3.1101 \(\int \frac{(a+b x)^{10} (A+B x)}{(d+e x)^{13}} \, dx\)

Optimal. Leaf size=86 \[ \frac{(a+b x)^{11} (-12 a B e+A b e+11 b B d)}{132 e (d+e x)^{11} (b d-a e)^2}-\frac{(a+b x)^{11} (B d-A e)}{12 e (d+e x)^{12} (b d-a e)} \]

[Out]

-((B*d - A*e)*(a + b*x)^11)/(12*e*(b*d - a*e)*(d + e*x)^12) + ((11*b*B*d + A*b*e - 12*a*B*e)*(a + b*x)^11)/(13
2*e*(b*d - a*e)^2*(d + e*x)^11)

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Rubi [A]  time = 0.0319486, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {78, 37} \[ \frac{(a+b x)^{11} (-12 a B e+A b e+11 b B d)}{132 e (d+e x)^{11} (b d-a e)^2}-\frac{(a+b x)^{11} (B d-A e)}{12 e (d+e x)^{12} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^10*(A + B*x))/(d + e*x)^13,x]

[Out]

-((B*d - A*e)*(a + b*x)^11)/(12*e*(b*d - a*e)*(d + e*x)^12) + ((11*b*B*d + A*b*e - 12*a*B*e)*(a + b*x)^11)/(13
2*e*(b*d - a*e)^2*(d + e*x)^11)

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

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

Mathematica [B]  time = 0.7442, size = 1421, normalized size = 16.52 \[ -\frac{\left (A e \left (d^{10}+12 e x d^9+66 e^2 x^2 d^8+220 e^3 x^3 d^7+495 e^4 x^4 d^6+792 e^5 x^5 d^5+924 e^6 x^6 d^4+792 e^7 x^7 d^3+495 e^8 x^8 d^2+220 e^9 x^9 d+66 e^{10} x^{10}\right )+11 B \left (d^{11}+12 e x d^{10}+66 e^2 x^2 d^9+220 e^3 x^3 d^8+495 e^4 x^4 d^7+792 e^5 x^5 d^6+924 e^6 x^6 d^5+792 e^7 x^7 d^4+495 e^8 x^8 d^3+220 e^9 x^9 d^2+66 e^{10} x^{10} d+12 e^{11} x^{11}\right )\right ) b^{10}+2 a e \left (A e \left (d^9+12 e x d^8+66 e^2 x^2 d^7+220 e^3 x^3 d^6+495 e^4 x^4 d^5+792 e^5 x^5 d^4+924 e^6 x^6 d^3+792 e^7 x^7 d^2+495 e^8 x^8 d+220 e^9 x^9\right )+5 B \left (d^{10}+12 e x d^9+66 e^2 x^2 d^8+220 e^3 x^3 d^7+495 e^4 x^4 d^6+792 e^5 x^5 d^5+924 e^6 x^6 d^4+792 e^7 x^7 d^3+495 e^8 x^8 d^2+220 e^9 x^9 d+66 e^{10} x^{10}\right )\right ) b^9+3 a^2 e^2 \left (A e \left (d^8+12 e x d^7+66 e^2 x^2 d^6+220 e^3 x^3 d^5+495 e^4 x^4 d^4+792 e^5 x^5 d^3+924 e^6 x^6 d^2+792 e^7 x^7 d+495 e^8 x^8\right )+3 B \left (d^9+12 e x d^8+66 e^2 x^2 d^7+220 e^3 x^3 d^6+495 e^4 x^4 d^5+792 e^5 x^5 d^4+924 e^6 x^6 d^3+792 e^7 x^7 d^2+495 e^8 x^8 d+220 e^9 x^9\right )\right ) b^8+4 a^3 e^3 \left (A e \left (d^7+12 e x d^6+66 e^2 x^2 d^5+220 e^3 x^3 d^4+495 e^4 x^4 d^3+792 e^5 x^5 d^2+924 e^6 x^6 d+792 e^7 x^7\right )+2 B \left (d^8+12 e x d^7+66 e^2 x^2 d^6+220 e^3 x^3 d^5+495 e^4 x^4 d^4+792 e^5 x^5 d^3+924 e^6 x^6 d^2+792 e^7 x^7 d+495 e^8 x^8\right )\right ) b^7+a^4 e^4 \left (5 A e \left (d^6+12 e x d^5+66 e^2 x^2 d^4+220 e^3 x^3 d^3+495 e^4 x^4 d^2+792 e^5 x^5 d+924 e^6 x^6\right )+7 B \left (d^7+12 e x d^6+66 e^2 x^2 d^5+220 e^3 x^3 d^4+495 e^4 x^4 d^3+792 e^5 x^5 d^2+924 e^6 x^6 d+792 e^7 x^7\right )\right ) b^6+6 a^5 e^5 \left (A e \left (d^5+12 e x d^4+66 e^2 x^2 d^3+220 e^3 x^3 d^2+495 e^4 x^4 d+792 e^5 x^5\right )+B \left (d^6+12 e x d^5+66 e^2 x^2 d^4+220 e^3 x^3 d^3+495 e^4 x^4 d^2+792 e^5 x^5 d+924 e^6 x^6\right )\right ) b^5+a^6 e^6 \left (7 A e \left (d^4+12 e x d^3+66 e^2 x^2 d^2+220 e^3 x^3 d+495 e^4 x^4\right )+5 B \left (d^5+12 e x d^4+66 e^2 x^2 d^3+220 e^3 x^3 d^2+495 e^4 x^4 d+792 e^5 x^5\right )\right ) b^4+4 a^7 e^7 \left (2 A e \left (d^3+12 e x d^2+66 e^2 x^2 d+220 e^3 x^3\right )+B \left (d^4+12 e x d^3+66 e^2 x^2 d^2+220 e^3 x^3 d+495 e^4 x^4\right )\right ) b^3+3 a^8 e^8 \left (3 A e \left (d^2+12 e x d+66 e^2 x^2\right )+B \left (d^3+12 e x d^2+66 e^2 x^2 d+220 e^3 x^3\right )\right ) b^2+2 a^9 e^9 \left (5 A e (d+12 e x)+B \left (d^2+12 e x d+66 e^2 x^2\right )\right ) b+a^{10} e^{10} (11 A e+B (d+12 e x))}{132 e^{12} (d+e x)^{12}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^10*e^10*(11*A*e + B*(d + 12*e*x)) + 2*a^9*b*e^9*(5*A*e*(d + 12*e*x) + B*(d^2 + 12*d*e*x + 66*e^2*x^2)) + 3
*a^8*b^2*e^8*(3*A*e*(d^2 + 12*d*e*x + 66*e^2*x^2) + B*(d^3 + 12*d^2*e*x + 66*d*e^2*x^2 + 220*e^3*x^3)) + 4*a^7
*b^3*e^7*(2*A*e*(d^3 + 12*d^2*e*x + 66*d*e^2*x^2 + 220*e^3*x^3) + B*(d^4 + 12*d^3*e*x + 66*d^2*e^2*x^2 + 220*d
*e^3*x^3 + 495*e^4*x^4)) + a^6*b^4*e^6*(7*A*e*(d^4 + 12*d^3*e*x + 66*d^2*e^2*x^2 + 220*d*e^3*x^3 + 495*e^4*x^4
) + 5*B*(d^5 + 12*d^4*e*x + 66*d^3*e^2*x^2 + 220*d^2*e^3*x^3 + 495*d*e^4*x^4 + 792*e^5*x^5)) + 6*a^5*b^5*e^5*(
A*e*(d^5 + 12*d^4*e*x + 66*d^3*e^2*x^2 + 220*d^2*e^3*x^3 + 495*d*e^4*x^4 + 792*e^5*x^5) + B*(d^6 + 12*d^5*e*x
+ 66*d^4*e^2*x^2 + 220*d^3*e^3*x^3 + 495*d^2*e^4*x^4 + 792*d*e^5*x^5 + 924*e^6*x^6)) + a^4*b^6*e^4*(5*A*e*(d^6
+ 12*d^5*e*x + 66*d^4*e^2*x^2 + 220*d^3*e^3*x^3 + 495*d^2*e^4*x^4 + 792*d*e^5*x^5 + 924*e^6*x^6) + 7*B*(d^7 +
12*d^6*e*x + 66*d^5*e^2*x^2 + 220*d^4*e^3*x^3 + 495*d^3*e^4*x^4 + 792*d^2*e^5*x^5 + 924*d*e^6*x^6 + 792*e^7*x
^7)) + 4*a^3*b^7*e^3*(A*e*(d^7 + 12*d^6*e*x + 66*d^5*e^2*x^2 + 220*d^4*e^3*x^3 + 495*d^3*e^4*x^4 + 792*d^2*e^5
*x^5 + 924*d*e^6*x^6 + 792*e^7*x^7) + 2*B*(d^8 + 12*d^7*e*x + 66*d^6*e^2*x^2 + 220*d^5*e^3*x^3 + 495*d^4*e^4*x
^4 + 792*d^3*e^5*x^5 + 924*d^2*e^6*x^6 + 792*d*e^7*x^7 + 495*e^8*x^8)) + 3*a^2*b^8*e^2*(A*e*(d^8 + 12*d^7*e*x
+ 66*d^6*e^2*x^2 + 220*d^5*e^3*x^3 + 495*d^4*e^4*x^4 + 792*d^3*e^5*x^5 + 924*d^2*e^6*x^6 + 792*d*e^7*x^7 + 495
*e^8*x^8) + 3*B*(d^9 + 12*d^8*e*x + 66*d^7*e^2*x^2 + 220*d^6*e^3*x^3 + 495*d^5*e^4*x^4 + 792*d^4*e^5*x^5 + 924
*d^3*e^6*x^6 + 792*d^2*e^7*x^7 + 495*d*e^8*x^8 + 220*e^9*x^9)) + 2*a*b^9*e*(A*e*(d^9 + 12*d^8*e*x + 66*d^7*e^2
*x^2 + 220*d^6*e^3*x^3 + 495*d^5*e^4*x^4 + 792*d^4*e^5*x^5 + 924*d^3*e^6*x^6 + 792*d^2*e^7*x^7 + 495*d*e^8*x^8
+ 220*e^9*x^9) + 5*B*(d^10 + 12*d^9*e*x + 66*d^8*e^2*x^2 + 220*d^7*e^3*x^3 + 495*d^6*e^4*x^4 + 792*d^5*e^5*x^
5 + 924*d^4*e^6*x^6 + 792*d^3*e^7*x^7 + 495*d^2*e^8*x^8 + 220*d*e^9*x^9 + 66*e^10*x^10)) + b^10*(A*e*(d^10 + 1
2*d^9*e*x + 66*d^8*e^2*x^2 + 220*d^7*e^3*x^3 + 495*d^6*e^4*x^4 + 792*d^5*e^5*x^5 + 924*d^4*e^6*x^6 + 792*d^3*e
^7*x^7 + 495*d^2*e^8*x^8 + 220*d*e^9*x^9 + 66*e^10*x^10) + 11*B*(d^11 + 12*d^10*e*x + 66*d^9*e^2*x^2 + 220*d^8
*e^3*x^3 + 495*d^7*e^4*x^4 + 792*d^6*e^5*x^5 + 924*d^5*e^6*x^6 + 792*d^4*e^7*x^7 + 495*d^3*e^8*x^8 + 220*d^2*e
^9*x^9 + 66*d*e^10*x^10 + 12*e^11*x^11)))/(132*e^12*(d + e*x)^12)

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Maple [B]  time = 0.014, size = 1942, normalized size = 22.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10*(B*x+A)/(e*x+d)^13,x)

[Out]

-7*b^5*(5*A*a^4*b*e^5-20*A*a^3*b^2*d*e^4+30*A*a^2*b^3*d^2*e^3-20*A*a*b^4*d^3*e^2+5*A*b^5*d^4*e+6*B*a^5*e^5-35*
B*a^4*b*d*e^4+80*B*a^3*b^2*d^2*e^3-90*B*a^2*b^3*d^3*e^2+50*B*a*b^4*d^4*e-11*B*b^5*d^5)/e^12/(e*x+d)^6-5/3*b^8*
(2*A*a*b*e^2-2*A*b^2*d*e+9*B*a^2*e^2-20*B*a*b*d*e+11*B*b^2*d^2)/e^12/(e*x+d)^3-6*b^4*(6*A*a^5*b*e^6-30*A*a^4*b
^2*d*e^5+60*A*a^3*b^3*d^2*e^4-60*A*a^2*b^4*d^3*e^3+30*A*a*b^5*d^4*e^2-6*A*b^6*d^5*e+5*B*a^6*e^6-36*B*a^5*b*d*e
^5+105*B*a^4*b^2*d^2*e^4-160*B*a^3*b^3*d^3*e^3+135*B*a^2*b^4*d^4*e^2-60*B*a*b^5*d^5*e+11*B*b^6*d^6)/e^12/(e*x+
d)^7-5/3*b^2*(8*A*a^7*b*e^8-56*A*a^6*b^2*d*e^7+168*A*a^5*b^3*d^2*e^6-280*A*a^4*b^4*d^3*e^5+280*A*a^3*b^5*d^4*e
^4-168*A*a^2*b^6*d^5*e^3+56*A*a*b^7*d^6*e^2-8*A*b^8*d^7*e+3*B*a^8*e^8-32*B*a^7*b*d*e^7+140*B*a^6*b^2*d^2*e^6-3
36*B*a^5*b^3*d^3*e^5+490*B*a^4*b^4*d^4*e^4-448*B*a^3*b^5*d^5*e^3+252*B*a^2*b^6*d^6*e^2-80*B*a*b^7*d^7*e+11*B*b
^8*d^8)/e^12/(e*x+d)^9-1/2*b^9*(A*b*e+10*B*a*e-11*B*b*d)/e^12/(e*x+d)^2-1/12*(A*a^10*e^11-10*A*a^9*b*d*e^10+45
*A*a^8*b^2*d^2*e^9-120*A*a^7*b^3*d^3*e^8+210*A*a^6*b^4*d^4*e^7-252*A*a^5*b^5*d^5*e^6+210*A*a^4*b^6*d^6*e^5-120
*A*a^3*b^7*d^7*e^4+45*A*a^2*b^8*d^8*e^3-10*A*a*b^9*d^9*e^2+A*b^10*d^10*e-B*a^10*d*e^10+10*B*a^9*b*d^2*e^9-45*B
*a^8*b^2*d^3*e^8+120*B*a^7*b^3*d^4*e^7-210*B*a^6*b^4*d^5*e^6+252*B*a^5*b^5*d^6*e^5-210*B*a^4*b^6*d^7*e^4+120*B
*a^3*b^7*d^8*e^3-45*B*a^2*b^8*d^9*e^2+10*B*a*b^9*d^10*e-B*b^10*d^11)/e^12/(e*x+d)^12-B*b^10/e^12/(e*x+d)-6*b^6
*(4*A*a^3*b*e^4-12*A*a^2*b^2*d*e^3+12*A*a*b^3*d^2*e^2-4*A*b^4*d^3*e+7*B*a^4*e^4-32*B*a^3*b*d*e^3+54*B*a^2*b^2*
d^2*e^2-40*B*a*b^3*d^3*e+11*B*b^4*d^4)/e^12/(e*x+d)^5-15/4*b^3*(7*A*a^6*b*e^7-42*A*a^5*b^2*d*e^6+105*A*a^4*b^3
*d^2*e^5-140*A*a^3*b^4*d^3*e^4+105*A*a^2*b^5*d^4*e^3-42*A*a*b^6*d^5*e^2+7*A*b^7*d^6*e+4*B*a^7*e^7-35*B*a^6*b*d
*e^6+126*B*a^5*b^2*d^2*e^5-245*B*a^4*b^3*d^3*e^4+280*B*a^3*b^4*d^4*e^3-189*B*a^2*b^5*d^5*e^2+70*B*a*b^6*d^6*e-
11*B*b^7*d^7)/e^12/(e*x+d)^8-1/2*b*(9*A*a^8*b*e^9-72*A*a^7*b^2*d*e^8+252*A*a^6*b^3*d^2*e^7-504*A*a^5*b^4*d^3*e
^6+630*A*a^4*b^5*d^4*e^5-504*A*a^3*b^6*d^5*e^4+252*A*a^2*b^7*d^6*e^3-72*A*a*b^8*d^7*e^2+9*A*b^9*d^8*e+2*B*a^9*
e^9-27*B*a^8*b*d*e^8+144*B*a^7*b^2*d^2*e^7-420*B*a^6*b^3*d^3*e^6+756*B*a^5*b^4*d^4*e^5-882*B*a^4*b^5*d^5*e^4+6
72*B*a^3*b^6*d^6*e^3-324*B*a^2*b^7*d^7*e^2+90*B*a*b^8*d^8*e-11*B*b^9*d^9)/e^12/(e*x+d)^10-15/4*b^7*(3*A*a^2*b*
e^3-6*A*a*b^2*d*e^2+3*A*b^3*d^2*e+8*B*a^3*e^3-27*B*a^2*b*d*e^2+30*B*a*b^2*d^2*e-11*B*b^3*d^3)/e^12/(e*x+d)^4-1
/11*(10*A*a^9*b*e^10-90*A*a^8*b^2*d*e^9+360*A*a^7*b^3*d^2*e^8-840*A*a^6*b^4*d^3*e^7+1260*A*a^5*b^5*d^4*e^6-126
0*A*a^4*b^6*d^5*e^5+840*A*a^3*b^7*d^6*e^4-360*A*a^2*b^8*d^7*e^3+90*A*a*b^9*d^8*e^2-10*A*b^10*d^9*e+B*a^10*e^10
-20*B*a^9*b*d*e^9+135*B*a^8*b^2*d^2*e^8-480*B*a^7*b^3*d^3*e^7+1050*B*a^6*b^4*d^4*e^6-1512*B*a^5*b^5*d^5*e^5+14
70*B*a^4*b^6*d^6*e^4-960*B*a^3*b^7*d^7*e^3+405*B*a^2*b^8*d^8*e^2-100*B*a*b^9*d^9*e+11*B*b^10*d^10)/e^12/(e*x+d
)^11

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Maxima [B]  time = 1.80929, size = 2531, normalized size = 29.43 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/132*(132*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 11*A*a^10*e^11 + (10*B*a*b^9 + A*b^10)*d^10*e + (9*B*a^2*b^8 +
2*A*a*b^9)*d^9*e^2 + (8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + (7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 + (6*B*a^5*b^5
+ 5*A*a^4*b^6)*d^6*e^5 + (5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + (4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + (3*B*a^8
*b^2 + 8*A*a^7*b^3)*d^3*e^8 + (2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 + (B*a^10 + 10*A*a^9*b)*d*e^10 + 66*(11*B*b^10
*d*e^10 + (10*B*a*b^9 + A*b^10)*e^11)*x^10 + 220*(11*B*b^10*d^2*e^9 + (10*B*a*b^9 + A*b^10)*d*e^10 + (9*B*a^2*
b^8 + 2*A*a*b^9)*e^11)*x^9 + 495*(11*B*b^10*d^3*e^8 + (10*B*a*b^9 + A*b^10)*d^2*e^9 + (9*B*a^2*b^8 + 2*A*a*b^9
)*d*e^10 + (8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 792*(11*B*b^10*d^4*e^7 + (10*B*a*b^9 + A*b^10)*d^3*e^8 + (9
*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 + (8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 + (7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 +
924*(11*B*b^10*d^5*e^6 + (10*B*a*b^9 + A*b^10)*d^4*e^7 + (9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 + (8*B*a^3*b^7 + 3
*A*a^2*b^8)*d^2*e^9 + (7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 + (6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 792*(11*B*b
^10*d^6*e^5 + (10*B*a*b^9 + A*b^10)*d^5*e^6 + (9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 + (8*B*a^3*b^7 + 3*A*a^2*b^8)*
d^3*e^8 + (7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 + (6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + (5*B*a^6*b^4 + 6*A*a^5*b^
5)*e^11)*x^5 + 495*(11*B*b^10*d^7*e^4 + (10*B*a*b^9 + A*b^10)*d^6*e^5 + (9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 + (8
*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + (7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 + (6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9
+ (5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 + (4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 220*(11*B*b^10*d^8*e^3 + (10*B*
a*b^9 + A*b^10)*d^7*e^4 + (9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 + (8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + (7*B*a^4*b
^6 + 4*A*a^3*b^7)*d^4*e^7 + (6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + (5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 + (4*B*a
^7*b^3 + 7*A*a^6*b^4)*d*e^10 + (3*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 + 66*(11*B*b^10*d^9*e^2 + (10*B*a*b^9 + A
*b^10)*d^8*e^3 + (9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 + (8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + (7*B*a^4*b^6 + 4*A*
a^3*b^7)*d^5*e^6 + (6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 + (5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 + (4*B*a^7*b^3 +
7*A*a^6*b^4)*d^2*e^9 + (3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 + (2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 12*(11*B*b^1
0*d^10*e + (10*B*a*b^9 + A*b^10)*d^9*e^2 + (9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 + (8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7
*e^4 + (7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 + (6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + (5*B*a^6*b^4 + 6*A*a^5*b^5)
*d^4*e^7 + (4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 + (3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 + (2*B*a^9*b + 9*A*a^8*b^
2)*d*e^10 + (B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^24*x^12 + 12*d*e^23*x^11 + 66*d^2*e^22*x^10 + 220*d^3*e^21*x^9 +
495*d^4*e^20*x^8 + 792*d^5*e^19*x^7 + 924*d^6*e^18*x^6 + 792*d^7*e^17*x^5 + 495*d^8*e^16*x^4 + 220*d^9*e^15*x
^3 + 66*d^10*e^14*x^2 + 12*d^11*e^13*x + d^12*e^12)

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Fricas [B]  time = 2.03693, size = 3933, normalized size = 45.73 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/132*(132*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 11*A*a^10*e^11 + (10*B*a*b^9 + A*b^10)*d^10*e + (9*B*a^2*b^8 +
2*A*a*b^9)*d^9*e^2 + (8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + (7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 + (6*B*a^5*b^5
+ 5*A*a^4*b^6)*d^6*e^5 + (5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + (4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + (3*B*a^8
*b^2 + 8*A*a^7*b^3)*d^3*e^8 + (2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 + (B*a^10 + 10*A*a^9*b)*d*e^10 + 66*(11*B*b^10
*d*e^10 + (10*B*a*b^9 + A*b^10)*e^11)*x^10 + 220*(11*B*b^10*d^2*e^9 + (10*B*a*b^9 + A*b^10)*d*e^10 + (9*B*a^2*
b^8 + 2*A*a*b^9)*e^11)*x^9 + 495*(11*B*b^10*d^3*e^8 + (10*B*a*b^9 + A*b^10)*d^2*e^9 + (9*B*a^2*b^8 + 2*A*a*b^9
)*d*e^10 + (8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 792*(11*B*b^10*d^4*e^7 + (10*B*a*b^9 + A*b^10)*d^3*e^8 + (9
*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 + (8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 + (7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 +
924*(11*B*b^10*d^5*e^6 + (10*B*a*b^9 + A*b^10)*d^4*e^7 + (9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 + (8*B*a^3*b^7 + 3
*A*a^2*b^8)*d^2*e^9 + (7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 + (6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 792*(11*B*b
^10*d^6*e^5 + (10*B*a*b^9 + A*b^10)*d^5*e^6 + (9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 + (8*B*a^3*b^7 + 3*A*a^2*b^8)*
d^3*e^8 + (7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 + (6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + (5*B*a^6*b^4 + 6*A*a^5*b^
5)*e^11)*x^5 + 495*(11*B*b^10*d^7*e^4 + (10*B*a*b^9 + A*b^10)*d^6*e^5 + (9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 + (8
*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + (7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 + (6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9
+ (5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 + (4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 220*(11*B*b^10*d^8*e^3 + (10*B*
a*b^9 + A*b^10)*d^7*e^4 + (9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 + (8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + (7*B*a^4*b
^6 + 4*A*a^3*b^7)*d^4*e^7 + (6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + (5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 + (4*B*a
^7*b^3 + 7*A*a^6*b^4)*d*e^10 + (3*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 + 66*(11*B*b^10*d^9*e^2 + (10*B*a*b^9 + A
*b^10)*d^8*e^3 + (9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 + (8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + (7*B*a^4*b^6 + 4*A*
a^3*b^7)*d^5*e^6 + (6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 + (5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 + (4*B*a^7*b^3 +
7*A*a^6*b^4)*d^2*e^9 + (3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 + (2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 12*(11*B*b^1
0*d^10*e + (10*B*a*b^9 + A*b^10)*d^9*e^2 + (9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 + (8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7
*e^4 + (7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 + (6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + (5*B*a^6*b^4 + 6*A*a^5*b^5)
*d^4*e^7 + (4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 + (3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 + (2*B*a^9*b + 9*A*a^8*b^
2)*d*e^10 + (B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^24*x^12 + 12*d*e^23*x^11 + 66*d^2*e^22*x^10 + 220*d^3*e^21*x^9 +
495*d^4*e^20*x^8 + 792*d^5*e^19*x^7 + 924*d^6*e^18*x^6 + 792*d^7*e^17*x^5 + 495*d^8*e^16*x^4 + 220*d^9*e^15*x
^3 + 66*d^10*e^14*x^2 + 12*d^11*e^13*x + d^12*e^12)

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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((b*x+a)**10*(B*x+A)/(e*x+d)**13,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 2.1834, size = 2827, normalized size = 32.87 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/132*(132*B*b^10*x^11*e^11 + 726*B*b^10*d*x^10*e^10 + 2420*B*b^10*d^2*x^9*e^9 + 5445*B*b^10*d^3*x^8*e^8 + 87
12*B*b^10*d^4*x^7*e^7 + 10164*B*b^10*d^5*x^6*e^6 + 8712*B*b^10*d^6*x^5*e^5 + 5445*B*b^10*d^7*x^4*e^4 + 2420*B*
b^10*d^8*x^3*e^3 + 726*B*b^10*d^9*x^2*e^2 + 132*B*b^10*d^10*x*e + 11*B*b^10*d^11 + 660*B*a*b^9*x^10*e^11 + 66*
A*b^10*x^10*e^11 + 2200*B*a*b^9*d*x^9*e^10 + 220*A*b^10*d*x^9*e^10 + 4950*B*a*b^9*d^2*x^8*e^9 + 495*A*b^10*d^2
*x^8*e^9 + 7920*B*a*b^9*d^3*x^7*e^8 + 792*A*b^10*d^3*x^7*e^8 + 9240*B*a*b^9*d^4*x^6*e^7 + 924*A*b^10*d^4*x^6*e
^7 + 7920*B*a*b^9*d^5*x^5*e^6 + 792*A*b^10*d^5*x^5*e^6 + 4950*B*a*b^9*d^6*x^4*e^5 + 495*A*b^10*d^6*x^4*e^5 + 2
200*B*a*b^9*d^7*x^3*e^4 + 220*A*b^10*d^7*x^3*e^4 + 660*B*a*b^9*d^8*x^2*e^3 + 66*A*b^10*d^8*x^2*e^3 + 120*B*a*b
^9*d^9*x*e^2 + 12*A*b^10*d^9*x*e^2 + 10*B*a*b^9*d^10*e + A*b^10*d^10*e + 1980*B*a^2*b^8*x^9*e^11 + 440*A*a*b^9
*x^9*e^11 + 4455*B*a^2*b^8*d*x^8*e^10 + 990*A*a*b^9*d*x^8*e^10 + 7128*B*a^2*b^8*d^2*x^7*e^9 + 1584*A*a*b^9*d^2
*x^7*e^9 + 8316*B*a^2*b^8*d^3*x^6*e^8 + 1848*A*a*b^9*d^3*x^6*e^8 + 7128*B*a^2*b^8*d^4*x^5*e^7 + 1584*A*a*b^9*d
^4*x^5*e^7 + 4455*B*a^2*b^8*d^5*x^4*e^6 + 990*A*a*b^9*d^5*x^4*e^6 + 1980*B*a^2*b^8*d^6*x^3*e^5 + 440*A*a*b^9*d
^6*x^3*e^5 + 594*B*a^2*b^8*d^7*x^2*e^4 + 132*A*a*b^9*d^7*x^2*e^4 + 108*B*a^2*b^8*d^8*x*e^3 + 24*A*a*b^9*d^8*x*
e^3 + 9*B*a^2*b^8*d^9*e^2 + 2*A*a*b^9*d^9*e^2 + 3960*B*a^3*b^7*x^8*e^11 + 1485*A*a^2*b^8*x^8*e^11 + 6336*B*a^3
*b^7*d*x^7*e^10 + 2376*A*a^2*b^8*d*x^7*e^10 + 7392*B*a^3*b^7*d^2*x^6*e^9 + 2772*A*a^2*b^8*d^2*x^6*e^9 + 6336*B
*a^3*b^7*d^3*x^5*e^8 + 2376*A*a^2*b^8*d^3*x^5*e^8 + 3960*B*a^3*b^7*d^4*x^4*e^7 + 1485*A*a^2*b^8*d^4*x^4*e^7 +
1760*B*a^3*b^7*d^5*x^3*e^6 + 660*A*a^2*b^8*d^5*x^3*e^6 + 528*B*a^3*b^7*d^6*x^2*e^5 + 198*A*a^2*b^8*d^6*x^2*e^5
+ 96*B*a^3*b^7*d^7*x*e^4 + 36*A*a^2*b^8*d^7*x*e^4 + 8*B*a^3*b^7*d^8*e^3 + 3*A*a^2*b^8*d^8*e^3 + 5544*B*a^4*b^
6*x^7*e^11 + 3168*A*a^3*b^7*x^7*e^11 + 6468*B*a^4*b^6*d*x^6*e^10 + 3696*A*a^3*b^7*d*x^6*e^10 + 5544*B*a^4*b^6*
d^2*x^5*e^9 + 3168*A*a^3*b^7*d^2*x^5*e^9 + 3465*B*a^4*b^6*d^3*x^4*e^8 + 1980*A*a^3*b^7*d^3*x^4*e^8 + 1540*B*a^
4*b^6*d^4*x^3*e^7 + 880*A*a^3*b^7*d^4*x^3*e^7 + 462*B*a^4*b^6*d^5*x^2*e^6 + 264*A*a^3*b^7*d^5*x^2*e^6 + 84*B*a
^4*b^6*d^6*x*e^5 + 48*A*a^3*b^7*d^6*x*e^5 + 7*B*a^4*b^6*d^7*e^4 + 4*A*a^3*b^7*d^7*e^4 + 5544*B*a^5*b^5*x^6*e^1
1 + 4620*A*a^4*b^6*x^6*e^11 + 4752*B*a^5*b^5*d*x^5*e^10 + 3960*A*a^4*b^6*d*x^5*e^10 + 2970*B*a^5*b^5*d^2*x^4*e
^9 + 2475*A*a^4*b^6*d^2*x^4*e^9 + 1320*B*a^5*b^5*d^3*x^3*e^8 + 1100*A*a^4*b^6*d^3*x^3*e^8 + 396*B*a^5*b^5*d^4*
x^2*e^7 + 330*A*a^4*b^6*d^4*x^2*e^7 + 72*B*a^5*b^5*d^5*x*e^6 + 60*A*a^4*b^6*d^5*x*e^6 + 6*B*a^5*b^5*d^6*e^5 +
5*A*a^4*b^6*d^6*e^5 + 3960*B*a^6*b^4*x^5*e^11 + 4752*A*a^5*b^5*x^5*e^11 + 2475*B*a^6*b^4*d*x^4*e^10 + 2970*A*a
^5*b^5*d*x^4*e^10 + 1100*B*a^6*b^4*d^2*x^3*e^9 + 1320*A*a^5*b^5*d^2*x^3*e^9 + 330*B*a^6*b^4*d^3*x^2*e^8 + 396*
A*a^5*b^5*d^3*x^2*e^8 + 60*B*a^6*b^4*d^4*x*e^7 + 72*A*a^5*b^5*d^4*x*e^7 + 5*B*a^6*b^4*d^5*e^6 + 6*A*a^5*b^5*d^
5*e^6 + 1980*B*a^7*b^3*x^4*e^11 + 3465*A*a^6*b^4*x^4*e^11 + 880*B*a^7*b^3*d*x^3*e^10 + 1540*A*a^6*b^4*d*x^3*e^
10 + 264*B*a^7*b^3*d^2*x^2*e^9 + 462*A*a^6*b^4*d^2*x^2*e^9 + 48*B*a^7*b^3*d^3*x*e^8 + 84*A*a^6*b^4*d^3*x*e^8 +
4*B*a^7*b^3*d^4*e^7 + 7*A*a^6*b^4*d^4*e^7 + 660*B*a^8*b^2*x^3*e^11 + 1760*A*a^7*b^3*x^3*e^11 + 198*B*a^8*b^2*
d*x^2*e^10 + 528*A*a^7*b^3*d*x^2*e^10 + 36*B*a^8*b^2*d^2*x*e^9 + 96*A*a^7*b^3*d^2*x*e^9 + 3*B*a^8*b^2*d^3*e^8
+ 8*A*a^7*b^3*d^3*e^8 + 132*B*a^9*b*x^2*e^11 + 594*A*a^8*b^2*x^2*e^11 + 24*B*a^9*b*d*x*e^10 + 108*A*a^8*b^2*d*
x*e^10 + 2*B*a^9*b*d^2*e^9 + 9*A*a^8*b^2*d^2*e^9 + 12*B*a^10*x*e^11 + 120*A*a^9*b*x*e^11 + B*a^10*d*e^10 + 10*
A*a^9*b*d*e^10 + 11*A*a^10*e^11)*e^(-12)/(x*e + d)^12