3.1323 \(\int \frac{(c+d x)^{10}}{(a+b x)^{12}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0030368, antiderivative size = 28, normalized size of antiderivative = 1., 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} \[ -\frac{(c+d x)^{11}}{11 (a+b x)^{11} (b c-a d)} \]

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*}

Mathematica [B]  time = 0.321978, size = 665, normalized size = 23.75 \[ -\frac{a^2 b^8 d^2 \left (55 c^6 d^2 x^2+165 c^5 d^3 x^3+330 c^4 d^4 x^4+462 c^3 d^5 x^5+462 c^2 d^6 x^6+11 c^7 d x+c^8+330 c d^7 x^7+165 d^8 x^8\right )+a^3 b^7 d^3 \left (55 c^5 d^2 x^2+165 c^4 d^3 x^3+330 c^3 d^4 x^4+462 c^2 d^5 x^5+11 c^6 d x+c^7+462 c d^6 x^6+330 d^7 x^7\right )+a^4 b^6 d^4 \left (55 c^4 d^2 x^2+165 c^3 d^3 x^3+330 c^2 d^4 x^4+11 c^5 d x+c^6+462 c d^5 x^5+462 d^6 x^6\right )+a^5 b^5 d^5 \left (55 c^3 d^2 x^2+165 c^2 d^3 x^3+11 c^4 d x+c^5+330 c d^4 x^4+462 d^5 x^5\right )+a^6 b^4 d^6 \left (55 c^2 d^2 x^2+11 c^3 d x+c^4+165 c d^3 x^3+330 d^4 x^4\right )+a^7 b^3 d^7 \left (11 c^2 d x+c^3+55 c d^2 x^2+165 d^3 x^3\right )+a^8 b^2 d^8 \left (c^2+11 c d x+55 d^2 x^2\right )+a^9 b d^9 (c+11 d x)+a^{10} d^{10}+a b^9 d \left (55 c^7 d^2 x^2+165 c^6 d^3 x^3+330 c^5 d^4 x^4+462 c^4 d^5 x^5+462 c^3 d^6 x^6+330 c^2 d^7 x^7+11 c^8 d x+c^9+165 c d^8 x^8+55 d^9 x^9\right )+b^{10} \left (55 c^8 d^2 x^2+165 c^7 d^3 x^3+330 c^6 d^4 x^4+462 c^5 d^5 x^5+462 c^4 d^6 x^6+330 c^3 d^7 x^7+165 c^2 d^8 x^8+11 c^9 d x+c^{10}+55 c d^9 x^9+11 d^{10} x^{10}\right )}{11 b^{11} (a+b x)^{11}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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 + 46
2*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))/(11*b^11*(a + b*x)^11)

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Maple [B]  time = 0.01, size = 866, normalized size = 30.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.20962, size = 1242, normalized size = 44.36 \begin{align*} \text{result too large to display} \end{align*}

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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Fricas [B]  time = 1.64626, size = 1858, normalized size = 66.36 \begin{align*} \text{result too large to display} \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Giac [B]  time = 1.06017, size = 1284, normalized size = 45.86 \begin{align*} \text{result too large to display} \end{align*}

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)