3.14.63 \(\int \frac {(a+b x)^8}{(c+d x)^8} \, dx\) [1363]

Optimal. Leaf size=209 \[ \frac {b^8 x}{d^8}-\frac {(b c-a d)^8}{7 d^9 (c+d x)^7}+\frac {4 b (b c-a d)^7}{3 d^9 (c+d x)^6}-\frac {28 b^2 (b c-a d)^6}{5 d^9 (c+d x)^5}+\frac {14 b^3 (b c-a d)^5}{d^9 (c+d x)^4}-\frac {70 b^4 (b c-a d)^4}{3 d^9 (c+d x)^3}+\frac {28 b^5 (b c-a d)^3}{d^9 (c+d x)^2}-\frac {28 b^6 (b c-a d)^2}{d^9 (c+d x)}-\frac {8 b^7 (b c-a d) \log (c+d x)}{d^9} \]

[Out]

b^8*x/d^8-1/7*(-a*d+b*c)^8/d^9/(d*x+c)^7+4/3*b*(-a*d+b*c)^7/d^9/(d*x+c)^6-28/5*b^2*(-a*d+b*c)^6/d^9/(d*x+c)^5+
14*b^3*(-a*d+b*c)^5/d^9/(d*x+c)^4-70/3*b^4*(-a*d+b*c)^4/d^9/(d*x+c)^3+28*b^5*(-a*d+b*c)^3/d^9/(d*x+c)^2-28*b^6
*(-a*d+b*c)^2/d^9/(d*x+c)-8*b^7*(-a*d+b*c)*ln(d*x+c)/d^9

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Rubi [A]
time = 0.19, antiderivative size = 209, 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 {8 b^7 (b c-a d) \log (c+d x)}{d^9}-\frac {28 b^6 (b c-a d)^2}{d^9 (c+d x)}+\frac {28 b^5 (b c-a d)^3}{d^9 (c+d x)^2}-\frac {70 b^4 (b c-a d)^4}{3 d^9 (c+d x)^3}+\frac {14 b^3 (b c-a d)^5}{d^9 (c+d x)^4}-\frac {28 b^2 (b c-a d)^6}{5 d^9 (c+d x)^5}+\frac {4 b (b c-a d)^7}{3 d^9 (c+d x)^6}-\frac {(b c-a d)^8}{7 d^9 (c+d x)^7}+\frac {b^8 x}{d^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^8/(c + d*x)^8,x]

[Out]

(b^8*x)/d^8 - (b*c - a*d)^8/(7*d^9*(c + d*x)^7) + (4*b*(b*c - a*d)^7)/(3*d^9*(c + d*x)^6) - (28*b^2*(b*c - a*d
)^6)/(5*d^9*(c + d*x)^5) + (14*b^3*(b*c - a*d)^5)/(d^9*(c + d*x)^4) - (70*b^4*(b*c - a*d)^4)/(3*d^9*(c + d*x)^
3) + (28*b^5*(b*c - a*d)^3)/(d^9*(c + d*x)^2) - (28*b^6*(b*c - a*d)^2)/(d^9*(c + d*x)) - (8*b^7*(b*c - a*d)*Lo
g[c + d*x])/d^9

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 {(a+b x)^8}{(c+d x)^8} \, dx &=\int \left (\frac {b^8}{d^8}+\frac {(-b c+a d)^8}{d^8 (c+d x)^8}-\frac {8 b (b c-a d)^7}{d^8 (c+d x)^7}+\frac {28 b^2 (b c-a d)^6}{d^8 (c+d x)^6}-\frac {56 b^3 (b c-a d)^5}{d^8 (c+d x)^5}+\frac {70 b^4 (b c-a d)^4}{d^8 (c+d x)^4}-\frac {56 b^5 (b c-a d)^3}{d^8 (c+d x)^3}+\frac {28 b^6 (b c-a d)^2}{d^8 (c+d x)^2}-\frac {8 b^7 (b c-a d)}{d^8 (c+d x)}\right ) \, dx\\ &=\frac {b^8 x}{d^8}-\frac {(b c-a d)^8}{7 d^9 (c+d x)^7}+\frac {4 b (b c-a d)^7}{3 d^9 (c+d x)^6}-\frac {28 b^2 (b c-a d)^6}{5 d^9 (c+d x)^5}+\frac {14 b^3 (b c-a d)^5}{d^9 (c+d x)^4}-\frac {70 b^4 (b c-a d)^4}{3 d^9 (c+d x)^3}+\frac {28 b^5 (b c-a d)^3}{d^9 (c+d x)^2}-\frac {28 b^6 (b c-a d)^2}{d^9 (c+d x)}-\frac {8 b^7 (b c-a d) \log (c+d x)}{d^9}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(474\) vs. \(2(209)=418\).
time = 0.13, size = 474, normalized size = 2.27 \begin {gather*} -\frac {15 a^8 d^8+20 a^7 b d^7 (c+7 d x)+28 a^6 b^2 d^6 \left (c^2+7 c d x+21 d^2 x^2\right )+42 a^5 b^3 d^5 \left (c^3+7 c^2 d x+21 c d^2 x^2+35 d^3 x^3\right )+70 a^4 b^4 d^4 \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 )+140 a^3 b^5 d^3 \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 )+420 a^2 b^6 d^2 \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 )-2 a b^7 c d \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 )+b^8 \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 )+840 b^7 (b c-a d) (c+d x)^7 \log (c+d x)}{105 d^9 (c+d x)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^8/(c + d*x)^8,x]

[Out]

-1/105*(15*a^8*d^8 + 20*a^7*b*d^7*(c + 7*d*x) + 28*a^6*b^2*d^6*(c^2 + 7*c*d*x + 21*d^2*x^2) + 42*a^5*b^3*d^5*(
c^3 + 7*c^2*d*x + 21*c*d^2*x^2 + 35*d^3*x^3) + 70*a^4*b^4*d^4*(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) + 140*a^3*b^5*d^3*(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) + 420*a^2*b^6*d^2*(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) - 2*a*b^7*c*d*(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 + 13
230*c*d^5*x^5 + 2940*d^6*x^6) + b^8*(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) + 840*b^7*(b*c - a*d)*(c + d*x
)^7*Log[c + d*x])/(d^9*(c + d*x)^7)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(575\) vs. \(2(201)=402\).
time = 0.14, size = 576, normalized size = 2.76

method	result	size
risch	\(\frac {b^{8} x}{d^{8}}+\frac {\left (-28 a^{2} b^{6} d^{7}+56 a \,b^{7} c \,d^{6}-28 b^{8} c^{2} d^{5}\right ) x^{6}-28 b^{5} d^{4} \left (a^{3} d^{3}+3 a^{2} b c \,d^{2}-9 a \,b^{2} c^{2} d +5 b^{3} c^{3}\right ) x^{5}-\frac {70 b^{4} d^{3} \left (a^{4} d^{4}+2 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-22 a \,b^{3} c^{3} d +13 b^{4} c^{4}\right ) x^{4}}{3}-\frac {14 b^{3} d^{2} \left (3 a^{5} d^{5}+5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}+30 a^{2} b^{3} c^{3} d^{2}-125 a \,b^{4} c^{4} d +77 b^{5} c^{5}\right ) x^{3}}{3}-\frac {14 b^{2} d \left (2 a^{6} d^{6}+3 a^{5} b c \,d^{5}+5 a^{4} b^{2} c^{2} d^{4}+10 a^{3} b^{3} c^{3} d^{3}+30 a^{2} b^{4} c^{4} d^{2}-137 a \,b^{5} c^{5} d +87 b^{6} c^{6}\right ) x^{2}}{5}-\frac {2 b \left (10 a^{7} d^{7}+14 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}+35 a^{4} b^{3} c^{3} d^{4}+70 a^{3} b^{4} c^{4} d^{3}+210 a^{2} b^{5} c^{5} d^{2}-1029 a \,b^{6} c^{6} d +669 b^{7} c^{7}\right ) x}{15}-\frac {15 a^{8} d^{8}+20 a^{7} b c \,d^{7}+28 a^{6} b^{2} c^{2} d^{6}+42 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} c^{4} d^{4}+140 a^{3} b^{5} c^{5} d^{3}+420 a^{2} b^{6} c^{6} d^{2}-2178 a \,b^{7} c^{7} d +1443 b^{8} c^{8}}{105 d}}{d^{8} \left (d x +c \right )^{7}}+\frac {8 b^{7} \ln \left (d x +c \right ) a}{d^{8}}-\frac {8 b^{8} \ln \left (d x +c \right ) c}{d^{9}}\)	\(570\)
default	\(\frac {b^{8} x}{d^{8}}-\frac {4 b \left (a^{7} d^{7}-7 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}-35 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}-21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d -b^{7} c^{7}\right )}{3 d^{9} \left (d x +c \right )^{6}}-\frac {28 b^{6} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{d^{9} \left (d x +c \right )}-\frac {28 b^{2} \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right )}{5 d^{9} \left (d x +c \right )^{5}}+\frac {8 b^{7} \left (a d -b c \right ) \ln \left (d x +c \right )}{d^{9}}-\frac {70 b^{4} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{3 d^{9} \left (d x +c \right )^{3}}-\frac {14 b^{3} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}{d^{9} \left (d x +c \right )^{4}}-\frac {28 b^{5} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{d^{9} \left (d x +c \right )^{2}}-\frac {a^{8} d^{8}-8 a^{7} b c \,d^{7}+28 a^{6} b^{2} c^{2} d^{6}-56 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} c^{4} d^{4}-56 a^{3} b^{5} c^{5} d^{3}+28 a^{2} b^{6} c^{6} d^{2}-8 a \,b^{7} c^{7} d +b^{8} c^{8}}{7 d^{9} \left (d x +c \right )^{7}}\)	\(576\)
norman	\(\frac {\frac {b^{8} x^{8}}{d}-\frac {15 a^{8} d^{8}+20 a^{7} b c \,d^{7}+28 a^{6} b^{2} c^{2} d^{6}+42 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} c^{4} d^{4}+140 a^{3} b^{5} c^{5} d^{3}+420 a^{2} b^{6} c^{6} d^{2}-2178 a \,b^{7} c^{7} d +2178 b^{8} c^{8}}{105 d^{9}}-\frac {7 \left (4 a^{2} b^{6} d^{2}-8 a \,b^{7} c d +8 b^{8} c^{2}\right ) x^{6}}{d^{3}}-\frac {7 \left (4 a^{3} b^{5} d^{3}+12 a^{2} b^{6} c \,d^{2}-36 a \,b^{7} c^{2} d +36 b^{8} c^{3}\right ) x^{5}}{d^{4}}-\frac {35 \left (2 a^{4} b^{4} d^{4}+4 a^{3} b^{5} c \,d^{3}+12 a^{2} b^{6} c^{2} d^{2}-44 a \,b^{7} c^{3} d +44 b^{8} c^{4}\right ) x^{4}}{3 d^{5}}-\frac {7 \left (6 a^{5} b^{3} d^{5}+10 a^{4} b^{4} c \,d^{4}+20 a^{3} b^{5} c^{2} d^{3}+60 a^{2} b^{6} c^{3} d^{2}-250 a \,b^{7} c^{4} d +250 b^{8} c^{5}\right ) x^{3}}{3 d^{6}}-\frac {7 \left (4 a^{6} b^{2} d^{6}+6 a^{5} b^{3} c \,d^{5}+10 a^{4} b^{4} c^{2} d^{4}+20 a^{3} b^{5} c^{3} d^{3}+60 a^{2} b^{6} c^{4} d^{2}-274 a \,b^{7} c^{5} d +274 b^{8} c^{6}\right ) x^{2}}{5 d^{7}}-\frac {\left (20 a^{7} b \,d^{7}+28 a^{6} b^{2} c \,d^{6}+42 a^{5} b^{3} c^{2} d^{5}+70 a^{4} b^{4} c^{3} d^{4}+140 a^{3} b^{5} c^{4} d^{3}+420 a^{2} b^{6} c^{5} d^{2}-2058 a \,b^{7} c^{6} d +2058 b^{8} c^{7}\right ) x}{15 d^{8}}}{\left (d x +c \right )^{7}}+\frac {8 b^{7} \left (a d -b c \right ) \ln \left (d x +c \right )}{d^{9}}\)	\(577\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^8*x/d^8-4/3*b/d^9*(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)/(d*x+c)^6-28*b^6/d^9*(a^2*d^2-2*a*b*c*d+b^2*c^2)/(d*x+c)-28/5*b^2/d^9*(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)/(d*x+c)^5+8*b^7
/d^9*(a*d-b*c)*ln(d*x+c)-70/3*b^4/d^9*(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)/(d*x+c)^
3-14*b^3/d^9*(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)/(d*x+c)^4-28*
b^5/d^9*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/(d*x+c)^2-1/7/d^9*(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)/(d*x+c)
^7

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 649 vs. \(2 (201) = 402\).
time = 0.35, size = 649, normalized size = 3.11 \begin {gather*} \frac {b^{8} x}{d^{8}} - \frac {1443 \, b^{8} c^{8} - 2178 \, a b^{7} c^{7} d + 420 \, a^{2} b^{6} c^{6} d^{2} + 140 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} + 42 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} + 20 \, a^{7} b c d^{7} + 15 \, a^{8} d^{8} + 2940 \, {\left (b^{8} c^{2} d^{6} - 2 \, a b^{7} c d^{7} + a^{2} b^{6} d^{8}\right )} x^{6} + 2940 \, {\left (5 \, b^{8} c^{3} d^{5} - 9 \, a b^{7} c^{2} d^{6} + 3 \, a^{2} b^{6} c d^{7} + a^{3} b^{5} d^{8}\right )} x^{5} + 2450 \, {\left (13 \, b^{8} c^{4} d^{4} - 22 \, a b^{7} c^{3} d^{5} + 6 \, a^{2} b^{6} c^{2} d^{6} + 2 \, a^{3} b^{5} c d^{7} + a^{4} b^{4} d^{8}\right )} x^{4} + 490 \, {\left (77 \, b^{8} c^{5} d^{3} - 125 \, a b^{7} c^{4} d^{4} + 30 \, a^{2} b^{6} c^{3} d^{5} + 10 \, a^{3} b^{5} c^{2} d^{6} + 5 \, a^{4} b^{4} c d^{7} + 3 \, a^{5} b^{3} d^{8}\right )} x^{3} + 294 \, {\left (87 \, b^{8} c^{6} d^{2} - 137 \, a b^{7} c^{5} d^{3} + 30 \, a^{2} b^{6} c^{4} d^{4} + 10 \, a^{3} b^{5} c^{3} d^{5} + 5 \, a^{4} b^{4} c^{2} d^{6} + 3 \, a^{5} b^{3} c d^{7} + 2 \, a^{6} b^{2} d^{8}\right )} x^{2} + 14 \, {\left (669 \, b^{8} c^{7} d - 1029 \, a b^{7} c^{6} d^{2} + 210 \, a^{2} b^{6} c^{5} d^{3} + 70 \, a^{3} b^{5} c^{4} d^{4} + 35 \, a^{4} b^{4} c^{3} d^{5} + 21 \, a^{5} b^{3} c^{2} d^{6} + 14 \, a^{6} b^{2} c d^{7} + 10 \, a^{7} b d^{8}\right )} x}{105 \, {\left (d^{16} x^{7} + 7 \, c d^{15} x^{6} + 21 \, c^{2} d^{14} x^{5} + 35 \, c^{3} d^{13} x^{4} + 35 \, c^{4} d^{12} x^{3} + 21 \, c^{5} d^{11} x^{2} + 7 \, c^{6} d^{10} x + c^{7} d^{9}\right )}} - \frac {8 \, {\left (b^{8} c - a b^{7} d\right )} \log \left (d x + c\right )}{d^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^8*x/d^8 - 1/105*(1443*b^8*c^8 - 2178*a*b^7*c^7*d + 420*a^2*b^6*c^6*d^2 + 140*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^
4*d^4 + 42*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 + 20*a^7*b*c*d^7 + 15*a^8*d^8 + 2940*(b^8*c^2*d^6 - 2*a*b^7*c*
d^7 + a^2*b^6*d^8)*x^6 + 2940*(5*b^8*c^3*d^5 - 9*a*b^7*c^2*d^6 + 3*a^2*b^6*c*d^7 + a^3*b^5*d^8)*x^5 + 2450*(13
*b^8*c^4*d^4 - 22*a*b^7*c^3*d^5 + 6*a^2*b^6*c^2*d^6 + 2*a^3*b^5*c*d^7 + a^4*b^4*d^8)*x^4 + 490*(77*b^8*c^5*d^3
- 125*a*b^7*c^4*d^4 + 30*a^2*b^6*c^3*d^5 + 10*a^3*b^5*c^2*d^6 + 5*a^4*b^4*c*d^7 + 3*a^5*b^3*d^8)*x^3 + 294*(8
7*b^8*c^6*d^2 - 137*a*b^7*c^5*d^3 + 30*a^2*b^6*c^4*d^4 + 10*a^3*b^5*c^3*d^5 + 5*a^4*b^4*c^2*d^6 + 3*a^5*b^3*c*
d^7 + 2*a^6*b^2*d^8)*x^2 + 14*(669*b^8*c^7*d - 1029*a*b^7*c^6*d^2 + 210*a^2*b^6*c^5*d^3 + 70*a^3*b^5*c^4*d^4 +
35*a^4*b^4*c^3*d^5 + 21*a^5*b^3*c^2*d^6 + 14*a^6*b^2*c*d^7 + 10*a^7*b*d^8)*x)/(d^16*x^7 + 7*c*d^15*x^6 + 21*c
^2*d^14*x^5 + 35*c^3*d^13*x^4 + 35*c^4*d^12*x^3 + 21*c^5*d^11*x^2 + 7*c^6*d^10*x + c^7*d^9) - 8*(b^8*c - a*b^7
*d)*log(d*x + c)/d^9

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 852 vs. \(2 (201) = 402\).
time = 0.48, size = 852, normalized size = 4.08 \begin {gather*} \frac {105 \, b^{8} d^{8} x^{8} + 735 \, b^{8} c d^{7} x^{7} - 1443 \, b^{8} c^{8} + 2178 \, a b^{7} c^{7} d - 420 \, a^{2} b^{6} c^{6} d^{2} - 140 \, a^{3} b^{5} c^{5} d^{3} - 70 \, a^{4} b^{4} c^{4} d^{4} - 42 \, a^{5} b^{3} c^{3} d^{5} - 28 \, a^{6} b^{2} c^{2} d^{6} - 20 \, a^{7} b c d^{7} - 15 \, a^{8} d^{8} - 735 \, {\left (b^{8} c^{2} d^{6} - 8 \, a b^{7} c d^{7} + 4 \, a^{2} b^{6} d^{8}\right )} x^{6} - 735 \, {\left (15 \, b^{8} c^{3} d^{5} - 36 \, a b^{7} c^{2} d^{6} + 12 \, a^{2} b^{6} c d^{7} + 4 \, a^{3} b^{5} d^{8}\right )} x^{5} - 1225 \, {\left (23 \, b^{8} c^{4} d^{4} - 44 \, a b^{7} c^{3} d^{5} + 12 \, a^{2} b^{6} c^{2} d^{6} + 4 \, a^{3} b^{5} c d^{7} + 2 \, a^{4} b^{4} d^{8}\right )} x^{4} - 245 \, {\left (145 \, b^{8} c^{5} d^{3} - 250 \, a b^{7} c^{4} d^{4} + 60 \, a^{2} b^{6} c^{3} d^{5} + 20 \, a^{3} b^{5} c^{2} d^{6} + 10 \, a^{4} b^{4} c d^{7} + 6 \, a^{5} b^{3} d^{8}\right )} x^{3} - 147 \, {\left (169 \, b^{8} c^{6} d^{2} - 274 \, a b^{7} c^{5} d^{3} + 60 \, a^{2} b^{6} c^{4} d^{4} + 20 \, a^{3} b^{5} c^{3} d^{5} + 10 \, a^{4} b^{4} c^{2} d^{6} + 6 \, a^{5} b^{3} c d^{7} + 4 \, a^{6} b^{2} d^{8}\right )} x^{2} - 7 \, {\left (1323 \, b^{8} c^{7} d - 2058 \, a b^{7} c^{6} d^{2} + 420 \, a^{2} b^{6} c^{5} d^{3} + 140 \, a^{3} b^{5} c^{4} d^{4} + 70 \, a^{4} b^{4} c^{3} d^{5} + 42 \, a^{5} b^{3} c^{2} d^{6} + 28 \, a^{6} b^{2} c d^{7} + 20 \, a^{7} b d^{8}\right )} x - 840 \, {\left (b^{8} c^{8} - a b^{7} c^{7} d + {\left (b^{8} c d^{7} - a b^{7} d^{8}\right )} x^{7} + 7 \, {\left (b^{8} c^{2} d^{6} - a b^{7} c d^{7}\right )} x^{6} + 21 \, {\left (b^{8} c^{3} d^{5} - a b^{7} c^{2} d^{6}\right )} x^{5} + 35 \, {\left (b^{8} c^{4} d^{4} - a b^{7} c^{3} d^{5}\right )} x^{4} + 35 \, {\left (b^{8} c^{5} d^{3} - a b^{7} c^{4} d^{4}\right )} x^{3} + 21 \, {\left (b^{8} c^{6} d^{2} - a b^{7} c^{5} d^{3}\right )} x^{2} + 7 \, {\left (b^{8} c^{7} d - a b^{7} c^{6} d^{2}\right )} x\right )} \log \left (d x + c\right )}{105 \, {\left (d^{16} x^{7} + 7 \, c d^{15} x^{6} + 21 \, c^{2} d^{14} x^{5} + 35 \, c^{3} d^{13} x^{4} + 35 \, c^{4} d^{12} x^{3} + 21 \, c^{5} d^{11} x^{2} + 7 \, c^{6} d^{10} x + c^{7} d^{9}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(105*b^8*d^8*x^8 + 735*b^8*c*d^7*x^7 - 1443*b^8*c^8 + 2178*a*b^7*c^7*d - 420*a^2*b^6*c^6*d^2 - 140*a^3*b
^5*c^5*d^3 - 70*a^4*b^4*c^4*d^4 - 42*a^5*b^3*c^3*d^5 - 28*a^6*b^2*c^2*d^6 - 20*a^7*b*c*d^7 - 15*a^8*d^8 - 735*
(b^8*c^2*d^6 - 8*a*b^7*c*d^7 + 4*a^2*b^6*d^8)*x^6 - 735*(15*b^8*c^3*d^5 - 36*a*b^7*c^2*d^6 + 12*a^2*b^6*c*d^7
+ 4*a^3*b^5*d^8)*x^5 - 1225*(23*b^8*c^4*d^4 - 44*a*b^7*c^3*d^5 + 12*a^2*b^6*c^2*d^6 + 4*a^3*b^5*c*d^7 + 2*a^4*
b^4*d^8)*x^4 - 245*(145*b^8*c^5*d^3 - 250*a*b^7*c^4*d^4 + 60*a^2*b^6*c^3*d^5 + 20*a^3*b^5*c^2*d^6 + 10*a^4*b^4
*c*d^7 + 6*a^5*b^3*d^8)*x^3 - 147*(169*b^8*c^6*d^2 - 274*a*b^7*c^5*d^3 + 60*a^2*b^6*c^4*d^4 + 20*a^3*b^5*c^3*d
^5 + 10*a^4*b^4*c^2*d^6 + 6*a^5*b^3*c*d^7 + 4*a^6*b^2*d^8)*x^2 - 7*(1323*b^8*c^7*d - 2058*a*b^7*c^6*d^2 + 420*
a^2*b^6*c^5*d^3 + 140*a^3*b^5*c^4*d^4 + 70*a^4*b^4*c^3*d^5 + 42*a^5*b^3*c^2*d^6 + 28*a^6*b^2*c*d^7 + 20*a^7*b*
d^8)*x - 840*(b^8*c^8 - a*b^7*c^7*d + (b^8*c*d^7 - a*b^7*d^8)*x^7 + 7*(b^8*c^2*d^6 - a*b^7*c*d^7)*x^6 + 21*(b^
8*c^3*d^5 - a*b^7*c^2*d^6)*x^5 + 35*(b^8*c^4*d^4 - a*b^7*c^3*d^5)*x^4 + 35*(b^8*c^5*d^3 - a*b^7*c^4*d^4)*x^3 +
21*(b^8*c^6*d^2 - a*b^7*c^5*d^3)*x^2 + 7*(b^8*c^7*d - a*b^7*c^6*d^2)*x)*log(d*x + c))/(d^16*x^7 + 7*c*d^15*x^
6 + 21*c^2*d^14*x^5 + 35*c^3*d^13*x^4 + 35*c^4*d^12*x^3 + 21*c^5*d^11*x^2 + 7*c^6*d^10*x + c^7*d^9)

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Sympy [F(-1)] Timed out
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)**8/(d*x+c)**8,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 581 vs. \(2 (201) = 402\).
time = 1.69, size = 581, normalized size = 2.78 \begin {gather*} \frac {b^{8} x}{d^{8}} - \frac {8 \, {\left (b^{8} c - a b^{7} d\right )} \log \left ({\left | d x + c \right |}\right )}{d^{9}} - \frac {1443 \, b^{8} c^{8} - 2178 \, a b^{7} c^{7} d + 420 \, a^{2} b^{6} c^{6} d^{2} + 140 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} + 42 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} + 20 \, a^{7} b c d^{7} + 15 \, a^{8} d^{8} + 2940 \, {\left (b^{8} c^{2} d^{6} - 2 \, a b^{7} c d^{7} + a^{2} b^{6} d^{8}\right )} x^{6} + 2940 \, {\left (5 \, b^{8} c^{3} d^{5} - 9 \, a b^{7} c^{2} d^{6} + 3 \, a^{2} b^{6} c d^{7} + a^{3} b^{5} d^{8}\right )} x^{5} + 2450 \, {\left (13 \, b^{8} c^{4} d^{4} - 22 \, a b^{7} c^{3} d^{5} + 6 \, a^{2} b^{6} c^{2} d^{6} + 2 \, a^{3} b^{5} c d^{7} + a^{4} b^{4} d^{8}\right )} x^{4} + 490 \, {\left (77 \, b^{8} c^{5} d^{3} - 125 \, a b^{7} c^{4} d^{4} + 30 \, a^{2} b^{6} c^{3} d^{5} + 10 \, a^{3} b^{5} c^{2} d^{6} + 5 \, a^{4} b^{4} c d^{7} + 3 \, a^{5} b^{3} d^{8}\right )} x^{3} + 294 \, {\left (87 \, b^{8} c^{6} d^{2} - 137 \, a b^{7} c^{5} d^{3} + 30 \, a^{2} b^{6} c^{4} d^{4} + 10 \, a^{3} b^{5} c^{3} d^{5} + 5 \, a^{4} b^{4} c^{2} d^{6} + 3 \, a^{5} b^{3} c d^{7} + 2 \, a^{6} b^{2} d^{8}\right )} x^{2} + 14 \, {\left (669 \, b^{8} c^{7} d - 1029 \, a b^{7} c^{6} d^{2} + 210 \, a^{2} b^{6} c^{5} d^{3} + 70 \, a^{3} b^{5} c^{4} d^{4} + 35 \, a^{4} b^{4} c^{3} d^{5} + 21 \, a^{5} b^{3} c^{2} d^{6} + 14 \, a^{6} b^{2} c d^{7} + 10 \, a^{7} b d^{8}\right )} x}{105 \, {\left (d x + c\right )}^{7} d^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^8*x/d^8 - 8*(b^8*c - a*b^7*d)*log(abs(d*x + c))/d^9 - 1/105*(1443*b^8*c^8 - 2178*a*b^7*c^7*d + 420*a^2*b^6*c
^6*d^2 + 140*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 + 42*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 + 20*a^7*b*c*d^7 +
15*a^8*d^8 + 2940*(b^8*c^2*d^6 - 2*a*b^7*c*d^7 + a^2*b^6*d^8)*x^6 + 2940*(5*b^8*c^3*d^5 - 9*a*b^7*c^2*d^6 + 3
*a^2*b^6*c*d^7 + a^3*b^5*d^8)*x^5 + 2450*(13*b^8*c^4*d^4 - 22*a*b^7*c^3*d^5 + 6*a^2*b^6*c^2*d^6 + 2*a^3*b^5*c*
d^7 + a^4*b^4*d^8)*x^4 + 490*(77*b^8*c^5*d^3 - 125*a*b^7*c^4*d^4 + 30*a^2*b^6*c^3*d^5 + 10*a^3*b^5*c^2*d^6 + 5
*a^4*b^4*c*d^7 + 3*a^5*b^3*d^8)*x^3 + 294*(87*b^8*c^6*d^2 - 137*a*b^7*c^5*d^3 + 30*a^2*b^6*c^4*d^4 + 10*a^3*b^
5*c^3*d^5 + 5*a^4*b^4*c^2*d^6 + 3*a^5*b^3*c*d^7 + 2*a^6*b^2*d^8)*x^2 + 14*(669*b^8*c^7*d - 1029*a*b^7*c^6*d^2
+ 210*a^2*b^6*c^5*d^3 + 70*a^3*b^5*c^4*d^4 + 35*a^4*b^4*c^3*d^5 + 21*a^5*b^3*c^2*d^6 + 14*a^6*b^2*c*d^7 + 10*a
^7*b*d^8)*x)/((d*x + c)^7*d^9)

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Mupad [B]
time = 0.43, size = 649, normalized size = 3.11 \begin {gather*} \frac {b^8\,x}{d^8}-\frac {\ln \left (c+d\,x\right )\,\left (8\,b^8\,c-8\,a\,b^7\,d\right )}{d^9}-\frac {x^4\,\left (\frac {70\,a^4\,b^4\,d^7}{3}+\frac {140\,a^3\,b^5\,c\,d^6}{3}+140\,a^2\,b^6\,c^2\,d^5-\frac {1540\,a\,b^7\,c^3\,d^4}{3}+\frac {910\,b^8\,c^4\,d^3}{3}\right )+x^6\,\left (28\,a^2\,b^6\,d^7-56\,a\,b^7\,c\,d^6+28\,b^8\,c^2\,d^5\right )+\frac {15\,a^8\,d^8+20\,a^7\,b\,c\,d^7+28\,a^6\,b^2\,c^2\,d^6+42\,a^5\,b^3\,c^3\,d^5+70\,a^4\,b^4\,c^4\,d^4+140\,a^3\,b^5\,c^5\,d^3+420\,a^2\,b^6\,c^6\,d^2-2178\,a\,b^7\,c^7\,d+1443\,b^8\,c^8}{105\,d}+x\,\left (\frac {4\,a^7\,b\,d^7}{3}+\frac {28\,a^6\,b^2\,c\,d^6}{15}+\frac {14\,a^5\,b^3\,c^2\,d^5}{5}+\frac {14\,a^4\,b^4\,c^3\,d^4}{3}+\frac {28\,a^3\,b^5\,c^4\,d^3}{3}+28\,a^2\,b^6\,c^5\,d^2-\frac {686\,a\,b^7\,c^6\,d}{5}+\frac {446\,b^8\,c^7}{5}\right )+x^3\,\left (14\,a^5\,b^3\,d^7+\frac {70\,a^4\,b^4\,c\,d^6}{3}+\frac {140\,a^3\,b^5\,c^2\,d^5}{3}+140\,a^2\,b^6\,c^3\,d^4-\frac {1750\,a\,b^7\,c^4\,d^3}{3}+\frac {1078\,b^8\,c^5\,d^2}{3}\right )+x^2\,\left (\frac {28\,a^6\,b^2\,d^7}{5}+\frac {42\,a^5\,b^3\,c\,d^6}{5}+14\,a^4\,b^4\,c^2\,d^5+28\,a^3\,b^5\,c^3\,d^4+84\,a^2\,b^6\,c^4\,d^3-\frac {1918\,a\,b^7\,c^5\,d^2}{5}+\frac {1218\,b^8\,c^6\,d}{5}\right )+x^5\,\left (28\,a^3\,b^5\,d^7+84\,a^2\,b^6\,c\,d^6-252\,a\,b^7\,c^2\,d^5+140\,b^8\,c^3\,d^4\right )}{c^7\,d^8+7\,c^6\,d^9\,x+21\,c^5\,d^{10}\,x^2+35\,c^4\,d^{11}\,x^3+35\,c^3\,d^{12}\,x^4+21\,c^2\,d^{13}\,x^5+7\,c\,d^{14}\,x^6+d^{15}\,x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^8/(c + d*x)^8,x)

[Out]

(b^8*x)/d^8 - (log(c + d*x)*(8*b^8*c - 8*a*b^7*d))/d^9 - (x^4*((70*a^4*b^4*d^7)/3 + (910*b^8*c^4*d^3)/3 - (154
0*a*b^7*c^3*d^4)/3 + (140*a^3*b^5*c*d^6)/3 + 140*a^2*b^6*c^2*d^5) + x^6*(28*a^2*b^6*d^7 + 28*b^8*c^2*d^5 - 56*
a*b^7*c*d^6) + (15*a^8*d^8 + 1443*b^8*c^8 + 420*a^2*b^6*c^6*d^2 + 140*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 + 4
2*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 2178*a*b^7*c^7*d + 20*a^7*b*c*d^7)/(105*d) + x*((446*b^8*c^7)/5 + (4*
a^7*b*d^7)/3 + (28*a^6*b^2*c*d^6)/15 + 28*a^2*b^6*c^5*d^2 + (28*a^3*b^5*c^4*d^3)/3 + (14*a^4*b^4*c^3*d^4)/3 +
(14*a^5*b^3*c^2*d^5)/5 - (686*a*b^7*c^6*d)/5) + x^3*(14*a^5*b^3*d^7 + (1078*b^8*c^5*d^2)/3 - (1750*a*b^7*c^4*d
^3)/3 + (70*a^4*b^4*c*d^6)/3 + 140*a^2*b^6*c^3*d^4 + (140*a^3*b^5*c^2*d^5)/3) + x^2*((1218*b^8*c^6*d)/5 + (28*
a^6*b^2*d^7)/5 - (1918*a*b^7*c^5*d^2)/5 + (42*a^5*b^3*c*d^6)/5 + 84*a^2*b^6*c^4*d^3 + 28*a^3*b^5*c^3*d^4 + 14*
a^4*b^4*c^2*d^5) + x^5*(28*a^3*b^5*d^7 + 140*b^8*c^3*d^4 - 252*a*b^7*c^2*d^5 + 84*a^2*b^6*c*d^6))/(c^7*d^8 + d
^15*x^7 + 7*c^6*d^9*x + 7*c*d^14*x^6 + 21*c^5*d^10*x^2 + 35*c^4*d^11*x^3 + 35*c^3*d^12*x^4 + 21*c^2*d^13*x^5)

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