\(\int \frac {(c+d x)^7}{(a+b x)^{12}} \, dx\) [1294]

Optimal result

Integrand size = 15, antiderivative size = 120 \[ \int \frac {(c+d x)^7}{(a+b x)^{12}} \, dx=-\frac {(c+d x)^8}{11 (b c-a d) (a+b x)^{11}}+\frac {3 d (c+d x)^8}{110 (b c-a d)^2 (a+b x)^{10}}-\frac {d^2 (c+d x)^8}{165 (b c-a d)^3 (a+b x)^9}+\frac {d^3 (c+d x)^8}{1320 (b c-a d)^4 (a+b x)^8} \]

[Out]

-1/11*(d*x+c)^8/(-a*d+b*c)/(b*x+a)^11+3/110*d*(d*x+c)^8/(-a*d+b*c)^2/(b*x+a)^10-1/165*d^2*(d*x+c)^8/(-a*d+b*c)
^3/(b*x+a)^9+1/1320*d^3*(d*x+c)^8/(-a*d+b*c)^4/(b*x+a)^8

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {47, 37} \[ \int \frac {(c+d x)^7}{(a+b x)^{12}} \, dx=\frac {d^3 (c+d x)^8}{1320 (a+b x)^8 (b c-a d)^4}-\frac {d^2 (c+d x)^8}{165 (a+b x)^9 (b c-a d)^3}+\frac {3 d (c+d x)^8}{110 (a+b x)^{10} (b c-a d)^2}-\frac {(c+d x)^8}{11 (a+b x)^{11} (b c-a d)} \]

[In]

Int[(c + d*x)^7/(a + b*x)^12,x]

[Out]

-1/11*(c + d*x)^8/((b*c - a*d)*(a + b*x)^11) + (3*d*(c + d*x)^8)/(110*(b*c - a*d)^2*(a + b*x)^10) - (d^2*(c +
d*x)^8)/(165*(b*c - a*d)^3*(a + b*x)^9) + (d^3*(c + d*x)^8)/(1320*(b*c - a*d)^4*(a + b*x)^8)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rubi steps \begin{align*} \text {integral}& = -\frac {(c+d x)^8}{11 (b c-a d) (a+b x)^{11}}-\frac {(3 d) \int \frac {(c+d x)^7}{(a+b x)^{11}} \, dx}{11 (b c-a d)} \\ & = -\frac {(c+d x)^8}{11 (b c-a d) (a+b x)^{11}}+\frac {3 d (c+d x)^8}{110 (b c-a d)^2 (a+b x)^{10}}+\frac {\left (3 d^2\right ) \int \frac {(c+d x)^7}{(a+b x)^{10}} \, dx}{55 (b c-a d)^2} \\ & = -\frac {(c+d x)^8}{11 (b c-a d) (a+b x)^{11}}+\frac {3 d (c+d x)^8}{110 (b c-a d)^2 (a+b x)^{10}}-\frac {d^2 (c+d x)^8}{165 (b c-a d)^3 (a+b x)^9}-\frac {d^3 \int \frac {(c+d x)^7}{(a+b x)^9} \, dx}{165 (b c-a d)^3} \\ & = -\frac {(c+d x)^8}{11 (b c-a d) (a+b x)^{11}}+\frac {3 d (c+d x)^8}{110 (b c-a d)^2 (a+b x)^{10}}-\frac {d^2 (c+d x)^8}{165 (b c-a d)^3 (a+b x)^9}+\frac {d^3 (c+d x)^8}{1320 (b c-a d)^4 (a+b x)^8} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(369\) vs. \(2(120)=240\).

Time = 0.08 (sec) , antiderivative size = 369, normalized size of antiderivative = 3.08 \[ \int \frac {(c+d x)^7}{(a+b x)^{12}} \, dx=-\frac {a^7 d^7+a^6 b d^6 (4 c+11 d x)+a^5 b^2 d^5 \left (10 c^2+44 c d x+55 d^2 x^2\right )+5 a^4 b^3 d^4 \left (4 c^3+22 c^2 d x+44 c d^2 x^2+33 d^3 x^3\right )+5 a^3 b^4 d^3 \left (7 c^4+44 c^3 d x+110 c^2 d^2 x^2+132 c d^3 x^3+66 d^4 x^4\right )+a^2 b^5 d^2 \left (56 c^5+385 c^4 d x+1100 c^3 d^2 x^2+1650 c^2 d^3 x^3+1320 c d^4 x^4+462 d^5 x^5\right )+a b^6 d \left (84 c^6+616 c^5 d x+1925 c^4 d^2 x^2+3300 c^3 d^3 x^3+3300 c^2 d^4 x^4+1848 c d^5 x^5+462 d^6 x^6\right )+b^7 \left (120 c^7+924 c^6 d x+3080 c^5 d^2 x^2+5775 c^4 d^3 x^3+6600 c^3 d^4 x^4+4620 c^2 d^5 x^5+1848 c d^6 x^6+330 d^7 x^7\right )}{1320 b^8 (a+b x)^{11}} \]

[In]

Integrate[(c + d*x)^7/(a + b*x)^12,x]

[Out]

-1/1320*(a^7*d^7 + a^6*b*d^6*(4*c + 11*d*x) + a^5*b^2*d^5*(10*c^2 + 44*c*d*x + 55*d^2*x^2) + 5*a^4*b^3*d^4*(4*
c^3 + 22*c^2*d*x + 44*c*d^2*x^2 + 33*d^3*x^3) + 5*a^3*b^4*d^3*(7*c^4 + 44*c^3*d*x + 110*c^2*d^2*x^2 + 132*c*d^
3*x^3 + 66*d^4*x^4) + a^2*b^5*d^2*(56*c^5 + 385*c^4*d*x + 1100*c^3*d^2*x^2 + 1650*c^2*d^3*x^3 + 1320*c*d^4*x^4
+ 462*d^5*x^5) + a*b^6*d*(84*c^6 + 616*c^5*d*x + 1925*c^4*d^2*x^2 + 3300*c^3*d^3*x^3 + 3300*c^2*d^4*x^4 + 184
8*c*d^5*x^5 + 462*d^6*x^6) + b^7*(120*c^7 + 924*c^6*d*x + 3080*c^5*d^2*x^2 + 5775*c^4*d^3*x^3 + 6600*c^3*d^4*x
^4 + 4620*c^2*d^5*x^5 + 1848*c*d^6*x^6 + 330*d^7*x^7))/(b^8*(a + b*x)^11)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(437\) vs. \(2(112)=224\).

Time = 0.22 (sec) , antiderivative size = 438, normalized size of antiderivative = 3.65

method	result	size
risch	\(\frac {-\frac {d^{7} x^{7}}{4 b}-\frac {7 d^{6} \left (a d +4 b c \right ) x^{6}}{20 b^{2}}-\frac {7 d^{5} \left (a^{2} d^{2}+4 a b c d +10 b^{2} c^{2}\right ) x^{5}}{20 b^{3}}-\frac {d^{4} \left (a^{3} d^{3}+4 a^{2} b c \,d^{2}+10 a \,b^{2} c^{2} d +20 b^{3} c^{3}\right ) x^{4}}{4 b^{4}}-\frac {d^{3} \left (a^{4} d^{4}+4 a^{3} b c \,d^{3}+10 a^{2} b^{2} c^{2} d^{2}+20 a \,b^{3} c^{3} d +35 b^{4} c^{4}\right ) x^{3}}{8 b^{5}}-\frac {d^{2} \left (a^{5} d^{5}+4 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}+20 a^{2} b^{3} c^{3} d^{2}+35 a \,b^{4} c^{4} d +56 b^{5} c^{5}\right ) x^{2}}{24 b^{6}}-\frac {d \left (a^{6} d^{6}+4 a^{5} b c \,d^{5}+10 a^{4} b^{2} c^{2} d^{4}+20 a^{3} b^{3} c^{3} d^{3}+35 a^{2} b^{4} c^{4} d^{2}+56 a \,b^{5} c^{5} d +84 b^{6} c^{6}\right ) x}{120 b^{7}}-\frac {a^{7} d^{7}+4 a^{6} b c \,d^{6}+10 a^{5} b^{2} c^{2} d^{5}+20 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}+56 a^{2} b^{5} c^{5} d^{2}+84 a \,b^{6} c^{6} d +120 b^{7} c^{7}}{1320 b^{8}}}{\left (b x +a \right )^{11}}\)	\(438\)
default	\(-\frac {-a^{7} d^{7}+7 a^{6} b c \,d^{6}-21 a^{5} b^{2} c^{2} d^{5}+35 a^{4} b^{3} c^{3} d^{4}-35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d +b^{7} c^{7}}{11 b^{8} \left (b x +a \right )^{11}}+\frac {7 d^{2} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}{3 b^{8} \left (b x +a \right )^{9}}-\frac {7 d^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{2 b^{8} \left (b x +a \right )^{6}}-\frac {35 d^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{8 b^{8} \left (b x +a \right )^{8}}-\frac {d^{7}}{4 b^{8} \left (b x +a \right )^{4}}+\frac {5 d^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{b^{8} \left (b x +a \right )^{7}}-\frac {7 d \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right )}{10 b^{8} \left (b x +a \right )^{10}}+\frac {7 d^{6} \left (a d -b c \right )}{5 b^{8} \left (b x +a \right )^{5}}\)	\(464\)
norman	\(\frac {-\frac {d^{7} x^{7}}{4 b}+\frac {7 \left (-a \,b^{3} d^{7}-4 b^{4} c \,d^{6}\right ) x^{6}}{20 b^{5}}+\frac {7 \left (-a^{2} b^{3} d^{7}-4 a \,b^{4} c \,d^{6}-10 b^{5} c^{2} d^{5}\right ) x^{5}}{20 b^{6}}+\frac {\left (-a^{3} b^{3} d^{7}-4 a^{2} b^{4} c \,d^{6}-10 a \,b^{5} c^{2} d^{5}-20 b^{6} c^{3} d^{4}\right ) x^{4}}{4 b^{7}}+\frac {\left (-a^{4} b^{3} d^{7}-4 a^{3} b^{4} c \,d^{6}-10 a^{2} b^{5} c^{2} d^{5}-20 a \,b^{6} c^{3} d^{4}-35 b^{7} c^{4} d^{3}\right ) x^{3}}{8 b^{8}}+\frac {\left (-a^{5} b^{3} d^{7}-4 a^{4} b^{4} c \,d^{6}-10 a^{3} b^{5} c^{2} d^{5}-20 a^{2} b^{6} c^{3} d^{4}-35 a \,b^{7} c^{4} d^{3}-56 b^{8} c^{5} d^{2}\right ) x^{2}}{24 b^{9}}+\frac {\left (-a^{6} b^{3} d^{7}-4 a^{5} b^{4} c \,d^{6}-10 a^{4} b^{5} c^{2} d^{5}-20 a^{3} b^{6} c^{3} d^{4}-35 a^{2} b^{7} c^{4} d^{3}-56 a \,b^{8} c^{5} d^{2}-84 b^{9} c^{6} d \right ) x}{120 b^{10}}+\frac {-a^{7} b^{3} d^{7}-4 a^{6} b^{4} c \,d^{6}-10 a^{5} b^{5} c^{2} d^{5}-20 a^{4} b^{6} c^{3} d^{4}-35 a^{3} c^{4} d^{3} b^{7}-56 a^{2} b^{8} c^{5} d^{2}-84 a \,b^{9} c^{6} d -120 b^{10} c^{7}}{1320 b^{11}}}{\left (b x +a \right )^{11}}\)	\(492\)
gosper	\(-\frac {330 x^{7} d^{7} b^{7}+462 x^{6} a \,b^{6} d^{7}+1848 x^{6} b^{7} c \,d^{6}+462 x^{5} a^{2} b^{5} d^{7}+1848 x^{5} a \,b^{6} c \,d^{6}+4620 x^{5} b^{7} c^{2} d^{5}+330 x^{4} a^{3} b^{4} d^{7}+1320 x^{4} a^{2} b^{5} c \,d^{6}+3300 x^{4} a \,b^{6} c^{2} d^{5}+6600 x^{4} b^{7} c^{3} d^{4}+165 x^{3} a^{4} b^{3} d^{7}+660 x^{3} a^{3} b^{4} c \,d^{6}+1650 x^{3} a^{2} b^{5} c^{2} d^{5}+3300 x^{3} a \,b^{6} c^{3} d^{4}+5775 x^{3} b^{7} c^{4} d^{3}+55 x^{2} a^{5} b^{2} d^{7}+220 x^{2} a^{4} b^{3} c \,d^{6}+550 x^{2} a^{3} b^{4} c^{2} d^{5}+1100 x^{2} a^{2} b^{5} c^{3} d^{4}+1925 x^{2} a \,b^{6} c^{4} d^{3}+3080 x^{2} b^{7} c^{5} d^{2}+11 x \,a^{6} b \,d^{7}+44 x \,a^{5} b^{2} c \,d^{6}+110 x \,a^{4} b^{3} c^{2} d^{5}+220 x \,a^{3} b^{4} c^{3} d^{4}+385 x \,a^{2} b^{5} c^{4} d^{3}+616 x a \,b^{6} c^{5} d^{2}+924 x \,b^{7} c^{6} d +a^{7} d^{7}+4 a^{6} b c \,d^{6}+10 a^{5} b^{2} c^{2} d^{5}+20 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}+56 a^{2} b^{5} c^{5} d^{2}+84 a \,b^{6} c^{6} d +120 b^{7} c^{7}}{1320 b^{8} \left (b x +a \right )^{11}}\)	\(497\)
parallelrisch	\(\frac {-330 d^{7} x^{7} b^{10}-462 a \,b^{9} d^{7} x^{6}-1848 b^{10} c \,d^{6} x^{6}-462 a^{2} b^{8} d^{7} x^{5}-1848 a \,b^{9} c \,d^{6} x^{5}-4620 b^{10} c^{2} d^{5} x^{5}-330 a^{3} b^{7} d^{7} x^{4}-1320 a^{2} b^{8} c \,d^{6} x^{4}-3300 a \,b^{9} c^{2} d^{5} x^{4}-6600 b^{10} c^{3} d^{4} x^{4}-165 a^{4} b^{6} d^{7} x^{3}-660 a^{3} b^{7} c \,d^{6} x^{3}-1650 a^{2} b^{8} c^{2} d^{5} x^{3}-3300 a \,b^{9} c^{3} d^{4} x^{3}-5775 b^{10} c^{4} d^{3} x^{3}-55 a^{5} b^{5} d^{7} x^{2}-220 a^{4} b^{6} c \,d^{6} x^{2}-550 a^{3} b^{7} c^{2} d^{5} x^{2}-1100 a^{2} b^{8} c^{3} d^{4} x^{2}-1925 a \,b^{9} c^{4} d^{3} x^{2}-3080 b^{10} c^{5} d^{2} x^{2}-11 a^{6} b^{4} d^{7} x -44 a^{5} b^{5} c \,d^{6} x -110 a^{4} b^{6} c^{2} d^{5} x -220 a^{3} b^{7} c^{3} d^{4} x -385 a^{2} b^{8} c^{4} d^{3} x -616 a \,b^{9} c^{5} d^{2} x -924 b^{10} c^{6} d x -a^{7} b^{3} d^{7}-4 a^{6} b^{4} c \,d^{6}-10 a^{5} b^{5} c^{2} d^{5}-20 a^{4} b^{6} c^{3} d^{4}-35 a^{3} c^{4} d^{3} b^{7}-56 a^{2} b^{8} c^{5} d^{2}-84 a \,b^{9} c^{6} d -120 b^{10} c^{7}}{1320 b^{11} \left (b x +a \right )^{11}}\)	\(505\)

[In]

int((d*x+c)^7/(b*x+a)^12,x,method=_RETURNVERBOSE)

[Out]

(-1/4/b*d^7*x^7-7/20/b^2*d^6*(a*d+4*b*c)*x^6-7/20/b^3*d^5*(a^2*d^2+4*a*b*c*d+10*b^2*c^2)*x^5-1/4/b^4*d^4*(a^3*
d^3+4*a^2*b*c*d^2+10*a*b^2*c^2*d+20*b^3*c^3)*x^4-1/8/b^5*d^3*(a^4*d^4+4*a^3*b*c*d^3+10*a^2*b^2*c^2*d^2+20*a*b^
3*c^3*d+35*b^4*c^4)*x^3-1/24/b^6*d^2*(a^5*d^5+4*a^4*b*c*d^4+10*a^3*b^2*c^2*d^3+20*a^2*b^3*c^3*d^2+35*a*b^4*c^4
*d+56*b^5*c^5)*x^2-1/120/b^7*d*(a^6*d^6+4*a^5*b*c*d^5+10*a^4*b^2*c^2*d^4+20*a^3*b^3*c^3*d^3+35*a^2*b^4*c^4*d^2
+56*a*b^5*c^5*d+84*b^6*c^6)*x-1/1320/b^8*(a^7*d^7+4*a^6*b*c*d^6+10*a^5*b^2*c^2*d^5+20*a^4*b^3*c^3*d^4+35*a^3*b
^4*c^4*d^3+56*a^2*b^5*c^5*d^2+84*a*b^6*c^6*d+120*b^7*c^7))/(b*x+a)^11

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 570 vs. \(2 (112) = 224\).

Time = 0.22 (sec) , antiderivative size = 570, normalized size of antiderivative = 4.75 \[ \int \frac {(c+d x)^7}{(a+b x)^{12}} \, dx=-\frac {330 \, b^{7} d^{7} x^{7} + 120 \, b^{7} c^{7} + 84 \, a b^{6} c^{6} d + 56 \, a^{2} b^{5} c^{5} d^{2} + 35 \, a^{3} b^{4} c^{4} d^{3} + 20 \, a^{4} b^{3} c^{3} d^{4} + 10 \, a^{5} b^{2} c^{2} d^{5} + 4 \, a^{6} b c d^{6} + a^{7} d^{7} + 462 \, {\left (4 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 462 \, {\left (10 \, b^{7} c^{2} d^{5} + 4 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 330 \, {\left (20 \, b^{7} c^{3} d^{4} + 10 \, a b^{6} c^{2} d^{5} + 4 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 165 \, {\left (35 \, b^{7} c^{4} d^{3} + 20 \, a b^{6} c^{3} d^{4} + 10 \, a^{2} b^{5} c^{2} d^{5} + 4 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 55 \, {\left (56 \, b^{7} c^{5} d^{2} + 35 \, a b^{6} c^{4} d^{3} + 20 \, a^{2} b^{5} c^{3} d^{4} + 10 \, a^{3} b^{4} c^{2} d^{5} + 4 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 11 \, {\left (84 \, b^{7} c^{6} d + 56 \, a b^{6} c^{5} d^{2} + 35 \, a^{2} b^{5} c^{4} d^{3} + 20 \, a^{3} b^{4} c^{3} d^{4} + 10 \, a^{4} b^{3} c^{2} d^{5} + 4 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{1320 \, {\left (b^{19} x^{11} + 11 \, a b^{18} x^{10} + 55 \, a^{2} b^{17} x^{9} + 165 \, a^{3} b^{16} x^{8} + 330 \, a^{4} b^{15} x^{7} + 462 \, a^{5} b^{14} x^{6} + 462 \, a^{6} b^{13} x^{5} + 330 \, a^{7} b^{12} x^{4} + 165 \, a^{8} b^{11} x^{3} + 55 \, a^{9} b^{10} x^{2} + 11 \, a^{10} b^{9} x + a^{11} b^{8}\right )}} \]

[In]

integrate((d*x+c)^7/(b*x+a)^12,x, algorithm="fricas")

[Out]

-1/1320*(330*b^7*d^7*x^7 + 120*b^7*c^7 + 84*a*b^6*c^6*d + 56*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^3 + 20*a^4*b^3
*c^3*d^4 + 10*a^5*b^2*c^2*d^5 + 4*a^6*b*c*d^6 + a^7*d^7 + 462*(4*b^7*c*d^6 + a*b^6*d^7)*x^6 + 462*(10*b^7*c^2*
d^5 + 4*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 330*(20*b^7*c^3*d^4 + 10*a*b^6*c^2*d^5 + 4*a^2*b^5*c*d^6 + a^3*b^4*d^
7)*x^4 + 165*(35*b^7*c^4*d^3 + 20*a*b^6*c^3*d^4 + 10*a^2*b^5*c^2*d^5 + 4*a^3*b^4*c*d^6 + a^4*b^3*d^7)*x^3 + 55
*(56*b^7*c^5*d^2 + 35*a*b^6*c^4*d^3 + 20*a^2*b^5*c^3*d^4 + 10*a^3*b^4*c^2*d^5 + 4*a^4*b^3*c*d^6 + a^5*b^2*d^7)
*x^2 + 11*(84*b^7*c^6*d + 56*a*b^6*c^5*d^2 + 35*a^2*b^5*c^4*d^3 + 20*a^3*b^4*c^3*d^4 + 10*a^4*b^3*c^2*d^5 + 4*
a^5*b^2*c*d^6 + a^6*b*d^7)*x)/(b^19*x^11 + 11*a*b^18*x^10 + 55*a^2*b^17*x^9 + 165*a^3*b^16*x^8 + 330*a^4*b^15*
x^7 + 462*a^5*b^14*x^6 + 462*a^6*b^13*x^5 + 330*a^7*b^12*x^4 + 165*a^8*b^11*x^3 + 55*a^9*b^10*x^2 + 11*a^10*b^
9*x + a^11*b^8)

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d x)^7}{(a+b x)^{12}} \, dx=\text {Timed out} \]

[In]

integrate((d*x+c)**7/(b*x+a)**12,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 570 vs. \(2 (112) = 224\).

Time = 0.25 (sec) , antiderivative size = 570, normalized size of antiderivative = 4.75 \[ \int \frac {(c+d x)^7}{(a+b x)^{12}} \, dx=-\frac {330 \, b^{7} d^{7} x^{7} + 120 \, b^{7} c^{7} + 84 \, a b^{6} c^{6} d + 56 \, a^{2} b^{5} c^{5} d^{2} + 35 \, a^{3} b^{4} c^{4} d^{3} + 20 \, a^{4} b^{3} c^{3} d^{4} + 10 \, a^{5} b^{2} c^{2} d^{5} + 4 \, a^{6} b c d^{6} + a^{7} d^{7} + 462 \, {\left (4 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 462 \, {\left (10 \, b^{7} c^{2} d^{5} + 4 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 330 \, {\left (20 \, b^{7} c^{3} d^{4} + 10 \, a b^{6} c^{2} d^{5} + 4 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 165 \, {\left (35 \, b^{7} c^{4} d^{3} + 20 \, a b^{6} c^{3} d^{4} + 10 \, a^{2} b^{5} c^{2} d^{5} + 4 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 55 \, {\left (56 \, b^{7} c^{5} d^{2} + 35 \, a b^{6} c^{4} d^{3} + 20 \, a^{2} b^{5} c^{3} d^{4} + 10 \, a^{3} b^{4} c^{2} d^{5} + 4 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 11 \, {\left (84 \, b^{7} c^{6} d + 56 \, a b^{6} c^{5} d^{2} + 35 \, a^{2} b^{5} c^{4} d^{3} + 20 \, a^{3} b^{4} c^{3} d^{4} + 10 \, a^{4} b^{3} c^{2} d^{5} + 4 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{1320 \, {\left (b^{19} x^{11} + 11 \, a b^{18} x^{10} + 55 \, a^{2} b^{17} x^{9} + 165 \, a^{3} b^{16} x^{8} + 330 \, a^{4} b^{15} x^{7} + 462 \, a^{5} b^{14} x^{6} + 462 \, a^{6} b^{13} x^{5} + 330 \, a^{7} b^{12} x^{4} + 165 \, a^{8} b^{11} x^{3} + 55 \, a^{9} b^{10} x^{2} + 11 \, a^{10} b^{9} x + a^{11} b^{8}\right )}} \]

[In]

integrate((d*x+c)^7/(b*x+a)^12,x, algorithm="maxima")

[Out]

-1/1320*(330*b^7*d^7*x^7 + 120*b^7*c^7 + 84*a*b^6*c^6*d + 56*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^3 + 20*a^4*b^3
*c^3*d^4 + 10*a^5*b^2*c^2*d^5 + 4*a^6*b*c*d^6 + a^7*d^7 + 462*(4*b^7*c*d^6 + a*b^6*d^7)*x^6 + 462*(10*b^7*c^2*
d^5 + 4*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 330*(20*b^7*c^3*d^4 + 10*a*b^6*c^2*d^5 + 4*a^2*b^5*c*d^6 + a^3*b^4*d^
7)*x^4 + 165*(35*b^7*c^4*d^3 + 20*a*b^6*c^3*d^4 + 10*a^2*b^5*c^2*d^5 + 4*a^3*b^4*c*d^6 + a^4*b^3*d^7)*x^3 + 55
*(56*b^7*c^5*d^2 + 35*a*b^6*c^4*d^3 + 20*a^2*b^5*c^3*d^4 + 10*a^3*b^4*c^2*d^5 + 4*a^4*b^3*c*d^6 + a^5*b^2*d^7)
*x^2 + 11*(84*b^7*c^6*d + 56*a*b^6*c^5*d^2 + 35*a^2*b^5*c^4*d^3 + 20*a^3*b^4*c^3*d^4 + 10*a^4*b^3*c^2*d^5 + 4*
a^5*b^2*c*d^6 + a^6*b*d^7)*x)/(b^19*x^11 + 11*a*b^18*x^10 + 55*a^2*b^17*x^9 + 165*a^3*b^16*x^8 + 330*a^4*b^15*
x^7 + 462*a^5*b^14*x^6 + 462*a^6*b^13*x^5 + 330*a^7*b^12*x^4 + 165*a^8*b^11*x^3 + 55*a^9*b^10*x^2 + 11*a^10*b^
9*x + a^11*b^8)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 496 vs. \(2 (112) = 224\).

Time = 0.31 (sec) , antiderivative size = 496, normalized size of antiderivative = 4.13 \[ \int \frac {(c+d x)^7}{(a+b x)^{12}} \, dx=-\frac {330 \, b^{7} d^{7} x^{7} + 1848 \, b^{7} c d^{6} x^{6} + 462 \, a b^{6} d^{7} x^{6} + 4620 \, b^{7} c^{2} d^{5} x^{5} + 1848 \, a b^{6} c d^{6} x^{5} + 462 \, a^{2} b^{5} d^{7} x^{5} + 6600 \, b^{7} c^{3} d^{4} x^{4} + 3300 \, a b^{6} c^{2} d^{5} x^{4} + 1320 \, a^{2} b^{5} c d^{6} x^{4} + 330 \, a^{3} b^{4} d^{7} x^{4} + 5775 \, b^{7} c^{4} d^{3} x^{3} + 3300 \, a b^{6} c^{3} d^{4} x^{3} + 1650 \, a^{2} b^{5} c^{2} d^{5} x^{3} + 660 \, a^{3} b^{4} c d^{6} x^{3} + 165 \, a^{4} b^{3} d^{7} x^{3} + 3080 \, b^{7} c^{5} d^{2} x^{2} + 1925 \, a b^{6} c^{4} d^{3} x^{2} + 1100 \, a^{2} b^{5} c^{3} d^{4} x^{2} + 550 \, a^{3} b^{4} c^{2} d^{5} x^{2} + 220 \, a^{4} b^{3} c d^{6} x^{2} + 55 \, a^{5} b^{2} d^{7} x^{2} + 924 \, b^{7} c^{6} d x + 616 \, a b^{6} c^{5} d^{2} x + 385 \, a^{2} b^{5} c^{4} d^{3} x + 220 \, a^{3} b^{4} c^{3} d^{4} x + 110 \, a^{4} b^{3} c^{2} d^{5} x + 44 \, a^{5} b^{2} c d^{6} x + 11 \, a^{6} b d^{7} x + 120 \, b^{7} c^{7} + 84 \, a b^{6} c^{6} d + 56 \, a^{2} b^{5} c^{5} d^{2} + 35 \, a^{3} b^{4} c^{4} d^{3} + 20 \, a^{4} b^{3} c^{3} d^{4} + 10 \, a^{5} b^{2} c^{2} d^{5} + 4 \, a^{6} b c d^{6} + a^{7} d^{7}}{1320 \, {\left (b x + a\right )}^{11} b^{8}} \]

[In]

integrate((d*x+c)^7/(b*x+a)^12,x, algorithm="giac")

[Out]

-1/1320*(330*b^7*d^7*x^7 + 1848*b^7*c*d^6*x^6 + 462*a*b^6*d^7*x^6 + 4620*b^7*c^2*d^5*x^5 + 1848*a*b^6*c*d^6*x^
5 + 462*a^2*b^5*d^7*x^5 + 6600*b^7*c^3*d^4*x^4 + 3300*a*b^6*c^2*d^5*x^4 + 1320*a^2*b^5*c*d^6*x^4 + 330*a^3*b^4
*d^7*x^4 + 5775*b^7*c^4*d^3*x^3 + 3300*a*b^6*c^3*d^4*x^3 + 1650*a^2*b^5*c^2*d^5*x^3 + 660*a^3*b^4*c*d^6*x^3 +
165*a^4*b^3*d^7*x^3 + 3080*b^7*c^5*d^2*x^2 + 1925*a*b^6*c^4*d^3*x^2 + 1100*a^2*b^5*c^3*d^4*x^2 + 550*a^3*b^4*c
^2*d^5*x^2 + 220*a^4*b^3*c*d^6*x^2 + 55*a^5*b^2*d^7*x^2 + 924*b^7*c^6*d*x + 616*a*b^6*c^5*d^2*x + 385*a^2*b^5*
c^4*d^3*x + 220*a^3*b^4*c^3*d^4*x + 110*a^4*b^3*c^2*d^5*x + 44*a^5*b^2*c*d^6*x + 11*a^6*b*d^7*x + 120*b^7*c^7
+ 84*a*b^6*c^6*d + 56*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^3 + 20*a^4*b^3*c^3*d^4 + 10*a^5*b^2*c^2*d^5 + 4*a^6*b
*c*d^6 + a^7*d^7)/((b*x + a)^11*b^8)

Mupad [B] (verification not implemented)

Time = 0.55 (sec) , antiderivative size = 548, normalized size of antiderivative = 4.57 \[ \int \frac {(c+d x)^7}{(a+b x)^{12}} \, dx=-\frac {\frac {a^7\,d^7+4\,a^6\,b\,c\,d^6+10\,a^5\,b^2\,c^2\,d^5+20\,a^4\,b^3\,c^3\,d^4+35\,a^3\,b^4\,c^4\,d^3+56\,a^2\,b^5\,c^5\,d^2+84\,a\,b^6\,c^6\,d+120\,b^7\,c^7}{1320\,b^8}+\frac {d^7\,x^7}{4\,b}+\frac {d^2\,x^2\,\left (a^5\,d^5+4\,a^4\,b\,c\,d^4+10\,a^3\,b^2\,c^2\,d^3+20\,a^2\,b^3\,c^3\,d^2+35\,a\,b^4\,c^4\,d+56\,b^5\,c^5\right )}{24\,b^6}+\frac {d^4\,x^4\,\left (a^3\,d^3+4\,a^2\,b\,c\,d^2+10\,a\,b^2\,c^2\,d+20\,b^3\,c^3\right )}{4\,b^4}+\frac {7\,d^6\,x^6\,\left (a\,d+4\,b\,c\right )}{20\,b^2}+\frac {d^3\,x^3\,\left (a^4\,d^4+4\,a^3\,b\,c\,d^3+10\,a^2\,b^2\,c^2\,d^2+20\,a\,b^3\,c^3\,d+35\,b^4\,c^4\right )}{8\,b^5}+\frac {d\,x\,\left (a^6\,d^6+4\,a^5\,b\,c\,d^5+10\,a^4\,b^2\,c^2\,d^4+20\,a^3\,b^3\,c^3\,d^3+35\,a^2\,b^4\,c^4\,d^2+56\,a\,b^5\,c^5\,d+84\,b^6\,c^6\right )}{120\,b^7}+\frac {7\,d^5\,x^5\,\left (a^2\,d^2+4\,a\,b\,c\,d+10\,b^2\,c^2\right )}{20\,b^3}}{a^{11}+11\,a^{10}\,b\,x+55\,a^9\,b^2\,x^2+165\,a^8\,b^3\,x^3+330\,a^7\,b^4\,x^4+462\,a^6\,b^5\,x^5+462\,a^5\,b^6\,x^6+330\,a^4\,b^7\,x^7+165\,a^3\,b^8\,x^8+55\,a^2\,b^9\,x^9+11\,a\,b^{10}\,x^{10}+b^{11}\,x^{11}} \]

[In]

int((c + d*x)^7/(a + b*x)^12,x)

[Out]

-((a^7*d^7 + 120*b^7*c^7 + 56*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^3 + 20*a^4*b^3*c^3*d^4 + 10*a^5*b^2*c^2*d^5 +
84*a*b^6*c^6*d + 4*a^6*b*c*d^6)/(1320*b^8) + (d^7*x^7)/(4*b) + (d^2*x^2*(a^5*d^5 + 56*b^5*c^5 + 20*a^2*b^3*c^
3*d^2 + 10*a^3*b^2*c^2*d^3 + 35*a*b^4*c^4*d + 4*a^4*b*c*d^4))/(24*b^6) + (d^4*x^4*(a^3*d^3 + 20*b^3*c^3 + 10*a
*b^2*c^2*d + 4*a^2*b*c*d^2))/(4*b^4) + (7*d^6*x^6*(a*d + 4*b*c))/(20*b^2) + (d^3*x^3*(a^4*d^4 + 35*b^4*c^4 + 1
0*a^2*b^2*c^2*d^2 + 20*a*b^3*c^3*d + 4*a^3*b*c*d^3))/(8*b^5) + (d*x*(a^6*d^6 + 84*b^6*c^6 + 35*a^2*b^4*c^4*d^2
+ 20*a^3*b^3*c^3*d^3 + 10*a^4*b^2*c^2*d^4 + 56*a*b^5*c^5*d + 4*a^5*b*c*d^5))/(120*b^7) + (7*d^5*x^5*(a^2*d^2
+ 10*b^2*c^2 + 4*a*b*c*d))/(20*b^3))/(a^11 + b^11*x^11 + 11*a*b^10*x^10 + 55*a^9*b^2*x^2 + 165*a^8*b^3*x^3 + 3
30*a^7*b^4*x^4 + 462*a^6*b^5*x^5 + 462*a^5*b^6*x^6 + 330*a^4*b^7*x^7 + 165*a^3*b^8*x^8 + 55*a^2*b^9*x^9 + 11*a
^10*b*x)