3.14.16 \(\int \frac {(c+d x)^{10}}{(a+b x)^5} \, dx\) [1316]

Optimal. Leaf size=262 \[ \frac {252 d^5 (b c-a d)^5 x}{b^{10}}-\frac {(b c-a d)^{10}}{4 b^{11} (a+b x)^4}-\frac {10 d (b c-a d)^9}{3 b^{11} (a+b x)^3}-\frac {45 d^2 (b c-a d)^8}{2 b^{11} (a+b x)^2}-\frac {120 d^3 (b c-a d)^7}{b^{11} (a+b x)}+\frac {105 d^6 (b c-a d)^4 (a+b x)^2}{b^{11}}+\frac {40 d^7 (b c-a d)^3 (a+b x)^3}{b^{11}}+\frac {45 d^8 (b c-a d)^2 (a+b x)^4}{4 b^{11}}+\frac {2 d^9 (b c-a d) (a+b x)^5}{b^{11}}+\frac {d^{10} (a+b x)^6}{6 b^{11}}+\frac {210 d^4 (b c-a d)^6 \log (a+b x)}{b^{11}} \]

[Out]

252*d^5*(-a*d+b*c)^5*x/b^10-1/4*(-a*d+b*c)^10/b^11/(b*x+a)^4-10/3*d*(-a*d+b*c)^9/b^11/(b*x+a)^3-45/2*d^2*(-a*d
+b*c)^8/b^11/(b*x+a)^2-120*d^3*(-a*d+b*c)^7/b^11/(b*x+a)+105*d^6*(-a*d+b*c)^4*(b*x+a)^2/b^11+40*d^7*(-a*d+b*c)
^3*(b*x+a)^3/b^11+45/4*d^8*(-a*d+b*c)^2*(b*x+a)^4/b^11+2*d^9*(-a*d+b*c)*(b*x+a)^5/b^11+1/6*d^10*(b*x+a)^6/b^11
+210*d^4*(-a*d+b*c)^6*ln(b*x+a)/b^11

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Rubi [A]
time = 0.29, antiderivative size = 262, 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 {2 d^9 (a+b x)^5 (b c-a d)}{b^{11}}+\frac {45 d^8 (a+b x)^4 (b c-a d)^2}{4 b^{11}}+\frac {40 d^7 (a+b x)^3 (b c-a d)^3}{b^{11}}+\frac {105 d^6 (a+b x)^2 (b c-a d)^4}{b^{11}}+\frac {210 d^4 (b c-a d)^6 \log (a+b x)}{b^{11}}-\frac {120 d^3 (b c-a d)^7}{b^{11} (a+b x)}-\frac {45 d^2 (b c-a d)^8}{2 b^{11} (a+b x)^2}-\frac {10 d (b c-a d)^9}{3 b^{11} (a+b x)^3}-\frac {(b c-a d)^{10}}{4 b^{11} (a+b x)^4}+\frac {d^{10} (a+b x)^6}{6 b^{11}}+\frac {252 d^5 x (b c-a d)^5}{b^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^10/(a + b*x)^5,x]

[Out]

(252*d^5*(b*c - a*d)^5*x)/b^10 - (b*c - a*d)^10/(4*b^11*(a + b*x)^4) - (10*d*(b*c - a*d)^9)/(3*b^11*(a + b*x)^
3) - (45*d^2*(b*c - a*d)^8)/(2*b^11*(a + b*x)^2) - (120*d^3*(b*c - a*d)^7)/(b^11*(a + b*x)) + (105*d^6*(b*c -
a*d)^4*(a + b*x)^2)/b^11 + (40*d^7*(b*c - a*d)^3*(a + b*x)^3)/b^11 + (45*d^8*(b*c - a*d)^2*(a + b*x)^4)/(4*b^1
1) + (2*d^9*(b*c - a*d)*(a + b*x)^5)/b^11 + (d^10*(a + b*x)^6)/(6*b^11) + (210*d^4*(b*c - a*d)^6*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)^5} \, dx &=\int \left (\frac {252 d^5 (b c-a d)^5}{b^{10}}+\frac {(b c-a d)^{10}}{b^{10} (a+b x)^5}+\frac {10 d (b c-a d)^9}{b^{10} (a+b x)^4}+\frac {45 d^2 (b c-a d)^8}{b^{10} (a+b x)^3}+\frac {120 d^3 (b c-a d)^7}{b^{10} (a+b x)^2}+\frac {210 d^4 (b c-a d)^6}{b^{10} (a+b x)}+\frac {210 d^6 (b c-a d)^4 (a+b x)}{b^{10}}+\frac {120 d^7 (b c-a d)^3 (a+b x)^2}{b^{10}}+\frac {45 d^8 (b c-a d)^2 (a+b x)^3}{b^{10}}+\frac {10 d^9 (b c-a d) (a+b x)^4}{b^{10}}+\frac {d^{10} (a+b x)^5}{b^{10}}\right ) \, dx\\ &=\frac {252 d^5 (b c-a d)^5 x}{b^{10}}-\frac {(b c-a d)^{10}}{4 b^{11} (a+b x)^4}-\frac {10 d (b c-a d)^9}{3 b^{11} (a+b x)^3}-\frac {45 d^2 (b c-a d)^8}{2 b^{11} (a+b x)^2}-\frac {120 d^3 (b c-a d)^7}{b^{11} (a+b x)}+\frac {105 d^6 (b c-a d)^4 (a+b x)^2}{b^{11}}+\frac {40 d^7 (b c-a d)^3 (a+b x)^3}{b^{11}}+\frac {45 d^8 (b c-a d)^2 (a+b x)^4}{4 b^{11}}+\frac {2 d^9 (b c-a d) (a+b x)^5}{b^{11}}+\frac {d^{10} (a+b x)^6}{6 b^{11}}+\frac {210 d^4 (b c-a d)^6 \log (a+b x)}{b^{11}}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 359, normalized size = 1.37 \begin {gather*} \frac {12 b d^5 \left (252 b^5 c^5-1050 a b^4 c^4 d+1800 a^2 b^3 c^3 d^2-1575 a^3 b^2 c^2 d^3+700 a^4 b c d^4-126 a^5 d^5\right ) x+30 b^2 d^6 \left (42 b^4 c^4-120 a b^3 c^3 d+135 a^2 b^2 c^2 d^2-70 a^3 b c d^3+14 a^4 d^4\right ) x^2+20 b^3 d^7 \left (24 b^3 c^3-45 a b^2 c^2 d+30 a^2 b c d^2-7 a^3 d^3\right ) x^3+15 b^4 d^8 \left (9 b^2 c^2-10 a b c d+3 a^2 d^2\right ) x^4+12 b^5 d^9 (2 b c-a d) x^5+2 b^6 d^{10} x^6-\frac {3 (b c-a d)^{10}}{(a+b x)^4}+\frac {40 d (-b c+a d)^9}{(a+b x)^3}-\frac {270 d^2 (b c-a d)^8}{(a+b x)^2}+\frac {1440 d^3 (-b c+a d)^7}{a+b x}+2520 d^4 (b c-a d)^6 \log (a+b x)}{12 b^{11}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^10/(a + b*x)^5,x]

[Out]

(12*b*d^5*(252*b^5*c^5 - 1050*a*b^4*c^4*d + 1800*a^2*b^3*c^3*d^2 - 1575*a^3*b^2*c^2*d^3 + 700*a^4*b*c*d^4 - 12
6*a^5*d^5)*x + 30*b^2*d^6*(42*b^4*c^4 - 120*a*b^3*c^3*d + 135*a^2*b^2*c^2*d^2 - 70*a^3*b*c*d^3 + 14*a^4*d^4)*x
^2 + 20*b^3*d^7*(24*b^3*c^3 - 45*a*b^2*c^2*d + 30*a^2*b*c*d^2 - 7*a^3*d^3)*x^3 + 15*b^4*d^8*(9*b^2*c^2 - 10*a*
b*c*d + 3*a^2*d^2)*x^4 + 12*b^5*d^9*(2*b*c - a*d)*x^5 + 2*b^6*d^10*x^6 - (3*(b*c - a*d)^10)/(a + b*x)^4 + (40*
d*(-(b*c) + a*d)^9)/(a + b*x)^3 - (270*d^2*(b*c - a*d)^8)/(a + b*x)^2 + (1440*d^3*(-(b*c) + a*d)^7)/(a + b*x)
+ 2520*d^4*(b*c - a*d)^6*Log[a + b*x])/(12*b^11)

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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(1062\) vs. \(2(262)=524\).
time = 196.77, size = 1060, normalized size = 4.05 \begin {gather*} \frac {1207 a^{10} d^{10}-8250 a^9 b c d^9+23985 a^8 b^2 c^2 d^8-38280 a^7 b^3 c^3 d^7+35910 a^6 b^4 c^4 d^6-19404 a^5 b^5 c^5 d^5+5250 a^4 b^6 c^6 d^4-360 a^3 b^7 c^7 d^3-45 a^2 b^8 c^8 d^2-10 a b^9 c^9 d-3 b^{10} c^{10}-20 b d x \left (-191 a^9 d^9+1314 a^8 b c d^8-3852 a^7 b^2 c^2 d^7+6216 a^6 b^3 c^3 d^6-5922 a^5 b^4 c^4 d^5+3276 a^4 b^5 c^5 d^4-924 a^3 b^6 c^6 d^3+72 a^2 b^7 c^7 d^2+9 a b^8 c^8 d+2 b^9 c^9\right )+270 b^2 d^2 x^2 \left (15 a^8 d^8-104 a^7 b c d^7+308 a^6 b^2 c^2 d^6-504 a^5 b^3 c^3 d^5+490 a^4 b^4 c^4 d^4-280 a^3 b^5 c^5 d^3+84 a^2 b^6 c^6 d^2-8 a b^7 c^7 d-b^8 c^8\right )+1440 b^3 d^3 x^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 )+2520 d^4 \text {Log}\left [a+b x\right ] \left (a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4\right ) \left (a d-b c\right )^6-12 b d^5 x \left (a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4\right ) \left (126 a^5 d^5-700 a^4 b c d^4+1575 a^3 b^2 c^2 d^3-1800 a^2 b^3 c^3 d^2+1050 a b^4 c^4 d-252 b^5 c^5\right )+30 b^2 d^6 x^2 \left (a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4\right ) \left (14 a^4 d^4-70 a^3 b c d^3+135 a^2 b^2 c^2 d^2-120 a b^3 c^3 d+42 b^4 c^4\right )-20 b^3 d^7 x^3 \left (a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4\right ) \left (7 a^3 d^3-30 a^2 b c d^2+45 a b^2 c^2 d-24 b^3 c^3\right )+15 b^4 d^8 x^4 \left (a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4\right ) \left (3 a^2 d^2-10 a b c d+9 b^2 c^2\right )-12 b^5 d^9 x^5 \left (a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4\right ) \left (a d-2 b c\right )+2 b^6 d^{10} x^6 \left (a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4\right )}{12 b^{11} \left (a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[(c + d*x)^10/(a + b*x)^5,x]')

[Out]

(1207 a ^ 10 d ^ 10 - 8250 a ^ 9 b c d ^ 9 + 23985 a ^ 8 b ^ 2 c ^ 2 d ^ 8 - 38280 a ^ 7 b ^ 3 c ^ 3 d ^ 7 + 3
5910 a ^ 6 b ^ 4 c ^ 4 d ^ 6 - 19404 a ^ 5 b ^ 5 c ^ 5 d ^ 5 + 5250 a ^ 4 b ^ 6 c ^ 6 d ^ 4 - 360 a ^ 3 b ^ 7
c ^ 7 d ^ 3 - 45 a ^ 2 b ^ 8 c ^ 8 d ^ 2 - 10 a b ^ 9 c ^ 9 d - 3 b ^ 10 c ^ 10 - 20 b d x (-191 a ^ 9 d ^ 9 +
1314 a ^ 8 b c d ^ 8 - 3852 a ^ 7 b ^ 2 c ^ 2 d ^ 7 + 6216 a ^ 6 b ^ 3 c ^ 3 d ^ 6 - 5922 a ^ 5 b ^ 4 c ^ 4 d
^ 5 + 3276 a ^ 4 b ^ 5 c ^ 5 d ^ 4 - 924 a ^ 3 b ^ 6 c ^ 6 d ^ 3 + 72 a ^ 2 b ^ 7 c ^ 7 d ^ 2 + 9 a b ^ 8 c ^
8 d + 2 b ^ 9 c ^ 9) + 270 b ^ 2 d ^ 2 x ^ 2 (15 a ^ 8 d ^ 8 - 104 a ^ 7 b c d ^ 7 + 308 a ^ 6 b ^ 2 c ^ 2 d
^ 6 - 504 a ^ 5 b ^ 3 c ^ 3 d ^ 5 + 490 a ^ 4 b ^ 4 c ^ 4 d ^ 4 - 280 a ^ 3 b ^ 5 c ^ 5 d ^ 3 + 84 a ^ 2 b ^ 6
c ^ 6 d ^ 2 - 8 a b ^ 7 c ^ 7 d - b ^ 8 c ^ 8) + 1440 b ^ 3 d ^ 3 x ^ 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) + 2520 d ^ 4 Log[a + b x] (a ^ 4 + 4 a ^ 3 b x + 6 a ^ 2 b ^ 2 x ^ 2 + 4
a b ^ 3 x ^ 3 + b ^ 4 x ^ 4) (a d - b c) ^ 6 - 12 b d ^ 5 x (a ^ 4 + 4 a ^ 3 b x + 6 a ^ 2 b ^ 2 x ^ 2 + 4 a b
^ 3 x ^ 3 + b ^ 4 x ^ 4) (126 a ^ 5 d ^ 5 - 700 a ^ 4 b c d ^ 4 + 1575 a ^ 3 b ^ 2 c ^ 2 d ^ 3 - 1800 a ^ 2 b
^ 3 c ^ 3 d ^ 2 + 1050 a b ^ 4 c ^ 4 d - 252 b ^ 5 c ^ 5) + 30 b ^ 2 d ^ 6 x ^ 2 (a ^ 4 + 4 a ^ 3 b x + 6 a ^
2 b ^ 2 x ^ 2 + 4 a b ^ 3 x ^ 3 + b ^ 4 x ^ 4) (14 a ^ 4 d ^ 4 - 70 a ^ 3 b c d ^ 3 + 135 a ^ 2 b ^ 2 c ^ 2 d
^ 2 - 120 a b ^ 3 c ^ 3 d + 42 b ^ 4 c ^ 4) - 20 b ^ 3 d ^ 7 x ^ 3 (a ^ 4 + 4 a ^ 3 b x + 6 a ^ 2 b ^ 2 x ^ 2
+ 4 a b ^ 3 x ^ 3 + b ^ 4 x ^ 4) (7 a ^ 3 d ^ 3 - 30 a ^ 2 b c d ^ 2 + 45 a b ^ 2 c ^ 2 d - 24 b ^ 3 c ^ 3) +
15 b ^ 4 d ^ 8 x ^ 4 (a ^ 4 + 4 a ^ 3 b x + 6 a ^ 2 b ^ 2 x ^ 2 + 4 a b ^ 3 x ^ 3 + b ^ 4 x ^ 4) (3 a ^ 2 d ^
2 - 10 a b c d + 9 b ^ 2 c ^ 2) - 12 b ^ 5 d ^ 9 x ^ 5 (a ^ 4 + 4 a ^ 3 b x + 6 a ^ 2 b ^ 2 x ^ 2 + 4 a b ^ 3
x ^ 3 + b ^ 4 x ^ 4) (a d - 2 b c) + 2 b ^ 6 d ^ 10 x ^ 6 (a ^ 4 + 4 a ^ 3 b x + 6 a ^ 2 b ^ 2 x ^ 2 + 4 a b
^ 3 x ^ 3 + b ^ 4 x ^ 4)) / (12 b ^ 11 (a ^ 4 + 4 a ^ 3 b x + 6 a ^ 2 b ^ 2 x ^ 2 + 4 a b ^ 3 x ^ 3 + b ^ 4 x
^ 4))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(880\) vs. \(2(252)=504\).
time = 0.15, size = 881, normalized size = 3.36

method	result	size
norman	\(\frac {\frac {5250 a^{10} d^{10}-31500 a^{9} b c \,d^{9}+78750 a^{8} b^{2} c^{2} d^{8}-105000 a^{7} b^{3} c^{3} d^{7}+78750 a^{6} b^{4} c^{4} d^{6}-31500 a^{5} b^{5} c^{5} d^{5}+5250 a^{4} b^{6} c^{6} d^{4}-360 a^{3} b^{7} c^{7} d^{3}-45 a^{2} b^{8} c^{8} d^{2}-10 a \,b^{9} c^{9} d -3 b^{10} c^{10}}{12 b^{11}}+\frac {d^{10} x^{10}}{6 b}+\frac {4 \left (210 a^{7} d^{10}-1260 a^{6} b c \,d^{9}+3150 a^{5} b^{2} c^{2} d^{8}-4200 a^{4} b^{3} c^{3} d^{7}+3150 a^{3} b^{4} c^{4} d^{6}-1260 a^{2} b^{5} c^{5} d^{5}+210 a \,b^{6} c^{6} d^{4}-30 b^{7} c^{7} d^{3}\right ) x^{3}}{b^{8}}+\frac {3 \left (1260 a^{8} d^{10}-7560 a^{7} b c \,d^{9}+18900 a^{6} b^{2} c^{2} d^{8}-25200 a^{5} b^{3} c^{3} d^{7}+18900 a^{4} b^{4} c^{4} d^{6}-7560 a^{3} b^{5} c^{5} d^{5}+1260 b^{6} a^{2} c^{6} d^{4}-120 a \,b^{7} c^{7} d^{3}-15 b^{8} c^{8} d^{2}\right ) x^{2}}{2 b^{9}}+\frac {\left (4620 a^{9} d^{10}-27720 a^{8} b c \,d^{9}+69300 a^{7} b^{2} c^{2} d^{8}-92400 a^{6} b^{3} c^{3} d^{7}+69300 a^{5} b^{4} c^{4} d^{6}-27720 a^{4} b^{5} c^{5} d^{5}+4620 a^{3} b^{6} c^{6} d^{4}-360 a^{2} b^{7} c^{7} d^{3}-45 a \,b^{8} c^{8} d^{2}-10 b^{9} c^{9} d \right ) x}{3 b^{10}}-\frac {42 d^{5} \left (a^{5} d^{5}-6 a^{4} b c \,d^{4}+15 a^{3} b^{2} c^{2} d^{3}-20 a^{2} b^{3} c^{3} d^{2}+15 a \,b^{4} c^{4} d -6 b^{5} c^{5}\right ) x^{5}}{b^{6}}+\frac {7 d^{6} \left (a^{4} d^{4}-6 a^{3} b c \,d^{3}+15 a^{2} b^{2} c^{2} d^{2}-20 a \,b^{3} c^{3} d +15 b^{4} c^{4}\right ) x^{6}}{b^{5}}-\frac {2 d^{7} \left (a^{3} d^{3}-6 a^{2} b c \,d^{2}+15 a \,b^{2} c^{2} d -20 b^{3} c^{3}\right ) x^{7}}{b^{4}}+\frac {3 d^{8} \left (a^{2} d^{2}-6 a b c d +15 b^{2} c^{2}\right ) x^{8}}{4 b^{3}}-\frac {d^{9} \left (a d -6 b c \right ) x^{9}}{3 b^{2}}}{\left (b x +a \right )^{4}}+\frac {210 d^{4} \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right ) \ln \left (b x +a \right )}{b^{11}}\)	\(844\)
default	\(-\frac {d^{5} \left (-\frac {1}{6} d^{5} x^{6} b^{5}+a \,b^{4} d^{5} x^{5}-2 b^{5} c \,d^{4} x^{5}-\frac {15}{4} a^{2} b^{3} d^{5} x^{4}+\frac {25}{2} a \,b^{4} c \,d^{4} x^{4}-\frac {45}{4} b^{5} c^{2} d^{3} x^{4}+\frac {35}{3} a^{3} b^{2} d^{5} x^{3}-50 a^{2} b^{3} c \,d^{4} x^{3}+75 a \,b^{4} c^{2} d^{3} x^{3}-40 b^{5} c^{3} d^{2} x^{3}-35 a^{4} b \,d^{5} x^{2}+175 a^{3} b^{2} c \,d^{4} x^{2}-\frac {675}{2} a^{2} b^{3} c^{2} d^{3} x^{2}+300 a \,b^{4} c^{3} d^{2} x^{2}-105 b^{5} c^{4} d \,x^{2}+126 a^{5} d^{5} x -700 a^{4} b c \,d^{4} x +1575 a^{3} b^{2} c^{2} d^{3} x -1800 a^{2} b^{3} c^{3} d^{2} x +1050 a \,b^{4} c^{4} d x -252 b^{5} c^{5} x \right )}{b^{10}}+\frac {120 d^{3} \left (a^{7} d^{7}-7 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}-35 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}-21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d -b^{7} c^{7}\right )}{b^{11} \left (b x +a \right )}-\frac {a^{10} d^{10}-10 a^{9} b c \,d^{9}+45 a^{8} b^{2} c^{2} d^{8}-120 a^{7} b^{3} c^{3} d^{7}+210 a^{6} b^{4} c^{4} d^{6}-252 a^{5} b^{5} c^{5} d^{5}+210 a^{4} b^{6} c^{6} d^{4}-120 a^{3} b^{7} c^{7} d^{3}+45 a^{2} b^{8} c^{8} d^{2}-10 a \,b^{9} c^{9} d +b^{10} c^{10}}{4 b^{11} \left (b x +a \right )^{4}}-\frac {45 d^{2} \left (a^{8} d^{8}-8 a^{7} b c \,d^{7}+28 a^{6} b^{2} c^{2} d^{6}-56 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} c^{4} d^{4}-56 a^{3} b^{5} c^{5} d^{3}+28 a^{2} b^{6} c^{6} d^{2}-8 a \,b^{7} c^{7} d +b^{8} c^{8}\right )}{2 b^{11} \left (b x +a \right )^{2}}+\frac {210 d^{4} \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right ) \ln \left (b x +a \right )}{b^{11}}+\frac {10 d \left (a^{9} d^{9}-9 a^{8} b c \,d^{8}+36 a^{7} b^{2} c^{2} d^{7}-84 a^{6} b^{3} c^{3} d^{6}+126 a^{5} b^{4} c^{4} d^{5}-126 a^{4} b^{5} c^{5} d^{4}+84 a^{3} b^{6} c^{6} d^{3}-36 a^{2} b^{7} c^{7} d^{2}+9 a \,b^{8} c^{8} d -b^{9} c^{9}\right )}{3 b^{11} \left (b x +a \right )^{3}}\)	\(881\)
risch	\(\frac {d^{10} x^{6}}{6 b^{5}}-\frac {25 d^{9} a c \,x^{4}}{2 b^{6}}+\frac {\left (120 a^{7} b^{2} d^{10}-840 a^{6} b^{3} c \,d^{9}+2520 a^{5} b^{4} c^{2} d^{8}-4200 a^{4} b^{5} c^{3} d^{7}+4200 a^{3} b^{6} c^{4} d^{6}-2520 a^{2} b^{7} c^{5} d^{5}+840 a \,b^{8} c^{6} d^{4}-120 b^{9} c^{7} d^{3}\right ) x^{3}+\frac {45 b \,d^{2} \left (15 a^{8} d^{8}-104 a^{7} b c \,d^{7}+308 a^{6} b^{2} c^{2} d^{6}-504 a^{5} b^{3} c^{3} d^{5}+490 a^{4} b^{4} c^{4} d^{4}-280 a^{3} b^{5} c^{5} d^{3}+84 a^{2} b^{6} c^{6} d^{2}-8 a \,b^{7} c^{7} d -b^{8} c^{8}\right ) x^{2}}{2}+\frac {5 d \left (191 a^{9} d^{9}-1314 a^{8} b c \,d^{8}+3852 a^{7} b^{2} c^{2} d^{7}-6216 a^{6} b^{3} c^{3} d^{6}+5922 a^{5} b^{4} c^{4} d^{5}-3276 a^{4} b^{5} c^{5} d^{4}+924 a^{3} b^{6} c^{6} d^{3}-72 a^{2} b^{7} c^{7} d^{2}-9 a \,b^{8} c^{8} d -2 b^{9} c^{9}\right ) x}{3}+\frac {1207 a^{10} d^{10}-8250 a^{9} b c \,d^{9}+23985 a^{8} b^{2} c^{2} d^{8}-38280 a^{7} b^{3} c^{3} d^{7}+35910 a^{6} b^{4} c^{4} d^{6}-19404 a^{5} b^{5} c^{5} d^{5}+5250 a^{4} b^{6} c^{6} d^{4}-360 a^{3} b^{7} c^{7} d^{3}-45 a^{2} b^{8} c^{8} d^{2}-10 a \,b^{9} c^{9} d -3 b^{10} c^{10}}{12 b}}{b^{10} \left (b x +a \right )^{4}}+\frac {50 d^{9} a^{2} c \,x^{3}}{b^{7}}-\frac {75 d^{8} a \,c^{2} x^{3}}{b^{6}}-\frac {175 d^{9} a^{3} c \,x^{2}}{b^{8}}+\frac {675 d^{8} a^{2} c^{2} x^{2}}{2 b^{7}}-\frac {300 d^{7} a \,c^{3} x^{2}}{b^{6}}+\frac {700 d^{9} a^{4} c x}{b^{9}}-\frac {1575 d^{8} a^{3} c^{2} x}{b^{8}}+\frac {1800 d^{7} a^{2} c^{3} x}{b^{7}}-\frac {1050 d^{6} a \,c^{4} x}{b^{6}}-\frac {1260 d^{9} \ln \left (b x +a \right ) a^{5} c}{b^{10}}+\frac {3150 d^{8} \ln \left (b x +a \right ) a^{4} c^{2}}{b^{9}}-\frac {4200 d^{7} \ln \left (b x +a \right ) a^{3} c^{3}}{b^{8}}+\frac {3150 d^{6} \ln \left (b x +a \right ) a^{2} c^{4}}{b^{7}}-\frac {1260 d^{5} \ln \left (b x +a \right ) a \,c^{5}}{b^{6}}+\frac {210 d^{10} \ln \left (b x +a \right ) a^{6}}{b^{11}}+\frac {210 d^{4} \ln \left (b x +a \right ) c^{6}}{b^{5}}-\frac {d^{10} a \,x^{5}}{b^{6}}+\frac {2 d^{9} c \,x^{5}}{b^{5}}+\frac {15 d^{10} a^{2} x^{4}}{4 b^{7}}+\frac {45 d^{8} c^{2} x^{4}}{4 b^{5}}-\frac {35 d^{10} a^{3} x^{3}}{3 b^{8}}+\frac {40 d^{7} c^{3} x^{3}}{b^{5}}+\frac {35 d^{10} a^{4} x^{2}}{b^{9}}+\frac {105 d^{6} c^{4} x^{2}}{b^{5}}-\frac {126 d^{10} a^{5} x}{b^{10}}+\frac {252 d^{5} c^{5} x}{b^{5}}\)	\(921\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^10/(b*x+a)^5,x,method=_RETURNVERBOSE)

[Out]

-d^5/b^10*(-1/6*d^5*x^6*b^5+a*b^4*d^5*x^5-2*b^5*c*d^4*x^5-15/4*a^2*b^3*d^5*x^4+25/2*a*b^4*c*d^4*x^4-45/4*b^5*c
^2*d^3*x^4+35/3*a^3*b^2*d^5*x^3-50*a^2*b^3*c*d^4*x^3+75*a*b^4*c^2*d^3*x^3-40*b^5*c^3*d^2*x^3-35*a^4*b*d^5*x^2+
175*a^3*b^2*c*d^4*x^2-675/2*a^2*b^3*c^2*d^3*x^2+300*a*b^4*c^3*d^2*x^2-105*b^5*c^4*d*x^2+126*a^5*d^5*x-700*a^4*
b*c*d^4*x+1575*a^3*b^2*c^2*d^3*x-1800*a^2*b^3*c^3*d^2*x+1050*a*b^4*c^4*d*x-252*b^5*c^5*x)+120/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)-1/4*(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)^
4-45/2/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)^2+210/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)*ln(b*x+a)+10/3/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)^3

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 903 vs. \(2 (252) = 504\).
time = 0.29, size = 903, normalized size = 3.45 \begin {gather*} -\frac {3 \, b^{10} c^{10} + 10 \, a b^{9} c^{9} d + 45 \, a^{2} b^{8} c^{8} d^{2} + 360 \, a^{3} b^{7} c^{7} d^{3} - 5250 \, a^{4} b^{6} c^{6} d^{4} + 19404 \, a^{5} b^{5} c^{5} d^{5} - 35910 \, a^{6} b^{4} c^{4} d^{6} + 38280 \, a^{7} b^{3} c^{3} d^{7} - 23985 \, a^{8} b^{2} c^{2} d^{8} + 8250 \, a^{9} b c d^{9} - 1207 \, a^{10} d^{10} + 1440 \, {\left (b^{10} c^{7} d^{3} - 7 \, a b^{9} c^{6} d^{4} + 21 \, a^{2} b^{8} c^{5} d^{5} - 35 \, a^{3} b^{7} c^{4} d^{6} + 35 \, a^{4} b^{6} c^{3} d^{7} - 21 \, a^{5} b^{5} c^{2} d^{8} + 7 \, a^{6} b^{4} c d^{9} - a^{7} b^{3} d^{10}\right )} x^{3} + 270 \, {\left (b^{10} c^{8} d^{2} + 8 \, a b^{9} c^{7} d^{3} - 84 \, a^{2} b^{8} c^{6} d^{4} + 280 \, a^{3} b^{7} c^{5} d^{5} - 490 \, a^{4} b^{6} c^{4} d^{6} + 504 \, a^{5} b^{5} c^{3} d^{7} - 308 \, a^{6} b^{4} c^{2} d^{8} + 104 \, a^{7} b^{3} c d^{9} - 15 \, a^{8} b^{2} d^{10}\right )} x^{2} + 20 \, {\left (2 \, b^{10} c^{9} d + 9 \, a b^{9} c^{8} d^{2} + 72 \, a^{2} b^{8} c^{7} d^{3} - 924 \, a^{3} b^{7} c^{6} d^{4} + 3276 \, a^{4} b^{6} c^{5} d^{5} - 5922 \, a^{5} b^{5} c^{4} d^{6} + 6216 \, a^{6} b^{4} c^{3} d^{7} - 3852 \, a^{7} b^{3} c^{2} d^{8} + 1314 \, a^{8} b^{2} c d^{9} - 191 \, a^{9} b d^{10}\right )} x}{12 \, {\left (b^{15} x^{4} + 4 \, a b^{14} x^{3} + 6 \, a^{2} b^{13} x^{2} + 4 \, a^{3} b^{12} x + a^{4} b^{11}\right )}} + \frac {2 \, b^{5} d^{10} x^{6} + 12 \, {\left (2 \, b^{5} c d^{9} - a b^{4} d^{10}\right )} x^{5} + 15 \, {\left (9 \, b^{5} c^{2} d^{8} - 10 \, a b^{4} c d^{9} + 3 \, a^{2} b^{3} d^{10}\right )} x^{4} + 20 \, {\left (24 \, b^{5} c^{3} d^{7} - 45 \, a b^{4} c^{2} d^{8} + 30 \, a^{2} b^{3} c d^{9} - 7 \, a^{3} b^{2} d^{10}\right )} x^{3} + 30 \, {\left (42 \, b^{5} c^{4} d^{6} - 120 \, a b^{4} c^{3} d^{7} + 135 \, a^{2} b^{3} c^{2} d^{8} - 70 \, a^{3} b^{2} c d^{9} + 14 \, a^{4} b d^{10}\right )} x^{2} + 12 \, {\left (252 \, b^{5} c^{5} d^{5} - 1050 \, a b^{4} c^{4} d^{6} + 1800 \, a^{2} b^{3} c^{3} d^{7} - 1575 \, a^{3} b^{2} c^{2} d^{8} + 700 \, a^{4} b c d^{9} - 126 \, a^{5} d^{10}\right )} x}{12 \, b^{10}} + \frac {210 \, {\left (b^{6} c^{6} d^{4} - 6 \, a b^{5} c^{5} d^{5} + 15 \, a^{2} b^{4} c^{4} d^{6} - 20 \, a^{3} b^{3} c^{3} d^{7} + 15 \, a^{4} b^{2} c^{2} d^{8} - 6 \, a^{5} b c d^{9} + a^{6} 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)^5,x, algorithm="maxima")

[Out]

-1/12*(3*b^10*c^10 + 10*a*b^9*c^9*d + 45*a^2*b^8*c^8*d^2 + 360*a^3*b^7*c^7*d^3 - 5250*a^4*b^6*c^6*d^4 + 19404*
a^5*b^5*c^5*d^5 - 35910*a^6*b^4*c^4*d^6 + 38280*a^7*b^3*c^3*d^7 - 23985*a^8*b^2*c^2*d^8 + 8250*a^9*b*c*d^9 - 1
207*a^10*d^10 + 1440*(b^10*c^7*d^3 - 7*a*b^9*c^6*d^4 + 21*a^2*b^8*c^5*d^5 - 35*a^3*b^7*c^4*d^6 + 35*a^4*b^6*c^
3*d^7 - 21*a^5*b^5*c^2*d^8 + 7*a^6*b^4*c*d^9 - a^7*b^3*d^10)*x^3 + 270*(b^10*c^8*d^2 + 8*a*b^9*c^7*d^3 - 84*a^
2*b^8*c^6*d^4 + 280*a^3*b^7*c^5*d^5 - 490*a^4*b^6*c^4*d^6 + 504*a^5*b^5*c^3*d^7 - 308*a^6*b^4*c^2*d^8 + 104*a^
7*b^3*c*d^9 - 15*a^8*b^2*d^10)*x^2 + 20*(2*b^10*c^9*d + 9*a*b^9*c^8*d^2 + 72*a^2*b^8*c^7*d^3 - 924*a^3*b^7*c^6
*d^4 + 3276*a^4*b^6*c^5*d^5 - 5922*a^5*b^5*c^4*d^6 + 6216*a^6*b^4*c^3*d^7 - 3852*a^7*b^3*c^2*d^8 + 1314*a^8*b^
2*c*d^9 - 191*a^9*b*d^10)*x)/(b^15*x^4 + 4*a*b^14*x^3 + 6*a^2*b^13*x^2 + 4*a^3*b^12*x + a^4*b^11) + 1/12*(2*b^
5*d^10*x^6 + 12*(2*b^5*c*d^9 - a*b^4*d^10)*x^5 + 15*(9*b^5*c^2*d^8 - 10*a*b^4*c*d^9 + 3*a^2*b^3*d^10)*x^4 + 20
*(24*b^5*c^3*d^7 - 45*a*b^4*c^2*d^8 + 30*a^2*b^3*c*d^9 - 7*a^3*b^2*d^10)*x^3 + 30*(42*b^5*c^4*d^6 - 120*a*b^4*
c^3*d^7 + 135*a^2*b^3*c^2*d^8 - 70*a^3*b^2*c*d^9 + 14*a^4*b*d^10)*x^2 + 12*(252*b^5*c^5*d^5 - 1050*a*b^4*c^4*d
^6 + 1800*a^2*b^3*c^3*d^7 - 1575*a^3*b^2*c^2*d^8 + 700*a^4*b*c*d^9 - 126*a^5*d^10)*x)/b^10 + 210*(b^6*c^6*d^4
- 6*a*b^5*c^5*d^5 + 15*a^2*b^4*c^4*d^6 - 20*a^3*b^3*c^3*d^7 + 15*a^4*b^2*c^2*d^8 - 6*a^5*b*c*d^9 + a^6*d^10)*l
og(b*x + a)/b^11

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1365 vs. \(2 (252) = 504\).
time = 0.30, size = 1365, normalized size = 5.21

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^5,x, algorithm="fricas")

[Out]

1/12*(2*b^10*d^10*x^10 - 3*b^10*c^10 - 10*a*b^9*c^9*d - 45*a^2*b^8*c^8*d^2 - 360*a^3*b^7*c^7*d^3 + 5250*a^4*b^
6*c^6*d^4 - 19404*a^5*b^5*c^5*d^5 + 35910*a^6*b^4*c^4*d^6 - 38280*a^7*b^3*c^3*d^7 + 23985*a^8*b^2*c^2*d^8 - 82
50*a^9*b*c*d^9 + 1207*a^10*d^10 + 4*(6*b^10*c*d^9 - a*b^9*d^10)*x^9 + 9*(15*b^10*c^2*d^8 - 6*a*b^9*c*d^9 + a^2
*b^8*d^10)*x^8 + 24*(20*b^10*c^3*d^7 - 15*a*b^9*c^2*d^8 + 6*a^2*b^8*c*d^9 - a^3*b^7*d^10)*x^7 + 84*(15*b^10*c^
4*d^6 - 20*a*b^9*c^3*d^7 + 15*a^2*b^8*c^2*d^8 - 6*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 504*(6*b^10*c^5*d^5 - 15
*a*b^9*c^4*d^6 + 20*a^2*b^8*c^3*d^7 - 15*a^3*b^7*c^2*d^8 + 6*a^4*b^6*c*d^9 - a^5*b^5*d^10)*x^5 + (12096*a*b^9*
c^5*d^5 - 42840*a^2*b^8*c^4*d^6 + 66720*a^3*b^7*c^3*d^7 - 54765*a^4*b^6*c^2*d^8 + 23250*a^5*b^5*c*d^9 - 4043*a
^6*b^4*d^10)*x^4 - 4*(360*b^10*c^7*d^3 - 2520*a*b^9*c^6*d^4 + 3024*a^2*b^8*c^5*d^5 + 5040*a^3*b^7*c^4*d^6 - 16
320*a^4*b^6*c^3*d^7 + 16965*a^5*b^5*c^2*d^8 - 8130*a^6*b^4*c*d^9 + 1523*a^7*b^3*d^10)*x^3 - 6*(45*b^10*c^8*d^2
+ 360*a*b^9*c^7*d^3 - 3780*a^2*b^8*c^6*d^4 + 10584*a^3*b^7*c^5*d^5 - 13860*a^4*b^6*c^4*d^6 + 8880*a^5*b^5*c^3
*d^7 - 1935*a^6*b^4*c^2*d^8 - 570*a^7*b^3*c*d^9 + 263*a^8*b^2*d^10)*x^2 - 4*(10*b^10*c^9*d + 45*a*b^9*c^8*d^2
+ 360*a^2*b^8*c^7*d^3 - 4620*a^3*b^7*c^6*d^4 + 15624*a^4*b^6*c^5*d^5 - 26460*a^5*b^5*c^4*d^6 + 25680*a^6*b^4*c
^3*d^7 - 14535*a^7*b^3*c^2*d^8 + 4470*a^8*b^2*c*d^9 - 577*a^9*b*d^10)*x + 2520*(a^4*b^6*c^6*d^4 - 6*a^5*b^5*c^
5*d^5 + 15*a^6*b^4*c^4*d^6 - 20*a^7*b^3*c^3*d^7 + 15*a^8*b^2*c^2*d^8 - 6*a^9*b*c*d^9 + a^10*d^10 + (b^10*c^6*d
^4 - 6*a*b^9*c^5*d^5 + 15*a^2*b^8*c^4*d^6 - 20*a^3*b^7*c^3*d^7 + 15*a^4*b^6*c^2*d^8 - 6*a^5*b^5*c*d^9 + a^6*b^
4*d^10)*x^4 + 4*(a*b^9*c^6*d^4 - 6*a^2*b^8*c^5*d^5 + 15*a^3*b^7*c^4*d^6 - 20*a^4*b^6*c^3*d^7 + 15*a^5*b^5*c^2*
d^8 - 6*a^6*b^4*c*d^9 + a^7*b^3*d^10)*x^3 + 6*(a^2*b^8*c^6*d^4 - 6*a^3*b^7*c^5*d^5 + 15*a^4*b^6*c^4*d^6 - 20*a
^5*b^5*c^3*d^7 + 15*a^6*b^4*c^2*d^8 - 6*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 4*(a^3*b^7*c^6*d^4 - 6*a^4*b^6*c^5
*d^5 + 15*a^5*b^5*c^4*d^6 - 20*a^6*b^4*c^3*d^7 + 15*a^7*b^3*c^2*d^8 - 6*a^8*b^2*c*d^9 + a^9*b*d^10)*x)*log(b*x
+ a))/(b^15*x^4 + 4*a*b^14*x^3 + 6*a^2*b^13*x^2 + 4*a^3*b^12*x + a^4*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)**5,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 895 vs. \(2 (252) = 504\).
time = 0.00, size = 945, normalized size = 3.61 \begin {gather*} \frac {\frac {1}{6} x^{6} d^{10} b^{25}-x^{5} d^{10} b^{24} a+2 x^{5} d^{9} c b^{25}+\frac {15}{4} x^{4} d^{10} b^{23} a^{2}-\frac {25}{2} x^{4} d^{9} c b^{24} a+\frac {45}{4} x^{4} d^{8} c^{2} b^{25}-\frac {35}{3} x^{3} d^{10} b^{22} a^{3}+50 x^{3} d^{9} c b^{23} a^{2}-75 x^{3} d^{8} c^{2} b^{24} a+40 x^{3} d^{7} c^{3} b^{25}+35 x^{2} d^{10} b^{21} a^{4}-175 x^{2} d^{9} c b^{22} a^{3}+\frac {675}{2} x^{2} d^{8} c^{2} b^{23} a^{2}-300 x^{2} d^{7} c^{3} b^{24} a+105 x^{2} d^{6} c^{4} b^{25}-126 x d^{10} b^{20} a^{5}+700 x d^{9} c b^{21} a^{4}-1575 x d^{8} c^{2} b^{22} a^{3}+1800 x d^{7} c^{3} b^{23} a^{2}-1050 x d^{6} c^{4} b^{24} a+252 x d^{5} c^{5} b^{25}}{b^{30}}+\frac {\frac {1}{12} \left (\left (1440 d^{10} b^{3} a^{7}-10080 d^{9} b^{4} a^{6} c+30240 d^{8} b^{5} a^{5} c^{2}-50400 d^{7} b^{6} a^{4} c^{3}+50400 d^{6} b^{7} a^{3} c^{4}-30240 d^{5} b^{8} a^{2} c^{5}+10080 d^{4} b^{9} a c^{6}-1440 d^{3} b^{10} c^{7}\right ) x^{3}+\left (4050 d^{10} b^{2} a^{8}-28080 d^{9} b^{3} a^{7} c+83160 d^{8} b^{4} a^{6} c^{2}-136080 d^{7} b^{5} a^{5} c^{3}+132300 d^{6} b^{6} a^{4} c^{4}-75600 d^{5} b^{7} a^{3} c^{5}+22680 d^{4} b^{8} a^{2} c^{6}-2160 d^{3} b^{9} a c^{7}-270 d^{2} b^{10} c^{8}\right ) x^{2}+\left (3820 d^{10} b a^{9}-26280 d^{9} b^{2} a^{8} c+77040 d^{8} b^{3} a^{7} c^{2}-124320 d^{7} b^{4} a^{6} c^{3}+118440 d^{6} b^{5} a^{5} c^{4}-65520 d^{5} b^{6} a^{4} c^{5}+18480 d^{4} b^{7} a^{3} c^{6}-1440 d^{3} b^{8} a^{2} c^{7}-180 d^{2} b^{9} a c^{8}-40 d b^{10} c^{9}\right ) x+1207 d^{10} a^{10}-8250 d^{9} b a^{9} c+23985 d^{8} b^{2} a^{8} c^{2}-38280 d^{7} b^{3} a^{7} c^{3}+35910 d^{6} b^{4} a^{6} c^{4}-19404 d^{5} b^{5} a^{5} c^{5}+5250 d^{4} b^{6} a^{4} c^{6}-360 d^{3} b^{7} a^{3} c^{7}-45 d^{2} b^{8} a^{2} c^{8}-10 d b^{9} a c^{9}-3 b^{10} c^{10}\right )}{b^{11} \left (x b+a\right )^{4}}+\frac {\left (210 d^{10} a^{6}-1260 d^{9} c b a^{5}+3150 d^{8} c^{2} b^{2} a^{4}-4200 d^{7} c^{3} b^{3} a^{3}+3150 d^{6} c^{4} b^{4} a^{2}-1260 d^{5} c^{5} b^{5} a+210 d^{4} c^{6} b^{6}\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)^5,x)

[Out]

210*(b^6*c^6*d^4 - 6*a*b^5*c^5*d^5 + 15*a^2*b^4*c^4*d^6 - 20*a^3*b^3*c^3*d^7 + 15*a^4*b^2*c^2*d^8 - 6*a^5*b*c*
d^9 + a^6*d^10)*log(abs(b*x + a))/b^11 - 1/12*(3*b^10*c^10 + 10*a*b^9*c^9*d + 45*a^2*b^8*c^8*d^2 + 360*a^3*b^7
*c^7*d^3 - 5250*a^4*b^6*c^6*d^4 + 19404*a^5*b^5*c^5*d^5 - 35910*a^6*b^4*c^4*d^6 + 38280*a^7*b^3*c^3*d^7 - 2398
5*a^8*b^2*c^2*d^8 + 8250*a^9*b*c*d^9 - 1207*a^10*d^10 + 1440*(b^10*c^7*d^3 - 7*a*b^9*c^6*d^4 + 21*a^2*b^8*c^5*
d^5 - 35*a^3*b^7*c^4*d^6 + 35*a^4*b^6*c^3*d^7 - 21*a^5*b^5*c^2*d^8 + 7*a^6*b^4*c*d^9 - a^7*b^3*d^10)*x^3 + 270
*(b^10*c^8*d^2 + 8*a*b^9*c^7*d^3 - 84*a^2*b^8*c^6*d^4 + 280*a^3*b^7*c^5*d^5 - 490*a^4*b^6*c^4*d^6 + 504*a^5*b^
5*c^3*d^7 - 308*a^6*b^4*c^2*d^8 + 104*a^7*b^3*c*d^9 - 15*a^8*b^2*d^10)*x^2 + 20*(2*b^10*c^9*d + 9*a*b^9*c^8*d^
2 + 72*a^2*b^8*c^7*d^3 - 924*a^3*b^7*c^6*d^4 + 3276*a^4*b^6*c^5*d^5 - 5922*a^5*b^5*c^4*d^6 + 6216*a^6*b^4*c^3*
d^7 - 3852*a^7*b^3*c^2*d^8 + 1314*a^8*b^2*c*d^9 - 191*a^9*b*d^10)*x)/((b*x + a)^4*b^11) + 1/12*(2*b^25*d^10*x^
6 + 24*b^25*c*d^9*x^5 - 12*a*b^24*d^10*x^5 + 135*b^25*c^2*d^8*x^4 - 150*a*b^24*c*d^9*x^4 + 45*a^2*b^23*d^10*x^
4 + 480*b^25*c^3*d^7*x^3 - 900*a*b^24*c^2*d^8*x^3 + 600*a^2*b^23*c*d^9*x^3 - 140*a^3*b^22*d^10*x^3 + 1260*b^25
*c^4*d^6*x^2 - 3600*a*b^24*c^3*d^7*x^2 + 4050*a^2*b^23*c^2*d^8*x^2 - 2100*a^3*b^22*c*d^9*x^2 + 420*a^4*b^21*d^
10*x^2 + 3024*b^25*c^5*d^5*x - 12600*a*b^24*c^4*d^6*x + 21600*a^2*b^23*c^3*d^7*x - 18900*a^3*b^22*c^2*d^8*x +
8400*a^4*b^21*c*d^9*x - 1512*a^5*b^20*d^10*x)/b^30

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Mupad [B]
time = 0.38, size = 1494, normalized size = 5.70

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^10/(a + b*x)^5,x)

[Out]

x^2*((5*a*((5*a*((5*a*((5*a*d^10)/b^6 - (10*c*d^9)/b^5))/b - (10*a^2*d^10)/b^7 + (45*c^2*d^8)/b^5))/b + (10*a^
3*d^10)/b^8 - (120*c^3*d^7)/b^5 - (10*a^2*((5*a*d^10)/b^6 - (10*c*d^9)/b^5))/b^2))/(2*b) - (5*a^4*d^10)/(2*b^9
) + (105*c^4*d^6)/b^5 + (5*a^3*((5*a*d^10)/b^6 - (10*c*d^9)/b^5))/b^3 - (5*a^2*((5*a*((5*a*d^10)/b^6 - (10*c*d
^9)/b^5))/b - (10*a^2*d^10)/b^7 + (45*c^2*d^8)/b^5))/b^2) - x^5*((a*d^10)/b^6 - (2*c*d^9)/b^5) - x^3*((5*a*((5
*a*((5*a*d^10)/b^6 - (10*c*d^9)/b^5))/b - (10*a^2*d^10)/b^7 + (45*c^2*d^8)/b^5))/(3*b) + (10*a^3*d^10)/(3*b^8)
- (40*c^3*d^7)/b^5 - (10*a^2*((5*a*d^10)/b^6 - (10*c*d^9)/b^5))/(3*b^2)) + x^4*((5*a*((5*a*d^10)/b^6 - (10*c*
d^9)/b^5))/(4*b) - (5*a^2*d^10)/(2*b^7) + (45*c^2*d^8)/(4*b^5)) - x*((5*a*((5*a*((5*a*((5*a*((5*a*d^10)/b^6 -
(10*c*d^9)/b^5))/b - (10*a^2*d^10)/b^7 + (45*c^2*d^8)/b^5))/b + (10*a^3*d^10)/b^8 - (120*c^3*d^7)/b^5 - (10*a^
2*((5*a*d^10)/b^6 - (10*c*d^9)/b^5))/b^2))/b - (5*a^4*d^10)/b^9 + (210*c^4*d^6)/b^5 + (10*a^3*((5*a*d^10)/b^6
- (10*c*d^9)/b^5))/b^3 - (10*a^2*((5*a*((5*a*d^10)/b^6 - (10*c*d^9)/b^5))/b - (10*a^2*d^10)/b^7 + (45*c^2*d^8)
/b^5))/b^2))/b + (a^5*d^10)/b^10 - (252*c^5*d^5)/b^5 - (5*a^4*((5*a*d^10)/b^6 - (10*c*d^9)/b^5))/b^4 - (10*a^2
*((5*a*((5*a*((5*a*d^10)/b^6 - (10*c*d^9)/b^5))/b - (10*a^2*d^10)/b^7 + (45*c^2*d^8)/b^5))/b + (10*a^3*d^10)/b
^8 - (120*c^3*d^7)/b^5 - (10*a^2*((5*a*d^10)/b^6 - (10*c*d^9)/b^5))/b^2))/b^2 + (10*a^3*((5*a*((5*a*d^10)/b^6
- (10*c*d^9)/b^5))/b - (10*a^2*d^10)/b^7 + (45*c^2*d^8)/b^5))/b^3) - ((3*b^10*c^10 - 1207*a^10*d^10 + 45*a^2*b
^8*c^8*d^2 + 360*a^3*b^7*c^7*d^3 - 5250*a^4*b^6*c^6*d^4 + 19404*a^5*b^5*c^5*d^5 - 35910*a^6*b^4*c^4*d^6 + 3828
0*a^7*b^3*c^3*d^7 - 23985*a^8*b^2*c^2*d^8 + 10*a*b^9*c^9*d + 8250*a^9*b*c*d^9)/(12*b) + x*((10*b^9*c^9*d)/3 -
(955*a^9*d^10)/3 + 15*a*b^8*c^8*d^2 + 120*a^2*b^7*c^7*d^3 - 1540*a^3*b^6*c^6*d^4 + 5460*a^4*b^5*c^5*d^5 - 9870
*a^5*b^4*c^4*d^6 + 10360*a^6*b^3*c^3*d^7 - 6420*a^7*b^2*c^2*d^8 + 2190*a^8*b*c*d^9) - x^3*(120*a^7*b^2*d^10 -
120*b^9*c^7*d^3 + 840*a*b^8*c^6*d^4 - 840*a^6*b^3*c*d^9 - 2520*a^2*b^7*c^5*d^5 + 4200*a^3*b^6*c^4*d^6 - 4200*a
^4*b^5*c^3*d^7 + 2520*a^5*b^4*c^2*d^8) + x^2*((45*b^9*c^8*d^2)/2 - (675*a^8*b*d^10)/2 + 180*a*b^8*c^7*d^3 + 23
40*a^7*b^2*c*d^9 - 1890*a^2*b^7*c^6*d^4 + 6300*a^3*b^6*c^5*d^5 - 11025*a^4*b^5*c^4*d^6 + 11340*a^5*b^4*c^3*d^7
- 6930*a^6*b^3*c^2*d^8))/(a^4*b^10 + b^14*x^4 + 4*a^3*b^11*x + 4*a*b^13*x^3 + 6*a^2*b^12*x^2) + (log(a + b*x)
*(210*a^6*d^10 + 210*b^6*c^6*d^4 - 1260*a*b^5*c^5*d^5 + 3150*a^2*b^4*c^4*d^6 - 4200*a^3*b^3*c^3*d^7 + 3150*a^4
*b^2*c^2*d^8 - 1260*a^5*b*c*d^9))/b^11 + (d^10*x^6)/(6*b^5)

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