3.1324 \(\int \frac{(c+d x)^{10}}{(a+b x)^{13}} \, dx\)

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

[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)

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Rubi [A]  time = 0.0101382, antiderivative size = 58, normalized size of antiderivative = 1., 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} \[ \frac{d (c+d x)^{11}}{132 (a+b x)^{11} (b c-a d)^2}-\frac{(c+d x)^{11}}{12 (a+b x)^{12} (b c-a d)} \]

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 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])

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)^{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*}

Mathematica [B]  time = 0.300956, size = 684, normalized size = 11.79 \[ -\frac{3 a^2 b^8 d^2 \left (154 c^6 d^2 x^2+440 c^5 d^3 x^3+825 c^4 d^4 x^4+1056 c^3 d^5 x^5+924 c^2 d^6 x^6+32 c^7 d x+3 c^8+528 c d^7 x^7+165 d^8 x^8\right )+4 a^3 b^7 d^3 \left (99 c^5 d^2 x^2+275 c^4 d^3 x^3+495 c^3 d^4 x^4+594 c^2 d^5 x^5+21 c^6 d x+2 c^7+462 c d^6 x^6+198 d^7 x^7\right )+a^4 b^6 d^4 \left (330 c^4 d^2 x^2+880 c^3 d^3 x^3+1485 c^2 d^4 x^4+72 c^5 d x+7 c^6+1584 c d^5 x^5+924 d^6 x^6\right )+6 a^5 b^5 d^5 \left (44 c^3 d^2 x^2+110 c^2 d^3 x^3+10 c^4 d x+c^5+165 c d^4 x^4+132 d^5 x^5\right )+a^6 b^4 d^6 \left (198 c^2 d^2 x^2+48 c^3 d x+5 c^4+440 c d^3 x^3+495 d^4 x^4\right )+4 a^7 b^3 d^7 \left (9 c^2 d x+c^3+33 c d^2 x^2+55 d^3 x^3\right )+3 a^8 b^2 d^8 \left (c^2+8 c d x+22 d^2 x^2\right )+2 a^9 b d^9 (c+6 d x)+a^{10} d^{10}+2 a b^9 d \left (264 c^7 d^2 x^2+770 c^6 d^3 x^3+1485 c^5 d^4 x^4+1980 c^4 d^5 x^5+1848 c^3 d^6 x^6+1188 c^2 d^7 x^7+54 c^8 d x+5 c^9+495 c d^8 x^8+110 d^9 x^9\right )+b^{10} \left (594 c^8 d^2 x^2+1760 c^7 d^3 x^3+3465 c^6 d^4 x^4+4752 c^5 d^5 x^5+4620 c^4 d^6 x^6+3168 c^3 d^7 x^7+1485 c^2 d^8 x^8+120 c^9 d x+11 c^{10}+440 c d^9 x^9+66 d^{10} x^{10}\right )}{132 b^{11} (a+b x)^{12}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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 + 1188*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 + 3
465*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))/(132*b^11*(a + b*x)^12)

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Maple [B]  time = 0.011, size = 867, normalized size = 15. \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)^13,x)

[Out]

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

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Maxima [B]  time = 1.24292, size = 1331, normalized size = 22.95 \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)^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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Fricas [B]  time = 1.86512, size = 2041, normalized size = 35.19 \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)^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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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)**13,x)

[Out]

Timed out

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Giac [B]  time = 1.07204, size = 1297, normalized size = 22.36 \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)^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)