3.13.56 \(\int \frac {(a+b x)^9}{(c+d x)^8} \, dx\)

Optimal. Leaf size=232 \[ -\frac {b^8 x (8 b c-9 a d)}{d^9}+\frac {36 b^7 (b c-a d)^2 \log (c+d x)}{d^{10}}+\frac {84 b^6 (b c-a d)^3}{d^{10} (c+d x)}-\frac {63 b^5 (b c-a d)^4}{d^{10} (c+d x)^2}+\frac {42 b^4 (b c-a d)^5}{d^{10} (c+d x)^3}-\frac {21 b^3 (b c-a d)^6}{d^{10} (c+d x)^4}+\frac {36 b^2 (b c-a d)^7}{5 d^{10} (c+d x)^5}-\frac {3 b (b c-a d)^8}{2 d^{10} (c+d x)^6}+\frac {(b c-a d)^9}{7 d^{10} (c+d x)^7}+\frac {b^9 x^2}{2 d^8} \]

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Rubi [A]  time = 0.36, antiderivative size = 232, 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 = {43} \begin {gather*} -\frac {b^8 x (8 b c-9 a d)}{d^9}+\frac {84 b^6 (b c-a d)^3}{d^{10} (c+d x)}-\frac {63 b^5 (b c-a d)^4}{d^{10} (c+d x)^2}+\frac {42 b^4 (b c-a d)^5}{d^{10} (c+d x)^3}-\frac {21 b^3 (b c-a d)^6}{d^{10} (c+d x)^4}+\frac {36 b^2 (b c-a d)^7}{5 d^{10} (c+d x)^5}+\frac {36 b^7 (b c-a d)^2 \log (c+d x)}{d^{10}}-\frac {3 b (b c-a d)^8}{2 d^{10} (c+d x)^6}+\frac {(b c-a d)^9}{7 d^{10} (c+d x)^7}+\frac {b^9 x^2}{2 d^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^9/(c + d*x)^8,x]

[Out]

-((b^8*(8*b*c - 9*a*d)*x)/d^9) + (b^9*x^2)/(2*d^8) + (b*c - a*d)^9/(7*d^10*(c + d*x)^7) - (3*b*(b*c - a*d)^8)/
(2*d^10*(c + d*x)^6) + (36*b^2*(b*c - a*d)^7)/(5*d^10*(c + d*x)^5) - (21*b^3*(b*c - a*d)^6)/(d^10*(c + d*x)^4)
+ (42*b^4*(b*c - a*d)^5)/(d^10*(c + d*x)^3) - (63*b^5*(b*c - a*d)^4)/(d^10*(c + d*x)^2) + (84*b^6*(b*c - a*d)
^3)/(d^10*(c + d*x)) + (36*b^7*(b*c - a*d)^2*Log[c + d*x])/d^10

Rule 43

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 {(a+b x)^9}{(c+d x)^8} \, dx &=\int \left (-\frac {b^8 (8 b c-9 a d)}{d^9}+\frac {b^9 x}{d^8}+\frac {(-b c+a d)^9}{d^9 (c+d x)^8}+\frac {9 b (b c-a d)^8}{d^9 (c+d x)^7}-\frac {36 b^2 (b c-a d)^7}{d^9 (c+d x)^6}+\frac {84 b^3 (b c-a d)^6}{d^9 (c+d x)^5}-\frac {126 b^4 (b c-a d)^5}{d^9 (c+d x)^4}+\frac {126 b^5 (b c-a d)^4}{d^9 (c+d x)^3}-\frac {84 b^6 (b c-a d)^3}{d^9 (c+d x)^2}+\frac {36 b^7 (b c-a d)^2}{d^9 (c+d x)}\right ) \, dx\\ &=-\frac {b^8 (8 b c-9 a d) x}{d^9}+\frac {b^9 x^2}{2 d^8}+\frac {(b c-a d)^9}{7 d^{10} (c+d x)^7}-\frac {3 b (b c-a d)^8}{2 d^{10} (c+d x)^6}+\frac {36 b^2 (b c-a d)^7}{5 d^{10} (c+d x)^5}-\frac {21 b^3 (b c-a d)^6}{d^{10} (c+d x)^4}+\frac {42 b^4 (b c-a d)^5}{d^{10} (c+d x)^3}-\frac {63 b^5 (b c-a d)^4}{d^{10} (c+d x)^2}+\frac {84 b^6 (b c-a d)^3}{d^{10} (c+d x)}+\frac {36 b^7 (b c-a d)^2 \log (c+d x)}{d^{10}}\\ \end {align*}

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Mathematica [B]  time = 0.27, size = 584, normalized size = 2.52 \begin {gather*} -\frac {10 a^9 d^9+15 a^8 b d^8 (c+7 d x)+24 a^7 b^2 d^7 \left (c^2+7 c d x+21 d^2 x^2\right )+42 a^6 b^3 d^6 \left (c^3+7 c^2 d x+21 c d^2 x^2+35 d^3 x^3\right )+84 a^5 b^4 d^5 \left (c^4+7 c^3 d x+21 c^2 d^2 x^2+35 c d^3 x^3+35 d^4 x^4\right )+210 a^4 b^5 d^4 \left (c^5+7 c^4 d x+21 c^3 d^2 x^2+35 c^2 d^3 x^3+35 c d^4 x^4+21 d^5 x^5\right )+840 a^3 b^6 d^3 \left (c^6+7 c^5 d x+21 c^4 d^2 x^2+35 c^3 d^3 x^3+35 c^2 d^4 x^4+21 c d^5 x^5+7 d^6 x^6\right )-6 a^2 b^7 c d^2 \left (1089 c^6+7203 c^5 d x+20139 c^4 d^2 x^2+30625 c^3 d^3 x^3+26950 c^2 d^4 x^4+13230 c d^5 x^5+2940 d^6 x^6\right )+6 a b^8 d \left (1443 c^8+9261 c^7 d x+24843 c^6 d^2 x^2+35525 c^5 d^3 x^3+28175 c^4 d^4 x^4+11025 c^3 d^5 x^5+735 c^2 d^6 x^6-735 c d^7 x^7-105 d^8 x^8\right )-2520 b^7 (c+d x)^7 (b c-a d)^2 \log (c+d x)-\left (b^9 \left (3349 c^9+20923 c^8 d x+53949 c^7 d^2 x^2+72275 c^6 d^3 x^3+50225 c^5 d^4 x^4+12495 c^4 d^5 x^5-4655 c^3 d^6 x^6-3185 c^2 d^7 x^7-315 c d^8 x^8+35 d^9 x^9\right )\right )}{70 d^{10} (c+d x)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^9/(c + d*x)^8,x]

[Out]

-1/70*(10*a^9*d^9 + 15*a^8*b*d^8*(c + 7*d*x) + 24*a^7*b^2*d^7*(c^2 + 7*c*d*x + 21*d^2*x^2) + 42*a^6*b^3*d^6*(c
^3 + 7*c^2*d*x + 21*c*d^2*x^2 + 35*d^3*x^3) + 84*a^5*b^4*d^5*(c^4 + 7*c^3*d*x + 21*c^2*d^2*x^2 + 35*c*d^3*x^3
+ 35*d^4*x^4) + 210*a^4*b^5*d^4*(c^5 + 7*c^4*d*x + 21*c^3*d^2*x^2 + 35*c^2*d^3*x^3 + 35*c*d^4*x^4 + 21*d^5*x^5
) + 840*a^3*b^6*d^3*(c^6 + 7*c^5*d*x + 21*c^4*d^2*x^2 + 35*c^3*d^3*x^3 + 35*c^2*d^4*x^4 + 21*c*d^5*x^5 + 7*d^6
*x^6) - 6*a^2*b^7*c*d^2*(1089*c^6 + 7203*c^5*d*x + 20139*c^4*d^2*x^2 + 30625*c^3*d^3*x^3 + 26950*c^2*d^4*x^4 +
13230*c*d^5*x^5 + 2940*d^6*x^6) + 6*a*b^8*d*(1443*c^8 + 9261*c^7*d*x + 24843*c^6*d^2*x^2 + 35525*c^5*d^3*x^3
+ 28175*c^4*d^4*x^4 + 11025*c^3*d^5*x^5 + 735*c^2*d^6*x^6 - 735*c*d^7*x^7 - 105*d^8*x^8) - b^9*(3349*c^9 + 209
23*c^8*d*x + 53949*c^7*d^2*x^2 + 72275*c^6*d^3*x^3 + 50225*c^5*d^4*x^4 + 12495*c^4*d^5*x^5 - 4655*c^3*d^6*x^6
- 3185*c^2*d^7*x^7 - 315*c*d^8*x^8 + 35*d^9*x^9) - 2520*b^7*(b*c - a*d)^2*(c + d*x)^7*Log[c + d*x])/(d^10*(c +
d*x)^7)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^9}{(c+d x)^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)^9/(c + d*x)^8,x]

[Out]

IntegrateAlgebraic[(a + b*x)^9/(c + d*x)^8, x]

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fricas [B]  time = 0.96, size = 1093, normalized size = 4.71 \begin {gather*} \frac {35 \, b^{9} d^{9} x^{9} + 3349 \, b^{9} c^{9} - 8658 \, a b^{8} c^{8} d + 6534 \, a^{2} b^{7} c^{7} d^{2} - 840 \, a^{3} b^{6} c^{6} d^{3} - 210 \, a^{4} b^{5} c^{5} d^{4} - 84 \, a^{5} b^{4} c^{4} d^{5} - 42 \, a^{6} b^{3} c^{3} d^{6} - 24 \, a^{7} b^{2} c^{2} d^{7} - 15 \, a^{8} b c d^{8} - 10 \, a^{9} d^{9} - 315 \, {\left (b^{9} c d^{8} - 2 \, a b^{8} d^{9}\right )} x^{8} - 245 \, {\left (13 \, b^{9} c^{2} d^{7} - 18 \, a b^{8} c d^{8}\right )} x^{7} - 245 \, {\left (19 \, b^{9} c^{3} d^{6} + 18 \, a b^{8} c^{2} d^{7} - 72 \, a^{2} b^{7} c d^{8} + 24 \, a^{3} b^{6} d^{9}\right )} x^{6} + 735 \, {\left (17 \, b^{9} c^{4} d^{5} - 90 \, a b^{8} c^{3} d^{6} + 108 \, a^{2} b^{7} c^{2} d^{7} - 24 \, a^{3} b^{6} c d^{8} - 6 \, a^{4} b^{5} d^{9}\right )} x^{5} + 245 \, {\left (205 \, b^{9} c^{5} d^{4} - 690 \, a b^{8} c^{4} d^{5} + 660 \, a^{2} b^{7} c^{3} d^{6} - 120 \, a^{3} b^{6} c^{2} d^{7} - 30 \, a^{4} b^{5} c d^{8} - 12 \, a^{5} b^{4} d^{9}\right )} x^{4} + 245 \, {\left (295 \, b^{9} c^{6} d^{3} - 870 \, a b^{8} c^{5} d^{4} + 750 \, a^{2} b^{7} c^{4} d^{5} - 120 \, a^{3} b^{6} c^{3} d^{6} - 30 \, a^{4} b^{5} c^{2} d^{7} - 12 \, a^{5} b^{4} c d^{8} - 6 \, a^{6} b^{3} d^{9}\right )} x^{3} + 21 \, {\left (2569 \, b^{9} c^{7} d^{2} - 7098 \, a b^{8} c^{6} d^{3} + 5754 \, a^{2} b^{7} c^{5} d^{4} - 840 \, a^{3} b^{6} c^{4} d^{5} - 210 \, a^{4} b^{5} c^{3} d^{6} - 84 \, a^{5} b^{4} c^{2} d^{7} - 42 \, a^{6} b^{3} c d^{8} - 24 \, a^{7} b^{2} d^{9}\right )} x^{2} + 7 \, {\left (2989 \, b^{9} c^{8} d - 7938 \, a b^{8} c^{7} d^{2} + 6174 \, a^{2} b^{7} c^{6} d^{3} - 840 \, a^{3} b^{6} c^{5} d^{4} - 210 \, a^{4} b^{5} c^{4} d^{5} - 84 \, a^{5} b^{4} c^{3} d^{6} - 42 \, a^{6} b^{3} c^{2} d^{7} - 24 \, a^{7} b^{2} c d^{8} - 15 \, a^{8} b d^{9}\right )} x + 2520 \, {\left (b^{9} c^{9} - 2 \, a b^{8} c^{8} d + a^{2} b^{7} c^{7} d^{2} + {\left (b^{9} c^{2} d^{7} - 2 \, a b^{8} c d^{8} + a^{2} b^{7} d^{9}\right )} x^{7} + 7 \, {\left (b^{9} c^{3} d^{6} - 2 \, a b^{8} c^{2} d^{7} + a^{2} b^{7} c d^{8}\right )} x^{6} + 21 \, {\left (b^{9} c^{4} d^{5} - 2 \, a b^{8} c^{3} d^{6} + a^{2} b^{7} c^{2} d^{7}\right )} x^{5} + 35 \, {\left (b^{9} c^{5} d^{4} - 2 \, a b^{8} c^{4} d^{5} + a^{2} b^{7} c^{3} d^{6}\right )} x^{4} + 35 \, {\left (b^{9} c^{6} d^{3} - 2 \, a b^{8} c^{5} d^{4} + a^{2} b^{7} c^{4} d^{5}\right )} x^{3} + 21 \, {\left (b^{9} c^{7} d^{2} - 2 \, a b^{8} c^{6} d^{3} + a^{2} b^{7} c^{5} d^{4}\right )} x^{2} + 7 \, {\left (b^{9} c^{8} d - 2 \, a b^{8} c^{7} d^{2} + a^{2} b^{7} c^{6} d^{3}\right )} x\right )} \log \left (d x + c\right )}{70 \, {\left (d^{17} x^{7} + 7 \, c d^{16} x^{6} + 21 \, c^{2} d^{15} x^{5} + 35 \, c^{3} d^{14} x^{4} + 35 \, c^{4} d^{13} x^{3} + 21 \, c^{5} d^{12} x^{2} + 7 \, c^{6} d^{11} x + c^{7} d^{10}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^9/(d*x+c)^8,x, algorithm="fricas")

[Out]

1/70*(35*b^9*d^9*x^9 + 3349*b^9*c^9 - 8658*a*b^8*c^8*d + 6534*a^2*b^7*c^7*d^2 - 840*a^3*b^6*c^6*d^3 - 210*a^4*
b^5*c^5*d^4 - 84*a^5*b^4*c^4*d^5 - 42*a^6*b^3*c^3*d^6 - 24*a^7*b^2*c^2*d^7 - 15*a^8*b*c*d^8 - 10*a^9*d^9 - 315
*(b^9*c*d^8 - 2*a*b^8*d^9)*x^8 - 245*(13*b^9*c^2*d^7 - 18*a*b^8*c*d^8)*x^7 - 245*(19*b^9*c^3*d^6 + 18*a*b^8*c^
2*d^7 - 72*a^2*b^7*c*d^8 + 24*a^3*b^6*d^9)*x^6 + 735*(17*b^9*c^4*d^5 - 90*a*b^8*c^3*d^6 + 108*a^2*b^7*c^2*d^7
- 24*a^3*b^6*c*d^8 - 6*a^4*b^5*d^9)*x^5 + 245*(205*b^9*c^5*d^4 - 690*a*b^8*c^4*d^5 + 660*a^2*b^7*c^3*d^6 - 120
*a^3*b^6*c^2*d^7 - 30*a^4*b^5*c*d^8 - 12*a^5*b^4*d^9)*x^4 + 245*(295*b^9*c^6*d^3 - 870*a*b^8*c^5*d^4 + 750*a^2
*b^7*c^4*d^5 - 120*a^3*b^6*c^3*d^6 - 30*a^4*b^5*c^2*d^7 - 12*a^5*b^4*c*d^8 - 6*a^6*b^3*d^9)*x^3 + 21*(2569*b^9
*c^7*d^2 - 7098*a*b^8*c^6*d^3 + 5754*a^2*b^7*c^5*d^4 - 840*a^3*b^6*c^4*d^5 - 210*a^4*b^5*c^3*d^6 - 84*a^5*b^4*
c^2*d^7 - 42*a^6*b^3*c*d^8 - 24*a^7*b^2*d^9)*x^2 + 7*(2989*b^9*c^8*d - 7938*a*b^8*c^7*d^2 + 6174*a^2*b^7*c^6*d
^3 - 840*a^3*b^6*c^5*d^4 - 210*a^4*b^5*c^4*d^5 - 84*a^5*b^4*c^3*d^6 - 42*a^6*b^3*c^2*d^7 - 24*a^7*b^2*c*d^8 -
15*a^8*b*d^9)*x + 2520*(b^9*c^9 - 2*a*b^8*c^8*d + a^2*b^7*c^7*d^2 + (b^9*c^2*d^7 - 2*a*b^8*c*d^8 + a^2*b^7*d^9
)*x^7 + 7*(b^9*c^3*d^6 - 2*a*b^8*c^2*d^7 + a^2*b^7*c*d^8)*x^6 + 21*(b^9*c^4*d^5 - 2*a*b^8*c^3*d^6 + a^2*b^7*c^
2*d^7)*x^5 + 35*(b^9*c^5*d^4 - 2*a*b^8*c^4*d^5 + a^2*b^7*c^3*d^6)*x^4 + 35*(b^9*c^6*d^3 - 2*a*b^8*c^5*d^4 + a^
2*b^7*c^4*d^5)*x^3 + 21*(b^9*c^7*d^2 - 2*a*b^8*c^6*d^3 + a^2*b^7*c^5*d^4)*x^2 + 7*(b^9*c^8*d - 2*a*b^8*c^7*d^2
+ a^2*b^7*c^6*d^3)*x)*log(d*x + c))/(d^17*x^7 + 7*c*d^16*x^6 + 21*c^2*d^15*x^5 + 35*c^3*d^14*x^4 + 35*c^4*d^1
3*x^3 + 21*c^5*d^12*x^2 + 7*c^6*d^11*x + c^7*d^10)

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giac [B]  time = 1.32, size = 723, normalized size = 3.12 \begin {gather*} \frac {36 \, {\left (b^{9} c^{2} - 2 \, a b^{8} c d + a^{2} b^{7} d^{2}\right )} \log \left ({\left | d x + c \right |}\right )}{d^{10}} + \frac {b^{9} d^{8} x^{2} - 16 \, b^{9} c d^{7} x + 18 \, a b^{8} d^{8} x}{2 \, d^{16}} + \frac {3349 \, b^{9} c^{9} - 8658 \, a b^{8} c^{8} d + 6534 \, a^{2} b^{7} c^{7} d^{2} - 840 \, a^{3} b^{6} c^{6} d^{3} - 210 \, a^{4} b^{5} c^{5} d^{4} - 84 \, a^{5} b^{4} c^{4} d^{5} - 42 \, a^{6} b^{3} c^{3} d^{6} - 24 \, a^{7} b^{2} c^{2} d^{7} - 15 \, a^{8} b c d^{8} - 10 \, a^{9} d^{9} + 5880 \, {\left (b^{9} c^{3} d^{6} - 3 \, a b^{8} c^{2} d^{7} + 3 \, a^{2} b^{7} c d^{8} - a^{3} b^{6} d^{9}\right )} x^{6} + 4410 \, {\left (7 \, b^{9} c^{4} d^{5} - 20 \, a b^{8} c^{3} d^{6} + 18 \, a^{2} b^{7} c^{2} d^{7} - 4 \, a^{3} b^{6} c d^{8} - a^{4} b^{5} d^{9}\right )} x^{5} + 1470 \, {\left (47 \, b^{9} c^{5} d^{4} - 130 \, a b^{8} c^{4} d^{5} + 110 \, a^{2} b^{7} c^{3} d^{6} - 20 \, a^{3} b^{6} c^{2} d^{7} - 5 \, a^{4} b^{5} c d^{8} - 2 \, a^{5} b^{4} d^{9}\right )} x^{4} + 1470 \, {\left (57 \, b^{9} c^{6} d^{3} - 154 \, a b^{8} c^{5} d^{4} + 125 \, a^{2} b^{7} c^{4} d^{5} - 20 \, a^{3} b^{6} c^{3} d^{6} - 5 \, a^{4} b^{5} c^{2} d^{7} - 2 \, a^{5} b^{4} c d^{8} - a^{6} b^{3} d^{9}\right )} x^{3} + 126 \, {\left (459 \, b^{9} c^{7} d^{2} - 1218 \, a b^{8} c^{6} d^{3} + 959 \, a^{2} b^{7} c^{5} d^{4} - 140 \, a^{3} b^{6} c^{4} d^{5} - 35 \, a^{4} b^{5} c^{3} d^{6} - 14 \, a^{5} b^{4} c^{2} d^{7} - 7 \, a^{6} b^{3} c d^{8} - 4 \, a^{7} b^{2} d^{9}\right )} x^{2} + 21 \, {\left (1023 \, b^{9} c^{8} d - 2676 \, a b^{8} c^{7} d^{2} + 2058 \, a^{2} b^{7} c^{6} d^{3} - 280 \, a^{3} b^{6} c^{5} d^{4} - 70 \, a^{4} b^{5} c^{4} d^{5} - 28 \, a^{5} b^{4} c^{3} d^{6} - 14 \, a^{6} b^{3} c^{2} d^{7} - 8 \, a^{7} b^{2} c d^{8} - 5 \, a^{8} b d^{9}\right )} x}{70 \, {\left (d x + c\right )}^{7} d^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^9/(d*x+c)^8,x, algorithm="giac")

[Out]

36*(b^9*c^2 - 2*a*b^8*c*d + a^2*b^7*d^2)*log(abs(d*x + c))/d^10 + 1/2*(b^9*d^8*x^2 - 16*b^9*c*d^7*x + 18*a*b^8
*d^8*x)/d^16 + 1/70*(3349*b^9*c^9 - 8658*a*b^8*c^8*d + 6534*a^2*b^7*c^7*d^2 - 840*a^3*b^6*c^6*d^3 - 210*a^4*b^
5*c^5*d^4 - 84*a^5*b^4*c^4*d^5 - 42*a^6*b^3*c^3*d^6 - 24*a^7*b^2*c^2*d^7 - 15*a^8*b*c*d^8 - 10*a^9*d^9 + 5880*
(b^9*c^3*d^6 - 3*a*b^8*c^2*d^7 + 3*a^2*b^7*c*d^8 - a^3*b^6*d^9)*x^6 + 4410*(7*b^9*c^4*d^5 - 20*a*b^8*c^3*d^6 +
18*a^2*b^7*c^2*d^7 - 4*a^3*b^6*c*d^8 - a^4*b^5*d^9)*x^5 + 1470*(47*b^9*c^5*d^4 - 130*a*b^8*c^4*d^5 + 110*a^2*
b^7*c^3*d^6 - 20*a^3*b^6*c^2*d^7 - 5*a^4*b^5*c*d^8 - 2*a^5*b^4*d^9)*x^4 + 1470*(57*b^9*c^6*d^3 - 154*a*b^8*c^5
*d^4 + 125*a^2*b^7*c^4*d^5 - 20*a^3*b^6*c^3*d^6 - 5*a^4*b^5*c^2*d^7 - 2*a^5*b^4*c*d^8 - a^6*b^3*d^9)*x^3 + 126
*(459*b^9*c^7*d^2 - 1218*a*b^8*c^6*d^3 + 959*a^2*b^7*c^5*d^4 - 140*a^3*b^6*c^4*d^5 - 35*a^4*b^5*c^3*d^6 - 14*a
^5*b^4*c^2*d^7 - 7*a^6*b^3*c*d^8 - 4*a^7*b^2*d^9)*x^2 + 21*(1023*b^9*c^8*d - 2676*a*b^8*c^7*d^2 + 2058*a^2*b^7
*c^6*d^3 - 280*a^3*b^6*c^5*d^4 - 70*a^4*b^5*c^4*d^5 - 28*a^5*b^4*c^3*d^6 - 14*a^6*b^3*c^2*d^7 - 8*a^7*b^2*c*d^
8 - 5*a^8*b*d^9)*x)/((d*x + c)^7*d^10)

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maple [B]  time = 0.02, size = 1035, normalized size = 4.46

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^9/(d*x+c)^8,x)

[Out]

1/2*b^9*x^2/d^8-21*b^9/d^10/(d*x+c)^4*c^6+36/5*b^9/d^10/(d*x+c)^5*c^7+1/7/d^10/(d*x+c)^7*b^9*c^9-36/5*b^2/d^3/
(d*x+c)^5*a^7+9*b^8/d^8*a*x-8*b^9/d^9*x*c-84*b^6/d^7/(d*x+c)*a^3+84*b^9/d^10/(d*x+c)*c^3+36*b^7/d^8*ln(d*x+c)*
a^2+36*b^9/d^10*ln(d*x+c)*c^2-3/2*b/d^2/(d*x+c)^6*a^8-3/2*b^9/d^10/(d*x+c)^6*c^8-42*b^4/d^5/(d*x+c)^3*a^5+42*b
^9/d^10/(d*x+c)^3*c^5-63*b^5/d^6/(d*x+c)^2*a^4-63*b^9/d^10/(d*x+c)^2*c^4-21*b^3/d^4/(d*x+c)^4*a^6-1/7/d/(d*x+c
)^7*a^9+252*b^6/d^7/(d*x+c)^2*a^3*c-378*b^7/d^8/(d*x+c)^2*a^2*c^2+252*b^8/d^9/(d*x+c)^2*a*c^3-36/7/d^3/(d*x+c)
^7*a^7*b^2*c^2+126*b^4/d^5/(d*x+c)^4*a^5*c-315*b^5/d^6/(d*x+c)^4*a^4*c^2+420*b^6/d^7/(d*x+c)^4*a^3*c^3-315*b^7
/d^8/(d*x+c)^4*a^2*c^4+126*b^8/d^9/(d*x+c)^4*a*c^5+9/7/d^2/(d*x+c)^7*a^8*b*c+252/5*b^3/d^4/(d*x+c)^5*a^6*c-756
/5*b^4/d^5/(d*x+c)^5*a^5*c^2+252*b^5/d^6/(d*x+c)^5*a^4*c^3-252*b^6/d^7/(d*x+c)^5*a^3*c^4+756/5*b^7/d^8/(d*x+c)
^5*a^2*c^5-252/5*b^8/d^9/(d*x+c)^5*a*c^6+252*b^7/d^8/(d*x+c)*a^2*c-252*b^8/d^9/(d*x+c)*a*c^2-72*b^8/d^9*ln(d*x
+c)*a*c+12*b^2/d^3/(d*x+c)^6*a^7*c-42*b^3/d^4/(d*x+c)^6*a^6*c^2+84*b^4/d^5/(d*x+c)^6*a^5*c^3-105*b^5/d^6/(d*x+
c)^6*a^4*c^4+84*b^6/d^7/(d*x+c)^6*a^3*c^5+12/d^4/(d*x+c)^7*a^6*b^3*c^3+18/d^6/(d*x+c)^7*a^4*b^5*c^5-12/d^7/(d*
x+c)^7*a^3*b^6*c^6+36/7/d^8/(d*x+c)^7*a^2*b^7*c^7-9/7/d^9/(d*x+c)^7*a*b^8*c^8-42*b^7/d^8/(d*x+c)^6*a^2*c^6+12*
b^8/d^9/(d*x+c)^6*a*c^7+210*b^5/d^6/(d*x+c)^3*a^4*c-420*b^6/d^7/(d*x+c)^3*a^3*c^2+420*b^7/d^8/(d*x+c)^3*a^2*c^
3-210*b^8/d^9/(d*x+c)^3*a*c^4-18/d^5/(d*x+c)^7*a^5*b^4*c^4

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maxima [B]  time = 2.20, size = 786, normalized size = 3.39 \begin {gather*} \frac {3349 \, b^{9} c^{9} - 8658 \, a b^{8} c^{8} d + 6534 \, a^{2} b^{7} c^{7} d^{2} - 840 \, a^{3} b^{6} c^{6} d^{3} - 210 \, a^{4} b^{5} c^{5} d^{4} - 84 \, a^{5} b^{4} c^{4} d^{5} - 42 \, a^{6} b^{3} c^{3} d^{6} - 24 \, a^{7} b^{2} c^{2} d^{7} - 15 \, a^{8} b c d^{8} - 10 \, a^{9} d^{9} + 5880 \, {\left (b^{9} c^{3} d^{6} - 3 \, a b^{8} c^{2} d^{7} + 3 \, a^{2} b^{7} c d^{8} - a^{3} b^{6} d^{9}\right )} x^{6} + 4410 \, {\left (7 \, b^{9} c^{4} d^{5} - 20 \, a b^{8} c^{3} d^{6} + 18 \, a^{2} b^{7} c^{2} d^{7} - 4 \, a^{3} b^{6} c d^{8} - a^{4} b^{5} d^{9}\right )} x^{5} + 1470 \, {\left (47 \, b^{9} c^{5} d^{4} - 130 \, a b^{8} c^{4} d^{5} + 110 \, a^{2} b^{7} c^{3} d^{6} - 20 \, a^{3} b^{6} c^{2} d^{7} - 5 \, a^{4} b^{5} c d^{8} - 2 \, a^{5} b^{4} d^{9}\right )} x^{4} + 1470 \, {\left (57 \, b^{9} c^{6} d^{3} - 154 \, a b^{8} c^{5} d^{4} + 125 \, a^{2} b^{7} c^{4} d^{5} - 20 \, a^{3} b^{6} c^{3} d^{6} - 5 \, a^{4} b^{5} c^{2} d^{7} - 2 \, a^{5} b^{4} c d^{8} - a^{6} b^{3} d^{9}\right )} x^{3} + 126 \, {\left (459 \, b^{9} c^{7} d^{2} - 1218 \, a b^{8} c^{6} d^{3} + 959 \, a^{2} b^{7} c^{5} d^{4} - 140 \, a^{3} b^{6} c^{4} d^{5} - 35 \, a^{4} b^{5} c^{3} d^{6} - 14 \, a^{5} b^{4} c^{2} d^{7} - 7 \, a^{6} b^{3} c d^{8} - 4 \, a^{7} b^{2} d^{9}\right )} x^{2} + 21 \, {\left (1023 \, b^{9} c^{8} d - 2676 \, a b^{8} c^{7} d^{2} + 2058 \, a^{2} b^{7} c^{6} d^{3} - 280 \, a^{3} b^{6} c^{5} d^{4} - 70 \, a^{4} b^{5} c^{4} d^{5} - 28 \, a^{5} b^{4} c^{3} d^{6} - 14 \, a^{6} b^{3} c^{2} d^{7} - 8 \, a^{7} b^{2} c d^{8} - 5 \, a^{8} b d^{9}\right )} x}{70 \, {\left (d^{17} x^{7} + 7 \, c d^{16} x^{6} + 21 \, c^{2} d^{15} x^{5} + 35 \, c^{3} d^{14} x^{4} + 35 \, c^{4} d^{13} x^{3} + 21 \, c^{5} d^{12} x^{2} + 7 \, c^{6} d^{11} x + c^{7} d^{10}\right )}} + \frac {b^{9} d x^{2} - 2 \, {\left (8 \, b^{9} c - 9 \, a b^{8} d\right )} x}{2 \, d^{9}} + \frac {36 \, {\left (b^{9} c^{2} - 2 \, a b^{8} c d + a^{2} b^{7} d^{2}\right )} \log \left (d x + c\right )}{d^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^9/(d*x+c)^8,x, algorithm="maxima")

[Out]

1/70*(3349*b^9*c^9 - 8658*a*b^8*c^8*d + 6534*a^2*b^7*c^7*d^2 - 840*a^3*b^6*c^6*d^3 - 210*a^4*b^5*c^5*d^4 - 84*
a^5*b^4*c^4*d^5 - 42*a^6*b^3*c^3*d^6 - 24*a^7*b^2*c^2*d^7 - 15*a^8*b*c*d^8 - 10*a^9*d^9 + 5880*(b^9*c^3*d^6 -
3*a*b^8*c^2*d^7 + 3*a^2*b^7*c*d^8 - a^3*b^6*d^9)*x^6 + 4410*(7*b^9*c^4*d^5 - 20*a*b^8*c^3*d^6 + 18*a^2*b^7*c^2
*d^7 - 4*a^3*b^6*c*d^8 - a^4*b^5*d^9)*x^5 + 1470*(47*b^9*c^5*d^4 - 130*a*b^8*c^4*d^5 + 110*a^2*b^7*c^3*d^6 - 2
0*a^3*b^6*c^2*d^7 - 5*a^4*b^5*c*d^8 - 2*a^5*b^4*d^9)*x^4 + 1470*(57*b^9*c^6*d^3 - 154*a*b^8*c^5*d^4 + 125*a^2*
b^7*c^4*d^5 - 20*a^3*b^6*c^3*d^6 - 5*a^4*b^5*c^2*d^7 - 2*a^5*b^4*c*d^8 - a^6*b^3*d^9)*x^3 + 126*(459*b^9*c^7*d
^2 - 1218*a*b^8*c^6*d^3 + 959*a^2*b^7*c^5*d^4 - 140*a^3*b^6*c^4*d^5 - 35*a^4*b^5*c^3*d^6 - 14*a^5*b^4*c^2*d^7
- 7*a^6*b^3*c*d^8 - 4*a^7*b^2*d^9)*x^2 + 21*(1023*b^9*c^8*d - 2676*a*b^8*c^7*d^2 + 2058*a^2*b^7*c^6*d^3 - 280*
a^3*b^6*c^5*d^4 - 70*a^4*b^5*c^4*d^5 - 28*a^5*b^4*c^3*d^6 - 14*a^6*b^3*c^2*d^7 - 8*a^7*b^2*c*d^8 - 5*a^8*b*d^9
)*x)/(d^17*x^7 + 7*c*d^16*x^6 + 21*c^2*d^15*x^5 + 35*c^3*d^14*x^4 + 35*c^4*d^13*x^3 + 21*c^5*d^12*x^2 + 7*c^6*
d^11*x + c^7*d^10) + 1/2*(b^9*d*x^2 - 2*(8*b^9*c - 9*a*b^8*d)*x)/d^9 + 36*(b^9*c^2 - 2*a*b^8*c*d + a^2*b^7*d^2
)*log(d*x + c)/d^10

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mupad [B]  time = 0.26, size = 784, normalized size = 3.38 \begin {gather*} x\,\left (\frac {9\,a\,b^8}{d^8}-\frac {8\,b^9\,c}{d^9}\right )-\frac {\frac {10\,a^9\,d^9+15\,a^8\,b\,c\,d^8+24\,a^7\,b^2\,c^2\,d^7+42\,a^6\,b^3\,c^3\,d^6+84\,a^5\,b^4\,c^4\,d^5+210\,a^4\,b^5\,c^5\,d^4+840\,a^3\,b^6\,c^6\,d^3-6534\,a^2\,b^7\,c^7\,d^2+8658\,a\,b^8\,c^8\,d-3349\,b^9\,c^9}{70\,d}+x\,\left (\frac {3\,a^8\,b\,d^8}{2}+\frac {12\,a^7\,b^2\,c\,d^7}{5}+\frac {21\,a^6\,b^3\,c^2\,d^6}{5}+\frac {42\,a^5\,b^4\,c^3\,d^5}{5}+21\,a^4\,b^5\,c^4\,d^4+84\,a^3\,b^6\,c^5\,d^3-\frac {3087\,a^2\,b^7\,c^6\,d^2}{5}+\frac {4014\,a\,b^8\,c^7\,d}{5}-\frac {3069\,b^9\,c^8}{10}\right )+x^3\,\left (21\,a^6\,b^3\,d^8+42\,a^5\,b^4\,c\,d^7+105\,a^4\,b^5\,c^2\,d^6+420\,a^3\,b^6\,c^3\,d^5-2625\,a^2\,b^7\,c^4\,d^4+3234\,a\,b^8\,c^5\,d^3-1197\,b^9\,c^6\,d^2\right )+x^2\,\left (\frac {36\,a^7\,b^2\,d^8}{5}+\frac {63\,a^6\,b^3\,c\,d^7}{5}+\frac {126\,a^5\,b^4\,c^2\,d^6}{5}+63\,a^4\,b^5\,c^3\,d^5+252\,a^3\,b^6\,c^4\,d^4-\frac {8631\,a^2\,b^7\,c^5\,d^3}{5}+\frac {10962\,a\,b^8\,c^6\,d^2}{5}-\frac {4131\,b^9\,c^7\,d}{5}\right )+x^5\,\left (63\,a^4\,b^5\,d^8+252\,a^3\,b^6\,c\,d^7-1134\,a^2\,b^7\,c^2\,d^6+1260\,a\,b^8\,c^3\,d^5-441\,b^9\,c^4\,d^4\right )+x^4\,\left (42\,a^5\,b^4\,d^8+105\,a^4\,b^5\,c\,d^7+420\,a^3\,b^6\,c^2\,d^6-2310\,a^2\,b^7\,c^3\,d^5+2730\,a\,b^8\,c^4\,d^4-987\,b^9\,c^5\,d^3\right )+x^6\,\left (84\,a^3\,b^6\,d^8-252\,a^2\,b^7\,c\,d^7+252\,a\,b^8\,c^2\,d^6-84\,b^9\,c^3\,d^5\right )}{c^7\,d^9+7\,c^6\,d^{10}\,x+21\,c^5\,d^{11}\,x^2+35\,c^4\,d^{12}\,x^3+35\,c^3\,d^{13}\,x^4+21\,c^2\,d^{14}\,x^5+7\,c\,d^{15}\,x^6+d^{16}\,x^7}+\frac {b^9\,x^2}{2\,d^8}+\frac {\ln \left (c+d\,x\right )\,\left (36\,a^2\,b^7\,d^2-72\,a\,b^8\,c\,d+36\,b^9\,c^2\right )}{d^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^9/(c + d*x)^8,x)

[Out]

x*((9*a*b^8)/d^8 - (8*b^9*c)/d^9) - ((10*a^9*d^9 - 3349*b^9*c^9 - 6534*a^2*b^7*c^7*d^2 + 840*a^3*b^6*c^6*d^3 +
210*a^4*b^5*c^5*d^4 + 84*a^5*b^4*c^4*d^5 + 42*a^6*b^3*c^3*d^6 + 24*a^7*b^2*c^2*d^7 + 8658*a*b^8*c^8*d + 15*a^
8*b*c*d^8)/(70*d) + x*((3*a^8*b*d^8)/2 - (3069*b^9*c^8)/10 + (12*a^7*b^2*c*d^7)/5 - (3087*a^2*b^7*c^6*d^2)/5 +
84*a^3*b^6*c^5*d^3 + 21*a^4*b^5*c^4*d^4 + (42*a^5*b^4*c^3*d^5)/5 + (21*a^6*b^3*c^2*d^6)/5 + (4014*a*b^8*c^7*d
)/5) + x^3*(21*a^6*b^3*d^8 - 1197*b^9*c^6*d^2 + 3234*a*b^8*c^5*d^3 + 42*a^5*b^4*c*d^7 - 2625*a^2*b^7*c^4*d^4 +
420*a^3*b^6*c^3*d^5 + 105*a^4*b^5*c^2*d^6) + x^2*((36*a^7*b^2*d^8)/5 - (4131*b^9*c^7*d)/5 + (10962*a*b^8*c^6*
d^2)/5 + (63*a^6*b^3*c*d^7)/5 - (8631*a^2*b^7*c^5*d^3)/5 + 252*a^3*b^6*c^4*d^4 + 63*a^4*b^5*c^3*d^5 + (126*a^5
*b^4*c^2*d^6)/5) + x^5*(63*a^4*b^5*d^8 - 441*b^9*c^4*d^4 + 1260*a*b^8*c^3*d^5 + 252*a^3*b^6*c*d^7 - 1134*a^2*b
^7*c^2*d^6) + x^4*(42*a^5*b^4*d^8 - 987*b^9*c^5*d^3 + 2730*a*b^8*c^4*d^4 + 105*a^4*b^5*c*d^7 - 2310*a^2*b^7*c^
3*d^5 + 420*a^3*b^6*c^2*d^6) + x^6*(84*a^3*b^6*d^8 - 84*b^9*c^3*d^5 + 252*a*b^8*c^2*d^6 - 252*a^2*b^7*c*d^7))/
(c^7*d^9 + d^16*x^7 + 7*c^6*d^10*x + 7*c*d^15*x^6 + 21*c^5*d^11*x^2 + 35*c^4*d^12*x^3 + 35*c^3*d^13*x^4 + 21*c
^2*d^14*x^5) + (b^9*x^2)/(2*d^8) + (log(c + d*x)*(36*b^9*c^2 + 36*a^2*b^7*d^2 - 72*a*b^8*c*d))/d^10

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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((b*x+a)**9/(d*x+c)**8,x)

[Out]

Timed out

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