3.14.27 \(\int \frac {(c+d x)^{10}}{(a+b x)^{16}} \, dx\) [1327]

Optimal. Leaf size=151 \[ -\frac {(c+d x)^{11}}{15 (b c-a d) (a+b x)^{15}}+\frac {2 d (c+d x)^{11}}{105 (b c-a d)^2 (a+b x)^{14}}-\frac {2 d^2 (c+d x)^{11}}{455 (b c-a d)^3 (a+b x)^{13}}+\frac {d^3 (c+d x)^{11}}{1365 (b c-a d)^4 (a+b x)^{12}}-\frac {d^4 (c+d x)^{11}}{15015 (b c-a d)^5 (a+b x)^{11}} \]

[Out]

-1/15*(d*x+c)^11/(-a*d+b*c)/(b*x+a)^15+2/105*d*(d*x+c)^11/(-a*d+b*c)^2/(b*x+a)^14-2/455*d^2*(d*x+c)^11/(-a*d+b
*c)^3/(b*x+a)^13+1/1365*d^3*(d*x+c)^11/(-a*d+b*c)^4/(b*x+a)^12-1/15015*d^4*(d*x+c)^11/(-a*d+b*c)^5/(b*x+a)^11

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Rubi [A]
time = 0.03, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {47, 37} \begin {gather*} -\frac {d^4 (c+d x)^{11}}{15015 (a+b x)^{11} (b c-a d)^5}+\frac {d^3 (c+d x)^{11}}{1365 (a+b x)^{12} (b c-a d)^4}-\frac {2 d^2 (c+d x)^{11}}{455 (a+b x)^{13} (b c-a d)^3}+\frac {2 d (c+d x)^{11}}{105 (a+b x)^{14} (b c-a d)^2}-\frac {(c+d x)^{11}}{15 (a+b x)^{15} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^10/(a + b*x)^16,x]

[Out]

-1/15*(c + d*x)^11/((b*c - a*d)*(a + b*x)^15) + (2*d*(c + d*x)^11)/(105*(b*c - a*d)^2*(a + b*x)^14) - (2*d^2*(
c + d*x)^11)/(455*(b*c - a*d)^3*(a + b*x)^13) + (d^3*(c + d*x)^11)/(1365*(b*c - a*d)^4*(a + b*x)^12) - (d^4*(c
+ d*x)^11)/(15015*(b*c - a*d)^5*(a + b*x)^11)

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*} \int \frac {(c+d x)^{10}}{(a+b x)^{16}} \, dx &=-\frac {(c+d x)^{11}}{15 (b c-a d) (a+b x)^{15}}-\frac {(4 d) \int \frac {(c+d x)^{10}}{(a+b x)^{15}} \, dx}{15 (b c-a d)}\\ &=-\frac {(c+d x)^{11}}{15 (b c-a d) (a+b x)^{15}}+\frac {2 d (c+d x)^{11}}{105 (b c-a d)^2 (a+b x)^{14}}+\frac {\left (2 d^2\right ) \int \frac {(c+d x)^{10}}{(a+b x)^{14}} \, dx}{35 (b c-a d)^2}\\ &=-\frac {(c+d x)^{11}}{15 (b c-a d) (a+b x)^{15}}+\frac {2 d (c+d x)^{11}}{105 (b c-a d)^2 (a+b x)^{14}}-\frac {2 d^2 (c+d x)^{11}}{455 (b c-a d)^3 (a+b x)^{13}}-\frac {\left (4 d^3\right ) \int \frac {(c+d x)^{10}}{(a+b x)^{13}} \, dx}{455 (b c-a d)^3}\\ &=-\frac {(c+d x)^{11}}{15 (b c-a d) (a+b x)^{15}}+\frac {2 d (c+d x)^{11}}{105 (b c-a d)^2 (a+b x)^{14}}-\frac {2 d^2 (c+d x)^{11}}{455 (b c-a d)^3 (a+b x)^{13}}+\frac {d^3 (c+d x)^{11}}{1365 (b c-a d)^4 (a+b x)^{12}}+\frac {d^4 \int \frac {(c+d x)^{10}}{(a+b x)^{12}} \, dx}{1365 (b c-a d)^4}\\ &=-\frac {(c+d x)^{11}}{15 (b c-a d) (a+b x)^{15}}+\frac {2 d (c+d x)^{11}}{105 (b c-a d)^2 (a+b x)^{14}}-\frac {2 d^2 (c+d x)^{11}}{455 (b c-a d)^3 (a+b x)^{13}}+\frac {d^3 (c+d x)^{11}}{1365 (b c-a d)^4 (a+b x)^{12}}-\frac {d^4 (c+d x)^{11}}{15015 (b c-a d)^5 (a+b x)^{11}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(690\) vs. \(2(151)=302\).
time = 0.18, size = 690, normalized size = 4.57 \begin {gather*} -\frac {a^{10} d^{10}+5 a^9 b d^9 (c+3 d x)+15 a^8 b^2 d^8 \left (c^2+5 c d x+7 d^2 x^2\right )+5 a^7 b^3 d^7 \left (7 c^3+45 c^2 d x+105 c d^2 x^2+91 d^3 x^3\right )+35 a^6 b^4 d^6 \left (2 c^4+15 c^3 d x+45 c^2 d^2 x^2+65 c d^3 x^3+39 d^4 x^4\right )+21 a^5 b^5 d^5 \left (6 c^5+50 c^4 d x+175 c^3 d^2 x^2+325 c^2 d^3 x^3+325 c d^4 x^4+143 d^5 x^5\right )+35 a^4 b^6 d^4 \left (6 c^6+54 c^5 d x+210 c^4 d^2 x^2+455 c^3 d^3 x^3+585 c^2 d^4 x^4+429 c d^5 x^5+143 d^6 x^6\right )+5 a^3 b^7 d^3 \left (66 c^7+630 c^6 d x+2646 c^5 d^2 x^2+6370 c^4 d^3 x^3+9555 c^3 d^4 x^4+9009 c^2 d^5 x^5+5005 c d^6 x^6+1287 d^7 x^7\right )+15 a^2 b^8 d^2 \left (33 c^8+330 c^7 d x+1470 c^6 d^2 x^2+3822 c^5 d^3 x^3+6370 c^4 d^4 x^4+7007 c^3 d^5 x^5+5005 c^2 d^6 x^6+2145 c d^7 x^7+429 d^8 x^8\right )+5 a b^9 d \left (143 c^9+1485 c^8 d x+6930 c^7 d^2 x^2+19110 c^6 d^3 x^3+34398 c^5 d^4 x^4+42042 c^4 d^5 x^5+35035 c^3 d^6 x^6+19305 c^2 d^7 x^7+6435 c d^8 x^8+1001 d^9 x^9\right )+b^{10} \left (1001 c^{10}+10725 c^9 d x+51975 c^8 d^2 x^2+150150 c^7 d^3 x^3+286650 c^6 d^4 x^4+378378 c^5 d^5 x^5+350350 c^4 d^6 x^6+225225 c^3 d^7 x^7+96525 c^2 d^8 x^8+25025 c d^9 x^9+3003 d^{10} x^{10}\right )}{15015 b^{11} (a+b x)^{15}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^10/(a + b*x)^16,x]

[Out]

-1/15015*(a^10*d^10 + 5*a^9*b*d^9*(c + 3*d*x) + 15*a^8*b^2*d^8*(c^2 + 5*c*d*x + 7*d^2*x^2) + 5*a^7*b^3*d^7*(7*
c^3 + 45*c^2*d*x + 105*c*d^2*x^2 + 91*d^3*x^3) + 35*a^6*b^4*d^6*(2*c^4 + 15*c^3*d*x + 45*c^2*d^2*x^2 + 65*c*d^
3*x^3 + 39*d^4*x^4) + 21*a^5*b^5*d^5*(6*c^5 + 50*c^4*d*x + 175*c^3*d^2*x^2 + 325*c^2*d^3*x^3 + 325*c*d^4*x^4 +
143*d^5*x^5) + 35*a^4*b^6*d^4*(6*c^6 + 54*c^5*d*x + 210*c^4*d^2*x^2 + 455*c^3*d^3*x^3 + 585*c^2*d^4*x^4 + 429
*c*d^5*x^5 + 143*d^6*x^6) + 5*a^3*b^7*d^3*(66*c^7 + 630*c^6*d*x + 2646*c^5*d^2*x^2 + 6370*c^4*d^3*x^3 + 9555*c
^3*d^4*x^4 + 9009*c^2*d^5*x^5 + 5005*c*d^6*x^6 + 1287*d^7*x^7) + 15*a^2*b^8*d^2*(33*c^8 + 330*c^7*d*x + 1470*c
^6*d^2*x^2 + 3822*c^5*d^3*x^3 + 6370*c^4*d^4*x^4 + 7007*c^3*d^5*x^5 + 5005*c^2*d^6*x^6 + 2145*c*d^7*x^7 + 429*
d^8*x^8) + 5*a*b^9*d*(143*c^9 + 1485*c^8*d*x + 6930*c^7*d^2*x^2 + 19110*c^6*d^3*x^3 + 34398*c^5*d^4*x^4 + 4204
2*c^4*d^5*x^5 + 35035*c^3*d^6*x^6 + 19305*c^2*d^7*x^7 + 6435*c*d^8*x^8 + 1001*d^9*x^9) + b^10*(1001*c^10 + 107
25*c^9*d*x + 51975*c^8*d^2*x^2 + 150150*c^7*d^3*x^3 + 286650*c^6*d^4*x^4 + 378378*c^5*d^5*x^5 + 350350*c^4*d^6
*x^6 + 225225*c^3*d^7*x^7 + 96525*c^2*d^8*x^8 + 25025*c*d^9*x^9 + 3003*d^10*x^10))/(b^11*(a + b*x)^15)

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Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[(c + d*x)^10/(a + b*x)^16,x]')

[Out]

Timed out

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(866\) vs. \(2(141)=282\).
time = 0.14, size = 867, normalized size = 5.74

method	result	size
risch	\(\frac {-\frac {a^{10} d^{10}+5 a^{9} b c \,d^{9}+15 a^{8} b^{2} c^{2} d^{8}+35 a^{7} b^{3} c^{3} d^{7}+70 a^{6} b^{4} c^{4} d^{6}+126 a^{5} b^{5} c^{5} d^{5}+210 a^{4} b^{6} c^{6} d^{4}+330 a^{3} b^{7} c^{7} d^{3}+495 a^{2} b^{8} c^{8} d^{2}+715 a \,b^{9} c^{9} d +1001 b^{10} c^{10}}{15015 b^{11}}-\frac {d \left (a^{9} d^{9}+5 a^{8} b c \,d^{8}+15 a^{7} b^{2} c^{2} d^{7}+35 a^{6} b^{3} c^{3} d^{6}+70 a^{5} b^{4} c^{4} d^{5}+126 a^{4} b^{5} c^{5} d^{4}+210 a^{3} b^{6} c^{6} d^{3}+330 a^{2} b^{7} c^{7} d^{2}+495 a \,b^{8} c^{8} d +715 b^{9} c^{9}\right ) x}{1001 b^{10}}-\frac {d^{2} \left (a^{8} d^{8}+5 a^{7} b c \,d^{7}+15 a^{6} b^{2} c^{2} d^{6}+35 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} c^{4} d^{4}+126 a^{3} b^{5} c^{5} d^{3}+210 a^{2} b^{6} c^{6} d^{2}+330 a \,b^{7} c^{7} d +495 b^{8} c^{8}\right ) x^{2}}{143 b^{9}}-\frac {d^{3} \left (a^{7} d^{7}+5 a^{6} b c \,d^{6}+15 a^{5} b^{2} c^{2} d^{5}+35 a^{4} b^{3} c^{3} d^{4}+70 a^{3} b^{4} c^{4} d^{3}+126 a^{2} b^{5} c^{5} d^{2}+210 a \,b^{6} c^{6} d +330 b^{7} c^{7}\right ) x^{3}}{33 b^{8}}-\frac {d^{4} \left (a^{6} d^{6}+5 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}+35 a^{3} b^{3} c^{3} d^{3}+70 a^{2} b^{4} c^{4} d^{2}+126 a \,b^{5} c^{5} d +210 b^{6} c^{6}\right ) x^{4}}{11 b^{7}}-\frac {d^{5} \left (a^{5} d^{5}+5 a^{4} b c \,d^{4}+15 a^{3} b^{2} c^{2} d^{3}+35 a^{2} b^{3} c^{3} d^{2}+70 a \,b^{4} c^{4} d +126 b^{5} c^{5}\right ) x^{5}}{5 b^{6}}-\frac {d^{6} \left (a^{4} d^{4}+5 a^{3} b c \,d^{3}+15 a^{2} b^{2} c^{2} d^{2}+35 a \,b^{3} c^{3} d +70 b^{4} c^{4}\right ) x^{6}}{3 b^{5}}-\frac {3 d^{7} \left (a^{3} d^{3}+5 a^{2} b c \,d^{2}+15 a \,b^{2} c^{2} d +35 b^{3} c^{3}\right ) x^{7}}{7 b^{4}}-\frac {3 d^{8} \left (a^{2} d^{2}+5 a b c d +15 b^{2} c^{2}\right ) x^{8}}{7 b^{3}}-\frac {d^{9} \left (a d +5 b c \right ) x^{9}}{3 b^{2}}-\frac {d^{10} x^{10}}{5 b}}{\left (b x +a \right )^{15}}\)	\(831\)
default	\(-\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 )}{13 b^{11} \left (b x +a \right )^{13}}+\frac {10 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 )^{12}}+\frac {5 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 )}{7 b^{11} \left (b x +a \right )^{14}}-\frac {70 d^{6} \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 )}{3 b^{11} \left (b x +a \right )^{9}}+\frac {15 d^{7} \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^{11} \left (b x +a \right )^{8}}+\frac {126 d^{5} \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 )}{5 b^{11} \left (b x +a \right )^{10}}-\frac {d^{10}}{5 b^{11} \left (b x +a \right )^{5}}-\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 )}{11 b^{11} \left (b x +a \right )^{11}}-\frac {45 d^{8} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{7 b^{11} \left (b x +a \right )^{7}}+\frac {5 d^{9} \left (a d -b c \right )}{3 b^{11} \left (b x +a \right )^{6}}-\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}}{15 b^{11} \left (b x +a \right )^{15}}\)	\(867\)
norman	\(\frac {\frac {-a^{10} b^{4} d^{10}-5 a^{9} b^{5} c \,d^{9}-15 a^{8} b^{6} c^{2} d^{8}-35 a^{7} b^{7} c^{3} d^{7}-70 a^{6} b^{8} c^{4} d^{6}-126 a^{5} b^{9} c^{5} d^{5}-210 a^{4} b^{10} c^{6} d^{4}-330 a^{3} b^{11} c^{7} d^{3}-495 a^{2} b^{12} c^{8} d^{2}-715 a \,b^{13} c^{9} d -1001 b^{14} c^{10}}{15015 b^{15}}+\frac {\left (-a^{9} b^{4} d^{10}-5 a^{8} b^{5} c \,d^{9}-15 a^{7} b^{6} c^{2} d^{8}-35 a^{6} b^{7} c^{3} d^{7}-70 a^{5} b^{8} c^{4} d^{6}-126 a^{4} b^{9} c^{5} d^{5}-210 b^{10} a^{3} c^{6} d^{4}-330 a^{2} b^{11} c^{7} d^{3}-495 a \,b^{12} c^{8} d^{2}-715 b^{13} c^{9} d \right ) x}{1001 b^{14}}+\frac {\left (-a^{8} b^{4} d^{10}-5 a^{7} b^{5} c \,d^{9}-15 a^{6} b^{6} c^{2} d^{8}-35 a^{5} b^{7} c^{3} d^{7}-70 a^{4} b^{8} c^{4} d^{6}-126 a^{3} b^{9} c^{5} d^{5}-210 a^{2} b^{10} c^{6} d^{4}-330 a \,b^{11} c^{7} d^{3}-495 b^{12} c^{8} d^{2}\right ) x^{2}}{143 b^{13}}+\frac {\left (-a^{7} b^{4} d^{10}-5 a^{6} b^{5} c \,d^{9}-15 a^{5} b^{6} c^{2} d^{8}-35 a^{4} b^{7} c^{3} d^{7}-70 a^{3} b^{8} c^{4} d^{6}-126 a^{2} b^{9} c^{5} d^{5}-210 a \,b^{10} c^{6} d^{4}-330 b^{11} c^{7} d^{3}\right ) x^{3}}{33 b^{12}}+\frac {\left (-a^{6} b^{4} d^{10}-5 a^{5} b^{5} c \,d^{9}-15 a^{4} b^{6} c^{2} d^{8}-35 a^{3} b^{7} c^{3} d^{7}-70 a^{2} b^{8} c^{4} d^{6}-126 a \,b^{9} c^{5} d^{5}-210 b^{10} c^{6} d^{4}\right ) x^{4}}{11 b^{11}}+\frac {\left (-a^{5} b^{4} d^{10}-5 a^{4} b^{5} c \,d^{9}-15 a^{3} b^{6} c^{2} d^{8}-35 a^{2} b^{7} c^{3} d^{7}-70 a \,b^{8} c^{4} d^{6}-126 b^{9} c^{5} d^{5}\right ) x^{5}}{5 b^{10}}+\frac {\left (-a^{4} b^{4} d^{10}-5 a^{3} b^{5} c \,d^{9}-15 b^{6} a^{2} c^{2} d^{8}-35 a \,b^{7} c^{3} d^{7}-70 b^{8} c^{4} d^{6}\right ) x^{6}}{3 b^{9}}+\frac {3 \left (-a^{3} b^{4} d^{10}-5 a^{2} b^{5} c \,d^{9}-15 a \,b^{6} c^{2} d^{8}-35 b^{7} c^{3} d^{7}\right ) x^{7}}{7 b^{8}}+\frac {3 \left (-a^{2} b^{4} d^{10}-5 a \,b^{5} c \,d^{9}-15 b^{6} c^{2} d^{8}\right ) x^{8}}{7 b^{7}}+\frac {\left (-a \,b^{4} d^{10}-5 b^{5} c \,d^{9}\right ) x^{9}}{3 b^{6}}-\frac {d^{10} x^{10}}{5 b}}{\left (b x +a \right )^{15}}\)	\(909\)
gosper	\(-\frac {3003 d^{10} x^{10} b^{10}+5005 a \,b^{9} d^{10} x^{9}+25025 b^{10} c \,d^{9} x^{9}+6435 a^{2} b^{8} d^{10} x^{8}+32175 a \,b^{9} c \,d^{9} x^{8}+96525 b^{10} c^{2} d^{8} x^{8}+6435 a^{3} b^{7} d^{10} x^{7}+32175 a^{2} b^{8} c \,d^{9} x^{7}+96525 a \,b^{9} c^{2} d^{8} x^{7}+225225 b^{10} c^{3} d^{7} x^{7}+5005 a^{4} b^{6} d^{10} x^{6}+25025 a^{3} b^{7} c \,d^{9} x^{6}+75075 a^{2} b^{8} c^{2} d^{8} x^{6}+175175 a \,b^{9} c^{3} d^{7} x^{6}+350350 b^{10} c^{4} d^{6} x^{6}+3003 a^{5} b^{5} d^{10} x^{5}+15015 a^{4} b^{6} c \,d^{9} x^{5}+45045 a^{3} b^{7} c^{2} d^{8} x^{5}+105105 a^{2} b^{8} c^{3} d^{7} x^{5}+210210 a \,b^{9} c^{4} d^{6} x^{5}+378378 b^{10} c^{5} d^{5} x^{5}+1365 a^{6} b^{4} d^{10} x^{4}+6825 a^{5} b^{5} c \,d^{9} x^{4}+20475 a^{4} b^{6} c^{2} d^{8} x^{4}+47775 a^{3} b^{7} c^{3} d^{7} x^{4}+95550 a^{2} b^{8} c^{4} d^{6} x^{4}+171990 a \,b^{9} c^{5} d^{5} x^{4}+286650 b^{10} c^{6} d^{4} x^{4}+455 a^{7} b^{3} d^{10} x^{3}+2275 a^{6} b^{4} c \,d^{9} x^{3}+6825 a^{5} b^{5} c^{2} d^{8} x^{3}+15925 a^{4} b^{6} c^{3} d^{7} x^{3}+31850 a^{3} b^{7} c^{4} d^{6} x^{3}+57330 a^{2} b^{8} c^{5} d^{5} x^{3}+95550 a \,b^{9} c^{6} d^{4} x^{3}+150150 b^{10} c^{7} d^{3} x^{3}+105 a^{8} b^{2} d^{10} x^{2}+525 a^{7} b^{3} c \,d^{9} x^{2}+1575 a^{6} b^{4} c^{2} d^{8} x^{2}+3675 a^{5} b^{5} c^{3} d^{7} x^{2}+7350 a^{4} b^{6} c^{4} d^{6} x^{2}+13230 a^{3} b^{7} c^{5} d^{5} x^{2}+22050 a^{2} b^{8} c^{6} d^{4} x^{2}+34650 a \,b^{9} c^{7} d^{3} x^{2}+51975 b^{10} c^{8} d^{2} x^{2}+15 a^{9} b \,d^{10} x +75 a^{8} b^{2} c \,d^{9} x +225 a^{7} b^{3} c^{2} d^{8} x +525 a^{6} b^{4} c^{3} d^{7} x +1050 a^{5} b^{5} c^{4} d^{6} x +1890 a^{4} b^{6} c^{5} d^{5} x +3150 a^{3} b^{7} c^{6} d^{4} x +4950 a^{2} b^{8} c^{7} d^{3} x +7425 a \,b^{9} c^{8} d^{2} x +10725 b^{10} c^{9} d x +a^{10} d^{10}+5 a^{9} b c \,d^{9}+15 a^{8} b^{2} c^{2} d^{8}+35 a^{7} b^{3} c^{3} d^{7}+70 a^{6} b^{4} c^{4} d^{6}+126 a^{5} b^{5} c^{5} d^{5}+210 a^{4} b^{6} c^{6} d^{4}+330 a^{3} b^{7} c^{7} d^{3}+495 a^{2} b^{8} c^{8} d^{2}+715 a \,b^{9} c^{9} d +1001 b^{10} c^{10}}{15015 b^{11} \left (b x +a \right )^{15}}\)	\(962\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^10/(b*x+a)^16,x,method=_RETURNVERBOSE)

[Out]

-45/13/b^11*d^2*(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)/(b*x+a)^13+10/b^11*d^3*(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)/(b*x+a)^12+5/7/b^11*d*(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)/(b*x+a)^14-70/3/b^11*d^6*(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)/(b*x+a)^9+15/b^11*d^7*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/(b*x+a)^8+126/5/b
^11*d^5*(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)/(b*x+a)^10-1/5*d^1
0/b^11/(b*x+a)^5-210/11/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)/(b*x+a)^11-45/7/b^11*d^8*(a^2*d^2-2*a*b*c*d+b^2*c^2)/(b*x+a)^7+5/3/b^11*d^9*(a*d-b*c
)/(b*x+a)^6-1/15*(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)/b^11/(b*x+a)^
15

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1019 vs. \(2 (141) = 282\).
time = 0.33, size = 1019, normalized size = 6.75 \begin {gather*} -\frac {3003 \, b^{10} d^{10} x^{10} + 1001 \, b^{10} c^{10} + 715 \, a b^{9} c^{9} d + 495 \, a^{2} b^{8} c^{8} d^{2} + 330 \, a^{3} b^{7} c^{7} d^{3} + 210 \, a^{4} b^{6} c^{6} d^{4} + 126 \, a^{5} b^{5} c^{5} d^{5} + 70 \, a^{6} b^{4} c^{4} d^{6} + 35 \, a^{7} b^{3} c^{3} d^{7} + 15 \, a^{8} b^{2} c^{2} d^{8} + 5 \, a^{9} b c d^{9} + a^{10} d^{10} + 5005 \, {\left (5 \, b^{10} c d^{9} + a b^{9} d^{10}\right )} x^{9} + 6435 \, {\left (15 \, b^{10} c^{2} d^{8} + 5 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 6435 \, {\left (35 \, b^{10} c^{3} d^{7} + 15 \, a b^{9} c^{2} d^{8} + 5 \, a^{2} b^{8} c d^{9} + a^{3} b^{7} d^{10}\right )} x^{7} + 5005 \, {\left (70 \, b^{10} c^{4} d^{6} + 35 \, a b^{9} c^{3} d^{7} + 15 \, a^{2} b^{8} c^{2} d^{8} + 5 \, a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 3003 \, {\left (126 \, b^{10} c^{5} d^{5} + 70 \, a b^{9} c^{4} d^{6} + 35 \, a^{2} b^{8} c^{3} d^{7} + 15 \, a^{3} b^{7} c^{2} d^{8} + 5 \, a^{4} b^{6} c d^{9} + a^{5} b^{5} d^{10}\right )} x^{5} + 1365 \, {\left (210 \, b^{10} c^{6} d^{4} + 126 \, a b^{9} c^{5} d^{5} + 70 \, a^{2} b^{8} c^{4} d^{6} + 35 \, a^{3} b^{7} c^{3} d^{7} + 15 \, a^{4} b^{6} c^{2} d^{8} + 5 \, a^{5} b^{5} c d^{9} + a^{6} b^{4} d^{10}\right )} x^{4} + 455 \, {\left (330 \, b^{10} c^{7} d^{3} + 210 \, a b^{9} c^{6} d^{4} + 126 \, a^{2} b^{8} c^{5} d^{5} + 70 \, a^{3} b^{7} c^{4} d^{6} + 35 \, a^{4} b^{6} c^{3} d^{7} + 15 \, a^{5} b^{5} c^{2} d^{8} + 5 \, a^{6} b^{4} c d^{9} + a^{7} b^{3} d^{10}\right )} x^{3} + 105 \, {\left (495 \, b^{10} c^{8} d^{2} + 330 \, a b^{9} c^{7} d^{3} + 210 \, a^{2} b^{8} c^{6} d^{4} + 126 \, a^{3} b^{7} c^{5} d^{5} + 70 \, a^{4} b^{6} c^{4} d^{6} + 35 \, a^{5} b^{5} c^{3} d^{7} + 15 \, a^{6} b^{4} c^{2} d^{8} + 5 \, a^{7} b^{3} c d^{9} + a^{8} b^{2} d^{10}\right )} x^{2} + 15 \, {\left (715 \, b^{10} c^{9} d + 495 \, a b^{9} c^{8} d^{2} + 330 \, a^{2} b^{8} c^{7} d^{3} + 210 \, a^{3} b^{7} c^{6} d^{4} + 126 \, a^{4} b^{6} c^{5} d^{5} + 70 \, a^{5} b^{5} c^{4} d^{6} + 35 \, a^{6} b^{4} c^{3} d^{7} + 15 \, a^{7} b^{3} c^{2} d^{8} + 5 \, a^{8} b^{2} c d^{9} + a^{9} b d^{10}\right )} x}{15015 \, {\left (b^{26} x^{15} + 15 \, a b^{25} x^{14} + 105 \, a^{2} b^{24} x^{13} + 455 \, a^{3} b^{23} x^{12} + 1365 \, a^{4} b^{22} x^{11} + 3003 \, a^{5} b^{21} x^{10} + 5005 \, a^{6} b^{20} x^{9} + 6435 \, a^{7} b^{19} x^{8} + 6435 \, a^{8} b^{18} x^{7} + 5005 \, a^{9} b^{17} x^{6} + 3003 \, a^{10} b^{16} x^{5} + 1365 \, a^{11} b^{15} x^{4} + 455 \, a^{12} b^{14} x^{3} + 105 \, a^{13} b^{13} x^{2} + 15 \, a^{14} b^{12} x + a^{15} b^{11}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^16,x, algorithm="maxima")

[Out]

-1/15015*(3003*b^10*d^10*x^10 + 1001*b^10*c^10 + 715*a*b^9*c^9*d + 495*a^2*b^8*c^8*d^2 + 330*a^3*b^7*c^7*d^3 +
210*a^4*b^6*c^6*d^4 + 126*a^5*b^5*c^5*d^5 + 70*a^6*b^4*c^4*d^6 + 35*a^7*b^3*c^3*d^7 + 15*a^8*b^2*c^2*d^8 + 5*
a^9*b*c*d^9 + a^10*d^10 + 5005*(5*b^10*c*d^9 + a*b^9*d^10)*x^9 + 6435*(15*b^10*c^2*d^8 + 5*a*b^9*c*d^9 + a^2*b
^8*d^10)*x^8 + 6435*(35*b^10*c^3*d^7 + 15*a*b^9*c^2*d^8 + 5*a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7 + 5005*(70*b^10*
c^4*d^6 + 35*a*b^9*c^3*d^7 + 15*a^2*b^8*c^2*d^8 + 5*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 3003*(126*b^10*c^5*d^5
+ 70*a*b^9*c^4*d^6 + 35*a^2*b^8*c^3*d^7 + 15*a^3*b^7*c^2*d^8 + 5*a^4*b^6*c*d^9 + a^5*b^5*d^10)*x^5 + 1365*(21
0*b^10*c^6*d^4 + 126*a*b^9*c^5*d^5 + 70*a^2*b^8*c^4*d^6 + 35*a^3*b^7*c^3*d^7 + 15*a^4*b^6*c^2*d^8 + 5*a^5*b^5*
c*d^9 + a^6*b^4*d^10)*x^4 + 455*(330*b^10*c^7*d^3 + 210*a*b^9*c^6*d^4 + 126*a^2*b^8*c^5*d^5 + 70*a^3*b^7*c^4*d
^6 + 35*a^4*b^6*c^3*d^7 + 15*a^5*b^5*c^2*d^8 + 5*a^6*b^4*c*d^9 + a^7*b^3*d^10)*x^3 + 105*(495*b^10*c^8*d^2 + 3
30*a*b^9*c^7*d^3 + 210*a^2*b^8*c^6*d^4 + 126*a^3*b^7*c^5*d^5 + 70*a^4*b^6*c^4*d^6 + 35*a^5*b^5*c^3*d^7 + 15*a^
6*b^4*c^2*d^8 + 5*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 15*(715*b^10*c^9*d + 495*a*b^9*c^8*d^2 + 330*a^2*b^8*c^7
*d^3 + 210*a^3*b^7*c^6*d^4 + 126*a^4*b^6*c^5*d^5 + 70*a^5*b^5*c^4*d^6 + 35*a^6*b^4*c^3*d^7 + 15*a^7*b^3*c^2*d^
8 + 5*a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^26*x^15 + 15*a*b^25*x^14 + 105*a^2*b^24*x^13 + 455*a^3*b^23*x^12 + 136
5*a^4*b^22*x^11 + 3003*a^5*b^21*x^10 + 5005*a^6*b^20*x^9 + 6435*a^7*b^19*x^8 + 6435*a^8*b^18*x^7 + 5005*a^9*b^
17*x^6 + 3003*a^10*b^16*x^5 + 1365*a^11*b^15*x^4 + 455*a^12*b^14*x^3 + 105*a^13*b^13*x^2 + 15*a^14*b^12*x + a^
15*b^11)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1019 vs. \(2 (141) = 282\).
time = 0.30, size = 1019, normalized size = 6.75 \begin {gather*} -\frac {3003 \, b^{10} d^{10} x^{10} + 1001 \, b^{10} c^{10} + 715 \, a b^{9} c^{9} d + 495 \, a^{2} b^{8} c^{8} d^{2} + 330 \, a^{3} b^{7} c^{7} d^{3} + 210 \, a^{4} b^{6} c^{6} d^{4} + 126 \, a^{5} b^{5} c^{5} d^{5} + 70 \, a^{6} b^{4} c^{4} d^{6} + 35 \, a^{7} b^{3} c^{3} d^{7} + 15 \, a^{8} b^{2} c^{2} d^{8} + 5 \, a^{9} b c d^{9} + a^{10} d^{10} + 5005 \, {\left (5 \, b^{10} c d^{9} + a b^{9} d^{10}\right )} x^{9} + 6435 \, {\left (15 \, b^{10} c^{2} d^{8} + 5 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 6435 \, {\left (35 \, b^{10} c^{3} d^{7} + 15 \, a b^{9} c^{2} d^{8} + 5 \, a^{2} b^{8} c d^{9} + a^{3} b^{7} d^{10}\right )} x^{7} + 5005 \, {\left (70 \, b^{10} c^{4} d^{6} + 35 \, a b^{9} c^{3} d^{7} + 15 \, a^{2} b^{8} c^{2} d^{8} + 5 \, a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 3003 \, {\left (126 \, b^{10} c^{5} d^{5} + 70 \, a b^{9} c^{4} d^{6} + 35 \, a^{2} b^{8} c^{3} d^{7} + 15 \, a^{3} b^{7} c^{2} d^{8} + 5 \, a^{4} b^{6} c d^{9} + a^{5} b^{5} d^{10}\right )} x^{5} + 1365 \, {\left (210 \, b^{10} c^{6} d^{4} + 126 \, a b^{9} c^{5} d^{5} + 70 \, a^{2} b^{8} c^{4} d^{6} + 35 \, a^{3} b^{7} c^{3} d^{7} + 15 \, a^{4} b^{6} c^{2} d^{8} + 5 \, a^{5} b^{5} c d^{9} + a^{6} b^{4} d^{10}\right )} x^{4} + 455 \, {\left (330 \, b^{10} c^{7} d^{3} + 210 \, a b^{9} c^{6} d^{4} + 126 \, a^{2} b^{8} c^{5} d^{5} + 70 \, a^{3} b^{7} c^{4} d^{6} + 35 \, a^{4} b^{6} c^{3} d^{7} + 15 \, a^{5} b^{5} c^{2} d^{8} + 5 \, a^{6} b^{4} c d^{9} + a^{7} b^{3} d^{10}\right )} x^{3} + 105 \, {\left (495 \, b^{10} c^{8} d^{2} + 330 \, a b^{9} c^{7} d^{3} + 210 \, a^{2} b^{8} c^{6} d^{4} + 126 \, a^{3} b^{7} c^{5} d^{5} + 70 \, a^{4} b^{6} c^{4} d^{6} + 35 \, a^{5} b^{5} c^{3} d^{7} + 15 \, a^{6} b^{4} c^{2} d^{8} + 5 \, a^{7} b^{3} c d^{9} + a^{8} b^{2} d^{10}\right )} x^{2} + 15 \, {\left (715 \, b^{10} c^{9} d + 495 \, a b^{9} c^{8} d^{2} + 330 \, a^{2} b^{8} c^{7} d^{3} + 210 \, a^{3} b^{7} c^{6} d^{4} + 126 \, a^{4} b^{6} c^{5} d^{5} + 70 \, a^{5} b^{5} c^{4} d^{6} + 35 \, a^{6} b^{4} c^{3} d^{7} + 15 \, a^{7} b^{3} c^{2} d^{8} + 5 \, a^{8} b^{2} c d^{9} + a^{9} b d^{10}\right )} x}{15015 \, {\left (b^{26} x^{15} + 15 \, a b^{25} x^{14} + 105 \, a^{2} b^{24} x^{13} + 455 \, a^{3} b^{23} x^{12} + 1365 \, a^{4} b^{22} x^{11} + 3003 \, a^{5} b^{21} x^{10} + 5005 \, a^{6} b^{20} x^{9} + 6435 \, a^{7} b^{19} x^{8} + 6435 \, a^{8} b^{18} x^{7} + 5005 \, a^{9} b^{17} x^{6} + 3003 \, a^{10} b^{16} x^{5} + 1365 \, a^{11} b^{15} x^{4} + 455 \, a^{12} b^{14} x^{3} + 105 \, a^{13} b^{13} x^{2} + 15 \, a^{14} b^{12} x + a^{15} b^{11}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^16,x, algorithm="fricas")

[Out]

-1/15015*(3003*b^10*d^10*x^10 + 1001*b^10*c^10 + 715*a*b^9*c^9*d + 495*a^2*b^8*c^8*d^2 + 330*a^3*b^7*c^7*d^3 +
210*a^4*b^6*c^6*d^4 + 126*a^5*b^5*c^5*d^5 + 70*a^6*b^4*c^4*d^6 + 35*a^7*b^3*c^3*d^7 + 15*a^8*b^2*c^2*d^8 + 5*
a^9*b*c*d^9 + a^10*d^10 + 5005*(5*b^10*c*d^9 + a*b^9*d^10)*x^9 + 6435*(15*b^10*c^2*d^8 + 5*a*b^9*c*d^9 + a^2*b
^8*d^10)*x^8 + 6435*(35*b^10*c^3*d^7 + 15*a*b^9*c^2*d^8 + 5*a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7 + 5005*(70*b^10*
c^4*d^6 + 35*a*b^9*c^3*d^7 + 15*a^2*b^8*c^2*d^8 + 5*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 3003*(126*b^10*c^5*d^5
+ 70*a*b^9*c^4*d^6 + 35*a^2*b^8*c^3*d^7 + 15*a^3*b^7*c^2*d^8 + 5*a^4*b^6*c*d^9 + a^5*b^5*d^10)*x^5 + 1365*(21
0*b^10*c^6*d^4 + 126*a*b^9*c^5*d^5 + 70*a^2*b^8*c^4*d^6 + 35*a^3*b^7*c^3*d^7 + 15*a^4*b^6*c^2*d^8 + 5*a^5*b^5*
c*d^9 + a^6*b^4*d^10)*x^4 + 455*(330*b^10*c^7*d^3 + 210*a*b^9*c^6*d^4 + 126*a^2*b^8*c^5*d^5 + 70*a^3*b^7*c^4*d
^6 + 35*a^4*b^6*c^3*d^7 + 15*a^5*b^5*c^2*d^8 + 5*a^6*b^4*c*d^9 + a^7*b^3*d^10)*x^3 + 105*(495*b^10*c^8*d^2 + 3
30*a*b^9*c^7*d^3 + 210*a^2*b^8*c^6*d^4 + 126*a^3*b^7*c^5*d^5 + 70*a^4*b^6*c^4*d^6 + 35*a^5*b^5*c^3*d^7 + 15*a^
6*b^4*c^2*d^8 + 5*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 15*(715*b^10*c^9*d + 495*a*b^9*c^8*d^2 + 330*a^2*b^8*c^7
*d^3 + 210*a^3*b^7*c^6*d^4 + 126*a^4*b^6*c^5*d^5 + 70*a^5*b^5*c^4*d^6 + 35*a^6*b^4*c^3*d^7 + 15*a^7*b^3*c^2*d^
8 + 5*a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^26*x^15 + 15*a*b^25*x^14 + 105*a^2*b^24*x^13 + 455*a^3*b^23*x^12 + 136
5*a^4*b^22*x^11 + 3003*a^5*b^21*x^10 + 5005*a^6*b^20*x^9 + 6435*a^7*b^19*x^8 + 6435*a^8*b^18*x^7 + 5005*a^9*b^
17*x^6 + 3003*a^10*b^16*x^5 + 1365*a^11*b^15*x^4 + 455*a^12*b^14*x^3 + 105*a^13*b^13*x^2 + 15*a^14*b^12*x + a^
15*b^11)

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Sympy [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**10/(b*x+a)**16,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 961 vs. \(2 (141) = 282\).
time = 0.00, size = 1029, normalized size = 6.81 \begin {gather*} \frac {-3003 x^{10} d^{10} b^{10}-5005 x^{9} d^{10} b^{9} a-25025 x^{9} d^{9} c b^{10}-6435 x^{8} d^{10} b^{8} a^{2}-32175 x^{8} d^{9} c b^{9} a-96525 x^{8} d^{8} c^{2} b^{10}-6435 x^{7} d^{10} b^{7} a^{3}-32175 x^{7} d^{9} c b^{8} a^{2}-96525 x^{7} d^{8} c^{2} b^{9} a-225225 x^{7} d^{7} c^{3} b^{10}-5005 x^{6} d^{10} b^{6} a^{4}-25025 x^{6} d^{9} c b^{7} a^{3}-75075 x^{6} d^{8} c^{2} b^{8} a^{2}-175175 x^{6} d^{7} c^{3} b^{9} a-350350 x^{6} d^{6} c^{4} b^{10}-3003 x^{5} d^{10} b^{5} a^{5}-15015 x^{5} d^{9} c b^{6} a^{4}-45045 x^{5} d^{8} c^{2} b^{7} a^{3}-105105 x^{5} d^{7} c^{3} b^{8} a^{2}-210210 x^{5} d^{6} c^{4} b^{9} a-378378 x^{5} d^{5} c^{5} b^{10}-1365 x^{4} d^{10} b^{4} a^{6}-6825 x^{4} d^{9} c b^{5} a^{5}-20475 x^{4} d^{8} c^{2} b^{6} a^{4}-47775 x^{4} d^{7} c^{3} b^{7} a^{3}-95550 x^{4} d^{6} c^{4} b^{8} a^{2}-171990 x^{4} d^{5} c^{5} b^{9} a-286650 x^{4} d^{4} c^{6} b^{10}-455 x^{3} d^{10} b^{3} a^{7}-2275 x^{3} d^{9} c b^{4} a^{6}-6825 x^{3} d^{8} c^{2} b^{5} a^{5}-15925 x^{3} d^{7} c^{3} b^{6} a^{4}-31850 x^{3} d^{6} c^{4} b^{7} a^{3}-57330 x^{3} d^{5} c^{5} b^{8} a^{2}-95550 x^{3} d^{4} c^{6} b^{9} a-150150 x^{3} d^{3} c^{7} b^{10}-105 x^{2} d^{10} b^{2} a^{8}-525 x^{2} d^{9} c b^{3} a^{7}-1575 x^{2} d^{8} c^{2} b^{4} a^{6}-3675 x^{2} d^{7} c^{3} b^{5} a^{5}-7350 x^{2} d^{6} c^{4} b^{6} a^{4}-13230 x^{2} d^{5} c^{5} b^{7} a^{3}-22050 x^{2} d^{4} c^{6} b^{8} a^{2}-34650 x^{2} d^{3} c^{7} b^{9} a-51975 x^{2} d^{2} c^{8} b^{10}-15 x d^{10} b a^{9}-75 x d^{9} c b^{2} a^{8}-225 x d^{8} c^{2} b^{3} a^{7}-525 x d^{7} c^{3} b^{4} a^{6}-1050 x d^{6} c^{4} b^{5} a^{5}-1890 x d^{5} c^{5} b^{6} a^{4}-3150 x d^{4} c^{6} b^{7} a^{3}-4950 x d^{3} c^{7} b^{8} a^{2}-7425 x d^{2} c^{8} b^{9} a-10725 x d c^{9} b^{10}-d^{10} a^{10}-5 d^{9} c b a^{9}-15 d^{8} c^{2} b^{2} a^{8}-35 d^{7} c^{3} b^{3} a^{7}-70 d^{6} c^{4} b^{4} a^{6}-126 d^{5} c^{5} b^{5} a^{5}-210 d^{4} c^{6} b^{6} a^{4}-330 d^{3} c^{7} b^{7} a^{3}-495 d^{2} c^{8} b^{8} a^{2}-715 d c^{9} b^{9} a-1001 c^{10} b^{10}}{15015 b^{11} \left (x b+a\right )^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^16,x)

[Out]

-1/15015*(3003*b^10*d^10*x^10 + 25025*b^10*c*d^9*x^9 + 5005*a*b^9*d^10*x^9 + 96525*b^10*c^2*d^8*x^8 + 32175*a*
b^9*c*d^9*x^8 + 6435*a^2*b^8*d^10*x^8 + 225225*b^10*c^3*d^7*x^7 + 96525*a*b^9*c^2*d^8*x^7 + 32175*a^2*b^8*c*d^
9*x^7 + 6435*a^3*b^7*d^10*x^7 + 350350*b^10*c^4*d^6*x^6 + 175175*a*b^9*c^3*d^7*x^6 + 75075*a^2*b^8*c^2*d^8*x^6
+ 25025*a^3*b^7*c*d^9*x^6 + 5005*a^4*b^6*d^10*x^6 + 378378*b^10*c^5*d^5*x^5 + 210210*a*b^9*c^4*d^6*x^5 + 1051
05*a^2*b^8*c^3*d^7*x^5 + 45045*a^3*b^7*c^2*d^8*x^5 + 15015*a^4*b^6*c*d^9*x^5 + 3003*a^5*b^5*d^10*x^5 + 286650*
b^10*c^6*d^4*x^4 + 171990*a*b^9*c^5*d^5*x^4 + 95550*a^2*b^8*c^4*d^6*x^4 + 47775*a^3*b^7*c^3*d^7*x^4 + 20475*a^
4*b^6*c^2*d^8*x^4 + 6825*a^5*b^5*c*d^9*x^4 + 1365*a^6*b^4*d^10*x^4 + 150150*b^10*c^7*d^3*x^3 + 95550*a*b^9*c^6
*d^4*x^3 + 57330*a^2*b^8*c^5*d^5*x^3 + 31850*a^3*b^7*c^4*d^6*x^3 + 15925*a^4*b^6*c^3*d^7*x^3 + 6825*a^5*b^5*c^
2*d^8*x^3 + 2275*a^6*b^4*c*d^9*x^3 + 455*a^7*b^3*d^10*x^3 + 51975*b^10*c^8*d^2*x^2 + 34650*a*b^9*c^7*d^3*x^2 +
22050*a^2*b^8*c^6*d^4*x^2 + 13230*a^3*b^7*c^5*d^5*x^2 + 7350*a^4*b^6*c^4*d^6*x^2 + 3675*a^5*b^5*c^3*d^7*x^2 +
1575*a^6*b^4*c^2*d^8*x^2 + 525*a^7*b^3*c*d^9*x^2 + 105*a^8*b^2*d^10*x^2 + 10725*b^10*c^9*d*x + 7425*a*b^9*c^8
*d^2*x + 4950*a^2*b^8*c^7*d^3*x + 3150*a^3*b^7*c^6*d^4*x + 1890*a^4*b^6*c^5*d^5*x + 1050*a^5*b^5*c^4*d^6*x + 5
25*a^6*b^4*c^3*d^7*x + 225*a^7*b^3*c^2*d^8*x + 75*a^8*b^2*c*d^9*x + 15*a^9*b*d^10*x + 1001*b^10*c^10 + 715*a*b
^9*c^9*d + 495*a^2*b^8*c^8*d^2 + 330*a^3*b^7*c^7*d^3 + 210*a^4*b^6*c^6*d^4 + 126*a^5*b^5*c^5*d^5 + 70*a^6*b^4*
c^4*d^6 + 35*a^7*b^3*c^3*d^7 + 15*a^8*b^2*c^2*d^8 + 5*a^9*b*c*d^9 + a^10*d^10)/((b*x + a)^15*b^11)

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Mupad [B]
time = 2.28, size = 1120, normalized size = 7.42 \begin {gather*} -\frac {a^{10}\,d^{10}+5\,a^9\,b\,c\,d^9+15\,a^9\,b\,d^{10}\,x+15\,a^8\,b^2\,c^2\,d^8+75\,a^8\,b^2\,c\,d^9\,x+105\,a^8\,b^2\,d^{10}\,x^2+35\,a^7\,b^3\,c^3\,d^7+225\,a^7\,b^3\,c^2\,d^8\,x+525\,a^7\,b^3\,c\,d^9\,x^2+455\,a^7\,b^3\,d^{10}\,x^3+70\,a^6\,b^4\,c^4\,d^6+525\,a^6\,b^4\,c^3\,d^7\,x+1575\,a^6\,b^4\,c^2\,d^8\,x^2+2275\,a^6\,b^4\,c\,d^9\,x^3+1365\,a^6\,b^4\,d^{10}\,x^4+126\,a^5\,b^5\,c^5\,d^5+1050\,a^5\,b^5\,c^4\,d^6\,x+3675\,a^5\,b^5\,c^3\,d^7\,x^2+6825\,a^5\,b^5\,c^2\,d^8\,x^3+6825\,a^5\,b^5\,c\,d^9\,x^4+3003\,a^5\,b^5\,d^{10}\,x^5+210\,a^4\,b^6\,c^6\,d^4+1890\,a^4\,b^6\,c^5\,d^5\,x+7350\,a^4\,b^6\,c^4\,d^6\,x^2+15925\,a^4\,b^6\,c^3\,d^7\,x^3+20475\,a^4\,b^6\,c^2\,d^8\,x^4+15015\,a^4\,b^6\,c\,d^9\,x^5+5005\,a^4\,b^6\,d^{10}\,x^6+330\,a^3\,b^7\,c^7\,d^3+3150\,a^3\,b^7\,c^6\,d^4\,x+13230\,a^3\,b^7\,c^5\,d^5\,x^2+31850\,a^3\,b^7\,c^4\,d^6\,x^3+47775\,a^3\,b^7\,c^3\,d^7\,x^4+45045\,a^3\,b^7\,c^2\,d^8\,x^5+25025\,a^3\,b^7\,c\,d^9\,x^6+6435\,a^3\,b^7\,d^{10}\,x^7+495\,a^2\,b^8\,c^8\,d^2+4950\,a^2\,b^8\,c^7\,d^3\,x+22050\,a^2\,b^8\,c^6\,d^4\,x^2+57330\,a^2\,b^8\,c^5\,d^5\,x^3+95550\,a^2\,b^8\,c^4\,d^6\,x^4+105105\,a^2\,b^8\,c^3\,d^7\,x^5+75075\,a^2\,b^8\,c^2\,d^8\,x^6+32175\,a^2\,b^8\,c\,d^9\,x^7+6435\,a^2\,b^8\,d^{10}\,x^8+715\,a\,b^9\,c^9\,d+7425\,a\,b^9\,c^8\,d^2\,x+34650\,a\,b^9\,c^7\,d^3\,x^2+95550\,a\,b^9\,c^6\,d^4\,x^3+171990\,a\,b^9\,c^5\,d^5\,x^4+210210\,a\,b^9\,c^4\,d^6\,x^5+175175\,a\,b^9\,c^3\,d^7\,x^6+96525\,a\,b^9\,c^2\,d^8\,x^7+32175\,a\,b^9\,c\,d^9\,x^8+5005\,a\,b^9\,d^{10}\,x^9+1001\,b^{10}\,c^{10}+10725\,b^{10}\,c^9\,d\,x+51975\,b^{10}\,c^8\,d^2\,x^2+150150\,b^{10}\,c^7\,d^3\,x^3+286650\,b^{10}\,c^6\,d^4\,x^4+378378\,b^{10}\,c^5\,d^5\,x^5+350350\,b^{10}\,c^4\,d^6\,x^6+225225\,b^{10}\,c^3\,d^7\,x^7+96525\,b^{10}\,c^2\,d^8\,x^8+25025\,b^{10}\,c\,d^9\,x^9+3003\,b^{10}\,d^{10}\,x^{10}}{15015\,a^{15}\,b^{11}+225225\,a^{14}\,b^{12}\,x+1576575\,a^{13}\,b^{13}\,x^2+6831825\,a^{12}\,b^{14}\,x^3+20495475\,a^{11}\,b^{15}\,x^4+45090045\,a^{10}\,b^{16}\,x^5+75150075\,a^9\,b^{17}\,x^6+96621525\,a^8\,b^{18}\,x^7+96621525\,a^7\,b^{19}\,x^8+75150075\,a^6\,b^{20}\,x^9+45090045\,a^5\,b^{21}\,x^{10}[phone]\,a^4\,b^{22}\,x^{11}+6831825\,a^3\,b^{23}\,x^{12}+1576575\,a^2\,b^{24}\,x^{13}+225225\,a\,b^{25}\,x^{14}+15015\,b^{26}\,x^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^10/(a + b*x)^16,x)

[Out]

-(a^10*d^10 + 1001*b^10*c^10 + 3003*b^10*d^10*x^10 + 5005*a*b^9*d^10*x^9 + 25025*b^10*c*d^9*x^9 + 495*a^2*b^8*
c^8*d^2 + 330*a^3*b^7*c^7*d^3 + 210*a^4*b^6*c^6*d^4 + 126*a^5*b^5*c^5*d^5 + 70*a^6*b^4*c^4*d^6 + 35*a^7*b^3*c^
3*d^7 + 15*a^8*b^2*c^2*d^8 + 105*a^8*b^2*d^10*x^2 + 455*a^7*b^3*d^10*x^3 + 1365*a^6*b^4*d^10*x^4 + 3003*a^5*b^
5*d^10*x^5 + 5005*a^4*b^6*d^10*x^6 + 6435*a^3*b^7*d^10*x^7 + 6435*a^2*b^8*d^10*x^8 + 51975*b^10*c^8*d^2*x^2 +
150150*b^10*c^7*d^3*x^3 + 286650*b^10*c^6*d^4*x^4 + 378378*b^10*c^5*d^5*x^5 + 350350*b^10*c^4*d^6*x^6 + 225225
*b^10*c^3*d^7*x^7 + 96525*b^10*c^2*d^8*x^8 + 715*a*b^9*c^9*d + 5*a^9*b*c*d^9 + 15*a^9*b*d^10*x + 10725*b^10*c^
9*d*x + 22050*a^2*b^8*c^6*d^4*x^2 + 13230*a^3*b^7*c^5*d^5*x^2 + 7350*a^4*b^6*c^4*d^6*x^2 + 3675*a^5*b^5*c^3*d^
7*x^2 + 1575*a^6*b^4*c^2*d^8*x^2 + 57330*a^2*b^8*c^5*d^5*x^3 + 31850*a^3*b^7*c^4*d^6*x^3 + 15925*a^4*b^6*c^3*d
^7*x^3 + 6825*a^5*b^5*c^2*d^8*x^3 + 95550*a^2*b^8*c^4*d^6*x^4 + 47775*a^3*b^7*c^3*d^7*x^4 + 20475*a^4*b^6*c^2*
d^8*x^4 + 105105*a^2*b^8*c^3*d^7*x^5 + 45045*a^3*b^7*c^2*d^8*x^5 + 75075*a^2*b^8*c^2*d^8*x^6 + 7425*a*b^9*c^8*
d^2*x + 75*a^8*b^2*c*d^9*x + 32175*a*b^9*c*d^9*x^8 + 4950*a^2*b^8*c^7*d^3*x + 3150*a^3*b^7*c^6*d^4*x + 1890*a^
4*b^6*c^5*d^5*x + 1050*a^5*b^5*c^4*d^6*x + 525*a^6*b^4*c^3*d^7*x + 225*a^7*b^3*c^2*d^8*x + 34650*a*b^9*c^7*d^3
*x^2 + 525*a^7*b^3*c*d^9*x^2 + 95550*a*b^9*c^6*d^4*x^3 + 2275*a^6*b^4*c*d^9*x^3 + 171990*a*b^9*c^5*d^5*x^4 + 6
825*a^5*b^5*c*d^9*x^4 + 210210*a*b^9*c^4*d^6*x^5 + 15015*a^4*b^6*c*d^9*x^5 + 175175*a*b^9*c^3*d^7*x^6 + 25025*
a^3*b^7*c*d^9*x^6 + 96525*a*b^9*c^2*d^8*x^7 + 32175*a^2*b^8*c*d^9*x^7)/(15015*a^15*b^11 + 15015*b^26*x^15 + 22
5225*a^14*b^12*x + 225225*a*b^25*x^14 + 1576575*a^13*b^13*x^2 + 6831825*a^12*b^14*x^3 + 20495475*a^11*b^15*x^4
+ 45090045*a^10*b^16*x^5 + 75150075*a^9*b^17*x^6 + 96621525*a^8*b^18*x^7 + 96621525*a^7*b^19*x^8 + 75150075*a
^6*b^20*x^9 + 45090045*a^5*b^21*x^10 + 20495475*a^4*b^22*x^11 + 6831825*a^3*b^23*x^12 + 1576575*a^2*b^24*x^13)

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