3.13.18 \(\int \frac {(c+d x)^{10}}{(a+b x)^{13}} \, dx\)

Optimal. Leaf size=58 \[ \frac {d (c+d x)^{11}}{132 (a+b x)^{11} (b c-a d)^2}-\frac {(c+d x)^{11}}{12 (a+b x)^{12} (b c-a d)} \]

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Rubi [A]  time = 0.01, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {45, 37} \begin {gather*} \frac {d (c+d x)^{11}}{132 (a+b x)^{11} (b c-a d)^2}-\frac {(c+d x)^{11}}{12 (a+b x)^{12} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^10/(a + b*x)^13,x]

[Out]

-(c + d*x)^11/(12*(b*c - a*d)*(a + b*x)^12) + (d*(c + d*x)^11)/(132*(b*c - a*d)^2*(a + b*x)^11)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rubi steps

\begin {align*} \int \frac {(c+d x)^{10}}{(a+b x)^{13}} \, dx &=-\frac {(c+d x)^{11}}{12 (b c-a d) (a+b x)^{12}}-\frac {d \int \frac {(c+d x)^{10}}{(a+b x)^{12}} \, dx}{12 (b c-a d)}\\ &=-\frac {(c+d x)^{11}}{12 (b c-a d) (a+b x)^{12}}+\frac {d (c+d x)^{11}}{132 (b c-a d)^2 (a+b x)^{11}}\\ \end {align*}

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Mathematica [B]  time = 0.28, size = 684, normalized size = 11.79 \begin {gather*} -\frac {a^{10} d^{10}+2 a^9 b d^9 (c+6 d x)+3 a^8 b^2 d^8 \left (c^2+8 c d x+22 d^2 x^2\right )+4 a^7 b^3 d^7 \left (c^3+9 c^2 d x+33 c d^2 x^2+55 d^3 x^3\right )+a^6 b^4 d^6 \left (5 c^4+48 c^3 d x+198 c^2 d^2 x^2+440 c d^3 x^3+495 d^4 x^4\right )+6 a^5 b^5 d^5 \left (c^5+10 c^4 d x+44 c^3 d^2 x^2+110 c^2 d^3 x^3+165 c d^4 x^4+132 d^5 x^5\right )+a^4 b^6 d^4 \left (7 c^6+72 c^5 d x+330 c^4 d^2 x^2+880 c^3 d^3 x^3+1485 c^2 d^4 x^4+1584 c d^5 x^5+924 d^6 x^6\right )+4 a^3 b^7 d^3 \left (2 c^7+21 c^6 d x+99 c^5 d^2 x^2+275 c^4 d^3 x^3+495 c^3 d^4 x^4+594 c^2 d^5 x^5+462 c d^6 x^6+198 d^7 x^7\right )+3 a^2 b^8 d^2 \left (3 c^8+32 c^7 d x+154 c^6 d^2 x^2+440 c^5 d^3 x^3+825 c^4 d^4 x^4+1056 c^3 d^5 x^5+924 c^2 d^6 x^6+528 c d^7 x^7+165 d^8 x^8\right )+2 a b^9 d \left (5 c^9+54 c^8 d x+264 c^7 d^2 x^2+770 c^6 d^3 x^3+1485 c^5 d^4 x^4+1980 c^4 d^5 x^5+1848 c^3 d^6 x^6+1188 c^2 d^7 x^7+495 c d^8 x^8+110 d^9 x^9\right )+b^{10} \left (11 c^{10}+120 c^9 d x+594 c^8 d^2 x^2+1760 c^7 d^3 x^3+3465 c^6 d^4 x^4+4752 c^5 d^5 x^5+4620 c^4 d^6 x^6+3168 c^3 d^7 x^7+1485 c^2 d^8 x^8+440 c d^9 x^9+66 d^{10} x^{10}\right )}{132 b^{11} (a+b x)^{12}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^10/(a + b*x)^13,x]

[Out]

-1/132*(a^10*d^10 + 2*a^9*b*d^9*(c + 6*d*x) + 3*a^8*b^2*d^8*(c^2 + 8*c*d*x + 22*d^2*x^2) + 4*a^7*b^3*d^7*(c^3
+ 9*c^2*d*x + 33*c*d^2*x^2 + 55*d^3*x^3) + a^6*b^4*d^6*(5*c^4 + 48*c^3*d*x + 198*c^2*d^2*x^2 + 440*c*d^3*x^3 +
495*d^4*x^4) + 6*a^5*b^5*d^5*(c^5 + 10*c^4*d*x + 44*c^3*d^2*x^2 + 110*c^2*d^3*x^3 + 165*c*d^4*x^4 + 132*d^5*x
^5) + a^4*b^6*d^4*(7*c^6 + 72*c^5*d*x + 330*c^4*d^2*x^2 + 880*c^3*d^3*x^3 + 1485*c^2*d^4*x^4 + 1584*c*d^5*x^5
+ 924*d^6*x^6) + 4*a^3*b^7*d^3*(2*c^7 + 21*c^6*d*x + 99*c^5*d^2*x^2 + 275*c^4*d^3*x^3 + 495*c^3*d^4*x^4 + 594*
c^2*d^5*x^5 + 462*c*d^6*x^6 + 198*d^7*x^7) + 3*a^2*b^8*d^2*(3*c^8 + 32*c^7*d*x + 154*c^6*d^2*x^2 + 440*c^5*d^3
*x^3 + 825*c^4*d^4*x^4 + 1056*c^3*d^5*x^5 + 924*c^2*d^6*x^6 + 528*c*d^7*x^7 + 165*d^8*x^8) + 2*a*b^9*d*(5*c^9
+ 54*c^8*d*x + 264*c^7*d^2*x^2 + 770*c^6*d^3*x^3 + 1485*c^5*d^4*x^4 + 1980*c^4*d^5*x^5 + 1848*c^3*d^6*x^6 + 11
88*c^2*d^7*x^7 + 495*c*d^8*x^8 + 110*d^9*x^9) + b^10*(11*c^10 + 120*c^9*d*x + 594*c^8*d^2*x^2 + 1760*c^7*d^3*x
^3 + 3465*c^6*d^4*x^4 + 4752*c^5*d^5*x^5 + 4620*c^4*d^6*x^6 + 3168*c^3*d^7*x^7 + 1485*c^2*d^8*x^8 + 440*c*d^9*
x^9 + 66*d^10*x^10))/(b^11*(a + b*x)^12)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(c+d x)^{10}}{(a+b x)^{13}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(c + d*x)^10/(a + b*x)^13,x]

[Out]

IntegrateAlgebraic[(c + d*x)^10/(a + b*x)^13, x]

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fricas [B]  time = 1.29, size = 986, normalized size = 17.00 \begin {gather*} -\frac {66 \, b^{10} d^{10} x^{10} + 11 \, b^{10} c^{10} + 10 \, a b^{9} c^{9} d + 9 \, a^{2} b^{8} c^{8} d^{2} + 8 \, a^{3} b^{7} c^{7} d^{3} + 7 \, a^{4} b^{6} c^{6} d^{4} + 6 \, a^{5} b^{5} c^{5} d^{5} + 5 \, a^{6} b^{4} c^{4} d^{6} + 4 \, a^{7} b^{3} c^{3} d^{7} + 3 \, a^{8} b^{2} c^{2} d^{8} + 2 \, a^{9} b c d^{9} + a^{10} d^{10} + 220 \, {\left (2 \, b^{10} c d^{9} + a b^{9} d^{10}\right )} x^{9} + 495 \, {\left (3 \, b^{10} c^{2} d^{8} + 2 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 792 \, {\left (4 \, b^{10} c^{3} d^{7} + 3 \, a b^{9} c^{2} d^{8} + 2 \, a^{2} b^{8} c d^{9} + a^{3} b^{7} d^{10}\right )} x^{7} + 924 \, {\left (5 \, b^{10} c^{4} d^{6} + 4 \, a b^{9} c^{3} d^{7} + 3 \, a^{2} b^{8} c^{2} d^{8} + 2 \, a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 792 \, {\left (6 \, b^{10} c^{5} d^{5} + 5 \, a b^{9} c^{4} d^{6} + 4 \, a^{2} b^{8} c^{3} d^{7} + 3 \, a^{3} b^{7} c^{2} d^{8} + 2 \, a^{4} b^{6} c d^{9} + a^{5} b^{5} d^{10}\right )} x^{5} + 495 \, {\left (7 \, b^{10} c^{6} d^{4} + 6 \, a b^{9} c^{5} d^{5} + 5 \, a^{2} b^{8} c^{4} d^{6} + 4 \, a^{3} b^{7} c^{3} d^{7} + 3 \, a^{4} b^{6} c^{2} d^{8} + 2 \, a^{5} b^{5} c d^{9} + a^{6} b^{4} d^{10}\right )} x^{4} + 220 \, {\left (8 \, b^{10} c^{7} d^{3} + 7 \, a b^{9} c^{6} d^{4} + 6 \, a^{2} b^{8} c^{5} d^{5} + 5 \, a^{3} b^{7} c^{4} d^{6} + 4 \, a^{4} b^{6} c^{3} d^{7} + 3 \, a^{5} b^{5} c^{2} d^{8} + 2 \, a^{6} b^{4} c d^{9} + a^{7} b^{3} d^{10}\right )} x^{3} + 66 \, {\left (9 \, b^{10} c^{8} d^{2} + 8 \, a b^{9} c^{7} d^{3} + 7 \, a^{2} b^{8} c^{6} d^{4} + 6 \, a^{3} b^{7} c^{5} d^{5} + 5 \, a^{4} b^{6} c^{4} d^{6} + 4 \, a^{5} b^{5} c^{3} d^{7} + 3 \, a^{6} b^{4} c^{2} d^{8} + 2 \, a^{7} b^{3} c d^{9} + a^{8} b^{2} d^{10}\right )} x^{2} + 12 \, {\left (10 \, b^{10} c^{9} d + 9 \, a b^{9} c^{8} d^{2} + 8 \, a^{2} b^{8} c^{7} d^{3} + 7 \, a^{3} b^{7} c^{6} d^{4} + 6 \, a^{4} b^{6} c^{5} d^{5} + 5 \, a^{5} b^{5} c^{4} d^{6} + 4 \, a^{6} b^{4} c^{3} d^{7} + 3 \, a^{7} b^{3} c^{2} d^{8} + 2 \, a^{8} b^{2} c d^{9} + a^{9} b d^{10}\right )} x}{132 \, {\left (b^{23} x^{12} + 12 \, a b^{22} x^{11} + 66 \, a^{2} b^{21} x^{10} + 220 \, a^{3} b^{20} x^{9} + 495 \, a^{4} b^{19} x^{8} + 792 \, a^{5} b^{18} x^{7} + 924 \, a^{6} b^{17} x^{6} + 792 \, a^{7} b^{16} x^{5} + 495 \, a^{8} b^{15} x^{4} + 220 \, a^{9} b^{14} x^{3} + 66 \, a^{10} b^{13} x^{2} + 12 \, a^{11} b^{12} x + a^{12} b^{11}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^13,x, algorithm="fricas")

[Out]

-1/132*(66*b^10*d^10*x^10 + 11*b^10*c^10 + 10*a*b^9*c^9*d + 9*a^2*b^8*c^8*d^2 + 8*a^3*b^7*c^7*d^3 + 7*a^4*b^6*
c^6*d^4 + 6*a^5*b^5*c^5*d^5 + 5*a^6*b^4*c^4*d^6 + 4*a^7*b^3*c^3*d^7 + 3*a^8*b^2*c^2*d^8 + 2*a^9*b*c*d^9 + a^10
*d^10 + 220*(2*b^10*c*d^9 + a*b^9*d^10)*x^9 + 495*(3*b^10*c^2*d^8 + 2*a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 + 792*(4
*b^10*c^3*d^7 + 3*a*b^9*c^2*d^8 + 2*a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7 + 924*(5*b^10*c^4*d^6 + 4*a*b^9*c^3*d^7
+ 3*a^2*b^8*c^2*d^8 + 2*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 792*(6*b^10*c^5*d^5 + 5*a*b^9*c^4*d^6 + 4*a^2*b^8*
c^3*d^7 + 3*a^3*b^7*c^2*d^8 + 2*a^4*b^6*c*d^9 + a^5*b^5*d^10)*x^5 + 495*(7*b^10*c^6*d^4 + 6*a*b^9*c^5*d^5 + 5*
a^2*b^8*c^4*d^6 + 4*a^3*b^7*c^3*d^7 + 3*a^4*b^6*c^2*d^8 + 2*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 220*(8*b^10*c^
7*d^3 + 7*a*b^9*c^6*d^4 + 6*a^2*b^8*c^5*d^5 + 5*a^3*b^7*c^4*d^6 + 4*a^4*b^6*c^3*d^7 + 3*a^5*b^5*c^2*d^8 + 2*a^
6*b^4*c*d^9 + a^7*b^3*d^10)*x^3 + 66*(9*b^10*c^8*d^2 + 8*a*b^9*c^7*d^3 + 7*a^2*b^8*c^6*d^4 + 6*a^3*b^7*c^5*d^5
+ 5*a^4*b^6*c^4*d^6 + 4*a^5*b^5*c^3*d^7 + 3*a^6*b^4*c^2*d^8 + 2*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 12*(10*b^
10*c^9*d + 9*a*b^9*c^8*d^2 + 8*a^2*b^8*c^7*d^3 + 7*a^3*b^7*c^6*d^4 + 6*a^4*b^6*c^5*d^5 + 5*a^5*b^5*c^4*d^6 + 4
*a^6*b^4*c^3*d^7 + 3*a^7*b^3*c^2*d^8 + 2*a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^23*x^12 + 12*a*b^22*x^11 + 66*a^2*b
^21*x^10 + 220*a^3*b^20*x^9 + 495*a^4*b^19*x^8 + 792*a^5*b^18*x^7 + 924*a^6*b^17*x^6 + 792*a^7*b^16*x^5 + 495*
a^8*b^15*x^4 + 220*a^9*b^14*x^3 + 66*a^10*b^13*x^2 + 12*a^11*b^12*x + a^12*b^11)

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giac [B]  time = 1.32, size = 961, normalized size = 16.57 \begin {gather*} -\frac {66 \, b^{10} d^{10} x^{10} + 440 \, b^{10} c d^{9} x^{9} + 220 \, a b^{9} d^{10} x^{9} + 1485 \, b^{10} c^{2} d^{8} x^{8} + 990 \, a b^{9} c d^{9} x^{8} + 495 \, a^{2} b^{8} d^{10} x^{8} + 3168 \, b^{10} c^{3} d^{7} x^{7} + 2376 \, a b^{9} c^{2} d^{8} x^{7} + 1584 \, a^{2} b^{8} c d^{9} x^{7} + 792 \, a^{3} b^{7} d^{10} x^{7} + 4620 \, b^{10} c^{4} d^{6} x^{6} + 3696 \, a b^{9} c^{3} d^{7} x^{6} + 2772 \, a^{2} b^{8} c^{2} d^{8} x^{6} + 1848 \, a^{3} b^{7} c d^{9} x^{6} + 924 \, a^{4} b^{6} d^{10} x^{6} + 4752 \, b^{10} c^{5} d^{5} x^{5} + 3960 \, a b^{9} c^{4} d^{6} x^{5} + 3168 \, a^{2} b^{8} c^{3} d^{7} x^{5} + 2376 \, a^{3} b^{7} c^{2} d^{8} x^{5} + 1584 \, a^{4} b^{6} c d^{9} x^{5} + 792 \, a^{5} b^{5} d^{10} x^{5} + 3465 \, b^{10} c^{6} d^{4} x^{4} + 2970 \, a b^{9} c^{5} d^{5} x^{4} + 2475 \, a^{2} b^{8} c^{4} d^{6} x^{4} + 1980 \, a^{3} b^{7} c^{3} d^{7} x^{4} + 1485 \, a^{4} b^{6} c^{2} d^{8} x^{4} + 990 \, a^{5} b^{5} c d^{9} x^{4} + 495 \, a^{6} b^{4} d^{10} x^{4} + 1760 \, b^{10} c^{7} d^{3} x^{3} + 1540 \, a b^{9} c^{6} d^{4} x^{3} + 1320 \, a^{2} b^{8} c^{5} d^{5} x^{3} + 1100 \, a^{3} b^{7} c^{4} d^{6} x^{3} + 880 \, a^{4} b^{6} c^{3} d^{7} x^{3} + 660 \, a^{5} b^{5} c^{2} d^{8} x^{3} + 440 \, a^{6} b^{4} c d^{9} x^{3} + 220 \, a^{7} b^{3} d^{10} x^{3} + 594 \, b^{10} c^{8} d^{2} x^{2} + 528 \, a b^{9} c^{7} d^{3} x^{2} + 462 \, a^{2} b^{8} c^{6} d^{4} x^{2} + 396 \, a^{3} b^{7} c^{5} d^{5} x^{2} + 330 \, a^{4} b^{6} c^{4} d^{6} x^{2} + 264 \, a^{5} b^{5} c^{3} d^{7} x^{2} + 198 \, a^{6} b^{4} c^{2} d^{8} x^{2} + 132 \, a^{7} b^{3} c d^{9} x^{2} + 66 \, a^{8} b^{2} d^{10} x^{2} + 120 \, b^{10} c^{9} d x + 108 \, a b^{9} c^{8} d^{2} x + 96 \, a^{2} b^{8} c^{7} d^{3} x + 84 \, a^{3} b^{7} c^{6} d^{4} x + 72 \, a^{4} b^{6} c^{5} d^{5} x + 60 \, a^{5} b^{5} c^{4} d^{6} x + 48 \, a^{6} b^{4} c^{3} d^{7} x + 36 \, a^{7} b^{3} c^{2} d^{8} x + 24 \, a^{8} b^{2} c d^{9} x + 12 \, a^{9} b d^{10} x + 11 \, b^{10} c^{10} + 10 \, a b^{9} c^{9} d + 9 \, a^{2} b^{8} c^{8} d^{2} + 8 \, a^{3} b^{7} c^{7} d^{3} + 7 \, a^{4} b^{6} c^{6} d^{4} + 6 \, a^{5} b^{5} c^{5} d^{5} + 5 \, a^{6} b^{4} c^{4} d^{6} + 4 \, a^{7} b^{3} c^{3} d^{7} + 3 \, a^{8} b^{2} c^{2} d^{8} + 2 \, a^{9} b c d^{9} + a^{10} d^{10}}{132 \, {\left (b x + a\right )}^{12} b^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^13,x, algorithm="giac")

[Out]

-1/132*(66*b^10*d^10*x^10 + 440*b^10*c*d^9*x^9 + 220*a*b^9*d^10*x^9 + 1485*b^10*c^2*d^8*x^8 + 990*a*b^9*c*d^9*
x^8 + 495*a^2*b^8*d^10*x^8 + 3168*b^10*c^3*d^7*x^7 + 2376*a*b^9*c^2*d^8*x^7 + 1584*a^2*b^8*c*d^9*x^7 + 792*a^3
*b^7*d^10*x^7 + 4620*b^10*c^4*d^6*x^6 + 3696*a*b^9*c^3*d^7*x^6 + 2772*a^2*b^8*c^2*d^8*x^6 + 1848*a^3*b^7*c*d^9
*x^6 + 924*a^4*b^6*d^10*x^6 + 4752*b^10*c^5*d^5*x^5 + 3960*a*b^9*c^4*d^6*x^5 + 3168*a^2*b^8*c^3*d^7*x^5 + 2376
*a^3*b^7*c^2*d^8*x^5 + 1584*a^4*b^6*c*d^9*x^5 + 792*a^5*b^5*d^10*x^5 + 3465*b^10*c^6*d^4*x^4 + 2970*a*b^9*c^5*
d^5*x^4 + 2475*a^2*b^8*c^4*d^6*x^4 + 1980*a^3*b^7*c^3*d^7*x^4 + 1485*a^4*b^6*c^2*d^8*x^4 + 990*a^5*b^5*c*d^9*x
^4 + 495*a^6*b^4*d^10*x^4 + 1760*b^10*c^7*d^3*x^3 + 1540*a*b^9*c^6*d^4*x^3 + 1320*a^2*b^8*c^5*d^5*x^3 + 1100*a
^3*b^7*c^4*d^6*x^3 + 880*a^4*b^6*c^3*d^7*x^3 + 660*a^5*b^5*c^2*d^8*x^3 + 440*a^6*b^4*c*d^9*x^3 + 220*a^7*b^3*d
^10*x^3 + 594*b^10*c^8*d^2*x^2 + 528*a*b^9*c^7*d^3*x^2 + 462*a^2*b^8*c^6*d^4*x^2 + 396*a^3*b^7*c^5*d^5*x^2 + 3
30*a^4*b^6*c^4*d^6*x^2 + 264*a^5*b^5*c^3*d^7*x^2 + 198*a^6*b^4*c^2*d^8*x^2 + 132*a^7*b^3*c*d^9*x^2 + 66*a^8*b^
2*d^10*x^2 + 120*b^10*c^9*d*x + 108*a*b^9*c^8*d^2*x + 96*a^2*b^8*c^7*d^3*x + 84*a^3*b^7*c^6*d^4*x + 72*a^4*b^6
*c^5*d^5*x + 60*a^5*b^5*c^4*d^6*x + 48*a^6*b^4*c^3*d^7*x + 36*a^7*b^3*c^2*d^8*x + 24*a^8*b^2*c*d^9*x + 12*a^9*
b*d^10*x + 11*b^10*c^10 + 10*a*b^9*c^9*d + 9*a^2*b^8*c^8*d^2 + 8*a^3*b^7*c^7*d^3 + 7*a^4*b^6*c^6*d^4 + 6*a^5*b
^5*c^5*d^5 + 5*a^6*b^4*c^4*d^6 + 4*a^7*b^3*c^3*d^7 + 3*a^8*b^2*c^2*d^8 + 2*a^9*b*c*d^9 + a^10*d^10)/((b*x + a)
^12*b^11)

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maple [B]  time = 0.01, size = 867, normalized size = 14.95 \begin {gather*} -\frac {d^{10}}{2 \left (b x +a \right )^{2} b^{11}}+\frac {10 \left (a d -b c \right ) d^{9}}{3 \left (b x +a \right )^{3} b^{11}}-\frac {45 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) d^{8}}{4 \left (b x +a \right )^{4} b^{11}}+\frac {24 \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^{7}}{\left (b x +a \right )^{5} b^{11}}-\frac {35 \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 ) d^{6}}{\left (b x +a \right )^{6} b^{11}}+\frac {36 \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^{5}}{\left (b x +a \right )^{7} b^{11}}-\frac {105 \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 ) d^{4}}{4 \left (b x +a \right )^{8} b^{11}}+\frac {40 \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 ) d^{3}}{3 \left (b x +a \right )^{9} b^{11}}-\frac {9 \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 ) d^{2}}{2 \left (b x +a \right )^{10} b^{11}}+\frac {10 \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 ) d}{11 \left (b x +a \right )^{11} b^{11}}-\frac {d^{10} a^{10}-10 c \,d^{9} a^{9} b +45 c^{2} d^{8} a^{8} b^{2}-120 a^{7} c^{3} d^{7} b^{3}+210 c^{4} d^{6} a^{6} b^{4}-252 a^{5} c^{5} d^{5} b^{5}+210 a^{4} c^{6} d^{4} b^{6}-120 a^{3} c^{7} d^{3} b^{7}+45 a^{2} c^{8} d^{2} b^{8}-10 a \,b^{9} c^{9} d +c^{10} b^{10}}{12 \left (b x +a \right )^{12} b^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^10/(b*x+a)^13,x)

[Out]

-105/4*d^4*(a^6*d^6-6*a^5*b*c*d^5+15*a^4*b^2*c^2*d^4-20*a^3*b^3*c^3*d^3+15*a^2*b^4*c^4*d^2-6*a*b^5*c^5*d+b^6*c
^6)/b^11/(b*x+a)^8+10/3*d^9*(a*d-b*c)/b^11/(b*x+a)^3+36*d^5*(a^5*d^5-5*a^4*b*c*d^4+10*a^3*b^2*c^2*d^3-10*a^2*b
^3*c^3*d^2+5*a*b^4*c^4*d-b^5*c^5)/b^11/(b*x+a)^7+40/3*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^11/(b*x+a)^9+24*d^7*(a^3*d^3-3*a^2*b*c
*d^2+3*a*b^2*c^2*d-b^3*c^3)/b^11/(b*x+a)^5-1/2*d^10/b^11/(b*x+a)^2-45/4*d^8*(a^2*d^2-2*a*b*c*d+b^2*c^2)/b^11/(
b*x+a)^4+10/11*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^11/(b*x+a)^11-1/12*(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)^12-35*d^6*(a^4*d^4-4*a^3*b*c*d^3+6*a^
2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)/b^11/(b*x+a)^6-9/2*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^11/(b*x+a)^10

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maxima [B]  time = 2.17, size = 986, normalized size = 17.00 \begin {gather*} -\frac {66 \, b^{10} d^{10} x^{10} + 11 \, b^{10} c^{10} + 10 \, a b^{9} c^{9} d + 9 \, a^{2} b^{8} c^{8} d^{2} + 8 \, a^{3} b^{7} c^{7} d^{3} + 7 \, a^{4} b^{6} c^{6} d^{4} + 6 \, a^{5} b^{5} c^{5} d^{5} + 5 \, a^{6} b^{4} c^{4} d^{6} + 4 \, a^{7} b^{3} c^{3} d^{7} + 3 \, a^{8} b^{2} c^{2} d^{8} + 2 \, a^{9} b c d^{9} + a^{10} d^{10} + 220 \, {\left (2 \, b^{10} c d^{9} + a b^{9} d^{10}\right )} x^{9} + 495 \, {\left (3 \, b^{10} c^{2} d^{8} + 2 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 792 \, {\left (4 \, b^{10} c^{3} d^{7} + 3 \, a b^{9} c^{2} d^{8} + 2 \, a^{2} b^{8} c d^{9} + a^{3} b^{7} d^{10}\right )} x^{7} + 924 \, {\left (5 \, b^{10} c^{4} d^{6} + 4 \, a b^{9} c^{3} d^{7} + 3 \, a^{2} b^{8} c^{2} d^{8} + 2 \, a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 792 \, {\left (6 \, b^{10} c^{5} d^{5} + 5 \, a b^{9} c^{4} d^{6} + 4 \, a^{2} b^{8} c^{3} d^{7} + 3 \, a^{3} b^{7} c^{2} d^{8} + 2 \, a^{4} b^{6} c d^{9} + a^{5} b^{5} d^{10}\right )} x^{5} + 495 \, {\left (7 \, b^{10} c^{6} d^{4} + 6 \, a b^{9} c^{5} d^{5} + 5 \, a^{2} b^{8} c^{4} d^{6} + 4 \, a^{3} b^{7} c^{3} d^{7} + 3 \, a^{4} b^{6} c^{2} d^{8} + 2 \, a^{5} b^{5} c d^{9} + a^{6} b^{4} d^{10}\right )} x^{4} + 220 \, {\left (8 \, b^{10} c^{7} d^{3} + 7 \, a b^{9} c^{6} d^{4} + 6 \, a^{2} b^{8} c^{5} d^{5} + 5 \, a^{3} b^{7} c^{4} d^{6} + 4 \, a^{4} b^{6} c^{3} d^{7} + 3 \, a^{5} b^{5} c^{2} d^{8} + 2 \, a^{6} b^{4} c d^{9} + a^{7} b^{3} d^{10}\right )} x^{3} + 66 \, {\left (9 \, b^{10} c^{8} d^{2} + 8 \, a b^{9} c^{7} d^{3} + 7 \, a^{2} b^{8} c^{6} d^{4} + 6 \, a^{3} b^{7} c^{5} d^{5} + 5 \, a^{4} b^{6} c^{4} d^{6} + 4 \, a^{5} b^{5} c^{3} d^{7} + 3 \, a^{6} b^{4} c^{2} d^{8} + 2 \, a^{7} b^{3} c d^{9} + a^{8} b^{2} d^{10}\right )} x^{2} + 12 \, {\left (10 \, b^{10} c^{9} d + 9 \, a b^{9} c^{8} d^{2} + 8 \, a^{2} b^{8} c^{7} d^{3} + 7 \, a^{3} b^{7} c^{6} d^{4} + 6 \, a^{4} b^{6} c^{5} d^{5} + 5 \, a^{5} b^{5} c^{4} d^{6} + 4 \, a^{6} b^{4} c^{3} d^{7} + 3 \, a^{7} b^{3} c^{2} d^{8} + 2 \, a^{8} b^{2} c d^{9} + a^{9} b d^{10}\right )} x}{132 \, {\left (b^{23} x^{12} + 12 \, a b^{22} x^{11} + 66 \, a^{2} b^{21} x^{10} + 220 \, a^{3} b^{20} x^{9} + 495 \, a^{4} b^{19} x^{8} + 792 \, a^{5} b^{18} x^{7} + 924 \, a^{6} b^{17} x^{6} + 792 \, a^{7} b^{16} x^{5} + 495 \, a^{8} b^{15} x^{4} + 220 \, a^{9} b^{14} x^{3} + 66 \, a^{10} b^{13} x^{2} + 12 \, a^{11} b^{12} x + a^{12} b^{11}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^13,x, algorithm="maxima")

[Out]

-1/132*(66*b^10*d^10*x^10 + 11*b^10*c^10 + 10*a*b^9*c^9*d + 9*a^2*b^8*c^8*d^2 + 8*a^3*b^7*c^7*d^3 + 7*a^4*b^6*
c^6*d^4 + 6*a^5*b^5*c^5*d^5 + 5*a^6*b^4*c^4*d^6 + 4*a^7*b^3*c^3*d^7 + 3*a^8*b^2*c^2*d^8 + 2*a^9*b*c*d^9 + a^10
*d^10 + 220*(2*b^10*c*d^9 + a*b^9*d^10)*x^9 + 495*(3*b^10*c^2*d^8 + 2*a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 + 792*(4
*b^10*c^3*d^7 + 3*a*b^9*c^2*d^8 + 2*a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7 + 924*(5*b^10*c^4*d^6 + 4*a*b^9*c^3*d^7
+ 3*a^2*b^8*c^2*d^8 + 2*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 792*(6*b^10*c^5*d^5 + 5*a*b^9*c^4*d^6 + 4*a^2*b^8*
c^3*d^7 + 3*a^3*b^7*c^2*d^8 + 2*a^4*b^6*c*d^9 + a^5*b^5*d^10)*x^5 + 495*(7*b^10*c^6*d^4 + 6*a*b^9*c^5*d^5 + 5*
a^2*b^8*c^4*d^6 + 4*a^3*b^7*c^3*d^7 + 3*a^4*b^6*c^2*d^8 + 2*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 220*(8*b^10*c^
7*d^3 + 7*a*b^9*c^6*d^4 + 6*a^2*b^8*c^5*d^5 + 5*a^3*b^7*c^4*d^6 + 4*a^4*b^6*c^3*d^7 + 3*a^5*b^5*c^2*d^8 + 2*a^
6*b^4*c*d^9 + a^7*b^3*d^10)*x^3 + 66*(9*b^10*c^8*d^2 + 8*a*b^9*c^7*d^3 + 7*a^2*b^8*c^6*d^4 + 6*a^3*b^7*c^5*d^5
+ 5*a^4*b^6*c^4*d^6 + 4*a^5*b^5*c^3*d^7 + 3*a^6*b^4*c^2*d^8 + 2*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 12*(10*b^
10*c^9*d + 9*a*b^9*c^8*d^2 + 8*a^2*b^8*c^7*d^3 + 7*a^3*b^7*c^6*d^4 + 6*a^4*b^6*c^5*d^5 + 5*a^5*b^5*c^4*d^6 + 4
*a^6*b^4*c^3*d^7 + 3*a^7*b^3*c^2*d^8 + 2*a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^23*x^12 + 12*a*b^22*x^11 + 66*a^2*b
^21*x^10 + 220*a^3*b^20*x^9 + 495*a^4*b^19*x^8 + 792*a^5*b^18*x^7 + 924*a^6*b^17*x^6 + 792*a^7*b^16*x^5 + 495*
a^8*b^15*x^4 + 220*a^9*b^14*x^3 + 66*a^10*b^13*x^2 + 12*a^11*b^12*x + a^12*b^11)

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mupad [B]  time = 0.39, size = 39, normalized size = 0.67 \begin {gather*} \frac {{\left (c+d\,x\right )}^{11}\,\left (12\,a\,d-11\,b\,c+b\,d\,x\right )}{132\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^10/(a + b*x)^13,x)

[Out]

((c + d*x)^11*(12*a*d - 11*b*c + b*d*x))/(132*(a*d - b*c)^2*(a + b*x)^12)

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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)**13,x)

[Out]

Timed out

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