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 = 0.30, 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 = 0.13, 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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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 Too large to display

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 = 0.34, 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 = 0.93, size = 1365, normalized size = 5.21 \begin {gather*} \frac {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} - 8250 \, a^{9} b c d^{9} + 1207 \, a^{10} d^{10} + 4 \, {\left (6 \, b^{10} c d^{9} - a b^{9} d^{10}\right )} x^{9} + 9 \, {\left (15 \, b^{10} c^{2} d^{8} - 6 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 24 \, {\left (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}\right )} x^{7} + 84 \, {\left (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}\right )} x^{6} + 504 \, {\left (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}\right )} x^{5} + {\left (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}\right )} x^{4} - 4 \, {\left (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} - 16320 \, 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}\right )} x^{3} - 6 \, {\left (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}\right )} x^{2} - 4 \, {\left (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}\right )} x + 2520 \, {\left (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} + {\left (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}\right )} x^{4} + 4 \, {\left (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}\right )} x^{3} + 6 \, {\left (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}\right )} x^{2} + 4 \, {\left (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}\right )} x\right )} \log \left (b x + a\right )}{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 )}} \end {gather*}

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)] 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((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. 1168 vs. \(2 (252) = 504\).
time = 0.53, size = 1168, normalized size = 4.46 \begin {gather*} \frac {{\left (2 \, d^{10} + \frac {24 \, {\left (b^{2} c d^{9} - a b d^{10}\right )}}{{\left (b x + a\right )} b} + \frac {135 \, {\left (b^{4} c^{2} d^{8} - 2 \, a b^{3} c d^{9} + a^{2} b^{2} d^{10}\right )}}{{\left (b x + a\right )}^{2} b^{2}} + \frac {480 \, {\left (b^{6} c^{3} d^{7} - 3 \, a b^{5} c^{2} d^{8} + 3 \, a^{2} b^{4} c d^{9} - a^{3} b^{3} d^{10}\right )}}{{\left (b x + a\right )}^{3} b^{3}} + \frac {1260 \, {\left (b^{8} c^{4} d^{6} - 4 \, a b^{7} c^{3} d^{7} + 6 \, a^{2} b^{6} c^{2} d^{8} - 4 \, a^{3} b^{5} c d^{9} + a^{4} b^{4} d^{10}\right )}}{{\left (b x + a\right )}^{4} b^{4}} + \frac {3024 \, {\left (b^{10} c^{5} d^{5} - 5 \, a b^{9} c^{4} d^{6} + 10 \, a^{2} b^{8} c^{3} d^{7} - 10 \, a^{3} b^{7} c^{2} d^{8} + 5 \, a^{4} b^{6} c d^{9} - a^{5} b^{5} d^{10}\right )}}{{\left (b x + a\right )}^{5} b^{5}}\right )} {\left (b x + a\right )}^{6}}{12 \, b^{11}} - \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 (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{11}} - \frac {\frac {3 \, b^{67} c^{10}}{{\left (b x + a\right )}^{4}} + \frac {40 \, b^{66} c^{9} d}{{\left (b x + a\right )}^{3}} - \frac {30 \, a b^{66} c^{9} d}{{\left (b x + a\right )}^{4}} + \frac {270 \, b^{65} c^{8} d^{2}}{{\left (b x + a\right )}^{2}} - \frac {360 \, a b^{65} c^{8} d^{2}}{{\left (b x + a\right )}^{3}} + \frac {135 \, a^{2} b^{65} c^{8} d^{2}}{{\left (b x + a\right )}^{4}} + \frac {1440 \, b^{64} c^{7} d^{3}}{b x + a} - \frac {2160 \, a b^{64} c^{7} d^{3}}{{\left (b x + a\right )}^{2}} + \frac {1440 \, a^{2} b^{64} c^{7} d^{3}}{{\left (b x + a\right )}^{3}} - \frac {360 \, a^{3} b^{64} c^{7} d^{3}}{{\left (b x + a\right )}^{4}} - \frac {10080 \, a b^{63} c^{6} d^{4}}{b x + a} + \frac {7560 \, a^{2} b^{63} c^{6} d^{4}}{{\left (b x + a\right )}^{2}} - \frac {3360 \, a^{3} b^{63} c^{6} d^{4}}{{\left (b x + a\right )}^{3}} + \frac {630 \, a^{4} b^{63} c^{6} d^{4}}{{\left (b x + a\right )}^{4}} + \frac {30240 \, a^{2} b^{62} c^{5} d^{5}}{b x + a} - \frac {15120 \, a^{3} b^{62} c^{5} d^{5}}{{\left (b x + a\right )}^{2}} + \frac {5040 \, a^{4} b^{62} c^{5} d^{5}}{{\left (b x + a\right )}^{3}} - \frac {756 \, a^{5} b^{62} c^{5} d^{5}}{{\left (b x + a\right )}^{4}} - \frac {50400 \, a^{3} b^{61} c^{4} d^{6}}{b x + a} + \frac {18900 \, a^{4} b^{61} c^{4} d^{6}}{{\left (b x + a\right )}^{2}} - \frac {5040 \, a^{5} b^{61} c^{4} d^{6}}{{\left (b x + a\right )}^{3}} + \frac {630 \, a^{6} b^{61} c^{4} d^{6}}{{\left (b x + a\right )}^{4}} + \frac {50400 \, a^{4} b^{60} c^{3} d^{7}}{b x + a} - \frac {15120 \, a^{5} b^{60} c^{3} d^{7}}{{\left (b x + a\right )}^{2}} + \frac {3360 \, a^{6} b^{60} c^{3} d^{7}}{{\left (b x + a\right )}^{3}} - \frac {360 \, a^{7} b^{60} c^{3} d^{7}}{{\left (b x + a\right )}^{4}} - \frac {30240 \, a^{5} b^{59} c^{2} d^{8}}{b x + a} + \frac {7560 \, a^{6} b^{59} c^{2} d^{8}}{{\left (b x + a\right )}^{2}} - \frac {1440 \, a^{7} b^{59} c^{2} d^{8}}{{\left (b x + a\right )}^{3}} + \frac {135 \, a^{8} b^{59} c^{2} d^{8}}{{\left (b x + a\right )}^{4}} + \frac {10080 \, a^{6} b^{58} c d^{9}}{b x + a} - \frac {2160 \, a^{7} b^{58} c d^{9}}{{\left (b x + a\right )}^{2}} + \frac {360 \, a^{8} b^{58} c d^{9}}{{\left (b x + a\right )}^{3}} - \frac {30 \, a^{9} b^{58} c d^{9}}{{\left (b x + a\right )}^{4}} - \frac {1440 \, a^{7} b^{57} d^{10}}{b x + a} + \frac {270 \, a^{8} b^{57} d^{10}}{{\left (b x + a\right )}^{2}} - \frac {40 \, a^{9} b^{57} d^{10}}{{\left (b x + a\right )}^{3}} + \frac {3 \, a^{10} b^{57} d^{10}}{{\left (b x + a\right )}^{4}}}{12 \, b^{68}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(2*d^10 + 24*(b^2*c*d^9 - a*b*d^10)/((b*x + a)*b) + 135*(b^4*c^2*d^8 - 2*a*b^3*c*d^9 + a^2*b^2*d^10)/((b*
x + a)^2*b^2) + 480*(b^6*c^3*d^7 - 3*a*b^5*c^2*d^8 + 3*a^2*b^4*c*d^9 - a^3*b^3*d^10)/((b*x + a)^3*b^3) + 1260*
(b^8*c^4*d^6 - 4*a*b^7*c^3*d^7 + 6*a^2*b^6*c^2*d^8 - 4*a^3*b^5*c*d^9 + a^4*b^4*d^10)/((b*x + a)^4*b^4) + 3024*
(b^10*c^5*d^5 - 5*a*b^9*c^4*d^6 + 10*a^2*b^8*c^3*d^7 - 10*a^3*b^7*c^2*d^8 + 5*a^4*b^6*c*d^9 - a^5*b^5*d^10)/((
b*x + a)^5*b^5))*(b*x + a)^6/b^11 - 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*x + a)^2*abs(b)))/b^11 - 1/12*(3*b^67
*c^10/(b*x + a)^4 + 40*b^66*c^9*d/(b*x + a)^3 - 30*a*b^66*c^9*d/(b*x + a)^4 + 270*b^65*c^8*d^2/(b*x + a)^2 - 3
60*a*b^65*c^8*d^2/(b*x + a)^3 + 135*a^2*b^65*c^8*d^2/(b*x + a)^4 + 1440*b^64*c^7*d^3/(b*x + a) - 2160*a*b^64*c
^7*d^3/(b*x + a)^2 + 1440*a^2*b^64*c^7*d^3/(b*x + a)^3 - 360*a^3*b^64*c^7*d^3/(b*x + a)^4 - 10080*a*b^63*c^6*d
^4/(b*x + a) + 7560*a^2*b^63*c^6*d^4/(b*x + a)^2 - 3360*a^3*b^63*c^6*d^4/(b*x + a)^3 + 630*a^4*b^63*c^6*d^4/(b
*x + a)^4 + 30240*a^2*b^62*c^5*d^5/(b*x + a) - 15120*a^3*b^62*c^5*d^5/(b*x + a)^2 + 5040*a^4*b^62*c^5*d^5/(b*x
+ a)^3 - 756*a^5*b^62*c^5*d^5/(b*x + a)^4 - 50400*a^3*b^61*c^4*d^6/(b*x + a) + 18900*a^4*b^61*c^4*d^6/(b*x +
a)^2 - 5040*a^5*b^61*c^4*d^6/(b*x + a)^3 + 630*a^6*b^61*c^4*d^6/(b*x + a)^4 + 50400*a^4*b^60*c^3*d^7/(b*x + a)
- 15120*a^5*b^60*c^3*d^7/(b*x + a)^2 + 3360*a^6*b^60*c^3*d^7/(b*x + a)^3 - 360*a^7*b^60*c^3*d^7/(b*x + a)^4 -
30240*a^5*b^59*c^2*d^8/(b*x + a) + 7560*a^6*b^59*c^2*d^8/(b*x + a)^2 - 1440*a^7*b^59*c^2*d^8/(b*x + a)^3 + 13
5*a^8*b^59*c^2*d^8/(b*x + a)^4 + 10080*a^6*b^58*c*d^9/(b*x + a) - 2160*a^7*b^58*c*d^9/(b*x + a)^2 + 360*a^8*b^
58*c*d^9/(b*x + a)^3 - 30*a^9*b^58*c*d^9/(b*x + a)^4 - 1440*a^7*b^57*d^10/(b*x + a) + 270*a^8*b^57*d^10/(b*x +
a)^2 - 40*a^9*b^57*d^10/(b*x + a)^3 + 3*a^10*b^57*d^10/(b*x + a)^4)/b^68

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Mupad [B]
time = 0.38, size = 1494, normalized size = 5.70 \begin {gather*} x^2\,\left (\frac {5\,a\,\left (\frac {5\,a\,\left (\frac {5\,a\,\left (\frac {5\,a\,d^{10}}{b^6}-\frac {10\,c\,d^9}{b^5}\right )}{b}-\frac {10\,a^2\,d^{10}}{b^7}+\frac {45\,c^2\,d^8}{b^5}\right )}{b}+\frac {10\,a^3\,d^{10}}{b^8}-\frac {120\,c^3\,d^7}{b^5}-\frac {10\,a^2\,\left (\frac {5\,a\,d^{10}}{b^6}-\frac {10\,c\,d^9}{b^5}\right )}{b^2}\right )}{2\,b}-\frac {5\,a^4\,d^{10}}{2\,b^9}+\frac {105\,c^4\,d^6}{b^5}+\frac {5\,a^3\,\left (\frac {5\,a\,d^{10}}{b^6}-\frac {10\,c\,d^9}{b^5}\right )}{b^3}-\frac {5\,a^2\,\left (\frac {5\,a\,\left (\frac {5\,a\,d^{10}}{b^6}-\frac {10\,c\,d^9}{b^5}\right )}{b}-\frac {10\,a^2\,d^{10}}{b^7}+\frac {45\,c^2\,d^8}{b^5}\right )}{b^2}\right )-x^5\,\left (\frac {a\,d^{10}}{b^6}-\frac {2\,c\,d^9}{b^5}\right )-x^3\,\left (\frac {5\,a\,\left (\frac {5\,a\,\left (\frac {5\,a\,d^{10}}{b^6}-\frac {10\,c\,d^9}{b^5}\right )}{b}-\frac {10\,a^2\,d^{10}}{b^7}+\frac {45\,c^2\,d^8}{b^5}\right )}{3\,b}+\frac {10\,a^3\,d^{10}}{3\,b^8}-\frac {40\,c^3\,d^7}{b^5}-\frac {10\,a^2\,\left (\frac {5\,a\,d^{10}}{b^6}-\frac {10\,c\,d^9}{b^5}\right )}{3\,b^2}\right )+x^4\,\left (\frac {5\,a\,\left (\frac {5\,a\,d^{10}}{b^6}-\frac {10\,c\,d^9}{b^5}\right )}{4\,b}-\frac {5\,a^2\,d^{10}}{2\,b^7}+\frac {45\,c^2\,d^8}{4\,b^5}\right )-x\,\left (\frac {5\,a\,\left (\frac {5\,a\,\left (\frac {5\,a\,\left (\frac {5\,a\,\left (\frac {5\,a\,d^{10}}{b^6}-\frac {10\,c\,d^9}{b^5}\right )}{b}-\frac {10\,a^2\,d^{10}}{b^7}+\frac {45\,c^2\,d^8}{b^5}\right )}{b}+\frac {10\,a^3\,d^{10}}{b^8}-\frac {120\,c^3\,d^7}{b^5}-\frac {10\,a^2\,\left (\frac {5\,a\,d^{10}}{b^6}-\frac {10\,c\,d^9}{b^5}\right )}{b^2}\right )}{b}-\frac {5\,a^4\,d^{10}}{b^9}+\frac {210\,c^4\,d^6}{b^5}+\frac {10\,a^3\,\left (\frac {5\,a\,d^{10}}{b^6}-\frac {10\,c\,d^9}{b^5}\right )}{b^3}-\frac {10\,a^2\,\left (\frac {5\,a\,\left (\frac {5\,a\,d^{10}}{b^6}-\frac {10\,c\,d^9}{b^5}\right )}{b}-\frac {10\,a^2\,d^{10}}{b^7}+\frac {45\,c^2\,d^8}{b^5}\right )}{b^2}\right )}{b}+\frac {a^5\,d^{10}}{b^{10}}-\frac {252\,c^5\,d^5}{b^5}-\frac {5\,a^4\,\left (\frac {5\,a\,d^{10}}{b^6}-\frac {10\,c\,d^9}{b^5}\right )}{b^4}-\frac {10\,a^2\,\left (\frac {5\,a\,\left (\frac {5\,a\,\left (\frac {5\,a\,d^{10}}{b^6}-\frac {10\,c\,d^9}{b^5}\right )}{b}-\frac {10\,a^2\,d^{10}}{b^7}+\frac {45\,c^2\,d^8}{b^5}\right )}{b}+\frac {10\,a^3\,d^{10}}{b^8}-\frac {120\,c^3\,d^7}{b^5}-\frac {10\,a^2\,\left (\frac {5\,a\,d^{10}}{b^6}-\frac {10\,c\,d^9}{b^5}\right )}{b^2}\right )}{b^2}+\frac {10\,a^3\,\left (\frac {5\,a\,\left (\frac {5\,a\,d^{10}}{b^6}-\frac {10\,c\,d^9}{b^5}\right )}{b}-\frac {10\,a^2\,d^{10}}{b^7}+\frac {45\,c^2\,d^8}{b^5}\right )}{b^3}\right )-\frac {\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}+x\,\left (-\frac {955\,a^9\,d^{10}}{3}+2190\,a^8\,b\,c\,d^9-6420\,a^7\,b^2\,c^2\,d^8+10360\,a^6\,b^3\,c^3\,d^7-9870\,a^5\,b^4\,c^4\,d^6+5460\,a^4\,b^5\,c^5\,d^5-1540\,a^3\,b^6\,c^6\,d^4+120\,a^2\,b^7\,c^7\,d^3+15\,a\,b^8\,c^8\,d^2+\frac {10\,b^9\,c^9\,d}{3}\right )-x^3\,\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^2\,\left (-\frac {675\,a^8\,b\,d^{10}}{2}+2340\,a^7\,b^2\,c\,d^9-6930\,a^6\,b^3\,c^2\,d^8+11340\,a^5\,b^4\,c^3\,d^7-11025\,a^4\,b^5\,c^4\,d^6+6300\,a^3\,b^6\,c^5\,d^5-1890\,a^2\,b^7\,c^6\,d^4+180\,a\,b^8\,c^7\,d^3+\frac {45\,b^9\,c^8\,d^2}{2}\right )}{a^4\,b^{10}+4\,a^3\,b^{11}\,x+6\,a^2\,b^{12}\,x^2+4\,a\,b^{13}\,x^3+b^{14}\,x^4}+\frac {\ln \left (a+b\,x\right )\,\left (210\,a^6\,d^{10}-1260\,a^5\,b\,c\,d^9+3150\,a^4\,b^2\,c^2\,d^8-4200\,a^3\,b^3\,c^3\,d^7+3150\,a^2\,b^4\,c^4\,d^6-1260\,a\,b^5\,c^5\,d^5+210\,b^6\,c^6\,d^4\right )}{b^{11}}+\frac {d^{10}\,x^6}{6\,b^5} \end {gather*}

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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