3.1103 \(\int \frac{(a+b x)^{10} (A+B x)}{(d+e x)^{15}} \, dx\)

Optimal. Leaf size=185 \[ \frac{b^2 (a+b x)^{11} (-14 a B e+3 A b e+11 b B d)}{12012 e (d+e x)^{11} (b d-a e)^4}+\frac{b (a+b x)^{11} (-14 a B e+3 A b e+11 b B d)}{1092 e (d+e x)^{12} (b d-a e)^3}+\frac{(a+b x)^{11} (-14 a B e+3 A b e+11 b B d)}{182 e (d+e x)^{13} (b d-a e)^2}-\frac{(a+b x)^{11} (B d-A e)}{14 e (d+e x)^{14} (b d-a e)} \]

[Out]

-((B*d - A*e)*(a + b*x)^11)/(14*e*(b*d - a*e)*(d + e*x)^14) + ((11*b*B*d + 3*A*b*e - 14*a*B*e)*(a + b*x)^11)/(
182*e*(b*d - a*e)^2*(d + e*x)^13) + (b*(11*b*B*d + 3*A*b*e - 14*a*B*e)*(a + b*x)^11)/(1092*e*(b*d - a*e)^3*(d
+ e*x)^12) + (b^2*(11*b*B*d + 3*A*b*e - 14*a*B*e)*(a + b*x)^11)/(12012*e*(b*d - a*e)^4*(d + e*x)^11)

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Rubi [A]  time = 0.0850289, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {78, 45, 37} \[ \frac{b^2 (a+b x)^{11} (-14 a B e+3 A b e+11 b B d)}{12012 e (d+e x)^{11} (b d-a e)^4}+\frac{b (a+b x)^{11} (-14 a B e+3 A b e+11 b B d)}{1092 e (d+e x)^{12} (b d-a e)^3}+\frac{(a+b x)^{11} (-14 a B e+3 A b e+11 b B d)}{182 e (d+e x)^{13} (b d-a e)^2}-\frac{(a+b x)^{11} (B d-A e)}{14 e (d+e x)^{14} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^10*(A + B*x))/(d + e*x)^15,x]

[Out]

-((B*d - A*e)*(a + b*x)^11)/(14*e*(b*d - a*e)*(d + e*x)^14) + ((11*b*B*d + 3*A*b*e - 14*a*B*e)*(a + b*x)^11)/(
182*e*(b*d - a*e)^2*(d + e*x)^13) + (b*(11*b*B*d + 3*A*b*e - 14*a*B*e)*(a + b*x)^11)/(1092*e*(b*d - a*e)^3*(d
+ e*x)^12) + (b^2*(11*b*B*d + 3*A*b*e - 14*a*B*e)*(a + b*x)^11)/(12012*e*(b*d - a*e)^4*(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 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{(a+b x)^{10} (A+B x)}{(d+e x)^{15}} \, dx &=-\frac{(B d-A e) (a+b x)^{11}}{14 e (b d-a e) (d+e x)^{14}}+\frac{(11 b B d+3 A b e-14 a B e) \int \frac{(a+b x)^{10}}{(d+e x)^{14}} \, dx}{14 e (b d-a e)}\\ &=-\frac{(B d-A e) (a+b x)^{11}}{14 e (b d-a e) (d+e x)^{14}}+\frac{(11 b B d+3 A b e-14 a B e) (a+b x)^{11}}{182 e (b d-a e)^2 (d+e x)^{13}}+\frac{(b (11 b B d+3 A b e-14 a B e)) \int \frac{(a+b x)^{10}}{(d+e x)^{13}} \, dx}{91 e (b d-a e)^2}\\ &=-\frac{(B d-A e) (a+b x)^{11}}{14 e (b d-a e) (d+e x)^{14}}+\frac{(11 b B d+3 A b e-14 a B e) (a+b x)^{11}}{182 e (b d-a e)^2 (d+e x)^{13}}+\frac{b (11 b B d+3 A b e-14 a B e) (a+b x)^{11}}{1092 e (b d-a e)^3 (d+e x)^{12}}+\frac{\left (b^2 (11 b B d+3 A b e-14 a B e)\right ) \int \frac{(a+b x)^{10}}{(d+e x)^{12}} \, dx}{1092 e (b d-a e)^3}\\ &=-\frac{(B d-A e) (a+b x)^{11}}{14 e (b d-a e) (d+e x)^{14}}+\frac{(11 b B d+3 A b e-14 a B e) (a+b x)^{11}}{182 e (b d-a e)^2 (d+e x)^{13}}+\frac{b (11 b B d+3 A b e-14 a B e) (a+b x)^{11}}{1092 e (b d-a e)^3 (d+e x)^{12}}+\frac{b^2 (11 b B d+3 A b e-14 a B e) (a+b x)^{11}}{12012 e (b d-a e)^4 (d+e x)^{11}}\\ \end{align*}

Mathematica [B]  time = 0.753682, size = 1430, normalized size = 7.73 \[ -\frac{\left (3 A e \left (d^{10}+14 e x d^9+91 e^2 x^2 d^8+364 e^3 x^3 d^7+1001 e^4 x^4 d^6+2002 e^5 x^5 d^5+3003 e^6 x^6 d^4+3432 e^7 x^7 d^3+3003 e^8 x^8 d^2+2002 e^9 x^9 d+1001 e^{10} x^{10}\right )+11 B \left (d^{11}+14 e x d^{10}+91 e^2 x^2 d^9+364 e^3 x^3 d^8+1001 e^4 x^4 d^7+2002 e^5 x^5 d^6+3003 e^6 x^6 d^5+3432 e^7 x^7 d^4+3003 e^8 x^8 d^3+2002 e^9 x^9 d^2+1001 e^{10} x^{10} d+364 e^{11} x^{11}\right )\right ) b^{10}+6 a e \left (2 A e \left (d^9+14 e x d^8+91 e^2 x^2 d^7+364 e^3 x^3 d^6+1001 e^4 x^4 d^5+2002 e^5 x^5 d^4+3003 e^6 x^6 d^3+3432 e^7 x^7 d^2+3003 e^8 x^8 d+2002 e^9 x^9\right )+5 B \left (d^{10}+14 e x d^9+91 e^2 x^2 d^8+364 e^3 x^3 d^7+1001 e^4 x^4 d^6+2002 e^5 x^5 d^5+3003 e^6 x^6 d^4+3432 e^7 x^7 d^3+3003 e^8 x^8 d^2+2002 e^9 x^9 d+1001 e^{10} x^{10}\right )\right ) b^9+6 a^2 e^2 \left (5 A e \left (d^8+14 e x d^7+91 e^2 x^2 d^6+364 e^3 x^3 d^5+1001 e^4 x^4 d^4+2002 e^5 x^5 d^3+3003 e^6 x^6 d^2+3432 e^7 x^7 d+3003 e^8 x^8\right )+9 B \left (d^9+14 e x d^8+91 e^2 x^2 d^7+364 e^3 x^3 d^6+1001 e^4 x^4 d^5+2002 e^5 x^5 d^4+3003 e^6 x^6 d^3+3432 e^7 x^7 d^2+3003 e^8 x^8 d+2002 e^9 x^9\right )\right ) b^8+20 a^3 e^3 \left (3 A e \left (d^7+14 e x d^6+91 e^2 x^2 d^5+364 e^3 x^3 d^4+1001 e^4 x^4 d^3+2002 e^5 x^5 d^2+3003 e^6 x^6 d+3432 e^7 x^7\right )+4 B \left (d^8+14 e x d^7+91 e^2 x^2 d^6+364 e^3 x^3 d^5+1001 e^4 x^4 d^4+2002 e^5 x^5 d^3+3003 e^6 x^6 d^2+3432 e^7 x^7 d+3003 e^8 x^8\right )\right ) b^7+105 a^4 e^4 \left (A e \left (d^6+14 e x d^5+91 e^2 x^2 d^4+364 e^3 x^3 d^3+1001 e^4 x^4 d^2+2002 e^5 x^5 d+3003 e^6 x^6\right )+B \left (d^7+14 e x d^6+91 e^2 x^2 d^5+364 e^3 x^3 d^4+1001 e^4 x^4 d^3+2002 e^5 x^5 d^2+3003 e^6 x^6 d+3432 e^7 x^7\right )\right ) b^6+42 a^5 e^5 \left (4 A e \left (d^5+14 e x d^4+91 e^2 x^2 d^3+364 e^3 x^3 d^2+1001 e^4 x^4 d+2002 e^5 x^5\right )+3 B \left (d^6+14 e x d^5+91 e^2 x^2 d^4+364 e^3 x^3 d^3+1001 e^4 x^4 d^2+2002 e^5 x^5 d+3003 e^6 x^6\right )\right ) b^5+28 a^6 e^6 \left (9 A e \left (d^4+14 e x d^3+91 e^2 x^2 d^2+364 e^3 x^3 d+1001 e^4 x^4\right )+5 B \left (d^5+14 e x d^4+91 e^2 x^2 d^3+364 e^3 x^3 d^2+1001 e^4 x^4 d+2002 e^5 x^5\right )\right ) b^4+72 a^7 e^7 \left (5 A e \left (d^3+14 e x d^2+91 e^2 x^2 d+364 e^3 x^3\right )+2 B \left (d^4+14 e x d^3+91 e^2 x^2 d^2+364 e^3 x^3 d+1001 e^4 x^4\right )\right ) b^3+45 a^8 e^8 \left (11 A e \left (d^2+14 e x d+91 e^2 x^2\right )+3 B \left (d^3+14 e x d^2+91 e^2 x^2 d+364 e^3 x^3\right )\right ) b^2+110 a^9 e^9 \left (6 A e (d+14 e x)+B \left (d^2+14 e x d+91 e^2 x^2\right )\right ) b+66 a^{10} e^{10} (13 A e+B (d+14 e x))}{12012 e^{12} (d+e x)^{14}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^15,x]

[Out]

-(66*a^10*e^10*(13*A*e + B*(d + 14*e*x)) + 110*a^9*b*e^9*(6*A*e*(d + 14*e*x) + B*(d^2 + 14*d*e*x + 91*e^2*x^2)
) + 45*a^8*b^2*e^8*(11*A*e*(d^2 + 14*d*e*x + 91*e^2*x^2) + 3*B*(d^3 + 14*d^2*e*x + 91*d*e^2*x^2 + 364*e^3*x^3)
) + 72*a^7*b^3*e^7*(5*A*e*(d^3 + 14*d^2*e*x + 91*d*e^2*x^2 + 364*e^3*x^3) + 2*B*(d^4 + 14*d^3*e*x + 91*d^2*e^2
*x^2 + 364*d*e^3*x^3 + 1001*e^4*x^4)) + 28*a^6*b^4*e^6*(9*A*e*(d^4 + 14*d^3*e*x + 91*d^2*e^2*x^2 + 364*d*e^3*x
^3 + 1001*e^4*x^4) + 5*B*(d^5 + 14*d^4*e*x + 91*d^3*e^2*x^2 + 364*d^2*e^3*x^3 + 1001*d*e^4*x^4 + 2002*e^5*x^5)
) + 42*a^5*b^5*e^5*(4*A*e*(d^5 + 14*d^4*e*x + 91*d^3*e^2*x^2 + 364*d^2*e^3*x^3 + 1001*d*e^4*x^4 + 2002*e^5*x^5
) + 3*B*(d^6 + 14*d^5*e*x + 91*d^4*e^2*x^2 + 364*d^3*e^3*x^3 + 1001*d^2*e^4*x^4 + 2002*d*e^5*x^5 + 3003*e^6*x^
6)) + 105*a^4*b^6*e^4*(A*e*(d^6 + 14*d^5*e*x + 91*d^4*e^2*x^2 + 364*d^3*e^3*x^3 + 1001*d^2*e^4*x^4 + 2002*d*e^
5*x^5 + 3003*e^6*x^6) + B*(d^7 + 14*d^6*e*x + 91*d^5*e^2*x^2 + 364*d^4*e^3*x^3 + 1001*d^3*e^4*x^4 + 2002*d^2*e
^5*x^5 + 3003*d*e^6*x^6 + 3432*e^7*x^7)) + 20*a^3*b^7*e^3*(3*A*e*(d^7 + 14*d^6*e*x + 91*d^5*e^2*x^2 + 364*d^4*
e^3*x^3 + 1001*d^3*e^4*x^4 + 2002*d^2*e^5*x^5 + 3003*d*e^6*x^6 + 3432*e^7*x^7) + 4*B*(d^8 + 14*d^7*e*x + 91*d^
6*e^2*x^2 + 364*d^5*e^3*x^3 + 1001*d^4*e^4*x^4 + 2002*d^3*e^5*x^5 + 3003*d^2*e^6*x^6 + 3432*d*e^7*x^7 + 3003*e
^8*x^8)) + 6*a^2*b^8*e^2*(5*A*e*(d^8 + 14*d^7*e*x + 91*d^6*e^2*x^2 + 364*d^5*e^3*x^3 + 1001*d^4*e^4*x^4 + 2002
*d^3*e^5*x^5 + 3003*d^2*e^6*x^6 + 3432*d*e^7*x^7 + 3003*e^8*x^8) + 9*B*(d^9 + 14*d^8*e*x + 91*d^7*e^2*x^2 + 36
4*d^6*e^3*x^3 + 1001*d^5*e^4*x^4 + 2002*d^4*e^5*x^5 + 3003*d^3*e^6*x^6 + 3432*d^2*e^7*x^7 + 3003*d*e^8*x^8 + 2
002*e^9*x^9)) + 6*a*b^9*e*(2*A*e*(d^9 + 14*d^8*e*x + 91*d^7*e^2*x^2 + 364*d^6*e^3*x^3 + 1001*d^5*e^4*x^4 + 200
2*d^4*e^5*x^5 + 3003*d^3*e^6*x^6 + 3432*d^2*e^7*x^7 + 3003*d*e^8*x^8 + 2002*e^9*x^9) + 5*B*(d^10 + 14*d^9*e*x
+ 91*d^8*e^2*x^2 + 364*d^7*e^3*x^3 + 1001*d^6*e^4*x^4 + 2002*d^5*e^5*x^5 + 3003*d^4*e^6*x^6 + 3432*d^3*e^7*x^7
+ 3003*d^2*e^8*x^8 + 2002*d*e^9*x^9 + 1001*e^10*x^10)) + b^10*(3*A*e*(d^10 + 14*d^9*e*x + 91*d^8*e^2*x^2 + 36
4*d^7*e^3*x^3 + 1001*d^6*e^4*x^4 + 2002*d^5*e^5*x^5 + 3003*d^4*e^6*x^6 + 3432*d^3*e^7*x^7 + 3003*d^2*e^8*x^8 +
2002*d*e^9*x^9 + 1001*e^10*x^10) + 11*B*(d^11 + 14*d^10*e*x + 91*d^9*e^2*x^2 + 364*d^8*e^3*x^3 + 1001*d^7*e^4
*x^4 + 2002*d^6*e^5*x^5 + 3003*d^5*e^6*x^6 + 3432*d^4*e^7*x^7 + 3003*d^3*e^8*x^8 + 2002*d^2*e^9*x^9 + 1001*d*e
^10*x^10 + 364*e^11*x^11)))/(12012*e^12*(d + e*x)^14)

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Maple [B]  time = 0.011, size = 1942, normalized size = 10.5 \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)^15,x)

[Out]

-5/2*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)^6-1/13*(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^6-1260*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+1470*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)^13-1/3*B*b^10/e^12/(e*x+d)^3-30/7*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)
^7-14/3*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)^9-1/14*(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)^14-5/12*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+672*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)^12-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)^5-21/4*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)^8-3*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)^10-1/4*b^9*(A*b*e+10*B*a*e-11*B*b*d)/e^12/(e*x+d)^4-15/1
1*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^6-336*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)^11

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Maxima [B]  time = 2.09587, size = 2649, normalized size = 14.32 \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)^15,x, algorithm="maxima")

[Out]

-1/12012*(4004*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 858*A*a^10*e^11 + 3*(10*B*a*b^9 + A*b^10)*d^10*e + 6*(9*B*a
^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4
+ 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 36*(4*B*a^7*b^3 + 7*A*a^6*
b^4)*d^4*e^7 + 45*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 + 55*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 + 66*(B*a^10 + 10
*A*a^9*b)*d*e^10 + 1001*(11*B*b^10*d*e^10 + 3*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 2002*(11*B*b^10*d^2*e^9 + 3*(
10*B*a*b^9 + A*b^10)*d*e^10 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 3003*(11*B*b^10*d^3*e^8 + 3*(10*B*a*b^9
+ A*b^10)*d^2*e^9 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 + 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 3432*(11*B
*b^10*d^4*e^7 + 3*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 + 10*(8*B*a^3*b^7 + 3*A*
a^2*b^8)*d*e^10 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 3003*(11*B*b^10*d^5*e^6 + 3*(10*B*a*b^9 + A*b^10)
*d^4*e^7 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 + 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 15*(7*B*a^4*b^6 + 4*
A*a^3*b^7)*d*e^10 + 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 2002*(11*B*b^10*d^6*e^5 + 3*(10*B*a*b^9 + A*b^1
0)*d^5*e^6 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 + 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 15*(7*B*a^4*b^6 +
4*A*a^3*b^7)*d^2*e^9 + 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 1001
*(11*B*b^10*d^7*e^4 + 3*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 + 10*(8*B*a^3*b^7
+ 3*A*a^2*b^8)*d^4*e^7 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 + 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9 + 28*
(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 + 36*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 364*(11*B*b^10*d^8*e^3 + 3*(10
*B*a*b^9 + A*b^10)*d^7*e^4 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 + 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 15
*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 + 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)
*d^2*e^9 + 36*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 + 45*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 + 91*(11*B*b^10*d^
9*e^2 + 3*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 + 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)
*d^6*e^5 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 + 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 + 28*(5*B*a^6*b^4 +
6*A*a^5*b^5)*d^3*e^8 + 36*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9 + 45*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 + 55*(2
*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 14*(11*B*b^10*d^10*e + 3*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 6*(9*B*a^2*b^8 +
2*A*a*b^9)*d^8*e^3 + 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 + 21*(6*B
*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 + 36*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*
e^8 + 45*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 + 55*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 + 66*(B*a^10 + 10*A*a^9*b)*
e^11)*x)/(e^26*x^14 + 14*d*e^25*x^13 + 91*d^2*e^24*x^12 + 364*d^3*e^23*x^11 + 1001*d^4*e^22*x^10 + 2002*d^5*e^
21*x^9 + 3003*d^6*e^20*x^8 + 3432*d^7*e^19*x^7 + 3003*d^8*e^18*x^6 + 2002*d^9*e^17*x^5 + 1001*d^10*e^16*x^4 +
364*d^11*e^15*x^3 + 91*d^12*e^14*x^2 + 14*d^13*e^13*x + d^14*e^12)

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Fricas [B]  time = 2.31167, size = 4250, normalized size = 22.97 \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)^15,x, algorithm="fricas")

[Out]

-1/12012*(4004*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 858*A*a^10*e^11 + 3*(10*B*a*b^9 + A*b^10)*d^10*e + 6*(9*B*a
^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4
+ 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 36*(4*B*a^7*b^3 + 7*A*a^6*
b^4)*d^4*e^7 + 45*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 + 55*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 + 66*(B*a^10 + 10
*A*a^9*b)*d*e^10 + 1001*(11*B*b^10*d*e^10 + 3*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 2002*(11*B*b^10*d^2*e^9 + 3*(
10*B*a*b^9 + A*b^10)*d*e^10 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 3003*(11*B*b^10*d^3*e^8 + 3*(10*B*a*b^9
+ A*b^10)*d^2*e^9 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 + 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 3432*(11*B
*b^10*d^4*e^7 + 3*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 + 10*(8*B*a^3*b^7 + 3*A*
a^2*b^8)*d*e^10 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 3003*(11*B*b^10*d^5*e^6 + 3*(10*B*a*b^9 + A*b^10)
*d^4*e^7 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 + 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 15*(7*B*a^4*b^6 + 4*
A*a^3*b^7)*d*e^10 + 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 2002*(11*B*b^10*d^6*e^5 + 3*(10*B*a*b^9 + A*b^1
0)*d^5*e^6 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 + 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 15*(7*B*a^4*b^6 +
4*A*a^3*b^7)*d^2*e^9 + 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 1001
*(11*B*b^10*d^7*e^4 + 3*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 + 10*(8*B*a^3*b^7
+ 3*A*a^2*b^8)*d^4*e^7 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 + 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9 + 28*
(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 + 36*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 364*(11*B*b^10*d^8*e^3 + 3*(10
*B*a*b^9 + A*b^10)*d^7*e^4 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 + 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 15
*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 + 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)
*d^2*e^9 + 36*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 + 45*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 + 91*(11*B*b^10*d^
9*e^2 + 3*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 + 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)
*d^6*e^5 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 + 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 + 28*(5*B*a^6*b^4 +
6*A*a^5*b^5)*d^3*e^8 + 36*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9 + 45*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 + 55*(2
*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 14*(11*B*b^10*d^10*e + 3*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 6*(9*B*a^2*b^8 +
2*A*a*b^9)*d^8*e^3 + 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 + 21*(6*B
*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 + 36*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*
e^8 + 45*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 + 55*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 + 66*(B*a^10 + 10*A*a^9*b)*
e^11)*x)/(e^26*x^14 + 14*d*e^25*x^13 + 91*d^2*e^24*x^12 + 364*d^3*e^23*x^11 + 1001*d^4*e^22*x^10 + 2002*d^5*e^
21*x^9 + 3003*d^6*e^20*x^8 + 3432*d^7*e^19*x^7 + 3003*d^8*e^18*x^6 + 2002*d^9*e^17*x^5 + 1001*d^10*e^16*x^4 +
364*d^11*e^15*x^3 + 91*d^12*e^14*x^2 + 14*d^13*e^13*x + d^14*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)**15,x)

[Out]

Timed out

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Giac [B]  time = 2.03943, size = 2830, normalized size = 15.3 \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)^15,x, algorithm="giac")

[Out]

-1/12012*(4004*B*b^10*x^11*e^11 + 11011*B*b^10*d*x^10*e^10 + 22022*B*b^10*d^2*x^9*e^9 + 33033*B*b^10*d^3*x^8*e
^8 + 37752*B*b^10*d^4*x^7*e^7 + 33033*B*b^10*d^5*x^6*e^6 + 22022*B*b^10*d^6*x^5*e^5 + 11011*B*b^10*d^7*x^4*e^4
+ 4004*B*b^10*d^8*x^3*e^3 + 1001*B*b^10*d^9*x^2*e^2 + 154*B*b^10*d^10*x*e + 11*B*b^10*d^11 + 30030*B*a*b^9*x^
10*e^11 + 3003*A*b^10*x^10*e^11 + 60060*B*a*b^9*d*x^9*e^10 + 6006*A*b^10*d*x^9*e^10 + 90090*B*a*b^9*d^2*x^8*e^
9 + 9009*A*b^10*d^2*x^8*e^9 + 102960*B*a*b^9*d^3*x^7*e^8 + 10296*A*b^10*d^3*x^7*e^8 + 90090*B*a*b^9*d^4*x^6*e^
7 + 9009*A*b^10*d^4*x^6*e^7 + 60060*B*a*b^9*d^5*x^5*e^6 + 6006*A*b^10*d^5*x^5*e^6 + 30030*B*a*b^9*d^6*x^4*e^5
+ 3003*A*b^10*d^6*x^4*e^5 + 10920*B*a*b^9*d^7*x^3*e^4 + 1092*A*b^10*d^7*x^3*e^4 + 2730*B*a*b^9*d^8*x^2*e^3 + 2
73*A*b^10*d^8*x^2*e^3 + 420*B*a*b^9*d^9*x*e^2 + 42*A*b^10*d^9*x*e^2 + 30*B*a*b^9*d^10*e + 3*A*b^10*d^10*e + 10
8108*B*a^2*b^8*x^9*e^11 + 24024*A*a*b^9*x^9*e^11 + 162162*B*a^2*b^8*d*x^8*e^10 + 36036*A*a*b^9*d*x^8*e^10 + 18
5328*B*a^2*b^8*d^2*x^7*e^9 + 41184*A*a*b^9*d^2*x^7*e^9 + 162162*B*a^2*b^8*d^3*x^6*e^8 + 36036*A*a*b^9*d^3*x^6*
e^8 + 108108*B*a^2*b^8*d^4*x^5*e^7 + 24024*A*a*b^9*d^4*x^5*e^7 + 54054*B*a^2*b^8*d^5*x^4*e^6 + 12012*A*a*b^9*d
^5*x^4*e^6 + 19656*B*a^2*b^8*d^6*x^3*e^5 + 4368*A*a*b^9*d^6*x^3*e^5 + 4914*B*a^2*b^8*d^7*x^2*e^4 + 1092*A*a*b^
9*d^7*x^2*e^4 + 756*B*a^2*b^8*d^8*x*e^3 + 168*A*a*b^9*d^8*x*e^3 + 54*B*a^2*b^8*d^9*e^2 + 12*A*a*b^9*d^9*e^2 +
240240*B*a^3*b^7*x^8*e^11 + 90090*A*a^2*b^8*x^8*e^11 + 274560*B*a^3*b^7*d*x^7*e^10 + 102960*A*a^2*b^8*d*x^7*e^
10 + 240240*B*a^3*b^7*d^2*x^6*e^9 + 90090*A*a^2*b^8*d^2*x^6*e^9 + 160160*B*a^3*b^7*d^3*x^5*e^8 + 60060*A*a^2*b
^8*d^3*x^5*e^8 + 80080*B*a^3*b^7*d^4*x^4*e^7 + 30030*A*a^2*b^8*d^4*x^4*e^7 + 29120*B*a^3*b^7*d^5*x^3*e^6 + 109
20*A*a^2*b^8*d^5*x^3*e^6 + 7280*B*a^3*b^7*d^6*x^2*e^5 + 2730*A*a^2*b^8*d^6*x^2*e^5 + 1120*B*a^3*b^7*d^7*x*e^4
+ 420*A*a^2*b^8*d^7*x*e^4 + 80*B*a^3*b^7*d^8*e^3 + 30*A*a^2*b^8*d^8*e^3 + 360360*B*a^4*b^6*x^7*e^11 + 205920*A
*a^3*b^7*x^7*e^11 + 315315*B*a^4*b^6*d*x^6*e^10 + 180180*A*a^3*b^7*d*x^6*e^10 + 210210*B*a^4*b^6*d^2*x^5*e^9 +
120120*A*a^3*b^7*d^2*x^5*e^9 + 105105*B*a^4*b^6*d^3*x^4*e^8 + 60060*A*a^3*b^7*d^3*x^4*e^8 + 38220*B*a^4*b^6*d
^4*x^3*e^7 + 21840*A*a^3*b^7*d^4*x^3*e^7 + 9555*B*a^4*b^6*d^5*x^2*e^6 + 5460*A*a^3*b^7*d^5*x^2*e^6 + 1470*B*a^
4*b^6*d^6*x*e^5 + 840*A*a^3*b^7*d^6*x*e^5 + 105*B*a^4*b^6*d^7*e^4 + 60*A*a^3*b^7*d^7*e^4 + 378378*B*a^5*b^5*x^
6*e^11 + 315315*A*a^4*b^6*x^6*e^11 + 252252*B*a^5*b^5*d*x^5*e^10 + 210210*A*a^4*b^6*d*x^5*e^10 + 126126*B*a^5*
b^5*d^2*x^4*e^9 + 105105*A*a^4*b^6*d^2*x^4*e^9 + 45864*B*a^5*b^5*d^3*x^3*e^8 + 38220*A*a^4*b^6*d^3*x^3*e^8 + 1
1466*B*a^5*b^5*d^4*x^2*e^7 + 9555*A*a^4*b^6*d^4*x^2*e^7 + 1764*B*a^5*b^5*d^5*x*e^6 + 1470*A*a^4*b^6*d^5*x*e^6
+ 126*B*a^5*b^5*d^6*e^5 + 105*A*a^4*b^6*d^6*e^5 + 280280*B*a^6*b^4*x^5*e^11 + 336336*A*a^5*b^5*x^5*e^11 + 1401
40*B*a^6*b^4*d*x^4*e^10 + 168168*A*a^5*b^5*d*x^4*e^10 + 50960*B*a^6*b^4*d^2*x^3*e^9 + 61152*A*a^5*b^5*d^2*x^3*
e^9 + 12740*B*a^6*b^4*d^3*x^2*e^8 + 15288*A*a^5*b^5*d^3*x^2*e^8 + 1960*B*a^6*b^4*d^4*x*e^7 + 2352*A*a^5*b^5*d^
4*x*e^7 + 140*B*a^6*b^4*d^5*e^6 + 168*A*a^5*b^5*d^5*e^6 + 144144*B*a^7*b^3*x^4*e^11 + 252252*A*a^6*b^4*x^4*e^1
1 + 52416*B*a^7*b^3*d*x^3*e^10 + 91728*A*a^6*b^4*d*x^3*e^10 + 13104*B*a^7*b^3*d^2*x^2*e^9 + 22932*A*a^6*b^4*d^
2*x^2*e^9 + 2016*B*a^7*b^3*d^3*x*e^8 + 3528*A*a^6*b^4*d^3*x*e^8 + 144*B*a^7*b^3*d^4*e^7 + 252*A*a^6*b^4*d^4*e^
7 + 49140*B*a^8*b^2*x^3*e^11 + 131040*A*a^7*b^3*x^3*e^11 + 12285*B*a^8*b^2*d*x^2*e^10 + 32760*A*a^7*b^3*d*x^2*
e^10 + 1890*B*a^8*b^2*d^2*x*e^9 + 5040*A*a^7*b^3*d^2*x*e^9 + 135*B*a^8*b^2*d^3*e^8 + 360*A*a^7*b^3*d^3*e^8 + 1
0010*B*a^9*b*x^2*e^11 + 45045*A*a^8*b^2*x^2*e^11 + 1540*B*a^9*b*d*x*e^10 + 6930*A*a^8*b^2*d*x*e^10 + 110*B*a^9
*b*d^2*e^9 + 495*A*a^8*b^2*d^2*e^9 + 924*B*a^10*x*e^11 + 9240*A*a^9*b*x*e^11 + 66*B*a^10*d*e^10 + 660*A*a^9*b*
d*e^10 + 858*A*a^10*e^11)*e^(-12)/(x*e + d)^14