3.14.19 \(\int \frac {(c+d x)^{10}}{(a+b x)^8} \, dx\) [1319]

Optimal. Leaf size=258 \[ \frac {45 d^8 (b c-a d)^2 x}{b^{10}}-\frac {(b c-a d)^{10}}{7 b^{11} (a+b x)^7}-\frac {5 d (b c-a d)^9}{3 b^{11} (a+b x)^6}-\frac {9 d^2 (b c-a d)^8}{b^{11} (a+b x)^5}-\frac {30 d^3 (b c-a d)^7}{b^{11} (a+b x)^4}-\frac {70 d^4 (b c-a d)^6}{b^{11} (a+b x)^3}-\frac {126 d^5 (b c-a d)^5}{b^{11} (a+b x)^2}-\frac {210 d^6 (b c-a d)^4}{b^{11} (a+b x)}+\frac {5 d^9 (b c-a d) (a+b x)^2}{b^{11}}+\frac {d^{10} (a+b x)^3}{3 b^{11}}+\frac {120 d^7 (b c-a d)^3 \log (a+b x)}{b^{11}} \]

[Out]

45*d^8*(-a*d+b*c)^2*x/b^10-1/7*(-a*d+b*c)^10/b^11/(b*x+a)^7-5/3*d*(-a*d+b*c)^9/b^11/(b*x+a)^6-9*d^2*(-a*d+b*c)
^8/b^11/(b*x+a)^5-30*d^3*(-a*d+b*c)^7/b^11/(b*x+a)^4-70*d^4*(-a*d+b*c)^6/b^11/(b*x+a)^3-126*d^5*(-a*d+b*c)^5/b
^11/(b*x+a)^2-210*d^6*(-a*d+b*c)^4/b^11/(b*x+a)+5*d^9*(-a*d+b*c)*(b*x+a)^2/b^11+1/3*d^10*(b*x+a)^3/b^11+120*d^
7*(-a*d+b*c)^3*ln(b*x+a)/b^11

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Rubi [A]
time = 0.25, antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45} \begin {gather*} \frac {5 d^9 (a+b x)^2 (b c-a d)}{b^{11}}+\frac {120 d^7 (b c-a d)^3 \log (a+b x)}{b^{11}}-\frac {210 d^6 (b c-a d)^4}{b^{11} (a+b x)}-\frac {126 d^5 (b c-a d)^5}{b^{11} (a+b x)^2}-\frac {70 d^4 (b c-a d)^6}{b^{11} (a+b x)^3}-\frac {30 d^3 (b c-a d)^7}{b^{11} (a+b x)^4}-\frac {9 d^2 (b c-a d)^8}{b^{11} (a+b x)^5}-\frac {5 d (b c-a d)^9}{3 b^{11} (a+b x)^6}-\frac {(b c-a d)^{10}}{7 b^{11} (a+b x)^7}+\frac {d^{10} (a+b x)^3}{3 b^{11}}+\frac {45 d^8 x (b c-a d)^2}{b^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^10/(a + b*x)^8,x]

[Out]

(45*d^8*(b*c - a*d)^2*x)/b^10 - (b*c - a*d)^10/(7*b^11*(a + b*x)^7) - (5*d*(b*c - a*d)^9)/(3*b^11*(a + b*x)^6)
- (9*d^2*(b*c - a*d)^8)/(b^11*(a + b*x)^5) - (30*d^3*(b*c - a*d)^7)/(b^11*(a + b*x)^4) - (70*d^4*(b*c - a*d)^
6)/(b^11*(a + b*x)^3) - (126*d^5*(b*c - a*d)^5)/(b^11*(a + b*x)^2) - (210*d^6*(b*c - a*d)^4)/(b^11*(a + b*x))
+ (5*d^9*(b*c - a*d)*(a + b*x)^2)/b^11 + (d^10*(a + b*x)^3)/(3*b^11) + (120*d^7*(b*c - a*d)^3*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*} \int \frac {(c+d x)^{10}}{(a+b x)^8} \, dx &=\int \left (\frac {45 d^8 (b c-a d)^2}{b^{10}}+\frac {(b c-a d)^{10}}{b^{10} (a+b x)^8}+\frac {10 d (b c-a d)^9}{b^{10} (a+b x)^7}+\frac {45 d^2 (b c-a d)^8}{b^{10} (a+b x)^6}+\frac {120 d^3 (b c-a d)^7}{b^{10} (a+b x)^5}+\frac {210 d^4 (b c-a d)^6}{b^{10} (a+b x)^4}+\frac {252 d^5 (b c-a d)^5}{b^{10} (a+b x)^3}+\frac {210 d^6 (b c-a d)^4}{b^{10} (a+b x)^2}+\frac {120 d^7 (b c-a d)^3}{b^{10} (a+b x)}+\frac {10 d^9 (b c-a d) (a+b x)}{b^{10}}+\frac {d^{10} (a+b x)^2}{b^{10}}\right ) \, dx\\ &=\frac {45 d^8 (b c-a d)^2 x}{b^{10}}-\frac {(b c-a d)^{10}}{7 b^{11} (a+b x)^7}-\frac {5 d (b c-a d)^9}{3 b^{11} (a+b x)^6}-\frac {9 d^2 (b c-a d)^8}{b^{11} (a+b x)^5}-\frac {30 d^3 (b c-a d)^7}{b^{11} (a+b x)^4}-\frac {70 d^4 (b c-a d)^6}{b^{11} (a+b x)^3}-\frac {126 d^5 (b c-a d)^5}{b^{11} (a+b x)^2}-\frac {210 d^6 (b c-a d)^4}{b^{11} (a+b x)}+\frac {5 d^9 (b c-a d) (a+b x)^2}{b^{11}}+\frac {d^{10} (a+b x)^3}{3 b^{11}}+\frac {120 d^7 (b c-a d)^3 \log (a+b x)}{b^{11}}\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 239, normalized size = 0.93 \begin {gather*} \frac {21 b d^8 \left (45 b^2 c^2-80 a b c d+36 a^2 d^2\right ) x+21 b^2 d^9 (5 b c-4 a d) x^2+7 b^3 d^{10} x^3-\frac {3 (b c-a d)^{10}}{(a+b x)^7}+\frac {35 d (-b c+a d)^9}{(a+b x)^6}-\frac {189 d^2 (b c-a d)^8}{(a+b x)^5}+\frac {630 d^3 (-b c+a d)^7}{(a+b x)^4}-\frac {1470 d^4 (b c-a d)^6}{(a+b x)^3}+\frac {2646 d^5 (-b c+a d)^5}{(a+b x)^2}-\frac {4410 d^6 (b c-a d)^4}{a+b x}+2520 d^7 (b c-a d)^3 \log (a+b x)}{21 b^{11}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^10/(a + b*x)^8,x]

[Out]

(21*b*d^8*(45*b^2*c^2 - 80*a*b*c*d + 36*a^2*d^2)*x + 21*b^2*d^9*(5*b*c - 4*a*d)*x^2 + 7*b^3*d^10*x^3 - (3*(b*c
- a*d)^10)/(a + b*x)^7 + (35*d*(-(b*c) + a*d)^9)/(a + b*x)^6 - (189*d^2*(b*c - a*d)^8)/(a + b*x)^5 + (630*d^3
*(-(b*c) + a*d)^7)/(a + b*x)^4 - (1470*d^4*(b*c - a*d)^6)/(a + b*x)^3 + (2646*d^5*(-(b*c) + a*d)^5)/(a + b*x)^
2 - (4410*d^6*(b*c - a*d)^4)/(a + b*x) + 2520*d^7*(b*c - a*d)^3*Log[a + b*x])/(21*b^11)

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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)^8,x]')

[Out]

Timed out

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(855\) vs. \(2(252)=504\).
time = 0.14, size = 856, normalized size = 3.32

method	result	size
norman	\(\frac {-\frac {6534 a^{10} d^{10}-19602 a^{9} b c \,d^{9}+19602 a^{8} b^{2} c^{2} d^{8}-6534 a^{7} b^{3} c^{3} d^{7}+630 a^{6} b^{4} c^{4} d^{6}+126 a^{5} b^{5} c^{5} d^{5}+42 a^{4} b^{6} c^{6} d^{4}+18 a^{3} b^{7} c^{7} d^{3}+9 a^{2} b^{8} c^{8} d^{2}+5 a \,b^{9} c^{9} d +3 b^{10} c^{10}}{21 b^{11}}+\frac {d^{10} x^{10}}{3 b}-\frac {7 \left (120 a^{4} d^{10}-360 a^{3} b c \,d^{9}+360 b^{2} a^{2} c^{2} d^{8}-120 a \,b^{3} c^{3} d^{7}+30 b^{4} c^{4} d^{6}\right ) x^{6}}{b^{5}}-\frac {21 \left (180 a^{5} d^{10}-540 a^{4} b c \,d^{9}+540 a^{3} b^{2} c^{2} d^{8}-180 a^{2} b^{3} c^{3} d^{7}+30 a \,b^{4} c^{4} d^{6}+6 b^{5} c^{5} d^{5}\right ) x^{5}}{b^{6}}-\frac {35 \left (220 a^{6} d^{10}-660 a^{5} b c \,d^{9}+660 a^{4} b^{2} c^{2} d^{8}-220 a^{3} b^{3} c^{3} d^{7}+30 a^{2} b^{4} c^{4} d^{6}+6 a \,b^{5} c^{5} d^{5}+2 b^{6} c^{6} d^{4}\right ) x^{4}}{b^{7}}-\frac {5 \left (1750 a^{7} d^{10}-5250 a^{6} b c \,d^{9}+5250 a^{5} b^{2} c^{2} d^{8}-1750 a^{4} b^{3} c^{3} d^{7}+210 a^{3} b^{4} c^{4} d^{6}+42 a^{2} b^{5} c^{5} d^{5}+14 a \,b^{6} c^{6} d^{4}+6 b^{7} c^{7} d^{3}\right ) x^{3}}{b^{8}}-\frac {3 \left (1918 a^{8} d^{10}-5754 a^{7} b c \,d^{9}+5754 a^{6} b^{2} c^{2} d^{8}-1918 a^{5} b^{3} c^{3} d^{7}+210 a^{4} b^{4} c^{4} d^{6}+42 a^{3} b^{5} c^{5} d^{5}+14 b^{6} a^{2} c^{6} d^{4}+6 a \,b^{7} c^{7} d^{3}+3 b^{8} c^{8} d^{2}\right ) x^{2}}{b^{9}}-\frac {\left (6174 a^{9} d^{10}-18522 a^{8} b c \,d^{9}+18522 a^{7} b^{2} c^{2} d^{8}-6174 a^{6} b^{3} c^{3} d^{7}+630 a^{5} b^{4} c^{4} d^{6}+126 a^{4} b^{5} c^{5} d^{5}+42 a^{3} b^{6} c^{6} d^{4}+18 a^{2} b^{7} c^{7} d^{3}+9 a \,b^{8} c^{8} d^{2}+5 b^{9} c^{9} d \right ) x}{3 b^{10}}+\frac {15 d^{8} \left (a^{2} d^{2}-3 a b c d +3 b^{2} c^{2}\right ) x^{8}}{b^{3}}-\frac {5 d^{9} \left (a d -3 b c \right ) x^{9}}{3 b^{2}}}{\left (b x +a \right )^{7}}-\frac {120 d^{7} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b x +a \right )}{b^{11}}\)	\(854\)
default	\(\frac {d^{8} \left (\frac {1}{3} d^{2} x^{3} b^{2}-4 a b \,d^{2} x^{2}+5 b^{2} c d \,x^{2}+36 a^{2} d^{2} x -80 a b c d x +45 b^{2} c^{2} x \right )}{b^{10}}-\frac {210 d^{6} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{b^{11} \left (b x +a \right )}+\frac {30 d^{3} \left (a^{7} d^{7}-7 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}-35 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}-21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d -b^{7} c^{7}\right )}{b^{11} \left (b x +a \right )^{4}}-\frac {a^{10} d^{10}-10 a^{9} b c \,d^{9}+45 a^{8} b^{2} c^{2} d^{8}-120 a^{7} b^{3} c^{3} d^{7}+210 a^{6} b^{4} c^{4} d^{6}-252 a^{5} b^{5} c^{5} d^{5}+210 a^{4} b^{6} c^{6} d^{4}-120 a^{3} b^{7} c^{7} d^{3}+45 a^{2} b^{8} c^{8} d^{2}-10 a \,b^{9} c^{9} d +b^{10} c^{10}}{7 b^{11} \left (b x +a \right )^{7}}-\frac {9 d^{2} \left (a^{8} d^{8}-8 a^{7} b c \,d^{7}+28 a^{6} b^{2} c^{2} d^{6}-56 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} c^{4} d^{4}-56 a^{3} b^{5} c^{5} d^{3}+28 a^{2} b^{6} c^{6} d^{2}-8 a \,b^{7} c^{7} d +b^{8} c^{8}\right )}{b^{11} \left (b x +a \right )^{5}}+\frac {126 d^{5} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}{b^{11} \left (b x +a \right )^{2}}+\frac {5 d \left (a^{9} d^{9}-9 a^{8} b c \,d^{8}+36 a^{7} b^{2} c^{2} d^{7}-84 a^{6} b^{3} c^{3} d^{6}+126 a^{5} b^{4} c^{4} d^{5}-126 a^{4} b^{5} c^{5} d^{4}+84 a^{3} b^{6} c^{6} d^{3}-36 a^{2} b^{7} c^{7} d^{2}+9 a \,b^{8} c^{8} d -b^{9} c^{9}\right )}{3 b^{11} \left (b x +a \right )^{6}}-\frac {120 d^{7} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b x +a \right )}{b^{11}}-\frac {70 d^{4} \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right )}{b^{11} \left (b x +a \right )^{3}}\)	\(856\)
risch	\(\frac {d^{10} x^{3}}{3 b^{8}}-\frac {4 d^{10} a \,x^{2}}{b^{9}}+\frac {5 d^{9} c \,x^{2}}{b^{8}}+\frac {36 d^{10} a^{2} x}{b^{10}}-\frac {80 d^{9} a c x}{b^{9}}+\frac {45 d^{8} c^{2} x}{b^{8}}+\frac {\left (-210 a^{4} b^{5} d^{10}+840 a^{3} b^{6} c \,d^{9}-1260 a^{2} b^{7} c^{2} d^{8}+840 a \,b^{8} c^{3} d^{7}-210 b^{9} c^{4} d^{6}\right ) x^{6}-126 b^{4} d^{5} \left (9 a^{5} d^{5}-35 a^{4} b c \,d^{4}+50 a^{3} b^{2} c^{2} d^{3}-30 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d +b^{5} c^{5}\right ) x^{5}-70 b^{3} d^{4} \left (37 a^{6} d^{6}-141 a^{5} b c \,d^{5}+195 a^{4} b^{2} c^{2} d^{4}-110 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}+3 a \,b^{5} c^{5} d +b^{6} c^{6}\right ) x^{4}-10 b^{2} d^{3} \left (319 a^{7} d^{7}-1197 a^{6} b c \,d^{6}+1617 a^{5} b^{2} c^{2} d^{5}-875 a^{4} b^{3} c^{3} d^{4}+105 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d +3 b^{7} c^{7}\right ) x^{3}-3 b \,d^{2} \left (743 a^{8} d^{8}-2754 a^{7} b c \,d^{7}+3654 a^{6} b^{2} c^{2} d^{6}-1918 a^{5} b^{3} c^{3} d^{5}+210 a^{4} b^{4} c^{4} d^{4}+42 a^{3} b^{5} c^{5} d^{3}+14 a^{2} b^{6} c^{6} d^{2}+6 a \,b^{7} c^{7} d +3 b^{8} c^{8}\right ) x^{2}-\frac {d \left (2509 a^{9} d^{9}-9207 a^{8} b c \,d^{8}+12042 a^{7} b^{2} c^{2} d^{7}-6174 a^{6} b^{3} c^{3} d^{6}+630 a^{5} b^{4} c^{4} d^{5}+126 a^{4} b^{5} c^{5} d^{4}+42 a^{3} b^{6} c^{6} d^{3}+18 a^{2} b^{7} c^{7} d^{2}+9 a \,b^{8} c^{8} d +5 b^{9} c^{9}\right ) x}{3}-\frac {2761 a^{10} d^{10}-10047 a^{9} b c \,d^{9}+12987 a^{8} b^{2} c^{2} d^{8}-6534 a^{7} b^{3} c^{3} d^{7}+630 a^{6} b^{4} c^{4} d^{6}+126 a^{5} b^{5} c^{5} d^{5}+42 a^{4} b^{6} c^{6} d^{4}+18 a^{3} b^{7} c^{7} d^{3}+9 a^{2} b^{8} c^{8} d^{2}+5 a \,b^{9} c^{9} d +3 b^{10} c^{10}}{21 b}}{b^{10} \left (b x +a \right )^{7}}-\frac {120 d^{10} \ln \left (b x +a \right ) a^{3}}{b^{11}}+\frac {360 d^{9} \ln \left (b x +a \right ) a^{2} c}{b^{10}}-\frac {360 d^{8} \ln \left (b x +a \right ) a \,c^{2}}{b^{9}}+\frac {120 d^{7} \ln \left (b x +a \right ) c^{3}}{b^{8}}\)	\(868\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^10/(b*x+a)^8,x,method=_RETURNVERBOSE)

[Out]

d^8/b^10*(1/3*d^2*x^3*b^2-4*a*b*d^2*x^2+5*b^2*c*d*x^2+36*a^2*d^2*x-80*a*b*c*d*x+45*b^2*c^2*x)-210/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)+30/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)^4-1/7*(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)^7-9/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)^5+126/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)^2+5/3/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)^6-120/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)*ln(b*x+a)-70/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)^3

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 934 vs. \(2 (252) = 504\).
time = 0.35, size = 934, normalized size = 3.62 \begin {gather*} -\frac {3 \, b^{10} c^{10} + 5 \, a b^{9} c^{9} d + 9 \, a^{2} b^{8} c^{8} d^{2} + 18 \, a^{3} b^{7} c^{7} d^{3} + 42 \, a^{4} b^{6} c^{6} d^{4} + 126 \, a^{5} b^{5} c^{5} d^{5} + 630 \, a^{6} b^{4} c^{4} d^{6} - 6534 \, a^{7} b^{3} c^{3} d^{7} + 12987 \, a^{8} b^{2} c^{2} d^{8} - 10047 \, a^{9} b c d^{9} + 2761 \, a^{10} d^{10} + 4410 \, {\left (b^{10} c^{4} d^{6} - 4 \, a b^{9} c^{3} d^{7} + 6 \, a^{2} b^{8} c^{2} d^{8} - 4 \, a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 2646 \, {\left (b^{10} c^{5} d^{5} + 5 \, a b^{9} c^{4} d^{6} - 30 \, a^{2} b^{8} c^{3} d^{7} + 50 \, a^{3} b^{7} c^{2} d^{8} - 35 \, a^{4} b^{6} c d^{9} + 9 \, a^{5} b^{5} d^{10}\right )} x^{5} + 1470 \, {\left (b^{10} c^{6} d^{4} + 3 \, a b^{9} c^{5} d^{5} + 15 \, a^{2} b^{8} c^{4} d^{6} - 110 \, a^{3} b^{7} c^{3} d^{7} + 195 \, a^{4} b^{6} c^{2} d^{8} - 141 \, a^{5} b^{5} c d^{9} + 37 \, a^{6} b^{4} d^{10}\right )} x^{4} + 210 \, {\left (3 \, b^{10} c^{7} d^{3} + 7 \, a b^{9} c^{6} d^{4} + 21 \, a^{2} b^{8} c^{5} d^{5} + 105 \, a^{3} b^{7} c^{4} d^{6} - 875 \, a^{4} b^{6} c^{3} d^{7} + 1617 \, a^{5} b^{5} c^{2} d^{8} - 1197 \, a^{6} b^{4} c d^{9} + 319 \, a^{7} b^{3} d^{10}\right )} x^{3} + 63 \, {\left (3 \, b^{10} c^{8} d^{2} + 6 \, a b^{9} c^{7} d^{3} + 14 \, a^{2} b^{8} c^{6} d^{4} + 42 \, a^{3} b^{7} c^{5} d^{5} + 210 \, a^{4} b^{6} c^{4} d^{6} - 1918 \, a^{5} b^{5} c^{3} d^{7} + 3654 \, a^{6} b^{4} c^{2} d^{8} - 2754 \, a^{7} b^{3} c d^{9} + 743 \, a^{8} b^{2} d^{10}\right )} x^{2} + 7 \, {\left (5 \, b^{10} c^{9} d + 9 \, a b^{9} c^{8} d^{2} + 18 \, a^{2} b^{8} c^{7} d^{3} + 42 \, a^{3} b^{7} c^{6} d^{4} + 126 \, a^{4} b^{6} c^{5} d^{5} + 630 \, a^{5} b^{5} c^{4} d^{6} - 6174 \, a^{6} b^{4} c^{3} d^{7} + 12042 \, a^{7} b^{3} c^{2} d^{8} - 9207 \, a^{8} b^{2} c d^{9} + 2509 \, a^{9} b d^{10}\right )} x}{21 \, {\left (b^{18} x^{7} + 7 \, a b^{17} x^{6} + 21 \, a^{2} b^{16} x^{5} + 35 \, a^{3} b^{15} x^{4} + 35 \, a^{4} b^{14} x^{3} + 21 \, a^{5} b^{13} x^{2} + 7 \, a^{6} b^{12} x + a^{7} b^{11}\right )}} + \frac {b^{2} d^{10} x^{3} + 3 \, {\left (5 \, b^{2} c d^{9} - 4 \, a b d^{10}\right )} x^{2} + 3 \, {\left (45 \, b^{2} c^{2} d^{8} - 80 \, a b c d^{9} + 36 \, a^{2} d^{10}\right )} x}{3 \, b^{10}} + \frac {120 \, {\left (b^{3} c^{3} d^{7} - 3 \, a b^{2} c^{2} d^{8} + 3 \, a^{2} b c d^{9} - a^{3} d^{10}\right )} \log \left (b x + a\right )}{b^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^8,x, algorithm="maxima")

[Out]

-1/21*(3*b^10*c^10 + 5*a*b^9*c^9*d + 9*a^2*b^8*c^8*d^2 + 18*a^3*b^7*c^7*d^3 + 42*a^4*b^6*c^6*d^4 + 126*a^5*b^5
*c^5*d^5 + 630*a^6*b^4*c^4*d^6 - 6534*a^7*b^3*c^3*d^7 + 12987*a^8*b^2*c^2*d^8 - 10047*a^9*b*c*d^9 + 2761*a^10*
d^10 + 4410*(b^10*c^4*d^6 - 4*a*b^9*c^3*d^7 + 6*a^2*b^8*c^2*d^8 - 4*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 2646*(
b^10*c^5*d^5 + 5*a*b^9*c^4*d^6 - 30*a^2*b^8*c^3*d^7 + 50*a^3*b^7*c^2*d^8 - 35*a^4*b^6*c*d^9 + 9*a^5*b^5*d^10)*
x^5 + 1470*(b^10*c^6*d^4 + 3*a*b^9*c^5*d^5 + 15*a^2*b^8*c^4*d^6 - 110*a^3*b^7*c^3*d^7 + 195*a^4*b^6*c^2*d^8 -
141*a^5*b^5*c*d^9 + 37*a^6*b^4*d^10)*x^4 + 210*(3*b^10*c^7*d^3 + 7*a*b^9*c^6*d^4 + 21*a^2*b^8*c^5*d^5 + 105*a^
3*b^7*c^4*d^6 - 875*a^4*b^6*c^3*d^7 + 1617*a^5*b^5*c^2*d^8 - 1197*a^6*b^4*c*d^9 + 319*a^7*b^3*d^10)*x^3 + 63*(
3*b^10*c^8*d^2 + 6*a*b^9*c^7*d^3 + 14*a^2*b^8*c^6*d^4 + 42*a^3*b^7*c^5*d^5 + 210*a^4*b^6*c^4*d^6 - 1918*a^5*b^
5*c^3*d^7 + 3654*a^6*b^4*c^2*d^8 - 2754*a^7*b^3*c*d^9 + 743*a^8*b^2*d^10)*x^2 + 7*(5*b^10*c^9*d + 9*a*b^9*c^8*
d^2 + 18*a^2*b^8*c^7*d^3 + 42*a^3*b^7*c^6*d^4 + 126*a^4*b^6*c^5*d^5 + 630*a^5*b^5*c^4*d^6 - 6174*a^6*b^4*c^3*d
^7 + 12042*a^7*b^3*c^2*d^8 - 9207*a^8*b^2*c*d^9 + 2509*a^9*b*d^10)*x)/(b^18*x^7 + 7*a*b^17*x^6 + 21*a^2*b^16*x
^5 + 35*a^3*b^15*x^4 + 35*a^4*b^14*x^3 + 21*a^5*b^13*x^2 + 7*a^6*b^12*x + a^7*b^11) + 1/3*(b^2*d^10*x^3 + 3*(5
*b^2*c*d^9 - 4*a*b*d^10)*x^2 + 3*(45*b^2*c^2*d^8 - 80*a*b*c*d^9 + 36*a^2*d^10)*x)/b^10 + 120*(b^3*c^3*d^7 - 3*
a*b^2*c^2*d^8 + 3*a^2*b*c*d^9 - a^3*d^10)*log(b*x + a)/b^11

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1362 vs. \(2 (252) = 504\).
time = 0.31, size = 1362, normalized size = 5.28

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^8,x, algorithm="fricas")

[Out]

1/21*(7*b^10*d^10*x^10 - 3*b^10*c^10 - 5*a*b^9*c^9*d - 9*a^2*b^8*c^8*d^2 - 18*a^3*b^7*c^7*d^3 - 42*a^4*b^6*c^6
*d^4 - 126*a^5*b^5*c^5*d^5 - 630*a^6*b^4*c^4*d^6 + 6534*a^7*b^3*c^3*d^7 - 12987*a^8*b^2*c^2*d^8 + 10047*a^9*b*
c*d^9 - 2761*a^10*d^10 + 35*(3*b^10*c*d^9 - a*b^9*d^10)*x^9 + 315*(3*b^10*c^2*d^8 - 3*a*b^9*c*d^9 + a^2*b^8*d^
10)*x^8 + 49*(135*a*b^9*c^2*d^8 - 195*a^2*b^8*c*d^9 + 77*a^3*b^7*d^10)*x^7 - 49*(90*b^10*c^4*d^6 - 360*a*b^9*c
^3*d^7 + 135*a^2*b^8*c^2*d^8 + 285*a^3*b^7*c*d^9 - 179*a^4*b^6*d^10)*x^6 - 147*(18*b^10*c^5*d^5 + 90*a*b^9*c^4
*d^6 - 540*a^2*b^8*c^3*d^7 + 675*a^3*b^7*c^2*d^8 - 255*a^4*b^6*c*d^9 + a^5*b^5*d^10)*x^5 - 245*(6*b^10*c^6*d^4
+ 18*a*b^9*c^5*d^5 + 90*a^2*b^8*c^4*d^6 - 660*a^3*b^7*c^3*d^7 + 1035*a^4*b^6*c^2*d^8 - 615*a^5*b^5*c*d^9 + 12
1*a^6*b^4*d^10)*x^4 - 35*(18*b^10*c^7*d^3 + 42*a*b^9*c^6*d^4 + 126*a^2*b^8*c^5*d^5 + 630*a^3*b^7*c^4*d^6 - 525
0*a^4*b^6*c^3*d^7 + 9135*a^5*b^5*c^2*d^8 - 6195*a^6*b^4*c*d^9 + 1477*a^7*b^3*d^10)*x^3 - 21*(9*b^10*c^8*d^2 +
18*a*b^9*c^7*d^3 + 42*a^2*b^8*c^6*d^4 + 126*a^3*b^7*c^5*d^5 + 630*a^4*b^6*c^4*d^6 - 5754*a^5*b^5*c^3*d^7 + 106
47*a^6*b^4*c^2*d^8 - 7707*a^7*b^3*c*d^9 + 1981*a^8*b^2*d^10)*x^2 - 7*(5*b^10*c^9*d + 9*a*b^9*c^8*d^2 + 18*a^2*
b^8*c^7*d^3 + 42*a^3*b^7*c^6*d^4 + 126*a^4*b^6*c^5*d^5 + 630*a^5*b^5*c^4*d^6 - 6174*a^6*b^4*c^3*d^7 + 11907*a^
7*b^3*c^2*d^8 - 8967*a^8*b^2*c*d^9 + 2401*a^9*b*d^10)*x + 2520*(a^7*b^3*c^3*d^7 - 3*a^8*b^2*c^2*d^8 + 3*a^9*b*
c*d^9 - a^10*d^10 + (b^10*c^3*d^7 - 3*a*b^9*c^2*d^8 + 3*a^2*b^8*c*d^9 - a^3*b^7*d^10)*x^7 + 7*(a*b^9*c^3*d^7 -
3*a^2*b^8*c^2*d^8 + 3*a^3*b^7*c*d^9 - a^4*b^6*d^10)*x^6 + 21*(a^2*b^8*c^3*d^7 - 3*a^3*b^7*c^2*d^8 + 3*a^4*b^6
*c*d^9 - a^5*b^5*d^10)*x^5 + 35*(a^3*b^7*c^3*d^7 - 3*a^4*b^6*c^2*d^8 + 3*a^5*b^5*c*d^9 - a^6*b^4*d^10)*x^4 + 3
5*(a^4*b^6*c^3*d^7 - 3*a^5*b^5*c^2*d^8 + 3*a^6*b^4*c*d^9 - a^7*b^3*d^10)*x^3 + 21*(a^5*b^5*c^3*d^7 - 3*a^6*b^4
*c^2*d^8 + 3*a^7*b^3*c*d^9 - a^8*b^2*d^10)*x^2 + 7*(a^6*b^4*c^3*d^7 - 3*a^7*b^3*c^2*d^8 + 3*a^8*b^2*c*d^9 - a^
9*b*d^10)*x)*log(b*x + a))/(b^18*x^7 + 7*a*b^17*x^6 + 21*a^2*b^16*x^5 + 35*a^3*b^15*x^4 + 35*a^4*b^14*x^3 + 21
*a^5*b^13*x^2 + 7*a^6*b^12*x + a^7*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)**8,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 872 vs. \(2 (252) = 504\).
time = 0.00, size = 924, normalized size = 3.58 \begin {gather*} \frac {\frac {1}{3} x^{3} d^{10} b^{16}-4 x^{2} d^{10} b^{15} a+5 x^{2} d^{9} c b^{16}+36 x d^{10} b^{14} a^{2}-80 x d^{9} c b^{15} a+45 x d^{8} c^{2} b^{16}}{b^{24}}+\frac {\frac {1}{21} \left (\left (-4410 d^{10} b^{6} a^{4}+17640 d^{9} b^{7} a^{3} c-26460 d^{8} b^{8} a^{2} c^{2}+17640 d^{7} b^{9} a c^{3}-4410 d^{6} b^{10} c^{4}\right ) x^{6}+\left (-23814 d^{10} b^{5} a^{5}+92610 d^{9} b^{6} a^{4} c-132300 d^{8} b^{7} a^{3} c^{2}+79380 d^{7} b^{8} a^{2} c^{3}-13230 d^{6} b^{9} a c^{4}-2646 d^{5} b^{10} c^{5}\right ) x^{5}+\left (-54390 d^{10} b^{4} a^{6}+207270 d^{9} b^{5} a^{5} c-286650 d^{8} b^{6} a^{4} c^{2}+161700 d^{7} b^{7} a^{3} c^{3}-22050 d^{6} b^{8} a^{2} c^{4}-4410 d^{5} b^{9} a c^{5}-1470 d^{4} b^{10} c^{6}\right ) x^{4}+\left (-66990 d^{10} b^{3} a^{7}+251370 d^{9} b^{4} a^{6} c-339570 d^{8} b^{5} a^{5} c^{2}+183750 d^{7} b^{6} a^{4} c^{3}-22050 d^{6} b^{7} a^{3} c^{4}-4410 d^{5} b^{8} a^{2} c^{5}-1470 d^{4} b^{9} a c^{6}-630 d^{3} b^{10} c^{7}\right ) x^{3}+\left (-46809 d^{10} b^{2} a^{8}+173502 d^{9} b^{3} a^{7} c-230202 d^{8} b^{4} a^{6} c^{2}+120834 d^{7} b^{5} a^{5} c^{3}-13230 d^{6} b^{6} a^{4} c^{4}-2646 d^{5} b^{7} a^{3} c^{5}-882 d^{4} b^{8} a^{2} c^{6}-378 d^{3} b^{9} a c^{7}-189 d^{2} b^{10} c^{8}\right ) x^{2}+\left (-17563 d^{10} b a^{9}+64449 d^{9} b^{2} a^{8} c-84294 d^{8} b^{3} a^{7} c^{2}+43218 d^{7} b^{4} a^{6} c^{3}-4410 d^{6} b^{5} a^{5} c^{4}-882 d^{5} b^{6} a^{4} c^{5}-294 d^{4} b^{7} a^{3} c^{6}-126 d^{3} b^{8} a^{2} c^{7}-63 d^{2} b^{9} a c^{8}-35 d b^{10} c^{9}\right ) x-2761 d^{10} a^{10}+10047 d^{9} b a^{9} c-12987 d^{8} b^{2} a^{8} c^{2}+6534 d^{7} b^{3} a^{7} c^{3}-630 d^{6} b^{4} a^{6} c^{4}-126 d^{5} b^{5} a^{5} c^{5}-42 d^{4} b^{6} a^{4} c^{6}-18 d^{3} b^{7} a^{3} c^{7}-9 d^{2} b^{8} a^{2} c^{8}-5 d b^{9} a c^{9}-3 b^{10} c^{10}\right )}{b^{11} \left (x b+a\right )^{7}}+\frac {\left (-120 d^{10} a^{3}+360 d^{9} c b a^{2}-360 d^{8} c^{2} b^{2} a+120 d^{7} c^{3} b^{3}\right ) \ln \left |x b+a\right |}{b^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^8,x)

[Out]

120*(b^3*c^3*d^7 - 3*a*b^2*c^2*d^8 + 3*a^2*b*c*d^9 - a^3*d^10)*log(abs(b*x + a))/b^11 - 1/21*(3*b^10*c^10 + 5*
a*b^9*c^9*d + 9*a^2*b^8*c^8*d^2 + 18*a^3*b^7*c^7*d^3 + 42*a^4*b^6*c^6*d^4 + 126*a^5*b^5*c^5*d^5 + 630*a^6*b^4*
c^4*d^6 - 6534*a^7*b^3*c^3*d^7 + 12987*a^8*b^2*c^2*d^8 - 10047*a^9*b*c*d^9 + 2761*a^10*d^10 + 4410*(b^10*c^4*d
^6 - 4*a*b^9*c^3*d^7 + 6*a^2*b^8*c^2*d^8 - 4*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 2646*(b^10*c^5*d^5 + 5*a*b^9*
c^4*d^6 - 30*a^2*b^8*c^3*d^7 + 50*a^3*b^7*c^2*d^8 - 35*a^4*b^6*c*d^9 + 9*a^5*b^5*d^10)*x^5 + 1470*(b^10*c^6*d^
4 + 3*a*b^9*c^5*d^5 + 15*a^2*b^8*c^4*d^6 - 110*a^3*b^7*c^3*d^7 + 195*a^4*b^6*c^2*d^8 - 141*a^5*b^5*c*d^9 + 37*
a^6*b^4*d^10)*x^4 + 210*(3*b^10*c^7*d^3 + 7*a*b^9*c^6*d^4 + 21*a^2*b^8*c^5*d^5 + 105*a^3*b^7*c^4*d^6 - 875*a^4
*b^6*c^3*d^7 + 1617*a^5*b^5*c^2*d^8 - 1197*a^6*b^4*c*d^9 + 319*a^7*b^3*d^10)*x^3 + 63*(3*b^10*c^8*d^2 + 6*a*b^
9*c^7*d^3 + 14*a^2*b^8*c^6*d^4 + 42*a^3*b^7*c^5*d^5 + 210*a^4*b^6*c^4*d^6 - 1918*a^5*b^5*c^3*d^7 + 3654*a^6*b^
4*c^2*d^8 - 2754*a^7*b^3*c*d^9 + 743*a^8*b^2*d^10)*x^2 + 7*(5*b^10*c^9*d + 9*a*b^9*c^8*d^2 + 18*a^2*b^8*c^7*d^
3 + 42*a^3*b^7*c^6*d^4 + 126*a^4*b^6*c^5*d^5 + 630*a^5*b^5*c^4*d^6 - 6174*a^6*b^4*c^3*d^7 + 12042*a^7*b^3*c^2*
d^8 - 9207*a^8*b^2*c*d^9 + 2509*a^9*b*d^10)*x)/((b*x + a)^7*b^11) + 1/3*(b^16*d^10*x^3 + 15*b^16*c*d^9*x^2 - 1
2*a*b^15*d^10*x^2 + 135*b^16*c^2*d^8*x - 240*a*b^15*c*d^9*x + 108*a^2*b^14*d^10*x)/b^24

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Mupad [B]
time = 0.43, size = 950, normalized size = 3.68 \begin {gather*} x\,\left (\frac {8\,a\,\left (\frac {8\,a\,d^{10}}{b^9}-\frac {10\,c\,d^9}{b^8}\right )}{b}-\frac {28\,a^2\,d^{10}}{b^{10}}+\frac {45\,c^2\,d^8}{b^8}\right )-\frac {x^4\,\left (2590\,a^6\,b^3\,d^{10}-9870\,a^5\,b^4\,c\,d^9+13650\,a^4\,b^5\,c^2\,d^8-7700\,a^3\,b^6\,c^3\,d^7+1050\,a^2\,b^7\,c^4\,d^6+210\,a\,b^8\,c^5\,d^5+70\,b^9\,c^6\,d^4\right )+x^6\,\left (210\,a^4\,b^5\,d^{10}-840\,a^3\,b^6\,c\,d^9+1260\,a^2\,b^7\,c^2\,d^8-840\,a\,b^8\,c^3\,d^7+210\,b^9\,c^4\,d^6\right )+\frac {2761\,a^{10}\,d^{10}-10047\,a^9\,b\,c\,d^9+12987\,a^8\,b^2\,c^2\,d^8-6534\,a^7\,b^3\,c^3\,d^7+630\,a^6\,b^4\,c^4\,d^6+126\,a^5\,b^5\,c^5\,d^5+42\,a^4\,b^6\,c^6\,d^4+18\,a^3\,b^7\,c^7\,d^3+9\,a^2\,b^8\,c^8\,d^2+5\,a\,b^9\,c^9\,d+3\,b^{10}\,c^{10}}{21\,b}+x\,\left (\frac {2509\,a^9\,d^{10}}{3}-3069\,a^8\,b\,c\,d^9+4014\,a^7\,b^2\,c^2\,d^8-2058\,a^6\,b^3\,c^3\,d^7+210\,a^5\,b^4\,c^4\,d^6+42\,a^4\,b^5\,c^5\,d^5+14\,a^3\,b^6\,c^6\,d^4+6\,a^2\,b^7\,c^7\,d^3+3\,a\,b^8\,c^8\,d^2+\frac {5\,b^9\,c^9\,d}{3}\right )+x^3\,\left (3190\,a^7\,b^2\,d^{10}-11970\,a^6\,b^3\,c\,d^9+16170\,a^5\,b^4\,c^2\,d^8-8750\,a^4\,b^5\,c^3\,d^7+1050\,a^3\,b^6\,c^4\,d^6+210\,a^2\,b^7\,c^5\,d^5+70\,a\,b^8\,c^6\,d^4+30\,b^9\,c^7\,d^3\right )+x^2\,\left (2229\,a^8\,b\,d^{10}-8262\,a^7\,b^2\,c\,d^9+10962\,a^6\,b^3\,c^2\,d^8-5754\,a^5\,b^4\,c^3\,d^7+630\,a^4\,b^5\,c^4\,d^6+126\,a^3\,b^6\,c^5\,d^5+42\,a^2\,b^7\,c^6\,d^4+18\,a\,b^8\,c^7\,d^3+9\,b^9\,c^8\,d^2\right )+x^5\,\left (1134\,a^5\,b^4\,d^{10}-4410\,a^4\,b^5\,c\,d^9+6300\,a^3\,b^6\,c^2\,d^8-3780\,a^2\,b^7\,c^3\,d^7+630\,a\,b^8\,c^4\,d^6+126\,b^9\,c^5\,d^5\right )}{a^7\,b^{10}+7\,a^6\,b^{11}\,x+21\,a^5\,b^{12}\,x^2+35\,a^4\,b^{13}\,x^3+35\,a^3\,b^{14}\,x^4+21\,a^2\,b^{15}\,x^5+7\,a\,b^{16}\,x^6+b^{17}\,x^7}-x^2\,\left (\frac {4\,a\,d^{10}}{b^9}-\frac {5\,c\,d^9}{b^8}\right )-\frac {\ln \left (a+b\,x\right )\,\left (120\,a^3\,d^{10}-360\,a^2\,b\,c\,d^9+360\,a\,b^2\,c^2\,d^8-120\,b^3\,c^3\,d^7\right )}{b^{11}}+\frac {d^{10}\,x^3}{3\,b^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^10/(a + b*x)^8,x)

[Out]

x*((8*a*((8*a*d^10)/b^9 - (10*c*d^9)/b^8))/b - (28*a^2*d^10)/b^10 + (45*c^2*d^8)/b^8) - (x^4*(2590*a^6*b^3*d^1
0 + 70*b^9*c^6*d^4 + 210*a*b^8*c^5*d^5 - 9870*a^5*b^4*c*d^9 + 1050*a^2*b^7*c^4*d^6 - 7700*a^3*b^6*c^3*d^7 + 13
650*a^4*b^5*c^2*d^8) + x^6*(210*a^4*b^5*d^10 + 210*b^9*c^4*d^6 - 840*a*b^8*c^3*d^7 - 840*a^3*b^6*c*d^9 + 1260*
a^2*b^7*c^2*d^8) + (2761*a^10*d^10 + 3*b^10*c^10 + 9*a^2*b^8*c^8*d^2 + 18*a^3*b^7*c^7*d^3 + 42*a^4*b^6*c^6*d^4
+ 126*a^5*b^5*c^5*d^5 + 630*a^6*b^4*c^4*d^6 - 6534*a^7*b^3*c^3*d^7 + 12987*a^8*b^2*c^2*d^8 + 5*a*b^9*c^9*d -
10047*a^9*b*c*d^9)/(21*b) + x*((2509*a^9*d^10)/3 + (5*b^9*c^9*d)/3 + 3*a*b^8*c^8*d^2 + 6*a^2*b^7*c^7*d^3 + 14*
a^3*b^6*c^6*d^4 + 42*a^4*b^5*c^5*d^5 + 210*a^5*b^4*c^4*d^6 - 2058*a^6*b^3*c^3*d^7 + 4014*a^7*b^2*c^2*d^8 - 306
9*a^8*b*c*d^9) + x^3*(3190*a^7*b^2*d^10 + 30*b^9*c^7*d^3 + 70*a*b^8*c^6*d^4 - 11970*a^6*b^3*c*d^9 + 210*a^2*b^
7*c^5*d^5 + 1050*a^3*b^6*c^4*d^6 - 8750*a^4*b^5*c^3*d^7 + 16170*a^5*b^4*c^2*d^8) + x^2*(2229*a^8*b*d^10 + 9*b^
9*c^8*d^2 + 18*a*b^8*c^7*d^3 - 8262*a^7*b^2*c*d^9 + 42*a^2*b^7*c^6*d^4 + 126*a^3*b^6*c^5*d^5 + 630*a^4*b^5*c^4
*d^6 - 5754*a^5*b^4*c^3*d^7 + 10962*a^6*b^3*c^2*d^8) + x^5*(1134*a^5*b^4*d^10 + 126*b^9*c^5*d^5 + 630*a*b^8*c^
4*d^6 - 4410*a^4*b^5*c*d^9 - 3780*a^2*b^7*c^3*d^7 + 6300*a^3*b^6*c^2*d^8))/(a^7*b^10 + b^17*x^7 + 7*a^6*b^11*x
+ 7*a*b^16*x^6 + 21*a^5*b^12*x^2 + 35*a^4*b^13*x^3 + 35*a^3*b^14*x^4 + 21*a^2*b^15*x^5) - x^2*((4*a*d^10)/b^9
- (5*c*d^9)/b^8) - (log(a + b*x)*(120*a^3*d^10 - 120*b^3*c^3*d^7 + 360*a*b^2*c^2*d^8 - 360*a^2*b*c*d^9))/b^11
+ (d^10*x^3)/(3*b^8)

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