (a^{3} d^{3}+8 a^{2} b c \,d^{2}+36 a \,b^{2} c^{2} d +120 b^{3} c^{3}\right ) x^{4}}{264 b^{4}}-\frac {7 d^{5} \left (a^{2} d^{2}+8 a b c d +36 b^{2} c^{2}\right ) x^{5}}{120 b^{3}}-\frac {7 d^{6} \left (a d +8 b c \right ) x^{6}}{72 b^{2}}-\frac {d^{7} x^{7}}{8 b}}{\left (b x +a \right )^{15}}\)	\(438\)
default	\(\frac {35 d^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{11 b^{8} \left (b x +a \right )^{11}}+\frac {21 d^{2} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}{13 b^{8} \left (b x +a \right )^{13}}+\frac {7 d^{6} \left (a d -b c \right )}{9 b^{8} \left (b x +a \right )^{9}}-\frac {d^{7}}{8 b^{8} \left (b x +a \right )^{8}}-\frac {d \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right )}{2 b^{8} \left (b x +a \right )^{14}}-\frac {35 d^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{12 b^{8} \left (b x +a \right )^{12}}-\frac {21 d^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{10 b^{8} \left (b x +a \right )^{10}}-\frac {-a^{7} d^{7}+7 a^{6} b c \,d^{6}-21 a^{5} b^{2} c^{2} d^{5}+35 a^{4} b^{3} c^{3} d^{4}-35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d +b^{7} c^{7}}{15 b^{8} \left (b x +a \right )^{15}}\)	\(464\)
norman	\(\frac {\frac {-a^{7} b^{7} d^{7}-8 a^{6} b^{8} c \,d^{6}-36 a^{5} b^{9} c^{2} d^{5}-120 a^{4} b^{10} c^{3} d^{4}-330 a^{3} c^{4} d^{3} b^{11}-792 a^{2} b^{12} c^{5} d^{2}-1716 a \,b^{13} c^{6} d -3432 b^{14} c^{7}}{51480 b^{15}}+\frac {\left (-a^{6} b^{7} d^{7}-8 a^{5} b^{8} c \,d^{6}-36 a^{4} b^{9} c^{2} d^{5}-120 a^{3} b^{10} c^{3} d^{4}-330 a^{2} c^{4} d^{3} b^{11}-792 a \,b^{12} c^{5} d^{2}-1716 b^{13} c^{6} d \right ) x}{3432 b^{14}}+\frac {7 \left (-a^{5} b^{7} d^{7}-8 a^{4} b^{8} c \,d^{6}-36 a^{3} b^{9} c^{2} d^{5}-120 a^{2} b^{10} c^{3} d^{4}-330 a \,c^{4} d^{3} b^{11}-792 b^{12} c^{5} d^{2}\right ) x^{2}}{3432 b^{13}}+\frac {7 \left (-a^{4} b^{7} d^{7}-8 a^{3} b^{8} c \,d^{6}-36 a^{2} b^{9} c^{2} d^{5}-120 a \,b^{10} c^{3} d^{4}-330 c^{4} d^{3} b^{11}\right ) x^{3}}{792 b^{12}}+\frac {7 \left (-a^{3} b^{7} d^{7}-8 a^{2} b^{8} c \,d^{6}-36 a \,b^{9} c^{2} d^{5}-120 b^{10} c^{3} d^{4}\right ) x^{4}}{264 b^{11}}+\frac {7 \left (-a^{2} b^{7} d^{7}-8 a \,b^{8} c \,d^{6}-36 b^{9} c^{2} d^{5}\right ) x^{5}}{120 b^{10}}+\frac {7 \left (-a \,b^{7} d^{7}-8 b^{8} c \,d^{6}\right ) x^{6}}{72 b^{9}}-\frac {d^{7} x^{7}}{8 b}}{\left (b x +a \right )^{15}}\)	\(492\)
gosper	\(-\frac {6435 x^{7} d^{7} b^{7}+5005 x^{6} a \,b^{6} d^{7}+40040 x^{6} b^{7} c \,d^{6}+3003 x^{5} a^{2} b^{5} d^{7}+24024 x^{5} a \,b^{6} c \,d^{6}+108108 x^{5} b^{7} c^{2} d^{5}+1365 x^{4} a^{3} b^{4} d^{7}+10920 x^{4} a^{2} b^{5} c \,d^{6}+49140 x^{4} a \,b^{6} c^{2} d^{5}+163800 x^{4} b^{7} c^{3} d^{4}+455 x^{3} a^{4} b^{3} d^{7}+3640 x^{3} a^{3} b^{4} c \,d^{6}+16380 x^{3} a^{2} b^{5} c^{2} d^{5}+54600 x^{3} a \,b^{6} c^{3} d^{4}+150150 x^{3} b^{7} c^{4} d^{3}+105 x^{2} a^{5} b^{2} d^{7}+840 x^{2} a^{4} b^{3} c \,d^{6}+3780 x^{2} a^{3} b^{4} c^{2} d^{5}+12600 x^{2} a^{2} b^{5} c^{3} d^{4}+34650 x^{2} a \,b^{6} c^{4} d^{3}+83160 x^{2} b^{7} c^{5} d^{2}+15 x \,a^{6} b \,d^{7}+120 x \,a^{5} b^{2} c \,d^{6}+540 x \,a^{4} b^{3} c^{2} d^{5}+1800 x \,a^{3} b^{4} c^{3} d^{4}+4950 x \,a^{2} b^{5} c^{4} d^{3}+11880 x a \,b^{6} c^{5} d^{2}+25740 x \,b^{7} c^{6} d +a^{7} d^{7}+8 a^{6} b c \,d^{6}+36 a^{5} b^{2} c^{2} d^{5}+120 a^{4} b^{3} c^{3} d^{4}+330 a^{3} b^{4} c^{4} d^{3}+792 a^{2} b^{5} c^{5} d^{2}+1716 a \,b^{6} c^{6} d +3432 b^{7} c^{7}}{51480 b^{8} \left (b x +a \right )^{15}}\)	\(497\)
parallelrisch	\(\frac {-6435 d^{7} x^{7} b^{14}-5005 a \,b^{13} d^{7} x^{6}-40040 b^{14} c \,d^{6} x^{6}-3003 a^{2} b^{12} d^{7} x^{5}-24024 a \,b^{13} c \,d^{6} x^{5}-108108 b^{14} c^{2} d^{5} x^{5}-1365 a^{3} b^{11} d^{7} x^{4}-10920 a^{2} b^{12} c \,d^{6} x^{4}-49140 a \,b^{13} c^{2} d^{5} x^{4}-163800 b^{14} c^{3} d^{4} x^{4}-455 a^{4} b^{10} d^{7} x^{3}-3640 a^{3} b^{11} c \,d^{6} x^{3}-16380 a^{2} b^{12} c^{2} d^{5} x^{3}-54600 a \,b^{13} c^{3} d^{4} x^{3}-150150 b^{14} c^{4} d^{3} x^{3}-105 a^{5} b^{9} d^{7} x^{2}-840 a^{4} b^{10} c \,d^{6} x^{2}-3780 a^{3} b^{11} c^{2} d^{5} x^{2}-12600 a^{2} b^{12} c^{3} d^{4} x^{2}-34650 a \,b^{13} c^{4} d^{3} x^{2}-83160 b^{14} c^{5} d^{2} x^{2}-15 a^{6} b^{8} d^{7} x -120 a^{5} b^{9} c \,d^{6} x -540 a^{4} b^{10} c^{2} d^{5} x -1800 a^{3} b^{11} c^{3} d^{4} x -4950 a^{2} b^{12} c^{4} d^{3} x -11880 a \,b^{13} c^{5} d^{2} x -25740 b^{14} c^{6} d x -a^{7} b^{7} d^{7}-8 a^{6} b^{8} c \,d^{6}-36 a^{5} b^{9} c^{2} d^{5}-120 a^{4} b^{10} c^{3} d^{4}-330 a^{3} c^{4} d^{3} b^{11}-792 a^{2} b^{12} c^{5} d^{2}-1716 a \,b^{13} c^{6} d -3432 b^{14} c^{7}}{51480 b^{15} \left (b x +a \right )^{15}}\)	\(505\)

[In]

int((d*x+c)^7/(b*x+a)^16,x,method=_RETURNVERBOSE)

[Out]

(-1/51480/b^8*(a^7*d^7+8*a^6*b*c*d^6+36*a^5*b^2*c^2*d^5+120*a^4*b^3*c^3*d^4+330*a^3*b^4*c^4*d^3+792*a^2*b^5*c^
5*d^2+1716*a*b^6*c^6*d+3432*b^7*c^7)-1/3432/b^7*d*(a^6*d^6+8*a^5*b*c*d^5+36*a^4*b^2*c^2*d^4+120*a^3*b^3*c^3*d^
3+330*a^2*b^4*c^4*d^2+792*a*b^5*c^5*d+1716*b^6*c^6)*x-7/3432/b^6*d^2*(a^5*d^5+8*a^4*b*c*d^4+36*a^3*b^2*c^2*d^3
+120*a^2*b^3*c^3*d^2+330*a*b^4*c^4*d+792*b^5*c^5)*x^2-7/792/b^5*d^3*(a^4*d^4+8*a^3*b*c*d^3+36*a^2*b^2*c^2*d^2+
120*a*b^3*c^3*d+330*b^4*c^4)*x^3-7/264/b^4*d^4*(a^3*d^3+8*a^2*b*c*d^2+36*a*b^2*c^2*d+120*b^3*c^3)*x^4-7/120/b^
3*d^5*(a^2*d^2+8*a*b*c*d+36*b^2*c^2)*x^5-7/72/b^2*d^6*(a*d+8*b*c)*x^6-1/8/b*d^7*x^7)/(b*x+a)^15

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 614 vs. \(2 (184) = 368\).

Time = 0.23 (sec) , antiderivative size = 614, normalized size of antiderivative = 3.07 \[ \int \frac {(c+d x)^7}{(a+b x)^{16}} \, dx=-\frac {6435 \, b^{7} d^{7} x^{7} + 3432 \, b^{7} c^{7} + 1716 \, a b^{6} c^{6} d + 792 \, a^{2} b^{5} c^{5} d^{2} + 330 \, a^{3} b^{4} c^{4} d^{3} + 120 \, a^{4} b^{3} c^{3} d^{4} + 36 \, a^{5} b^{2} c^{2} d^{5} + 8 \, a^{6} b c d^{6} + a^{7} d^{7} + 5005 \, {\left (8 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 3003 \, {\left (36 \, b^{7} c^{2} d^{5} + 8 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 1365 \, {\left (120 \, b^{7} c^{3} d^{4} + 36 \, a b^{6} c^{2} d^{5} + 8 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 455 \, {\left (330 \, b^{7} c^{4} d^{3} + 120 \, a b^{6} c^{3} d^{4} + 36 \, a^{2} b^{5} c^{2} d^{5} + 8 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 105 \, {\left (792 \, b^{7} c^{5} d^{2} + 330 \, a b^{6} c^{4} d^{3} + 120 \, a^{2} b^{5} c^{3} d^{4} + 36 \, a^{3} b^{4} c^{2} d^{5} + 8 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 15 \, {\left (1716 \, b^{7} c^{6} d + 792 \, a b^{6} c^{5} d^{2} + 330 \, a^{2} b^{5} c^{4} d^{3} + 120 \, a^{3} b^{4} c^{3} d^{4} + 36 \, a^{4} b^{3} c^{2} d^{5} + 8 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{51480 \, {\left (b^{23} x^{15} + 15 \, a b^{22} x^{14} + 105 \, a^{2} b^{21} x^{13} + 455 \, a^{3} b^{20} x^{12} + 1365 \, a^{4} b^{19} x^{11} + 3003 \, a^{5} b^{18} x^{10} + 5005 \, a^{6} b^{17} x^{9} + 6435 \, a^{7} b^{16} x^{8} + 6435 \, a^{8} b^{15} x^{7} + 5005 \, a^{9} b^{14} x^{6} + 3003 \, a^{10} b^{13} x^{5} + 1365 \, a^{11} b^{12} x^{4} + 455 \, a^{12} b^{11} x^{3} + 105 \, a^{13} b^{10} x^{2} + 15 \, a^{14} b^{9} x + a^{15} b^{8}\right )}} \]

[In]

integrate((d*x+c)^7/(b*x+a)^16,x, algorithm="fricas")

[Out]

-1/51480*(6435*b^7*d^7*x^7 + 3432*b^7*c^7 + 1716*a*b^6*c^6*d + 792*a^2*b^5*c^5*d^2 + 330*a^3*b^4*c^4*d^3 + 120
*a^4*b^3*c^3*d^4 + 36*a^5*b^2*c^2*d^5 + 8*a^6*b*c*d^6 + a^7*d^7 + 5005*(8*b^7*c*d^6 + a*b^6*d^7)*x^6 + 3003*(3
6*b^7*c^2*d^5 + 8*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 1365*(120*b^7*c^3*d^4 + 36*a*b^6*c^2*d^5 + 8*a^2*b^5*c*d^6
+ a^3*b^4*d^7)*x^4 + 455*(330*b^7*c^4*d^3 + 120*a*b^6*c^3*d^4 + 36*a^2*b^5*c^2*d^5 + 8*a^3*b^4*c*d^6 + a^4*b^3
*d^7)*x^3 + 105*(792*b^7*c^5*d^2 + 330*a*b^6*c^4*d^3 + 120*a^2*b^5*c^3*d^4 + 36*a^3*b^4*c^2*d^5 + 8*a^4*b^3*c*
d^6 + a^5*b^2*d^7)*x^2 + 15*(1716*b^7*c^6*d + 792*a*b^6*c^5*d^2 + 330*a^2*b^5*c^4*d^3 + 120*a^3*b^4*c^3*d^4 +
36*a^4*b^3*c^2*d^5 + 8*a^5*b^2*c*d^6 + a^6*b*d^7)*x)/(b^23*x^15 + 15*a*b^22*x^14 + 105*a^2*b^21*x^13 + 455*a^3
*b^20*x^12 + 1365*a^4*b^19*x^11 + 3003*a^5*b^18*x^10 + 5005*a^6*b^17*x^9 + 6435*a^7*b^16*x^8 + 6435*a^8*b^15*x
^7 + 5005*a^9*b^14*x^6 + 3003*a^10*b^13*x^5 + 1365*a^11*b^12*x^4 + 455*a^12*b^11*x^3 + 105*a^13*b^10*x^2 + 15*
a^14*b^9*x + a^15*b^8)

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d x)^7}{(a+b x)^{16}} \, dx=\text {Timed out} \]

[In]

integrate((d*x+c)**7/(b*x+a)**16,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 614 vs. \(2 (184) = 368\).

Time = 0.25 (sec) , antiderivative size = 614, normalized size of antiderivative = 3.07 \[ \int \frac {(c+d x)^7}{(a+b x)^{16}} \, dx=-\frac {6435 \, b^{7} d^{7} x^{7} + 3432 \, b^{7} c^{7} + 1716 \, a b^{6} c^{6} d + 792 \, a^{2} b^{5} c^{5} d^{2} + 330 \, a^{3} b^{4} c^{4} d^{3} + 120 \, a^{4} b^{3} c^{3} d^{4} + 36 \, a^{5} b^{2} c^{2} d^{5} + 8 \, a^{6} b c d^{6} + a^{7} d^{7} + 5005 \, {\left (8 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 3003 \, {\left (36 \, b^{7} c^{2} d^{5} + 8 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 1365 \, {\left (120 \, b^{7} c^{3} d^{4} + 36 \, a b^{6} c^{2} d^{5} + 8 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 455 \, {\left (330 \, b^{7} c^{4} d^{3} + 120 \, a b^{6} c^{3} d^{4} + 36 \, a^{2} b^{5} c^{2} d^{5} + 8 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 105 \, {\left (792 \, b^{7} c^{5} d^{2} + 330 \, a b^{6} c^{4} d^{3} + 120 \, a^{2} b^{5} c^{3} d^{4} + 36 \, a^{3} b^{4} c^{2} d^{5} + 8 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 15 \, {\left (1716 \, b^{7} c^{6} d + 792 \, a b^{6} c^{5} d^{2} + 330 \, a^{2} b^{5} c^{4} d^{3} + 120 \, a^{3} b^{4} c^{3} d^{4} + 36 \, a^{4} b^{3} c^{2} d^{5} + 8 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{51480 \, {\left (b^{23} x^{15} + 15 \, a b^{22} x^{14} + 105 \, a^{2} b^{21} x^{13} + 455 \, a^{3} b^{20} x^{12} + 1365 \, a^{4} b^{19} x^{11} + 3003 \, a^{5} b^{18} x^{10} + 5005 \, a^{6} b^{17} x^{9} + 6435 \, a^{7} b^{16} x^{8} + 6435 \, a^{8} b^{15} x^{7} + 5005 \, a^{9} b^{14} x^{6} + 3003 \, a^{10} b^{13} x^{5} + 1365 \, a^{11} b^{12} x^{4} + 455 \, a^{12} b^{11} x^{3} + 105 \, a^{13} b^{10} x^{2} + 15 \, a^{14} b^{9} x + a^{15} b^{8}\right )}} \]

[In]

integrate((d*x+c)^7/(b*x+a)^16,x, algorithm="maxima")

[Out]

-1/51480*(6435*b^7*d^7*x^7 + 3432*b^7*c^7 + 1716*a*b^6*c^6*d + 792*a^2*b^5*c^5*d^2 + 330*a^3*b^4*c^4*d^3 + 120
*a^4*b^3*c^3*d^4 + 36*a^5*b^2*c^2*d^5 + 8*a^6*b*c*d^6 + a^7*d^7 + 5005*(8*b^7*c*d^6 + a*b^6*d^7)*x^6 + 3003*(3
6*b^7*c^2*d^5 + 8*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 1365*(120*b^7*c^3*d^4 + 36*a*b^6*c^2*d^5 + 8*a^2*b^5*c*d^6
+ a^3*b^4*d^7)*x^4 + 455*(330*b^7*c^4*d^3 + 120*a*b^6*c^3*d^4 + 36*a^2*b^5*c^2*d^5 + 8*a^3*b^4*c*d^6 + a^4*b^3
*d^7)*x^3 + 105*(792*b^7*c^5*d^2 + 330*a*b^6*c^4*d^3 + 120*a^2*b^5*c^3*d^4 + 36*a^3*b^4*c^2*d^5 + 8*a^4*b^3*c*
d^6 + a^5*b^2*d^7)*x^2 + 15*(1716*b^7*c^6*d + 792*a*b^6*c^5*d^2 + 330*a^2*b^5*c^4*d^3 + 120*a^3*b^4*c^3*d^4 +
36*a^4*b^3*c^2*d^5 + 8*a^5*b^2*c*d^6 + a^6*b*d^7)*x)/(b^23*x^15 + 15*a*b^22*x^14 + 105*a^2*b^21*x^13 + 455*a^3
*b^20*x^12 + 1365*a^4*b^19*x^11 + 3003*a^5*b^18*x^10 + 5005*a^6*b^17*x^9 + 6435*a^7*b^16*x^8 + 6435*a^8*b^15*x
^7 + 5005*a^9*b^14*x^6 + 3003*a^10*b^13*x^5 + 1365*a^11*b^12*x^4 + 455*a^12*b^11*x^3 + 105*a^13*b^10*x^2 + 15*
a^14*b^9*x + a^15*b^8)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 496 vs. \(2 (184) = 368\).

Time = 0.30 (sec) , antiderivative size = 496, normalized size of antiderivative = 2.48 \[ \int \frac {(c+d x)^7}{(a+b x)^{16}} \, dx=-\frac {6435 \, b^{7} d^{7} x^{7} + 40040 \, b^{7} c d^{6} x^{6} + 5005 \, a b^{6} d^{7} x^{6} + 108108 \, b^{7} c^{2} d^{5} x^{5} + 24024 \, a b^{6} c d^{6} x^{5} + 3003 \, a^{2} b^{5} d^{7} x^{5} + 163800 \, b^{7} c^{3} d^{4} x^{4} + 49140 \, a b^{6} c^{2} d^{5} x^{4} + 10920 \, a^{2} b^{5} c d^{6} x^{4} + 1365 \, a^{3} b^{4} d^{7} x^{4} + 150150 \, b^{7} c^{4} d^{3} x^{3} + 54600 \, a b^{6} c^{3} d^{4} x^{3} + 16380 \, a^{2} b^{5} c^{2} d^{5} x^{3} + 3640 \, a^{3} b^{4} c d^{6} x^{3} + 455 \, a^{4} b^{3} d^{7} x^{3} + 83160 \, b^{7} c^{5} d^{2} x^{2} + 34650 \, a b^{6} c^{4} d^{3} x^{2} + 12600 \, a^{2} b^{5} c^{3} d^{4} x^{2} + 3780 \, a^{3} b^{4} c^{2} d^{5} x^{2} + 840 \, a^{4} b^{3} c d^{6} x^{2} + 105 \, a^{5} b^{2} d^{7} x^{2} + 25740 \, b^{7} c^{6} d x + 11880 \, a b^{6} c^{5} d^{2} x + 4950 \, a^{2} b^{5} c^{4} d^{3} x + 1800 \, a^{3} b^{4} c^{3} d^{4} x + 540 \, a^{4} b^{3} c^{2} d^{5} x + 120 \, a^{5} b^{2} c d^{6} x + 15 \, a^{6} b d^{7} x + 3432 \, b^{7} c^{7} + 1716 \, a b^{6} c^{6} d + 792 \, a^{2} b^{5} c^{5} d^{2} + 330 \, a^{3} b^{4} c^{4} d^{3} + 120 \, a^{4} b^{3} c^{3} d^{4} + 36 \, a^{5} b^{2} c^{2} d^{5} + 8 \, a^{6} b c d^{6} + a^{7} d^{7}}{51480 \, {\left (b x + a\right )}^{15} b^{8}} \]

[In]

integrate((d*x+c)^7/(b*x+a)^16,x, algorithm="giac")

[Out]

-1/51480*(6435*b^7*d^7*x^7 + 40040*b^7*c*d^6*x^6 + 5005*a*b^6*d^7*x^6 + 108108*b^7*c^2*d^5*x^5 + 24024*a*b^6*c
*d^6*x^5 + 3003*a^2*b^5*d^7*x^5 + 163800*b^7*c^3*d^4*x^4 + 49140*a*b^6*c^2*d^5*x^4 + 10920*a^2*b^5*c*d^6*x^4 +
1365*a^3*b^4*d^7*x^4 + 150150*b^7*c^4*d^3*x^3 + 54600*a*b^6*c^3*d^4*x^3 + 16380*a^2*b^5*c^2*d^5*x^3 + 3640*a^
3*b^4*c*d^6*x^3 + 455*a^4*b^3*d^7*x^3 + 83160*b^7*c^5*d^2*x^2 + 34650*a*b^6*c^4*d^3*x^2 + 12600*a^2*b^5*c^3*d^
4*x^2 + 3780*a^3*b^4*c^2*d^5*x^2 + 840*a^4*b^3*c*d^6*x^2 + 105*a^5*b^2*d^7*x^2 + 25740*b^7*c^6*d*x + 11880*a*b
^6*c^5*d^2*x + 4950*a^2*b^5*c^4*d^3*x + 1800*a^3*b^4*c^3*d^4*x + 540*a^4*b^3*c^2*d^5*x + 120*a^5*b^2*c*d^6*x +
15*a^6*b*d^7*x + 3432*b^7*c^7 + 1716*a*b^6*c^6*d + 792*a^2*b^5*c^5*d^2 + 330*a^3*b^4*c^4*d^3 + 120*a^4*b^3*c^
3*d^4 + 36*a^5*b^2*c^2*d^5 + 8*a^6*b*c*d^6 + a^7*d^7)/((b*x + a)^15*b^8)

Mupad [B] (verification not implemented)

Time = 2.35 (sec) , antiderivative size = 592, normalized size of antiderivative = 2.96 \[ \int \frac {(c+d x)^7}{(a+b x)^{16}} \, dx=-\frac {\frac {a^7\,d^7+8\,a^6\,b\,c\,d^6+36\,a^5\,b^2\,c^2\,d^5+120\,a^4\,b^3\,c^3\,d^4+330\,a^3\,b^4\,c^4\,d^3+792\,a^2\,b^5\,c^5\,d^2+1716\,a\,b^6\,c^6\,d+3432\,b^7\,c^7}{51480\,b^8}+\frac {d^7\,x^7}{8\,b}+\frac {7\,d^2\,x^2\,\left (a^5\,d^5+8\,a^4\,b\,c\,d^4+36\,a^3\,b^2\,c^2\,d^3+120\,a^2\,b^3\,c^3\,d^2+330\,a\,b^4\,c^4\,d+792\,b^5\,c^5\right )}{3432\,b^6}+\frac {7\,d^4\,x^4\,\left (a^3\,d^3+8\,a^2\,b\,c\,d^2+36\,a\,b^2\,c^2\,d+120\,b^3\,c^3\right )}{264\,b^4}+\frac {7\,d^6\,x^6\,\left (a\,d+8\,b\,c\right )}{72\,b^2}+\frac {7\,d^3\,x^3\,\left (a^4\,d^4+8\,a^3\,b\,c\,d^3+36\,a^2\,b^2\,c^2\,d^2+120\,a\,b^3\,c^3\,d+330\,b^4\,c^4\right )}{792\,b^5}+\frac {d\,x\,\left (a^6\,d^6+8\,a^5\,b\,c\,d^5+36\,a^4\,b^2\,c^2\,d^4+120\,a^3\,b^3\,c^3\,d^3+330\,a^2\,b^4\,c^4\,d^2+792\,a\,b^5\,c^5\,d+1716\,b^6\,c^6\right )}{3432\,b^7}+\frac {7\,d^5\,x^5\,\left (a^2\,d^2+8\,a\,b\,c\,d+36\,b^2\,c^2\right )}{120\,b^3}}{a^{15}+15\,a^{14}\,b\,x+105\,a^{13}\,b^2\,x^2+455\,a^{12}\,b^3\,x^3+1365\,a^{11}\,b^4\,x^4+3003\,a^{10}\,b^5\,x^5+5005\,a^9\,b^6\,x^6+6435\,a^8\,b^7\,x^7+6435\,a^7\,b^8\,x^8+5005\,a^6\,b^9\,x^9+3003\,a^5\,b^{10}\,x^{10}+1365\,a^4\,b^{11}\,x^{11}+455\,a^3\,b^{12}\,x^{12}+105\,a^2\,b^{13}\,x^{13}+15\,a\,b^{14}\,x^{14}+b^{15}\,x^{15}} \]

[In]

int((c + d*x)^7/(a + b*x)^16,x)

[Out]

-((a^7*d^7 + 3432*b^7*c^7 + 792*a^2*b^5*c^5*d^2 + 330*a^3*b^4*c^4*d^3 + 120*a^4*b^3*c^3*d^4 + 36*a^5*b^2*c^2*d
^5 + 1716*a*b^6*c^6*d + 8*a^6*b*c*d^6)/(51480*b^8) + (d^7*x^7)/(8*b) + (7*d^2*x^2*(a^5*d^5 + 792*b^5*c^5 + 120
*a^2*b^3*c^3*d^2 + 36*a^3*b^2*c^2*d^3 + 330*a*b^4*c^4*d + 8*a^4*b*c*d^4))/(3432*b^6) + (7*d^4*x^4*(a^3*d^3 + 1
20*b^3*c^3 + 36*a*b^2*c^2*d + 8*a^2*b*c*d^2))/(264*b^4) + (7*d^6*x^6*(a*d + 8*b*c))/(72*b^2) + (7*d^3*x^3*(a^4
*d^4 + 330*b^4*c^4 + 36*a^2*b^2*c^2*d^2 + 120*a*b^3*c^3*d + 8*a^3*b*c*d^3))/(792*b^5) + (d*x*(a^6*d^6 + 1716*b
^6*c^6 + 330*a^2*b^4*c^4*d^2 + 120*a^3*b^3*c^3*d^3 + 36*a^4*b^2*c^2*d^4 + 792*a*b^5*c^5*d + 8*a^5*b*c*d^5))/(3
432*b^7) + (7*d^5*x^5*(a^2*d^2 + 36*b^2*c^2 + 8*a*b*c*d))/(120*b^3))/(a^15 + b^15*x^15 + 15*a*b^14*x^14 + 105*
a^13*b^2*x^2 + 455*a^12*b^3*x^3 + 1365*a^11*b^4*x^4 + 3003*a^10*b^5*x^5 + 5005*a^9*b^6*x^6 + 6435*a^8*b^7*x^7
+ 6435*a^7*b^8*x^8 + 5005*a^6*b^9*x^9 + 3003*a^5*b^10*x^10 + 1365*a^4*b^11*x^11 + 455*a^3*b^12*x^12 + 105*a^2*
b^13*x^13 + 15*a^14*b*x)