3.13.17 \(\int \frac {(c+d x)^{10}}{(a+b x)^{12}} \, dx\)

Optimal. Leaf size=28 \[ -\frac {(c+d x)^{11}}{11 (a+b x)^{11} (b c-a d)} \]

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Rubi [A]  time = 0.00, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {37} \begin {gather*} -\frac {(c+d x)^{11}}{11 (a+b x)^{11} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^10/(a + b*x)^12,x]

[Out]

-(c + d*x)^11/(11*(b*c - a*d)*(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]

Rubi steps

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

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Mathematica [B]  time = 0.28, size = 665, normalized size = 23.75 \begin {gather*} -\frac {a^{10} d^{10}+a^9 b d^9 (c+11 d x)+a^8 b^2 d^8 \left (c^2+11 c d x+55 d^2 x^2\right )+a^7 b^3 d^7 \left (c^3+11 c^2 d x+55 c d^2 x^2+165 d^3 x^3\right )+a^6 b^4 d^6 \left (c^4+11 c^3 d x+55 c^2 d^2 x^2+165 c d^3 x^3+330 d^4 x^4\right )+a^5 b^5 d^5 \left (c^5+11 c^4 d x+55 c^3 d^2 x^2+165 c^2 d^3 x^3+330 c d^4 x^4+462 d^5 x^5\right )+a^4 b^6 d^4 \left (c^6+11 c^5 d x+55 c^4 d^2 x^2+165 c^3 d^3 x^3+330 c^2 d^4 x^4+462 c d^5 x^5+462 d^6 x^6\right )+a^3 b^7 d^3 \left (c^7+11 c^6 d x+55 c^5 d^2 x^2+165 c^4 d^3 x^3+330 c^3 d^4 x^4+462 c^2 d^5 x^5+462 c d^6 x^6+330 d^7 x^7\right )+a^2 b^8 d^2 \left (c^8+11 c^7 d x+55 c^6 d^2 x^2+165 c^5 d^3 x^3+330 c^4 d^4 x^4+462 c^3 d^5 x^5+462 c^2 d^6 x^6+330 c d^7 x^7+165 d^8 x^8\right )+a b^9 d \left (c^9+11 c^8 d x+55 c^7 d^2 x^2+165 c^6 d^3 x^3+330 c^5 d^4 x^4+462 c^4 d^5 x^5+462 c^3 d^6 x^6+330 c^2 d^7 x^7+165 c d^8 x^8+55 d^9 x^9\right )+b^{10} \left (c^{10}+11 c^9 d x+55 c^8 d^2 x^2+165 c^7 d^3 x^3+330 c^6 d^4 x^4+462 c^5 d^5 x^5+462 c^4 d^6 x^6+330 c^3 d^7 x^7+165 c^2 d^8 x^8+55 c d^9 x^9+11 d^{10} x^{10}\right )}{11 b^{11} (a+b x)^{11}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^10/(a + b*x)^12,x]

[Out]

-1/11*(a^10*d^10 + a^9*b*d^9*(c + 11*d*x) + a^8*b^2*d^8*(c^2 + 11*c*d*x + 55*d^2*x^2) + a^7*b^3*d^7*(c^3 + 11*
c^2*d*x + 55*c*d^2*x^2 + 165*d^3*x^3) + a^6*b^4*d^6*(c^4 + 11*c^3*d*x + 55*c^2*d^2*x^2 + 165*c*d^3*x^3 + 330*d
^4*x^4) + a^5*b^5*d^5*(c^5 + 11*c^4*d*x + 55*c^3*d^2*x^2 + 165*c^2*d^3*x^3 + 330*c*d^4*x^4 + 462*d^5*x^5) + a^
4*b^6*d^4*(c^6 + 11*c^5*d*x + 55*c^4*d^2*x^2 + 165*c^3*d^3*x^3 + 330*c^2*d^4*x^4 + 462*c*d^5*x^5 + 462*d^6*x^6
) + a^3*b^7*d^3*(c^7 + 11*c^6*d*x + 55*c^5*d^2*x^2 + 165*c^4*d^3*x^3 + 330*c^3*d^4*x^4 + 462*c^2*d^5*x^5 + 462
*c*d^6*x^6 + 330*d^7*x^7) + a^2*b^8*d^2*(c^8 + 11*c^7*d*x + 55*c^6*d^2*x^2 + 165*c^5*d^3*x^3 + 330*c^4*d^4*x^4
+ 462*c^3*d^5*x^5 + 462*c^2*d^6*x^6 + 330*c*d^7*x^7 + 165*d^8*x^8) + a*b^9*d*(c^9 + 11*c^8*d*x + 55*c^7*d^2*x
^2 + 165*c^6*d^3*x^3 + 330*c^5*d^4*x^4 + 462*c^4*d^5*x^5 + 462*c^3*d^6*x^6 + 330*c^2*d^7*x^7 + 165*c*d^8*x^8 +
55*d^9*x^9) + b^10*(c^10 + 11*c^9*d*x + 55*c^8*d^2*x^2 + 165*c^7*d^3*x^3 + 330*c^6*d^4*x^4 + 462*c^5*d^5*x^5
+ 462*c^4*d^6*x^6 + 330*c^3*d^7*x^7 + 165*c^2*d^8*x^8 + 55*c*d^9*x^9 + 11*d^10*x^10))/(b^11*(a + b*x)^11)

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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)^{12}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(c + d*x)^10/(a + b*x)^12,x]

[Out]

IntegrateAlgebraic[(c + d*x)^10/(a + b*x)^12, x]

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fricas [B]  time = 1.35, size = 920, normalized size = 32.86 \begin {gather*} -\frac {11 \, b^{10} d^{10} x^{10} + b^{10} c^{10} + a b^{9} c^{9} d + a^{2} b^{8} c^{8} d^{2} + a^{3} b^{7} c^{7} d^{3} + a^{4} b^{6} c^{6} d^{4} + a^{5} b^{5} c^{5} d^{5} + a^{6} b^{4} c^{4} d^{6} + a^{7} b^{3} c^{3} d^{7} + a^{8} b^{2} c^{2} d^{8} + a^{9} b c d^{9} + a^{10} d^{10} + 55 \, {\left (b^{10} c d^{9} + a b^{9} d^{10}\right )} x^{9} + 165 \, {\left (b^{10} c^{2} d^{8} + a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 330 \, {\left (b^{10} c^{3} d^{7} + a b^{9} c^{2} d^{8} + a^{2} b^{8} c d^{9} + a^{3} b^{7} d^{10}\right )} x^{7} + 462 \, {\left (b^{10} c^{4} d^{6} + a b^{9} c^{3} d^{7} + a^{2} b^{8} c^{2} d^{8} + a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 462 \, {\left (b^{10} c^{5} d^{5} + a b^{9} c^{4} d^{6} + a^{2} b^{8} c^{3} d^{7} + a^{3} b^{7} c^{2} d^{8} + a^{4} b^{6} c d^{9} + a^{5} b^{5} d^{10}\right )} x^{5} + 330 \, {\left (b^{10} c^{6} d^{4} + a b^{9} c^{5} d^{5} + a^{2} b^{8} c^{4} d^{6} + a^{3} b^{7} c^{3} d^{7} + a^{4} b^{6} c^{2} d^{8} + a^{5} b^{5} c d^{9} + a^{6} b^{4} d^{10}\right )} x^{4} + 165 \, {\left (b^{10} c^{7} d^{3} + a b^{9} c^{6} d^{4} + a^{2} b^{8} c^{5} d^{5} + a^{3} b^{7} c^{4} d^{6} + a^{4} b^{6} c^{3} d^{7} + a^{5} b^{5} c^{2} d^{8} + a^{6} b^{4} c d^{9} + a^{7} b^{3} d^{10}\right )} x^{3} + 55 \, {\left (b^{10} c^{8} d^{2} + a b^{9} c^{7} d^{3} + a^{2} b^{8} c^{6} d^{4} + a^{3} b^{7} c^{5} d^{5} + a^{4} b^{6} c^{4} d^{6} + a^{5} b^{5} c^{3} d^{7} + a^{6} b^{4} c^{2} d^{8} + a^{7} b^{3} c d^{9} + a^{8} b^{2} d^{10}\right )} x^{2} + 11 \, {\left (b^{10} c^{9} d + a b^{9} c^{8} d^{2} + a^{2} b^{8} c^{7} d^{3} + a^{3} b^{7} c^{6} d^{4} + a^{4} b^{6} c^{5} d^{5} + a^{5} b^{5} c^{4} d^{6} + a^{6} b^{4} c^{3} d^{7} + a^{7} b^{3} c^{2} d^{8} + a^{8} b^{2} c d^{9} + a^{9} b d^{10}\right )} x}{11 \, {\left (b^{22} x^{11} + 11 \, a b^{21} x^{10} + 55 \, a^{2} b^{20} x^{9} + 165 \, a^{3} b^{19} x^{8} + 330 \, a^{4} b^{18} x^{7} + 462 \, a^{5} b^{17} x^{6} + 462 \, a^{6} b^{16} x^{5} + 330 \, a^{7} b^{15} x^{4} + 165 \, a^{8} b^{14} x^{3} + 55 \, a^{9} b^{13} x^{2} + 11 \, a^{10} b^{12} x + a^{11} b^{11}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^12,x, algorithm="fricas")

[Out]

-1/11*(11*b^10*d^10*x^10 + b^10*c^10 + a*b^9*c^9*d + a^2*b^8*c^8*d^2 + a^3*b^7*c^7*d^3 + a^4*b^6*c^6*d^4 + a^5
*b^5*c^5*d^5 + a^6*b^4*c^4*d^6 + a^7*b^3*c^3*d^7 + a^8*b^2*c^2*d^8 + a^9*b*c*d^9 + a^10*d^10 + 55*(b^10*c*d^9
+ a*b^9*d^10)*x^9 + 165*(b^10*c^2*d^8 + a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 + 330*(b^10*c^3*d^7 + a*b^9*c^2*d^8 +
a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7 + 462*(b^10*c^4*d^6 + a*b^9*c^3*d^7 + a^2*b^8*c^2*d^8 + a^3*b^7*c*d^9 + a^4*
b^6*d^10)*x^6 + 462*(b^10*c^5*d^5 + a*b^9*c^4*d^6 + a^2*b^8*c^3*d^7 + a^3*b^7*c^2*d^8 + a^4*b^6*c*d^9 + a^5*b^
5*d^10)*x^5 + 330*(b^10*c^6*d^4 + a*b^9*c^5*d^5 + a^2*b^8*c^4*d^6 + a^3*b^7*c^3*d^7 + a^4*b^6*c^2*d^8 + a^5*b^
5*c*d^9 + a^6*b^4*d^10)*x^4 + 165*(b^10*c^7*d^3 + a*b^9*c^6*d^4 + a^2*b^8*c^5*d^5 + a^3*b^7*c^4*d^6 + a^4*b^6*
c^3*d^7 + a^5*b^5*c^2*d^8 + a^6*b^4*c*d^9 + a^7*b^3*d^10)*x^3 + 55*(b^10*c^8*d^2 + a*b^9*c^7*d^3 + a^2*b^8*c^6
*d^4 + a^3*b^7*c^5*d^5 + a^4*b^6*c^4*d^6 + a^5*b^5*c^3*d^7 + a^6*b^4*c^2*d^8 + a^7*b^3*c*d^9 + a^8*b^2*d^10)*x
^2 + 11*(b^10*c^9*d + a*b^9*c^8*d^2 + a^2*b^8*c^7*d^3 + a^3*b^7*c^6*d^4 + a^4*b^6*c^5*d^5 + a^5*b^5*c^4*d^6 +
a^6*b^4*c^3*d^7 + a^7*b^3*c^2*d^8 + a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^22*x^11 + 11*a*b^21*x^10 + 55*a^2*b^20*x
^9 + 165*a^3*b^19*x^8 + 330*a^4*b^18*x^7 + 462*a^5*b^17*x^6 + 462*a^6*b^16*x^5 + 330*a^7*b^15*x^4 + 165*a^8*b^
14*x^3 + 55*a^9*b^13*x^2 + 11*a^10*b^12*x + a^11*b^11)

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giac [B]  time = 1.36, size = 951, normalized size = 33.96 \begin {gather*} -\frac {11 \, b^{10} d^{10} x^{10} + 55 \, b^{10} c d^{9} x^{9} + 55 \, a b^{9} d^{10} x^{9} + 165 \, b^{10} c^{2} d^{8} x^{8} + 165 \, a b^{9} c d^{9} x^{8} + 165 \, a^{2} b^{8} d^{10} x^{8} + 330 \, b^{10} c^{3} d^{7} x^{7} + 330 \, a b^{9} c^{2} d^{8} x^{7} + 330 \, a^{2} b^{8} c d^{9} x^{7} + 330 \, a^{3} b^{7} d^{10} x^{7} + 462 \, b^{10} c^{4} d^{6} x^{6} + 462 \, a b^{9} c^{3} d^{7} x^{6} + 462 \, a^{2} b^{8} c^{2} d^{8} x^{6} + 462 \, a^{3} b^{7} c d^{9} x^{6} + 462 \, a^{4} b^{6} d^{10} x^{6} + 462 \, b^{10} c^{5} d^{5} x^{5} + 462 \, a b^{9} c^{4} d^{6} x^{5} + 462 \, a^{2} b^{8} c^{3} d^{7} x^{5} + 462 \, a^{3} b^{7} c^{2} d^{8} x^{5} + 462 \, a^{4} b^{6} c d^{9} x^{5} + 462 \, a^{5} b^{5} d^{10} x^{5} + 330 \, b^{10} c^{6} d^{4} x^{4} + 330 \, a b^{9} c^{5} d^{5} x^{4} + 330 \, a^{2} b^{8} c^{4} d^{6} x^{4} + 330 \, a^{3} b^{7} c^{3} d^{7} x^{4} + 330 \, a^{4} b^{6} c^{2} d^{8} x^{4} + 330 \, a^{5} b^{5} c d^{9} x^{4} + 330 \, a^{6} b^{4} d^{10} x^{4} + 165 \, b^{10} c^{7} d^{3} x^{3} + 165 \, a b^{9} c^{6} d^{4} x^{3} + 165 \, a^{2} b^{8} c^{5} d^{5} x^{3} + 165 \, a^{3} b^{7} c^{4} d^{6} x^{3} + 165 \, a^{4} b^{6} c^{3} d^{7} x^{3} + 165 \, a^{5} b^{5} c^{2} d^{8} x^{3} + 165 \, a^{6} b^{4} c d^{9} x^{3} + 165 \, a^{7} b^{3} d^{10} x^{3} + 55 \, b^{10} c^{8} d^{2} x^{2} + 55 \, a b^{9} c^{7} d^{3} x^{2} + 55 \, a^{2} b^{8} c^{6} d^{4} x^{2} + 55 \, a^{3} b^{7} c^{5} d^{5} x^{2} + 55 \, a^{4} b^{6} c^{4} d^{6} x^{2} + 55 \, a^{5} b^{5} c^{3} d^{7} x^{2} + 55 \, a^{6} b^{4} c^{2} d^{8} x^{2} + 55 \, a^{7} b^{3} c d^{9} x^{2} + 55 \, a^{8} b^{2} d^{10} x^{2} + 11 \, b^{10} c^{9} d x + 11 \, a b^{9} c^{8} d^{2} x + 11 \, a^{2} b^{8} c^{7} d^{3} x + 11 \, a^{3} b^{7} c^{6} d^{4} x + 11 \, a^{4} b^{6} c^{5} d^{5} x + 11 \, a^{5} b^{5} c^{4} d^{6} x + 11 \, a^{6} b^{4} c^{3} d^{7} x + 11 \, a^{7} b^{3} c^{2} d^{8} x + 11 \, a^{8} b^{2} c d^{9} x + 11 \, a^{9} b d^{10} x + b^{10} c^{10} + a b^{9} c^{9} d + a^{2} b^{8} c^{8} d^{2} + a^{3} b^{7} c^{7} d^{3} + a^{4} b^{6} c^{6} d^{4} + a^{5} b^{5} c^{5} d^{5} + a^{6} b^{4} c^{4} d^{6} + a^{7} b^{3} c^{3} d^{7} + a^{8} b^{2} c^{2} d^{8} + a^{9} b c d^{9} + a^{10} d^{10}}{11 \, {\left (b x + a\right )}^{11} b^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^12,x, algorithm="giac")

[Out]

-1/11*(11*b^10*d^10*x^10 + 55*b^10*c*d^9*x^9 + 55*a*b^9*d^10*x^9 + 165*b^10*c^2*d^8*x^8 + 165*a*b^9*c*d^9*x^8
+ 165*a^2*b^8*d^10*x^8 + 330*b^10*c^3*d^7*x^7 + 330*a*b^9*c^2*d^8*x^7 + 330*a^2*b^8*c*d^9*x^7 + 330*a^3*b^7*d^
10*x^7 + 462*b^10*c^4*d^6*x^6 + 462*a*b^9*c^3*d^7*x^6 + 462*a^2*b^8*c^2*d^8*x^6 + 462*a^3*b^7*c*d^9*x^6 + 462*
a^4*b^6*d^10*x^6 + 462*b^10*c^5*d^5*x^5 + 462*a*b^9*c^4*d^6*x^5 + 462*a^2*b^8*c^3*d^7*x^5 + 462*a^3*b^7*c^2*d^
8*x^5 + 462*a^4*b^6*c*d^9*x^5 + 462*a^5*b^5*d^10*x^5 + 330*b^10*c^6*d^4*x^4 + 330*a*b^9*c^5*d^5*x^4 + 330*a^2*
b^8*c^4*d^6*x^4 + 330*a^3*b^7*c^3*d^7*x^4 + 330*a^4*b^6*c^2*d^8*x^4 + 330*a^5*b^5*c*d^9*x^4 + 330*a^6*b^4*d^10
*x^4 + 165*b^10*c^7*d^3*x^3 + 165*a*b^9*c^6*d^4*x^3 + 165*a^2*b^8*c^5*d^5*x^3 + 165*a^3*b^7*c^4*d^6*x^3 + 165*
a^4*b^6*c^3*d^7*x^3 + 165*a^5*b^5*c^2*d^8*x^3 + 165*a^6*b^4*c*d^9*x^3 + 165*a^7*b^3*d^10*x^3 + 55*b^10*c^8*d^2
*x^2 + 55*a*b^9*c^7*d^3*x^2 + 55*a^2*b^8*c^6*d^4*x^2 + 55*a^3*b^7*c^5*d^5*x^2 + 55*a^4*b^6*c^4*d^6*x^2 + 55*a^
5*b^5*c^3*d^7*x^2 + 55*a^6*b^4*c^2*d^8*x^2 + 55*a^7*b^3*c*d^9*x^2 + 55*a^8*b^2*d^10*x^2 + 11*b^10*c^9*d*x + 11
*a*b^9*c^8*d^2*x + 11*a^2*b^8*c^7*d^3*x + 11*a^3*b^7*c^6*d^4*x + 11*a^4*b^6*c^5*d^5*x + 11*a^5*b^5*c^4*d^6*x +
11*a^6*b^4*c^3*d^7*x + 11*a^7*b^3*c^2*d^8*x + 11*a^8*b^2*c*d^9*x + 11*a^9*b*d^10*x + b^10*c^10 + a*b^9*c^9*d
+ a^2*b^8*c^8*d^2 + a^3*b^7*c^7*d^3 + a^4*b^6*c^6*d^4 + a^5*b^5*c^5*d^5 + a^6*b^4*c^4*d^6 + a^7*b^3*c^3*d^7 +
a^8*b^2*c^2*d^8 + a^9*b*c*d^9 + a^10*d^10)/((b*x + a)^11*b^11)

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maple [B]  time = 0.01, size = 866, normalized size = 30.93 \begin {gather*} -\frac {d^{10}}{\left (b x +a \right ) b^{11}}+\frac {5 \left (a d -b c \right ) d^{9}}{\left (b x +a \right )^{2} b^{11}}-\frac {15 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) d^{8}}{\left (b x +a \right )^{3} b^{11}}+\frac {30 \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 )^{4} b^{11}}-\frac {42 \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 )^{5} b^{11}}+\frac {42 \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 )^{6} b^{11}}-\frac {30 \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}}{\left (b x +a \right )^{7} b^{11}}+\frac {15 \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}}{\left (b x +a \right )^{8} b^{11}}-\frac {5 \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}}{\left (b x +a \right )^{9} b^{11}}+\frac {\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}{\left (b x +a \right )^{10} 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}}{11 \left (b x +a \right )^{11} b^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^10/(b*x+a)^12,x)

[Out]

15*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)^8-15*d^8*(a^2*d^2-2*a*b*c*d+b^2*c^2)/b^11/(b*x+a)^3-30*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)^7-5*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)^9-42*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)^5+5*d^9*(a*d-b*c)/b^11/(b*x+a)^2+30*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)^4-1/11*(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)^11-d^1
0/b^11/(b*x+a)+42*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)^6+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)^10

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maxima [B]  time = 2.13, size = 920, normalized size = 32.86 \begin {gather*} -\frac {11 \, b^{10} d^{10} x^{10} + b^{10} c^{10} + a b^{9} c^{9} d + a^{2} b^{8} c^{8} d^{2} + a^{3} b^{7} c^{7} d^{3} + a^{4} b^{6} c^{6} d^{4} + a^{5} b^{5} c^{5} d^{5} + a^{6} b^{4} c^{4} d^{6} + a^{7} b^{3} c^{3} d^{7} + a^{8} b^{2} c^{2} d^{8} + a^{9} b c d^{9} + a^{10} d^{10} + 55 \, {\left (b^{10} c d^{9} + a b^{9} d^{10}\right )} x^{9} + 165 \, {\left (b^{10} c^{2} d^{8} + a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 330 \, {\left (b^{10} c^{3} d^{7} + a b^{9} c^{2} d^{8} + a^{2} b^{8} c d^{9} + a^{3} b^{7} d^{10}\right )} x^{7} + 462 \, {\left (b^{10} c^{4} d^{6} + a b^{9} c^{3} d^{7} + a^{2} b^{8} c^{2} d^{8} + a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 462 \, {\left (b^{10} c^{5} d^{5} + a b^{9} c^{4} d^{6} + a^{2} b^{8} c^{3} d^{7} + a^{3} b^{7} c^{2} d^{8} + a^{4} b^{6} c d^{9} + a^{5} b^{5} d^{10}\right )} x^{5} + 330 \, {\left (b^{10} c^{6} d^{4} + a b^{9} c^{5} d^{5} + a^{2} b^{8} c^{4} d^{6} + a^{3} b^{7} c^{3} d^{7} + a^{4} b^{6} c^{2} d^{8} + a^{5} b^{5} c d^{9} + a^{6} b^{4} d^{10}\right )} x^{4} + 165 \, {\left (b^{10} c^{7} d^{3} + a b^{9} c^{6} d^{4} + a^{2} b^{8} c^{5} d^{5} + a^{3} b^{7} c^{4} d^{6} + a^{4} b^{6} c^{3} d^{7} + a^{5} b^{5} c^{2} d^{8} + a^{6} b^{4} c d^{9} + a^{7} b^{3} d^{10}\right )} x^{3} + 55 \, {\left (b^{10} c^{8} d^{2} + a b^{9} c^{7} d^{3} + a^{2} b^{8} c^{6} d^{4} + a^{3} b^{7} c^{5} d^{5} + a^{4} b^{6} c^{4} d^{6} + a^{5} b^{5} c^{3} d^{7} + a^{6} b^{4} c^{2} d^{8} + a^{7} b^{3} c d^{9} + a^{8} b^{2} d^{10}\right )} x^{2} + 11 \, {\left (b^{10} c^{9} d + a b^{9} c^{8} d^{2} + a^{2} b^{8} c^{7} d^{3} + a^{3} b^{7} c^{6} d^{4} + a^{4} b^{6} c^{5} d^{5} + a^{5} b^{5} c^{4} d^{6} + a^{6} b^{4} c^{3} d^{7} + a^{7} b^{3} c^{2} d^{8} + a^{8} b^{2} c d^{9} + a^{9} b d^{10}\right )} x}{11 \, {\left (b^{22} x^{11} + 11 \, a b^{21} x^{10} + 55 \, a^{2} b^{20} x^{9} + 165 \, a^{3} b^{19} x^{8} + 330 \, a^{4} b^{18} x^{7} + 462 \, a^{5} b^{17} x^{6} + 462 \, a^{6} b^{16} x^{5} + 330 \, a^{7} b^{15} x^{4} + 165 \, a^{8} b^{14} x^{3} + 55 \, a^{9} b^{13} x^{2} + 11 \, a^{10} b^{12} x + a^{11} b^{11}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^12,x, algorithm="maxima")

[Out]

-1/11*(11*b^10*d^10*x^10 + b^10*c^10 + a*b^9*c^9*d + a^2*b^8*c^8*d^2 + a^3*b^7*c^7*d^3 + a^4*b^6*c^6*d^4 + a^5
*b^5*c^5*d^5 + a^6*b^4*c^4*d^6 + a^7*b^3*c^3*d^7 + a^8*b^2*c^2*d^8 + a^9*b*c*d^9 + a^10*d^10 + 55*(b^10*c*d^9
+ a*b^9*d^10)*x^9 + 165*(b^10*c^2*d^8 + a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 + 330*(b^10*c^3*d^7 + a*b^9*c^2*d^8 +
a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7 + 462*(b^10*c^4*d^6 + a*b^9*c^3*d^7 + a^2*b^8*c^2*d^8 + a^3*b^7*c*d^9 + a^4*
b^6*d^10)*x^6 + 462*(b^10*c^5*d^5 + a*b^9*c^4*d^6 + a^2*b^8*c^3*d^7 + a^3*b^7*c^2*d^8 + a^4*b^6*c*d^9 + a^5*b^
5*d^10)*x^5 + 330*(b^10*c^6*d^4 + a*b^9*c^5*d^5 + a^2*b^8*c^4*d^6 + a^3*b^7*c^3*d^7 + a^4*b^6*c^2*d^8 + a^5*b^
5*c*d^9 + a^6*b^4*d^10)*x^4 + 165*(b^10*c^7*d^3 + a*b^9*c^6*d^4 + a^2*b^8*c^5*d^5 + a^3*b^7*c^4*d^6 + a^4*b^6*
c^3*d^7 + a^5*b^5*c^2*d^8 + a^6*b^4*c*d^9 + a^7*b^3*d^10)*x^3 + 55*(b^10*c^8*d^2 + a*b^9*c^7*d^3 + a^2*b^8*c^6
*d^4 + a^3*b^7*c^5*d^5 + a^4*b^6*c^4*d^6 + a^5*b^5*c^3*d^7 + a^6*b^4*c^2*d^8 + a^7*b^3*c*d^9 + a^8*b^2*d^10)*x
^2 + 11*(b^10*c^9*d + a*b^9*c^8*d^2 + a^2*b^8*c^7*d^3 + a^3*b^7*c^6*d^4 + a^4*b^6*c^5*d^5 + a^5*b^5*c^4*d^6 +
a^6*b^4*c^3*d^7 + a^7*b^3*c^2*d^8 + a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^22*x^11 + 11*a*b^21*x^10 + 55*a^2*b^20*x
^9 + 165*a^3*b^19*x^8 + 330*a^4*b^18*x^7 + 462*a^5*b^17*x^6 + 462*a^6*b^16*x^5 + 330*a^7*b^15*x^4 + 165*a^8*b^
14*x^3 + 55*a^9*b^13*x^2 + 11*a^10*b^12*x + a^11*b^11)

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mupad [B]  time = 0.46, size = 1066, normalized size = 38.07 \begin {gather*} -\frac {a^{10}\,d^{10}+a^9\,b\,c\,d^9+11\,a^9\,b\,d^{10}\,x+a^8\,b^2\,c^2\,d^8+11\,a^8\,b^2\,c\,d^9\,x+55\,a^8\,b^2\,d^{10}\,x^2+a^7\,b^3\,c^3\,d^7+11\,a^7\,b^3\,c^2\,d^8\,x+55\,a^7\,b^3\,c\,d^9\,x^2+165\,a^7\,b^3\,d^{10}\,x^3+a^6\,b^4\,c^4\,d^6+11\,a^6\,b^4\,c^3\,d^7\,x+55\,a^6\,b^4\,c^2\,d^8\,x^2+165\,a^6\,b^4\,c\,d^9\,x^3+330\,a^6\,b^4\,d^{10}\,x^4+a^5\,b^5\,c^5\,d^5+11\,a^5\,b^5\,c^4\,d^6\,x+55\,a^5\,b^5\,c^3\,d^7\,x^2+165\,a^5\,b^5\,c^2\,d^8\,x^3+330\,a^5\,b^5\,c\,d^9\,x^4+462\,a^5\,b^5\,d^{10}\,x^5+a^4\,b^6\,c^6\,d^4+11\,a^4\,b^6\,c^5\,d^5\,x+55\,a^4\,b^6\,c^4\,d^6\,x^2+165\,a^4\,b^6\,c^3\,d^7\,x^3+330\,a^4\,b^6\,c^2\,d^8\,x^4+462\,a^4\,b^6\,c\,d^9\,x^5+462\,a^4\,b^6\,d^{10}\,x^6+a^3\,b^7\,c^7\,d^3+11\,a^3\,b^7\,c^6\,d^4\,x+55\,a^3\,b^7\,c^5\,d^5\,x^2+165\,a^3\,b^7\,c^4\,d^6\,x^3+330\,a^3\,b^7\,c^3\,d^7\,x^4+462\,a^3\,b^7\,c^2\,d^8\,x^5+462\,a^3\,b^7\,c\,d^9\,x^6+330\,a^3\,b^7\,d^{10}\,x^7+a^2\,b^8\,c^8\,d^2+11\,a^2\,b^8\,c^7\,d^3\,x+55\,a^2\,b^8\,c^6\,d^4\,x^2+165\,a^2\,b^8\,c^5\,d^5\,x^3+330\,a^2\,b^8\,c^4\,d^6\,x^4+462\,a^2\,b^8\,c^3\,d^7\,x^5+462\,a^2\,b^8\,c^2\,d^8\,x^6+330\,a^2\,b^8\,c\,d^9\,x^7+165\,a^2\,b^8\,d^{10}\,x^8+a\,b^9\,c^9\,d+11\,a\,b^9\,c^8\,d^2\,x+55\,a\,b^9\,c^7\,d^3\,x^2+165\,a\,b^9\,c^6\,d^4\,x^3+330\,a\,b^9\,c^5\,d^5\,x^4+462\,a\,b^9\,c^4\,d^6\,x^5+462\,a\,b^9\,c^3\,d^7\,x^6+330\,a\,b^9\,c^2\,d^8\,x^7+165\,a\,b^9\,c\,d^9\,x^8+55\,a\,b^9\,d^{10}\,x^9+b^{10}\,c^{10}+11\,b^{10}\,c^9\,d\,x+55\,b^{10}\,c^8\,d^2\,x^2+165\,b^{10}\,c^7\,d^3\,x^3+330\,b^{10}\,c^6\,d^4\,x^4+462\,b^{10}\,c^5\,d^5\,x^5+462\,b^{10}\,c^4\,d^6\,x^6+330\,b^{10}\,c^3\,d^7\,x^7+165\,b^{10}\,c^2\,d^8\,x^8+55\,b^{10}\,c\,d^9\,x^9+11\,b^{10}\,d^{10}\,x^{10}}{11\,a^{11}\,b^{11}+121\,a^{10}\,b^{12}\,x+605\,a^9\,b^{13}\,x^2+1815\,a^8\,b^{14}\,x^3+3630\,a^7\,b^{15}\,x^4+5082\,a^6\,b^{16}\,x^5+5082\,a^5\,b^{17}\,x^6+3630\,a^4\,b^{18}\,x^7+1815\,a^3\,b^{19}\,x^8+605\,a^2\,b^{20}\,x^9+121\,a\,b^{21}\,x^{10}+11\,b^{22}\,x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^10/(a + b*x)^12,x)

[Out]

-(a^10*d^10 + b^10*c^10 + 11*b^10*d^10*x^10 + 55*a*b^9*d^10*x^9 + 55*b^10*c*d^9*x^9 + a^2*b^8*c^8*d^2 + a^3*b^
7*c^7*d^3 + a^4*b^6*c^6*d^4 + a^5*b^5*c^5*d^5 + a^6*b^4*c^4*d^6 + a^7*b^3*c^3*d^7 + a^8*b^2*c^2*d^8 + 55*a^8*b
^2*d^10*x^2 + 165*a^7*b^3*d^10*x^3 + 330*a^6*b^4*d^10*x^4 + 462*a^5*b^5*d^10*x^5 + 462*a^4*b^6*d^10*x^6 + 330*
a^3*b^7*d^10*x^7 + 165*a^2*b^8*d^10*x^8 + 55*b^10*c^8*d^2*x^2 + 165*b^10*c^7*d^3*x^3 + 330*b^10*c^6*d^4*x^4 +
462*b^10*c^5*d^5*x^5 + 462*b^10*c^4*d^6*x^6 + 330*b^10*c^3*d^7*x^7 + 165*b^10*c^2*d^8*x^8 + a*b^9*c^9*d + a^9*
b*c*d^9 + 11*a^9*b*d^10*x + 11*b^10*c^9*d*x + 55*a^2*b^8*c^6*d^4*x^2 + 55*a^3*b^7*c^5*d^5*x^2 + 55*a^4*b^6*c^4
*d^6*x^2 + 55*a^5*b^5*c^3*d^7*x^2 + 55*a^6*b^4*c^2*d^8*x^2 + 165*a^2*b^8*c^5*d^5*x^3 + 165*a^3*b^7*c^4*d^6*x^3
+ 165*a^4*b^6*c^3*d^7*x^3 + 165*a^5*b^5*c^2*d^8*x^3 + 330*a^2*b^8*c^4*d^6*x^4 + 330*a^3*b^7*c^3*d^7*x^4 + 330
*a^4*b^6*c^2*d^8*x^4 + 462*a^2*b^8*c^3*d^7*x^5 + 462*a^3*b^7*c^2*d^8*x^5 + 462*a^2*b^8*c^2*d^8*x^6 + 11*a*b^9*
c^8*d^2*x + 11*a^8*b^2*c*d^9*x + 165*a*b^9*c*d^9*x^8 + 11*a^2*b^8*c^7*d^3*x + 11*a^3*b^7*c^6*d^4*x + 11*a^4*b^
6*c^5*d^5*x + 11*a^5*b^5*c^4*d^6*x + 11*a^6*b^4*c^3*d^7*x + 11*a^7*b^3*c^2*d^8*x + 55*a*b^9*c^7*d^3*x^2 + 55*a
^7*b^3*c*d^9*x^2 + 165*a*b^9*c^6*d^4*x^3 + 165*a^6*b^4*c*d^9*x^3 + 330*a*b^9*c^5*d^5*x^4 + 330*a^5*b^5*c*d^9*x
^4 + 462*a*b^9*c^4*d^6*x^5 + 462*a^4*b^6*c*d^9*x^5 + 462*a*b^9*c^3*d^7*x^6 + 462*a^3*b^7*c*d^9*x^6 + 330*a*b^9
*c^2*d^8*x^7 + 330*a^2*b^8*c*d^9*x^7)/(11*a^11*b^11 + 11*b^22*x^11 + 121*a^10*b^12*x + 121*a*b^21*x^10 + 605*a
^9*b^13*x^2 + 1815*a^8*b^14*x^3 + 3630*a^7*b^15*x^4 + 5082*a^6*b^16*x^5 + 5082*a^5*b^17*x^6 + 3630*a^4*b^18*x^
7 + 1815*a^3*b^19*x^8 + 605*a^2*b^20*x^9)

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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)**12,x)

[Out]

Timed out

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