3.1325 \(\int \frac{(c+d x)^{10}}{(a+b x)^{14}} \, dx\)

Optimal. Leaf size=89 \[ -\frac{d^2 (c+d x)^{11}}{858 (a+b x)^{11} (b c-a d)^3}+\frac{d (c+d x)^{11}}{78 (a+b x)^{12} (b c-a d)^2}-\frac{(c+d x)^{11}}{13 (a+b x)^{13} (b c-a d)} \]

[Out]

-(c + d*x)^11/(13*(b*c - a*d)*(a + b*x)^13) + (d*(c + d*x)^11)/(78*(b*c - a*d)^2*(a + b*x)^12) - (d^2*(c + d*x
)^11)/(858*(b*c - a*d)^3*(a + b*x)^11)

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Rubi [A]  time = 0.0199106, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {45, 37} \[ -\frac{d^2 (c+d x)^{11}}{858 (a+b x)^{11} (b c-a d)^3}+\frac{d (c+d x)^{11}}{78 (a+b x)^{12} (b c-a d)^2}-\frac{(c+d x)^{11}}{13 (a+b x)^{13} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^10/(a + b*x)^14,x]

[Out]

-(c + d*x)^11/(13*(b*c - a*d)*(a + b*x)^13) + (d*(c + d*x)^11)/(78*(b*c - a*d)^2*(a + b*x)^12) - (d^2*(c + d*x
)^11)/(858*(b*c - a*d)^3*(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)^{14}} \, dx &=-\frac{(c+d x)^{11}}{13 (b c-a d) (a+b x)^{13}}-\frac{(2 d) \int \frac{(c+d x)^{10}}{(a+b x)^{13}} \, dx}{13 (b c-a d)}\\ &=-\frac{(c+d x)^{11}}{13 (b c-a d) (a+b x)^{13}}+\frac{d (c+d x)^{11}}{78 (b c-a d)^2 (a+b x)^{12}}+\frac{d^2 \int \frac{(c+d x)^{10}}{(a+b x)^{12}} \, dx}{78 (b c-a d)^2}\\ &=-\frac{(c+d x)^{11}}{13 (b c-a d) (a+b x)^{13}}+\frac{d (c+d x)^{11}}{78 (b c-a d)^2 (a+b x)^{12}}-\frac{d^2 (c+d x)^{11}}{858 (b c-a d)^3 (a+b x)^{11}}\\ \end{align*}

Mathematica [B]  time = 0.287829, size = 690, normalized size = 7.75 \[ -\frac{3 a^2 b^8 d^2 \left (728 c^6 d^2 x^2+2002 c^5 d^3 x^3+3575 c^4 d^4 x^4+4290 c^3 d^5 x^5+3432 c^2 d^6 x^6+156 c^7 d x+15 c^8+1716 c d^7 x^7+429 d^8 x^8\right )+2 a^3 b^7 d^3 \left (819 c^5 d^2 x^2+2145 c^4 d^3 x^3+3575 c^3 d^4 x^4+3861 c^2 d^5 x^5+182 c^6 d x+18 c^7+2574 c d^6 x^6+858 d^7 x^7\right )+a^4 b^6 d^4 \left (1170 c^4 d^2 x^2+2860 c^3 d^3 x^3+4290 c^2 d^4 x^4+273 c^5 d x+28 c^6+3861 c d^5 x^5+1716 d^6 x^6\right )+3 a^5 b^5 d^5 \left (260 c^3 d^2 x^2+572 c^2 d^3 x^3+65 c^4 d x+7 c^5+715 c d^4 x^4+429 d^5 x^5\right )+a^6 b^4 d^6 \left (468 c^2 d^2 x^2+130 c^3 d x+15 c^4+858 c d^3 x^3+715 d^4 x^4\right )+2 a^7 b^3 d^7 \left (39 c^2 d x+5 c^3+117 c d^2 x^2+143 d^3 x^3\right )+3 a^8 b^2 d^8 \left (2 c^2+13 c d x+26 d^2 x^2\right )+a^9 b d^9 (3 c+13 d x)+a^{10} d^{10}+a b^9 d \left (2808 c^7 d^2 x^2+8008 c^6 d^3 x^3+15015 c^5 d^4 x^4+19305 c^4 d^5 x^5+17160 c^3 d^6 x^6+10296 c^2 d^7 x^7+585 c^8 d x+55 c^9+3861 c d^8 x^8+715 d^9 x^9\right )+b^{10} \left (3510 c^8 d^2 x^2+10296 c^7 d^3 x^3+20020 c^6 d^4 x^4+27027 c^5 d^5 x^5+25740 c^4 d^6 x^6+17160 c^3 d^7 x^7+7722 c^2 d^8 x^8+715 c^9 d x+66 c^{10}+2145 c d^9 x^9+286 d^{10} x^{10}\right )}{858 b^{11} (a+b x)^{13}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^10/(a + b*x)^14,x]

[Out]

-(a^10*d^10 + a^9*b*d^9*(3*c + 13*d*x) + 3*a^8*b^2*d^8*(2*c^2 + 13*c*d*x + 26*d^2*x^2) + 2*a^7*b^3*d^7*(5*c^3
+ 39*c^2*d*x + 117*c*d^2*x^2 + 143*d^3*x^3) + a^6*b^4*d^6*(15*c^4 + 130*c^3*d*x + 468*c^2*d^2*x^2 + 858*c*d^3*
x^3 + 715*d^4*x^4) + 3*a^5*b^5*d^5*(7*c^5 + 65*c^4*d*x + 260*c^3*d^2*x^2 + 572*c^2*d^3*x^3 + 715*c*d^4*x^4 + 4
29*d^5*x^5) + a^4*b^6*d^4*(28*c^6 + 273*c^5*d*x + 1170*c^4*d^2*x^2 + 2860*c^3*d^3*x^3 + 4290*c^2*d^4*x^4 + 386
1*c*d^5*x^5 + 1716*d^6*x^6) + 2*a^3*b^7*d^3*(18*c^7 + 182*c^6*d*x + 819*c^5*d^2*x^2 + 2145*c^4*d^3*x^3 + 3575*
c^3*d^4*x^4 + 3861*c^2*d^5*x^5 + 2574*c*d^6*x^6 + 858*d^7*x^7) + 3*a^2*b^8*d^2*(15*c^8 + 156*c^7*d*x + 728*c^6
*d^2*x^2 + 2002*c^5*d^3*x^3 + 3575*c^4*d^4*x^4 + 4290*c^3*d^5*x^5 + 3432*c^2*d^6*x^6 + 1716*c*d^7*x^7 + 429*d^
8*x^8) + a*b^9*d*(55*c^9 + 585*c^8*d*x + 2808*c^7*d^2*x^2 + 8008*c^6*d^3*x^3 + 15015*c^5*d^4*x^4 + 19305*c^4*d
^5*x^5 + 17160*c^3*d^6*x^6 + 10296*c^2*d^7*x^7 + 3861*c*d^8*x^8 + 715*d^9*x^9) + b^10*(66*c^10 + 715*c^9*d*x +
3510*c^8*d^2*x^2 + 10296*c^7*d^3*x^3 + 20020*c^6*d^4*x^4 + 27027*c^5*d^5*x^5 + 25740*c^4*d^6*x^6 + 17160*c^3*
d^7*x^7 + 7722*c^2*d^8*x^8 + 2145*c*d^9*x^9 + 286*d^10*x^10))/(858*b^11*(a + b*x)^13)

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Maple [B]  time = 0.008, size = 867, normalized size = 9.7 \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)^14,x)

[Out]

12*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)^10+63/2*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)^8+20*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)^6-45/11*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^11/(b*x+a)^11-1/13*(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)^13-1/3*d^10/b^11/(b*x+a)^3-9*d^8*(a^2*d^2-2*a*b*c*d+b^2*c^2)/b^11/(b*x+a)^
5+5/2*d^9*(a*d-b*c)/b^11/(b*x+a)^4+5/6*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)^12-7
0/3*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)^9-30*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)^7

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Maxima [B]  time = 1.26755, size = 1346, normalized size = 15.12 \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)^14,x, algorithm="maxima")

[Out]

-1/858*(286*b^10*d^10*x^10 + 66*b^10*c^10 + 55*a*b^9*c^9*d + 45*a^2*b^8*c^8*d^2 + 36*a^3*b^7*c^7*d^3 + 28*a^4*
b^6*c^6*d^4 + 21*a^5*b^5*c^5*d^5 + 15*a^6*b^4*c^4*d^6 + 10*a^7*b^3*c^3*d^7 + 6*a^8*b^2*c^2*d^8 + 3*a^9*b*c*d^9
+ a^10*d^10 + 715*(3*b^10*c*d^9 + a*b^9*d^10)*x^9 + 1287*(6*b^10*c^2*d^8 + 3*a*b^9*c*d^9 + a^2*b^8*d^10)*x^8
+ 1716*(10*b^10*c^3*d^7 + 6*a*b^9*c^2*d^8 + 3*a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7 + 1716*(15*b^10*c^4*d^6 + 10*a
*b^9*c^3*d^7 + 6*a^2*b^8*c^2*d^8 + 3*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 1287*(21*b^10*c^5*d^5 + 15*a*b^9*c^4*
d^6 + 10*a^2*b^8*c^3*d^7 + 6*a^3*b^7*c^2*d^8 + 3*a^4*b^6*c*d^9 + a^5*b^5*d^10)*x^5 + 715*(28*b^10*c^6*d^4 + 21
*a*b^9*c^5*d^5 + 15*a^2*b^8*c^4*d^6 + 10*a^3*b^7*c^3*d^7 + 6*a^4*b^6*c^2*d^8 + 3*a^5*b^5*c*d^9 + a^6*b^4*d^10)
*x^4 + 286*(36*b^10*c^7*d^3 + 28*a*b^9*c^6*d^4 + 21*a^2*b^8*c^5*d^5 + 15*a^3*b^7*c^4*d^6 + 10*a^4*b^6*c^3*d^7
+ 6*a^5*b^5*c^2*d^8 + 3*a^6*b^4*c*d^9 + a^7*b^3*d^10)*x^3 + 78*(45*b^10*c^8*d^2 + 36*a*b^9*c^7*d^3 + 28*a^2*b^
8*c^6*d^4 + 21*a^3*b^7*c^5*d^5 + 15*a^4*b^6*c^4*d^6 + 10*a^5*b^5*c^3*d^7 + 6*a^6*b^4*c^2*d^8 + 3*a^7*b^3*c*d^9
+ a^8*b^2*d^10)*x^2 + 13*(55*b^10*c^9*d + 45*a*b^9*c^8*d^2 + 36*a^2*b^8*c^7*d^3 + 28*a^3*b^7*c^6*d^4 + 21*a^4
*b^6*c^5*d^5 + 15*a^5*b^5*c^4*d^6 + 10*a^6*b^4*c^3*d^7 + 6*a^7*b^3*c^2*d^8 + 3*a^8*b^2*c*d^9 + a^9*b*d^10)*x)/
(b^24*x^13 + 13*a*b^23*x^12 + 78*a^2*b^22*x^11 + 286*a^3*b^21*x^10 + 715*a^4*b^20*x^9 + 1287*a^5*b^19*x^8 + 17
16*a^6*b^18*x^7 + 1716*a^7*b^17*x^6 + 1287*a^8*b^16*x^5 + 715*a^9*b^15*x^4 + 286*a^10*b^14*x^3 + 78*a^11*b^13*
x^2 + 13*a^12*b^12*x + a^13*b^11)

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Fricas [B]  time = 1.78129, size = 2126, normalized size = 23.89 \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)^14,x, algorithm="fricas")

[Out]

-1/858*(286*b^10*d^10*x^10 + 66*b^10*c^10 + 55*a*b^9*c^9*d + 45*a^2*b^8*c^8*d^2 + 36*a^3*b^7*c^7*d^3 + 28*a^4*
b^6*c^6*d^4 + 21*a^5*b^5*c^5*d^5 + 15*a^6*b^4*c^4*d^6 + 10*a^7*b^3*c^3*d^7 + 6*a^8*b^2*c^2*d^8 + 3*a^9*b*c*d^9
+ a^10*d^10 + 715*(3*b^10*c*d^9 + a*b^9*d^10)*x^9 + 1287*(6*b^10*c^2*d^8 + 3*a*b^9*c*d^9 + a^2*b^8*d^10)*x^8
+ 1716*(10*b^10*c^3*d^7 + 6*a*b^9*c^2*d^8 + 3*a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7 + 1716*(15*b^10*c^4*d^6 + 10*a
*b^9*c^3*d^7 + 6*a^2*b^8*c^2*d^8 + 3*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 1287*(21*b^10*c^5*d^5 + 15*a*b^9*c^4*
d^6 + 10*a^2*b^8*c^3*d^7 + 6*a^3*b^7*c^2*d^8 + 3*a^4*b^6*c*d^9 + a^5*b^5*d^10)*x^5 + 715*(28*b^10*c^6*d^4 + 21
*a*b^9*c^5*d^5 + 15*a^2*b^8*c^4*d^6 + 10*a^3*b^7*c^3*d^7 + 6*a^4*b^6*c^2*d^8 + 3*a^5*b^5*c*d^9 + a^6*b^4*d^10)
*x^4 + 286*(36*b^10*c^7*d^3 + 28*a*b^9*c^6*d^4 + 21*a^2*b^8*c^5*d^5 + 15*a^3*b^7*c^4*d^6 + 10*a^4*b^6*c^3*d^7
+ 6*a^5*b^5*c^2*d^8 + 3*a^6*b^4*c*d^9 + a^7*b^3*d^10)*x^3 + 78*(45*b^10*c^8*d^2 + 36*a*b^9*c^7*d^3 + 28*a^2*b^
8*c^6*d^4 + 21*a^3*b^7*c^5*d^5 + 15*a^4*b^6*c^4*d^6 + 10*a^5*b^5*c^3*d^7 + 6*a^6*b^4*c^2*d^8 + 3*a^7*b^3*c*d^9
+ a^8*b^2*d^10)*x^2 + 13*(55*b^10*c^9*d + 45*a*b^9*c^8*d^2 + 36*a^2*b^8*c^7*d^3 + 28*a^3*b^7*c^6*d^4 + 21*a^4
*b^6*c^5*d^5 + 15*a^5*b^5*c^4*d^6 + 10*a^6*b^4*c^3*d^7 + 6*a^7*b^3*c^2*d^8 + 3*a^8*b^2*c*d^9 + a^9*b*d^10)*x)/
(b^24*x^13 + 13*a*b^23*x^12 + 78*a^2*b^22*x^11 + 286*a^3*b^21*x^10 + 715*a^4*b^20*x^9 + 1287*a^5*b^19*x^8 + 17
16*a^6*b^18*x^7 + 1716*a^7*b^17*x^6 + 1287*a^8*b^16*x^5 + 715*a^9*b^15*x^4 + 286*a^10*b^14*x^3 + 78*a^11*b^13*
x^2 + 13*a^12*b^12*x + a^13*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)**14,x)

[Out]

Timed out

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Giac [B]  time = 1.05961, size = 1297, normalized size = 14.57 \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)^14,x, algorithm="giac")

[Out]

-1/858*(286*b^10*d^10*x^10 + 2145*b^10*c*d^9*x^9 + 715*a*b^9*d^10*x^9 + 7722*b^10*c^2*d^8*x^8 + 3861*a*b^9*c*d
^9*x^8 + 1287*a^2*b^8*d^10*x^8 + 17160*b^10*c^3*d^7*x^7 + 10296*a*b^9*c^2*d^8*x^7 + 5148*a^2*b^8*c*d^9*x^7 + 1
716*a^3*b^7*d^10*x^7 + 25740*b^10*c^4*d^6*x^6 + 17160*a*b^9*c^3*d^7*x^6 + 10296*a^2*b^8*c^2*d^8*x^6 + 5148*a^3
*b^7*c*d^9*x^6 + 1716*a^4*b^6*d^10*x^6 + 27027*b^10*c^5*d^5*x^5 + 19305*a*b^9*c^4*d^6*x^5 + 12870*a^2*b^8*c^3*
d^7*x^5 + 7722*a^3*b^7*c^2*d^8*x^5 + 3861*a^4*b^6*c*d^9*x^5 + 1287*a^5*b^5*d^10*x^5 + 20020*b^10*c^6*d^4*x^4 +
15015*a*b^9*c^5*d^5*x^4 + 10725*a^2*b^8*c^4*d^6*x^4 + 7150*a^3*b^7*c^3*d^7*x^4 + 4290*a^4*b^6*c^2*d^8*x^4 + 2
145*a^5*b^5*c*d^9*x^4 + 715*a^6*b^4*d^10*x^4 + 10296*b^10*c^7*d^3*x^3 + 8008*a*b^9*c^6*d^4*x^3 + 6006*a^2*b^8*
c^5*d^5*x^3 + 4290*a^3*b^7*c^4*d^6*x^3 + 2860*a^4*b^6*c^3*d^7*x^3 + 1716*a^5*b^5*c^2*d^8*x^3 + 858*a^6*b^4*c*d
^9*x^3 + 286*a^7*b^3*d^10*x^3 + 3510*b^10*c^8*d^2*x^2 + 2808*a*b^9*c^7*d^3*x^2 + 2184*a^2*b^8*c^6*d^4*x^2 + 16
38*a^3*b^7*c^5*d^5*x^2 + 1170*a^4*b^6*c^4*d^6*x^2 + 780*a^5*b^5*c^3*d^7*x^2 + 468*a^6*b^4*c^2*d^8*x^2 + 234*a^
7*b^3*c*d^9*x^2 + 78*a^8*b^2*d^10*x^2 + 715*b^10*c^9*d*x + 585*a*b^9*c^8*d^2*x + 468*a^2*b^8*c^7*d^3*x + 364*a
^3*b^7*c^6*d^4*x + 273*a^4*b^6*c^5*d^5*x + 195*a^5*b^5*c^4*d^6*x + 130*a^6*b^4*c^3*d^7*x + 78*a^7*b^3*c^2*d^8*
x + 39*a^8*b^2*c*d^9*x + 13*a^9*b*d^10*x + 66*b^10*c^10 + 55*a*b^9*c^9*d + 45*a^2*b^8*c^8*d^2 + 36*a^3*b^7*c^7
*d^3 + 28*a^4*b^6*c^6*d^4 + 21*a^5*b^5*c^5*d^5 + 15*a^6*b^4*c^4*d^6 + 10*a^7*b^3*c^3*d^7 + 6*a^8*b^2*c^2*d^8 +
3*a^9*b*c*d^9 + a^10*d^10)/((b*x + a)^13*b^11)