∫ (a+bx)10(A+Bx)___________

3.11.97 \(\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^9} \, dx\) [1097]

Optimal. Leaf size=445 \[ \frac {5 b^8 (b d-a e) (11 b B d-2 A b e-9 a B e) x}{e^{11}}+\frac {(b d-a e)^{10} (B d-A e)}{8 e^{12} (d+e x)^8}-\frac {(b d-a e)^9 (11 b B d-10 A b e-a B e)}{7 e^{12} (d+e x)^7}+\frac {5 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{6 e^{12} (d+e x)^6}-\frac {3 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e)}{e^{12} (d+e x)^5}+\frac {15 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e)}{2 e^{12} (d+e x)^4}-\frac {14 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e)}{e^{12} (d+e x)^3}+\frac {21 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e)}{e^{12} (d+e x)^2}-\frac {30 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e)}{e^{12} (d+e x)}-\frac {b^9 (11 b B d-A b e-10 a B e) (d+e x)^2}{2 e^{12}}+\frac {b^{10} B (d+e x)^3}{3 e^{12}}-\frac {15 b^7 (b d-a e)^2 (11 b B d-3 A b e-8 a B e) \log (d+e x)}{e^{12}} \]

[Out]

5*b^8*(-a*e+b*d)*(-2*A*b*e-9*B*a*e+11*B*b*d)*x/e^11+1/8*(-a*e+b*d)^10*(-A*e+B*d)/e^12/(e*x+d)^8-1/7*(-a*e+b*d)

^9*(-10*A*b*e-B*a*e+11*B*b*d)/e^12/(e*x+d)^7+5/6*b*(-a*e+b*d)^8*(-9*A*b*e-2*B*a*e+11*B*b*d)/e^12/(e*x+d)^6-3*b

^2*(-a*e+b*d)^7*(-8*A*b*e-3*B*a*e+11*B*b*d)/e^12/(e*x+d)^5+15/2*b^3*(-a*e+b*d)^6*(-7*A*b*e-4*B*a*e+11*B*b*d)/e

^12/(e*x+d)^4-14*b^4*(-a*e+b*d)^5*(-6*A*b*e-5*B*a*e+11*B*b*d)/e^12/(e*x+d)^3+21*b^5*(-a*e+b*d)^4*(-5*A*b*e-6*B

*a*e+11*B*b*d)/e^12/(e*x+d)^2-30*b^6*(-a*e+b*d)^3*(-4*A*b*e-7*B*a*e+11*B*b*d)/e^12/(e*x+d)-1/2*b^9*(-A*b*e-10*

B*a*e+11*B*b*d)*(e*x+d)^2/e^12+1/3*b^10*B*(e*x+d)^3/e^12-15*b^7*(-a*e+b*d)^2*(-3*A*b*e-8*B*a*e+11*B*b*d)*ln(e*

x+d)/e^12

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Rubi [A]
time = 0.63, antiderivative size = 445, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78} \begin {gather*} -\frac {b^9 (d+e x)^2 (-10 a B e-A b e+11 b B d)}{2 e^{12}}+\frac {5 b^8 x (b d-a e) (-9 a B e-2 A b e+11 b B d)}{e^{11}}-\frac {15 b^7 (b d-a e)^2 \log (d+e x) (-8 a B e-3 A b e+11 b B d)}{e^{12}}-\frac {30 b^6 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{e^{12} (d+e x)}+\frac {21 b^5 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{e^{12} (d+e x)^2}-\frac {14 b^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{e^{12} (d+e x)^3}+\frac {15 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{2 e^{12} (d+e x)^4}-\frac {3 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{e^{12} (d+e x)^5}+\frac {5 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{6 e^{12} (d+e x)^6}-\frac {(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{7 e^{12} (d+e x)^7}+\frac {(b d-a e)^{10} (B d-A e)}{8 e^{12} (d+e x)^8}+\frac {b^{10} B (d+e x)^3}{3 e^{12}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Int[((a + b*x)^10*(A + B*x))/(d + e*x)^9,x]

[Out]

(5*b^8*(b*d - a*e)*(11*b*B*d - 2*A*b*e - 9*a*B*e)*x)/e^11 + ((b*d - a*e)^10*(B*d - A*e))/(8*e^12*(d + e*x)^8)

- ((b*d - a*e)^9*(11*b*B*d - 10*A*b*e - a*B*e))/(7*e^12*(d + e*x)^7) + (5*b*(b*d - a*e)^8*(11*b*B*d - 9*A*b*e

- 2*a*B*e))/(6*e^12*(d + e*x)^6) - (3*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b*e - 3*a*B*e))/(e^12*(d + e*x)^5) + (

15*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b*e - 4*a*B*e))/(2*e^12*(d + e*x)^4) - (14*b^4*(b*d - a*e)^5*(11*b*B*d -

6*A*b*e - 5*a*B*e))/(e^12*(d + e*x)^3) + (21*b^5*(b*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*B*e))/(e^12*(d + e*x)

^2) - (30*b^6*(b*d - a*e)^3*(11*b*B*d - 4*A*b*e - 7*a*B*e))/(e^12*(d + e*x)) - (b^9*(11*b*B*d - A*b*e - 10*a*B

*e)*(d + e*x)^2)/(2*e^12) + (b^10*B*(d + e*x)^3)/(3*e^12) - (15*b^7*(b*d - a*e)^2*(11*b*B*d - 3*A*b*e - 8*a*B*

e)*Log[d + e*x])/e^12

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran

d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[

n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1

, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin {align*} \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^9} \, dx &=\int \left (-\frac {5 b^8 (b d-a e) (-11 b B d+2 A b e+9 a B e)}{e^{11}}+\frac {(-b d+a e)^{10} (-B d+A e)}{e^{11} (d+e x)^9}+\frac {(-b d+a e)^9 (-11 b B d+10 A b e+a B e)}{e^{11} (d+e x)^8}+\frac {5 b (b d-a e)^8 (-11 b B d+9 A b e+2 a B e)}{e^{11} (d+e x)^7}-\frac {15 b^2 (b d-a e)^7 (-11 b B d+8 A b e+3 a B e)}{e^{11} (d+e x)^6}+\frac {30 b^3 (b d-a e)^6 (-11 b B d+7 A b e+4 a B e)}{e^{11} (d+e x)^5}-\frac {42 b^4 (b d-a e)^5 (-11 b B d+6 A b e+5 a B e)}{e^{11} (d+e x)^4}+\frac {42 b^5 (b d-a e)^4 (-11 b B d+5 A b e+6 a B e)}{e^{11} (d+e x)^3}-\frac {30 b^6 (b d-a e)^3 (-11 b B d+4 A b e+7 a B e)}{e^{11} (d+e x)^2}+\frac {15 b^7 (b d-a e)^2 (-11 b B d+3 A b e+8 a B e)}{e^{11} (d+e x)}+\frac {b^9 (-11 b B d+A b e+10 a B e) (d+e x)}{e^{11}}+\frac {b^{10} B (d+e x)^2}{e^{11}}\right ) \, dx\\ &=\frac {5 b^8 (b d-a e) (11 b B d-2 A b e-9 a B e) x}{e^{11}}+\frac {(b d-a e)^{10} (B d-A e)}{8 e^{12} (d+e x)^8}-\frac {(b d-a e)^9 (11 b B d-10 A b e-a B e)}{7 e^{12} (d+e x)^7}+\frac {5 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{6 e^{12} (d+e x)^6}-\frac {3 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e)}{e^{12} (d+e x)^5}+\frac {15 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e)}{2 e^{12} (d+e x)^4}-\frac {14 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e)}{e^{12} (d+e x)^3}+\frac {21 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e)}{e^{12} (d+e x)^2}-\frac {30 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e)}{e^{12} (d+e x)}-\frac {b^9 (11 b B d-A b e-10 a B e) (d+e x)^2}{2 e^{12}}+\frac {b^{10} B (d+e x)^3}{3 e^{12}}-\frac {15 b^7 (b d-a e)^2 (11 b B d-3 A b e-8 a B e) \log (d+e x)}{e^{12}}\\ \end {align*}

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Mathematica [A]
time = 0.39, size = 415, normalized size = 0.93 \begin {gather*} \frac {-168 b^8 e \left (-45 a^2 B e^2-10 a b e (-9 B d+A e)+9 b^2 d (-5 B d+A e)\right ) x+84 b^9 e^2 (-9 b B d+A b e+10 a B e) x^2+56 b^{10} B e^3 x^3+\frac {21 (b d-a e)^{10} (B d-A e)}{(d+e x)^8}-\frac {24 (b d-a e)^9 (11 b B d-10 A b e-a B e)}{(d+e x)^7}+\frac {140 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{(d+e x)^6}-\frac {504 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e)}{(d+e x)^5}+\frac {1260 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e)}{(d+e x)^4}-\frac {2352 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e)}{(d+e x)^3}+\frac {3528 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e)}{(d+e x)^2}-\frac {5040 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e)}{d+e x}-2520 b^7 (b d-a e)^2 (11 b B d-3 A b e-8 a B e) \log (d+e x)}{168 e^{12}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^9,x]

[Out]

(-168*b^8*e*(-45*a^2*B*e^2 - 10*a*b*e*(-9*B*d + A*e) + 9*b^2*d*(-5*B*d + A*e))*x + 84*b^9*e^2*(-9*b*B*d + A*b*

e + 10*a*B*e)*x^2 + 56*b^10*B*e^3*x^3 + (21*(b*d - a*e)^10*(B*d - A*e))/(d + e*x)^8 - (24*(b*d - a*e)^9*(11*b*

B*d - 10*A*b*e - a*B*e))/(d + e*x)^7 + (140*b*(b*d - a*e)^8*(11*b*B*d - 9*A*b*e - 2*a*B*e))/(d + e*x)^6 - (504

*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b*e - 3*a*B*e))/(d + e*x)^5 + (1260*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b*e -

4*a*B*e))/(d + e*x)^4 - (2352*b^4*(b*d - a*e)^5*(11*b*B*d - 6*A*b*e - 5*a*B*e))/(d + e*x)^3 + (3528*b^5*(b*d

- a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*B*e))/(d + e*x)^2 - (5040*b^6*(b*d - a*e)^3*(11*b*B*d - 4*A*b*e - 7*a*B*e))

/(d + e*x) - 2520*b^7*(b*d - a*e)^2*(11*b*B*d - 3*A*b*e - 8*a*B*e)*Log[d + e*x])/(168*e^12)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1937\) vs. \(2(433)=866\).
time = 0.08, size = 1938, normalized size = 4.36


method	result	size

norman	\(\text {Expression too large to display}\)	\(1929\)
default	\(\text {Expression too large to display}\)	\(1938\)
risch	\(\text {Expression too large to display}\)	\(1968\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10*(B*x+A)/(e*x+d)^9,x,method=_RETURNVERBOSE)

[Out]

b^8/e^11*(1/3*b^2*B*x^3*e^2+1/2*A*b^2*e^2*x^2+5*B*a*b*e^2*x^2-9/2*B*b^2*d*e*x^2+10*A*a*b*e^2*x-9*A*b^2*d*e*x+4

5*B*a^2*e^2*x-90*B*a*b*d*e*x+45*B*b^2*d^2*x)-21*b^5/e^12*(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*x+d)^2-15/2*b^3/e^12*(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*x+d)^4-1/8*(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^1

0*d^11)/e^12/(e*x+d)^8-30*b^6/e^12*(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*x+d)-1/7/e^12*(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*x+d)^7-5/6*b/e^12*(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+25

2*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*x+d)^6-3*b^2/e^12*(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*x+d)^5-14*b^4/e^12*(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*x+d)^3+15*b^7/e^12*(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)*ln(e*x+d)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1922 vs. \(2 (463) = 926\).
time = 0.49, size = 1922, normalized size = 4.32 \begin {gather*} \text {Too large to display} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^9,x, algorithm="maxima")

[Out]

-15*(11*B*b^10*d^3 - 8*B*a^3*b^7*e^3 - 3*A*a^2*b^8*e^3 - 3*(10*B*a*b^9*e + A*b^10*e)*d^2 + 3*(9*B*a^2*b^8*e^2

+ 2*A*a*b^9*e^2)*d)*e^(-12)*log(x*e + d) + 1/6*(2*B*b^10*x^3*e^2 - 3*(9*B*b^10*d*e - 10*B*a*b^9*e^2 - A*b^10*e

^2)*x^2 + 6*(45*B*b^10*d^2 + 45*B*a^2*b^8*e^2 + 10*A*a*b^9*e^2 - 9*(10*B*a*b^9*e + A*b^10*e)*d)*x)*e^(-11) - 1

/168*(32891*B*b^10*d^11 + 21*A*a^10*e^11 - 10803*(10*B*a*b^9*e + A*b^10*e)*d^10 + 13827*(9*B*a^2*b^8*e^2 + 2*A

*a*b^9*e^2)*d^9 - 6849*(8*B*a^3*b^7*e^3 + 3*A*a^2*b^8*e^3)*d^8 + 630*(7*B*a^4*b^6*e^4 + 4*A*a^3*b^7*e^4)*d^7 +

5040*(11*B*b^10*d^4*e^7 + 7*B*a^4*b^6*e^11 + 4*A*a^3*b^7*e^11 - 4*(10*B*a*b^9*e^8 + A*b^10*e^8)*d^3 + 6*(9*B*

a^2*b^8*e^9 + 2*A*a*b^9*e^9)*d^2 - 4*(8*B*a^3*b^7*e^10 + 3*A*a^2*b^8*e^10)*d)*x^7 + 126*(6*B*a^5*b^5*e^5 + 5*A

*a^4*b^6*e^5)*d^6 + 3528*(99*B*b^10*d^5*e^6 + 6*B*a^5*b^5*e^11 + 5*A*a^4*b^6*e^11 - 35*(10*B*a*b^9*e^7 + A*b^1

0*e^7)*d^4 + 50*(9*B*a^2*b^8*e^8 + 2*A*a*b^9*e^8)*d^3 - 30*(8*B*a^3*b^7*e^9 + 3*A*a^2*b^8*e^9)*d^2 + 5*(7*B*a^

4*b^6*e^10 + 4*A*a^3*b^7*e^10)*d)*x^6 + 42*(5*B*a^6*b^4*e^6 + 6*A*a^5*b^5*e^6)*d^5 + 2352*(407*B*b^10*d^6*e^5

+ 5*B*a^6*b^4*e^11 + 6*A*a^5*b^5*e^11 - 141*(10*B*a*b^9*e^6 + A*b^10*e^6)*d^5 + 195*(9*B*a^2*b^8*e^7 + 2*A*a*b

^9*e^7)*d^4 - 110*(8*B*a^3*b^7*e^8 + 3*A*a^2*b^8*e^8)*d^3 + 15*(7*B*a^4*b^6*e^9 + 4*A*a^3*b^7*e^9)*d^2 + 3*(6*

B*a^5*b^5*e^10 + 5*A*a^4*b^6*e^10)*d)*x^5 + 18*(4*B*a^7*b^3*e^7 + 7*A*a^6*b^4*e^7)*d^4 + 420*(3509*B*b^10*d^7*

e^4 + 12*B*a^7*b^3*e^11 + 21*A*a^6*b^4*e^11 - 1197*(10*B*a*b^9*e^5 + A*b^10*e^5)*d^6 + 1617*(9*B*a^2*b^8*e^6 +

2*A*a*b^9*e^6)*d^5 - 875*(8*B*a^3*b^7*e^7 + 3*A*a^2*b^8*e^7)*d^4 + 105*(7*B*a^4*b^6*e^8 + 4*A*a^3*b^7*e^8)*d^

3 + 21*(6*B*a^5*b^5*e^9 + 5*A*a^4*b^6*e^9)*d^2 + 7*(5*B*a^6*b^4*e^10 + 6*A*a^5*b^5*e^10)*d)*x^4 + 9*(3*B*a^8*b

^2*e^8 + 8*A*a^7*b^3*e^8)*d^3 + 168*(8173*B*b^10*d^8*e^3 + 9*B*a^8*b^2*e^11 + 24*A*a^7*b^3*e^11 - 2754*(10*B*a

*b^9*e^4 + A*b^10*e^4)*d^7 + 3654*(9*B*a^2*b^8*e^5 + 2*A*a*b^9*e^5)*d^6 - 1918*(8*B*a^3*b^7*e^6 + 3*A*a^2*b^8*

e^6)*d^5 + 210*(7*B*a^4*b^6*e^7 + 4*A*a^3*b^7*e^7)*d^4 + 42*(6*B*a^5*b^5*e^8 + 5*A*a^4*b^6*e^8)*d^3 + 14*(5*B*

a^6*b^4*e^9 + 6*A*a^5*b^5*e^9)*d^2 + 6*(4*B*a^7*b^3*e^10 + 7*A*a^6*b^4*e^10)*d)*x^3 + 5*(2*B*a^9*b*e^9 + 9*A*a

^8*b^2*e^9)*d^2 + 28*(27599*B*b^10*d^9*e^2 + 10*B*a^9*b*e^11 + 45*A*a^8*b^2*e^11 - 9207*(10*B*a*b^9*e^3 + A*b^

10*e^3)*d^8 + 12042*(9*B*a^2*b^8*e^4 + 2*A*a*b^9*e^4)*d^7 - 6174*(8*B*a^3*b^7*e^5 + 3*A*a^2*b^8*e^5)*d^6 + 630

*(7*B*a^4*b^6*e^6 + 4*A*a^3*b^7*e^6)*d^5 + 126*(6*B*a^5*b^5*e^7 + 5*A*a^4*b^6*e^7)*d^4 + 42*(5*B*a^6*b^4*e^8 +

6*A*a^5*b^5*e^8)*d^3 + 18*(4*B*a^7*b^3*e^9 + 7*A*a^6*b^4*e^9)*d^2 + 9*(3*B*a^8*b^2*e^10 + 8*A*a^7*b^3*e^10)*d

)*x^2 + 3*(B*a^10*e^10 + 10*A*a^9*b*e^10)*d + 8*(30371*B*b^10*d^10*e + 3*B*a^10*e^11 + 30*A*a^9*b*e^11 - 10047

*(10*B*a*b^9*e^2 + A*b^10*e^2)*d^9 + 12987*(9*B*a^2*b^8*e^3 + 2*A*a*b^9*e^3)*d^8 - 6534*(8*B*a^3*b^7*e^4 + 3*A

*a^2*b^8*e^4)*d^7 + 630*(7*B*a^4*b^6*e^5 + 4*A*a^3*b^7*e^5)*d^6 + 126*(6*B*a^5*b^5*e^6 + 5*A*a^4*b^6*e^6)*d^5

+ 42*(5*B*a^6*b^4*e^7 + 6*A*a^5*b^5*e^7)*d^4 + 18*(4*B*a^7*b^3*e^8 + 7*A*a^6*b^4*e^8)*d^3 + 9*(3*B*a^8*b^2*e^9

+ 8*A*a^7*b^3*e^9)*d^2 + 5*(2*B*a^9*b*e^10 + 9*A*a^8*b^2*e^10)*d)*x)/(x^8*e^20 + 8*d*x^7*e^19 + 28*d^2*x^6*e^

18 + 56*d^3*x^5*e^17 + 70*d^4*x^4*e^16 + 56*d^5*x^3*e^15 + 28*d^6*x^2*e^14 + 8*d^7*x*e^13 + d^8*e^12)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2597 vs. \(2 (463) = 926\).
time = 1.05, size = 2597, normalized size = 5.84 \begin {gather*} \text {Too large to display} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^9,x, algorithm="fricas")

[Out]

-1/168*(32891*B*b^10*d^11 - (56*B*b^10*x^11 - 21*A*a^10 + 84*(10*B*a*b^9 + A*b^10)*x^10 + 840*(9*B*a^2*b^8 + 2

*A*a*b^9)*x^9 - 5040*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 - 3528*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 - 2352*(5*B*a^6*b^

4 + 6*A*a^5*b^5)*x^5 - 1260*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 - 504*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 - 140*(2*B*a

^9*b + 9*A*a^8*b^2)*x^2 - 24*(B*a^10 + 10*A*a^9*b)*x)*e^11 + (308*B*b^10*d*x^10 + 840*(10*B*a*b^9 + A*b^10)*d*

x^9 - 6720*(9*B*a^2*b^8 + 2*A*a*b^9)*d*x^8 - 20160*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*x^7 + 17640*(7*B*a^4*b^6 + 4*

A*a^3*b^7)*d*x^6 + 7056*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*x^5 + 2940*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*x^4 + 1008*(4*B

*a^7*b^3 + 7*A*a^6*b^4)*d*x^3 + 252*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*x^2 + 40*(2*B*a^9*b + 9*A*a^8*b^2)*d*x + 3*(

B*a^10 + 10*A*a^9*b)*d)*e^10 - (3080*B*b^10*d^2*x^9 - 9744*(10*B*a*b^9 + A*b^10)*d^2*x^8 - 6720*(9*B*a^2*b^8 +

2*A*a*b^9)*d^2*x^7 + 105840*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*x^6 - 35280*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*x^5 -

8820*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*x^4 - 2352*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*x^3 - 504*(4*B*a^7*b^3 + 7*A*

a^6*b^4)*d^2*x^2 - 72*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*x - 5*(2*B*a^9*b + 9*A*a^8*b^2)*d^2)*e^9 - (42448*B*b^10

*d^3*x^8 - 17472*(10*B*a*b^9 + A*b^10)*d^3*x^7 - 129360*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*x^6 + 258720*(8*B*a^3*b^

7 + 3*A*a^2*b^8)*d^3*x^5 - 44100*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*x^4 - 7056*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*x^

3 - 1176*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*x^2 - 144*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*x - 9*(3*B*a^8*b^2 + 8*A*a^

7*b^3)*d^3)*e^8 - 2*(58912*B*b^10*d^4*x^7 + 22344*(10*B*a*b^9 + A*b^10)*d^4*x^6 - 199920*(9*B*a^2*b^8 + 2*A*a*

b^9)*d^4*x^5 + 183750*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*x^4 - 17640*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*x^3 - 1764*(

6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*x^2 - 168*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*x - 9*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4

)*e^7 - 14*(1736*B*b^10*d^5*x^6 + 16464*(10*B*a*b^9 + A*b^10)*d^5*x^5 - 45150*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*x^

4 + 23016*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*x^3 - 1260*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*x^2 - 72*(6*B*a^5*b^5 + 5

*A*a^4*b^6)*d^5*x - 3*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5)*e^6 + 14*(33488*B*b^10*d^6*x^5 - 30030*(10*B*a*b^9 + A*

b^10)*d^6*x^4 + 42168*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*x^3 - 12348*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*x^2 + 360*(7*B

*a^4*b^6 + 4*A*a^3*b^7)*d^6*x + 9*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6)*e^5 + 2*(535570*B*b^10*d^7*x^4 - 210504*(10

*B*a*b^9 + A*b^10)*d^7*x^3 + 165228*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*x^2 - 26136*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*

x + 315*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7)*e^4 + (1167376*B*b^10*d^8*x^3 - 245784*(10*B*a*b^9 + A*b^10)*d^8*x^2

+ 103056*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*x - 6849*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8)*e^3 + (713048*B*b^10*d^9*x^2

- 78864*(10*B*a*b^9 + A*b^10)*d^9*x + 13827*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9)*e^2 + (235408*B*b^10*d^10*x - 10803

*(10*B*a*b^9 + A*b^10)*d^10)*e + 2520*(11*B*b^10*d^11 - (8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8*e^11 + (3*(9*B*a^2*b^8

+ 2*A*a*b^9)*d*x^8 - 8*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*x^7)*e^10 - (3*(10*B*a*b^9 + A*b^10)*d^2*x^8 - 24*(9*B*a

^2*b^8 + 2*A*a*b^9)*d^2*x^7 + 28*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*x^6)*e^9 + (11*B*b^10*d^3*x^8 - 24*(10*B*a*b^

9 + A*b^10)*d^3*x^7 + 84*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*x^6 - 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*x^5)*e^8 + 2*(

44*B*b^10*d^4*x^7 - 42*(10*B*a*b^9 + A*b^10)*d^4*x^6 + 84*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*x^5 - 35*(8*B*a^3*b^7

+ 3*A*a^2*b^8)*d^4*x^4)*e^7 + 14*(22*B*b^10*d^5*x^6 - 12*(10*B*a*b^9 + A*b^10)*d^5*x^5 + 15*(9*B*a^2*b^8 + 2*A

*a*b^9)*d^5*x^4 - 4*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*x^3)*e^6 + 14*(44*B*b^10*d^6*x^5 - 15*(10*B*a*b^9 + A*b^10

)*d^6*x^4 + 12*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*x^3 - 2*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*x^2)*e^5 + 2*(385*B*b^10*

d^7*x^4 - 84*(10*B*a*b^9 + A*b^10)*d^7*x^3 + 42*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*x^2 - 4*(8*B*a^3*b^7 + 3*A*a^2*b

^8)*d^7*x)*e^4 + (616*B*b^10*d^8*x^3 - 84*(10*B*a*b^9 + A*b^10)*d^8*x^2 + 24*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*x -

(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8)*e^3 + (308*B*b^10*d^9*x^2 - 24*(10*B*a*b^9 + A*b^10)*d^9*x + 3*(9*B*a^2*b^8

+ 2*A*a*b^9)*d^9)*e^2 + (88*B*b^10*d^10*x - 3*(10*B*a*b^9 + A*b^10)*d^10)*e)*log(x*e + d))/(x^8*e^20 + 8*d*x^7

*e^19 + 28*d^2*x^6*e^18 + 56*d^3*x^5*e^17 + 70*d^4*x^4*e^16 + 56*d^5*x^3*e^15 + 28*d^6*x^2*e^14 + 8*d^7*x*e^13

+ d^8*e^12)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**10*(B*x+A)/(e*x+d)**9,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1849 vs. \(2 (463) = 926\).
time = 1.23, size = 1849, normalized size = 4.16 \begin {gather*} -15 \, {\left (11 \, B b^{10} d^{3} - 30 \, B a b^{9} d^{2} e - 3 \, A b^{10} d^{2} e + 27 \, B a^{2} b^{8} d e^{2} + 6 \, A a b^{9} d e^{2} - 8 \, B a^{3} b^{7} e^{3} - 3 \, A a^{2} b^{8} e^{3}\right )} e^{\left (-12\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{6} \, {\left (2 \, B b^{10} x^{3} e^{18} - 27 \, B b^{10} d x^{2} e^{17} + 270 \, B b^{10} d^{2} x e^{16} + 30 \, B a b^{9} x^{2} e^{18} + 3 \, A b^{10} x^{2} e^{18} - 540 \, B a b^{9} d x e^{17} - 54 \, A b^{10} d x e^{17} + 270 \, B a^{2} b^{8} x e^{18} + 60 \, A a b^{9} x e^{18}\right )} e^{\left (-27\right )} - \frac {{\left (32891 \, B b^{10} d^{11} - 108030 \, B a b^{9} d^{10} e - 10803 \, A b^{10} d^{10} e + 124443 \, B a^{2} b^{8} d^{9} e^{2} + 27654 \, A a b^{9} d^{9} e^{2} - 54792 \, B a^{3} b^{7} d^{8} e^{3} - 20547 \, A a^{2} b^{8} d^{8} e^{3} + 4410 \, B a^{4} b^{6} d^{7} e^{4} + 2520 \, A a^{3} b^{7} d^{7} e^{4} + 756 \, B a^{5} b^{5} d^{6} e^{5} + 630 \, A a^{4} b^{6} d^{6} e^{5} + 210 \, B a^{6} b^{4} d^{5} e^{6} + 252 \, A a^{5} b^{5} d^{5} e^{6} + 72 \, B a^{7} b^{3} d^{4} e^{7} + 126 \, A a^{6} b^{4} d^{4} e^{7} + 27 \, B a^{8} b^{2} d^{3} e^{8} + 72 \, A a^{7} b^{3} d^{3} e^{8} + 10 \, B a^{9} b d^{2} e^{9} + 45 \, A a^{8} b^{2} d^{2} e^{9} + 3 \, B a^{10} d e^{10} + 30 \, A a^{9} b d e^{10} + 21 \, A a^{10} e^{11} + 5040 \, {\left (11 \, B b^{10} d^{4} e^{7} - 40 \, B a b^{9} d^{3} e^{8} - 4 \, A b^{10} d^{3} e^{8} + 54 \, B a^{2} b^{8} d^{2} e^{9} + 12 \, A a b^{9} d^{2} e^{9} - 32 \, B a^{3} b^{7} d e^{10} - 12 \, A a^{2} b^{8} d e^{10} + 7 \, B a^{4} b^{6} e^{11} + 4 \, A a^{3} b^{7} e^{11}\right )} x^{7} + 3528 \, {\left (99 \, B b^{10} d^{5} e^{6} - 350 \, B a b^{9} d^{4} e^{7} - 35 \, A b^{10} d^{4} e^{7} + 450 \, B a^{2} b^{8} d^{3} e^{8} + 100 \, A a b^{9} d^{3} e^{8} - 240 \, B a^{3} b^{7} d^{2} e^{9} - 90 \, A a^{2} b^{8} d^{2} e^{9} + 35 \, B a^{4} b^{6} d e^{10} + 20 \, A a^{3} b^{7} d e^{10} + 6 \, B a^{5} b^{5} e^{11} + 5 \, A a^{4} b^{6} e^{11}\right )} x^{6} + 2352 \, {\left (407 \, B b^{10} d^{6} e^{5} - 1410 \, B a b^{9} d^{5} e^{6} - 141 \, A b^{10} d^{5} e^{6} + 1755 \, B a^{2} b^{8} d^{4} e^{7} + 390 \, A a b^{9} d^{4} e^{7} - 880 \, B a^{3} b^{7} d^{3} e^{8} - 330 \, A a^{2} b^{8} d^{3} e^{8} + 105 \, B a^{4} b^{6} d^{2} e^{9} + 60 \, A a^{3} b^{7} d^{2} e^{9} + 18 \, B a^{5} b^{5} d e^{10} + 15 \, A a^{4} b^{6} d e^{10} + 5 \, B a^{6} b^{4} e^{11} + 6 \, A a^{5} b^{5} e^{11}\right )} x^{5} + 420 \, {\left (3509 \, B b^{10} d^{7} e^{4} - 11970 \, B a b^{9} d^{6} e^{5} - 1197 \, A b^{10} d^{6} e^{5} + 14553 \, B a^{2} b^{8} d^{5} e^{6} + 3234 \, A a b^{9} d^{5} e^{6} - 7000 \, B a^{3} b^{7} d^{4} e^{7} - 2625 \, A a^{2} b^{8} d^{4} e^{7} + 735 \, B a^{4} b^{6} d^{3} e^{8} + 420 \, A a^{3} b^{7} d^{3} e^{8} + 126 \, B a^{5} b^{5} d^{2} e^{9} + 105 \, A a^{4} b^{6} d^{2} e^{9} + 35 \, B a^{6} b^{4} d e^{10} + 42 \, A a^{5} b^{5} d e^{10} + 12 \, B a^{7} b^{3} e^{11} + 21 \, A a^{6} b^{4} e^{11}\right )} x^{4} + 168 \, {\left (8173 \, B b^{10} d^{8} e^{3} - 27540 \, B a b^{9} d^{7} e^{4} - 2754 \, A b^{10} d^{7} e^{4} + 32886 \, B a^{2} b^{8} d^{6} e^{5} + 7308 \, A a b^{9} d^{6} e^{5} - 15344 \, B a^{3} b^{7} d^{5} e^{6} - 5754 \, A a^{2} b^{8} d^{5} e^{6} + 1470 \, B a^{4} b^{6} d^{4} e^{7} + 840 \, A a^{3} b^{7} d^{4} e^{7} + 252 \, B a^{5} b^{5} d^{3} e^{8} + 210 \, A a^{4} b^{6} d^{3} e^{8} + 70 \, B a^{6} b^{4} d^{2} e^{9} + 84 \, A a^{5} b^{5} d^{2} e^{9} + 24 \, B a^{7} b^{3} d e^{10} + 42 \, A a^{6} b^{4} d e^{10} + 9 \, B a^{8} b^{2} e^{11} + 24 \, A a^{7} b^{3} e^{11}\right )} x^{3} + 28 \, {\left (27599 \, B b^{10} d^{9} e^{2} - 92070 \, B a b^{9} d^{8} e^{3} - 9207 \, A b^{10} d^{8} e^{3} + 108378 \, B a^{2} b^{8} d^{7} e^{4} + 24084 \, A a b^{9} d^{7} e^{4} - 49392 \, B a^{3} b^{7} d^{6} e^{5} - 18522 \, A a^{2} b^{8} d^{6} e^{5} + 4410 \, B a^{4} b^{6} d^{5} e^{6} + 2520 \, A a^{3} b^{7} d^{5} e^{6} + 756 \, B a^{5} b^{5} d^{4} e^{7} + 630 \, A a^{4} b^{6} d^{4} e^{7} + 210 \, B a^{6} b^{4} d^{3} e^{8} + 252 \, A a^{5} b^{5} d^{3} e^{8} + 72 \, B a^{7} b^{3} d^{2} e^{9} + 126 \, A a^{6} b^{4} d^{2} e^{9} + 27 \, B a^{8} b^{2} d e^{10} + 72 \, A a^{7} b^{3} d e^{10} + 10 \, B a^{9} b e^{11} + 45 \, A a^{8} b^{2} e^{11}\right )} x^{2} + 8 \, {\left (30371 \, B b^{10} d^{10} e - 100470 \, B a b^{9} d^{9} e^{2} - 10047 \, A b^{10} d^{9} e^{2} + 116883 \, B a^{2} b^{8} d^{8} e^{3} + 25974 \, A a b^{9} d^{8} e^{3} - 52272 \, B a^{3} b^{7} d^{7} e^{4} - 19602 \, A a^{2} b^{8} d^{7} e^{4} + 4410 \, B a^{4} b^{6} d^{6} e^{5} + 2520 \, A a^{3} b^{7} d^{6} e^{5} + 756 \, B a^{5} b^{5} d^{5} e^{6} + 630 \, A a^{4} b^{6} d^{5} e^{6} + 210 \, B a^{6} b^{4} d^{4} e^{7} + 252 \, A a^{5} b^{5} d^{4} e^{7} + 72 \, B a^{7} b^{3} d^{3} e^{8} + 126 \, A a^{6} b^{4} d^{3} e^{8} + 27 \, B a^{8} b^{2} d^{2} e^{9} + 72 \, A a^{7} b^{3} d^{2} e^{9} + 10 \, B a^{9} b d e^{10} + 45 \, A a^{8} b^{2} d e^{10} + 3 \, B a^{10} e^{11} + 30 \, A a^{9} b e^{11}\right )} x\right )} e^{\left (-12\right )}}{168 \, {\left (x e + d\right )}^{8}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^9,x, algorithm="giac")

[Out]

-15*(11*B*b^10*d^3 - 30*B*a*b^9*d^2*e - 3*A*b^10*d^2*e + 27*B*a^2*b^8*d*e^2 + 6*A*a*b^9*d*e^2 - 8*B*a^3*b^7*e^

3 - 3*A*a^2*b^8*e^3)*e^(-12)*log(abs(x*e + d)) + 1/6*(2*B*b^10*x^3*e^18 - 27*B*b^10*d*x^2*e^17 + 270*B*b^10*d^

2*x*e^16 + 30*B*a*b^9*x^2*e^18 + 3*A*b^10*x^2*e^18 - 540*B*a*b^9*d*x*e^17 - 54*A*b^10*d*x*e^17 + 270*B*a^2*b^8

*x*e^18 + 60*A*a*b^9*x*e^18)*e^(-27) - 1/168*(32891*B*b^10*d^11 - 108030*B*a*b^9*d^10*e - 10803*A*b^10*d^10*e

+ 124443*B*a^2*b^8*d^9*e^2 + 27654*A*a*b^9*d^9*e^2 - 54792*B*a^3*b^7*d^8*e^3 - 20547*A*a^2*b^8*d^8*e^3 + 4410*

B*a^4*b^6*d^7*e^4 + 2520*A*a^3*b^7*d^7*e^4 + 756*B*a^5*b^5*d^6*e^5 + 630*A*a^4*b^6*d^6*e^5 + 210*B*a^6*b^4*d^5

*e^6 + 252*A*a^5*b^5*d^5*e^6 + 72*B*a^7*b^3*d^4*e^7 + 126*A*a^6*b^4*d^4*e^7 + 27*B*a^8*b^2*d^3*e^8 + 72*A*a^7*

b^3*d^3*e^8 + 10*B*a^9*b*d^2*e^9 + 45*A*a^8*b^2*d^2*e^9 + 3*B*a^10*d*e^10 + 30*A*a^9*b*d*e^10 + 21*A*a^10*e^11

+ 5040*(11*B*b^10*d^4*e^7 - 40*B*a*b^9*d^3*e^8 - 4*A*b^10*d^3*e^8 + 54*B*a^2*b^8*d^2*e^9 + 12*A*a*b^9*d^2*e^9

- 32*B*a^3*b^7*d*e^10 - 12*A*a^2*b^8*d*e^10 + 7*B*a^4*b^6*e^11 + 4*A*a^3*b^7*e^11)*x^7 + 3528*(99*B*b^10*d^5*

e^6 - 350*B*a*b^9*d^4*e^7 - 35*A*b^10*d^4*e^7 + 450*B*a^2*b^8*d^3*e^8 + 100*A*a*b^9*d^3*e^8 - 240*B*a^3*b^7*d^

2*e^9 - 90*A*a^2*b^8*d^2*e^9 + 35*B*a^4*b^6*d*e^10 + 20*A*a^3*b^7*d*e^10 + 6*B*a^5*b^5*e^11 + 5*A*a^4*b^6*e^11

)*x^6 + 2352*(407*B*b^10*d^6*e^5 - 1410*B*a*b^9*d^5*e^6 - 141*A*b^10*d^5*e^6 + 1755*B*a^2*b^8*d^4*e^7 + 390*A*

a*b^9*d^4*e^7 - 880*B*a^3*b^7*d^3*e^8 - 330*A*a^2*b^8*d^3*e^8 + 105*B*a^4*b^6*d^2*e^9 + 60*A*a^3*b^7*d^2*e^9 +

18*B*a^5*b^5*d*e^10 + 15*A*a^4*b^6*d*e^10 + 5*B*a^6*b^4*e^11 + 6*A*a^5*b^5*e^11)*x^5 + 420*(3509*B*b^10*d^7*e

^4 - 11970*B*a*b^9*d^6*e^5 - 1197*A*b^10*d^6*e^5 + 14553*B*a^2*b^8*d^5*e^6 + 3234*A*a*b^9*d^5*e^6 - 7000*B*a^3

*b^7*d^4*e^7 - 2625*A*a^2*b^8*d^4*e^7 + 735*B*a^4*b^6*d^3*e^8 + 420*A*a^3*b^7*d^3*e^8 + 126*B*a^5*b^5*d^2*e^9

+ 105*A*a^4*b^6*d^2*e^9 + 35*B*a^6*b^4*d*e^10 + 42*A*a^5*b^5*d*e^10 + 12*B*a^7*b^3*e^11 + 21*A*a^6*b^4*e^11)*x

^4 + 168*(8173*B*b^10*d^8*e^3 - 27540*B*a*b^9*d^7*e^4 - 2754*A*b^10*d^7*e^4 + 32886*B*a^2*b^8*d^6*e^5 + 7308*A

*a*b^9*d^6*e^5 - 15344*B*a^3*b^7*d^5*e^6 - 5754*A*a^2*b^8*d^5*e^6 + 1470*B*a^4*b^6*d^4*e^7 + 840*A*a^3*b^7*d^4

*e^7 + 252*B*a^5*b^5*d^3*e^8 + 210*A*a^4*b^6*d^3*e^8 + 70*B*a^6*b^4*d^2*e^9 + 84*A*a^5*b^5*d^2*e^9 + 24*B*a^7*

b^3*d*e^10 + 42*A*a^6*b^4*d*e^10 + 9*B*a^8*b^2*e^11 + 24*A*a^7*b^3*e^11)*x^3 + 28*(27599*B*b^10*d^9*e^2 - 9207

0*B*a*b^9*d^8*e^3 - 9207*A*b^10*d^8*e^3 + 108378*B*a^2*b^8*d^7*e^4 + 24084*A*a*b^9*d^7*e^4 - 49392*B*a^3*b^7*d

^6*e^5 - 18522*A*a^2*b^8*d^6*e^5 + 4410*B*a^4*b^6*d^5*e^6 + 2520*A*a^3*b^7*d^5*e^6 + 756*B*a^5*b^5*d^4*e^7 + 6

30*A*a^4*b^6*d^4*e^7 + 210*B*a^6*b^4*d^3*e^8 + 252*A*a^5*b^5*d^3*e^8 + 72*B*a^7*b^3*d^2*e^9 + 126*A*a^6*b^4*d^

2*e^9 + 27*B*a^8*b^2*d*e^10 + 72*A*a^7*b^3*d*e^10 + 10*B*a^9*b*e^11 + 45*A*a^8*b^2*e^11)*x^2 + 8*(30371*B*b^10

*d^10*e - 100470*B*a*b^9*d^9*e^2 - 10047*A*b^10*d^9*e^2 + 116883*B*a^2*b^8*d^8*e^3 + 25974*A*a*b^9*d^8*e^3 - 5

2272*B*a^3*b^7*d^7*e^4 - 19602*A*a^2*b^8*d^7*e^4 + 4410*B*a^4*b^6*d^6*e^5 + 2520*A*a^3*b^7*d^6*e^5 + 756*B*a^5

*b^5*d^5*e^6 + 630*A*a^4*b^6*d^5*e^6 + 210*B*a^6*b^4*d^4*e^7 + 252*A*a^5*b^5*d^4*e^7 + 72*B*a^7*b^3*d^3*e^8 +

126*A*a^6*b^4*d^3*e^8 + 27*B*a^8*b^2*d^2*e^9 + 72*A*a^7*b^3*d^2*e^9 + 10*B*a^9*b*d*e^10 + 45*A*a^8*b^2*d*e^10

+ 3*B*a^10*e^11 + 30*A*a^9*b*e^11)*x)*e^(-12)/(x*e + d)^8

________________________________________________________________________________________

Mupad [B]
time = 0.49, size = 2048, normalized size = 4.60 \begin {gather*} x^2\,\left (\frac {A\,b^{10}+10\,B\,a\,b^9}{2\,e^9}-\frac {9\,B\,b^{10}\,d}{2\,e^{10}}\right )-\frac {x^7\,\left (210\,B\,a^4\,b^6\,e^{10}-960\,B\,a^3\,b^7\,d\,e^9+120\,A\,a^3\,b^7\,e^{10}+1620\,B\,a^2\,b^8\,d^2\,e^8-360\,A\,a^2\,b^8\,d\,e^9-1200\,B\,a\,b^9\,d^3\,e^7+360\,A\,a\,b^9\,d^2\,e^8+330\,B\,b^{10}\,d^4\,e^6-120\,A\,b^{10}\,d^3\,e^7\right )+x^4\,\left (30\,B\,a^7\,b^3\,e^{10}+\frac {175\,B\,a^6\,b^4\,d\,e^9}{2}+\frac {105\,A\,a^6\,b^4\,e^{10}}{2}+315\,B\,a^5\,b^5\,d^2\,e^8+105\,A\,a^5\,b^5\,d\,e^9+\frac {3675\,B\,a^4\,b^6\,d^3\,e^7}{2}+\frac {525\,A\,a^4\,b^6\,d^2\,e^8}{2}-17500\,B\,a^3\,b^7\,d^4\,e^6+1050\,A\,a^3\,b^7\,d^3\,e^7+\frac {72765\,B\,a^2\,b^8\,d^5\,e^5}{2}-\frac {13125\,A\,a^2\,b^8\,d^4\,e^6}{2}-29925\,B\,a\,b^9\,d^6\,e^4+8085\,A\,a\,b^9\,d^5\,e^5+\frac {17545\,B\,b^{10}\,d^7\,e^3}{2}-\frac {5985\,A\,b^{10}\,d^6\,e^4}{2}\right )+x^6\,\left (126\,B\,a^5\,b^5\,e^{10}+735\,B\,a^4\,b^6\,d\,e^9+105\,A\,a^4\,b^6\,e^{10}-5040\,B\,a^3\,b^7\,d^2\,e^8+420\,A\,a^3\,b^7\,d\,e^9+9450\,B\,a^2\,b^8\,d^3\,e^7-1890\,A\,a^2\,b^8\,d^2\,e^8-7350\,B\,a\,b^9\,d^4\,e^6+2100\,A\,a\,b^9\,d^3\,e^7+2079\,B\,b^{10}\,d^5\,e^5-735\,A\,b^{10}\,d^4\,e^6\right )+x^3\,\left (9\,B\,a^8\,b^2\,e^{10}+24\,B\,a^7\,b^3\,d\,e^9+24\,A\,a^7\,b^3\,e^{10}+70\,B\,a^6\,b^4\,d^2\,e^8+42\,A\,a^6\,b^4\,d\,e^9+252\,B\,a^5\,b^5\,d^3\,e^7+84\,A\,a^5\,b^5\,d^2\,e^8+1470\,B\,a^4\,b^6\,d^4\,e^6+210\,A\,a^4\,b^6\,d^3\,e^7-15344\,B\,a^3\,b^7\,d^5\,e^5+840\,A\,a^3\,b^7\,d^4\,e^6+32886\,B\,a^2\,b^8\,d^6\,e^4-5754\,A\,a^2\,b^8\,d^5\,e^5-27540\,B\,a\,b^9\,d^7\,e^3+7308\,A\,a\,b^9\,d^6\,e^4+8173\,B\,b^{10}\,d^8\,e^2-2754\,A\,b^{10}\,d^7\,e^3\right )+\frac {3\,B\,a^{10}\,d\,e^{10}+21\,A\,a^{10}\,e^{11}+10\,B\,a^9\,b\,d^2\,e^9+30\,A\,a^9\,b\,d\,e^{10}+27\,B\,a^8\,b^2\,d^3\,e^8+45\,A\,a^8\,b^2\,d^2\,e^9+72\,B\,a^7\,b^3\,d^4\,e^7+72\,A\,a^7\,b^3\,d^3\,e^8+210\,B\,a^6\,b^4\,d^5\,e^6+126\,A\,a^6\,b^4\,d^4\,e^7+756\,B\,a^5\,b^5\,d^6\,e^5+252\,A\,a^5\,b^5\,d^5\,e^6+4410\,B\,a^4\,b^6\,d^7\,e^4+630\,A\,a^4\,b^6\,d^6\,e^5-54792\,B\,a^3\,b^7\,d^8\,e^3+2520\,A\,a^3\,b^7\,d^7\,e^4+124443\,B\,a^2\,b^8\,d^9\,e^2-20547\,A\,a^2\,b^8\,d^8\,e^3-108030\,B\,a\,b^9\,d^{10}\,e+27654\,A\,a\,b^9\,d^9\,e^2+32891\,B\,b^{10}\,d^{11}-10803\,A\,b^{10}\,d^{10}\,e}{168\,e}+x\,\left (\frac {B\,a^{10}\,e^{10}}{7}+\frac {10\,B\,a^9\,b\,d\,e^9}{21}+\frac {10\,A\,a^9\,b\,e^{10}}{7}+\frac {9\,B\,a^8\,b^2\,d^2\,e^8}{7}+\frac {15\,A\,a^8\,b^2\,d\,e^9}{7}+\frac {24\,B\,a^7\,b^3\,d^3\,e^7}{7}+\frac {24\,A\,a^7\,b^3\,d^2\,e^8}{7}+10\,B\,a^6\,b^4\,d^4\,e^6+6\,A\,a^6\,b^4\,d^3\,e^7+36\,B\,a^5\,b^5\,d^5\,e^5+12\,A\,a^5\,b^5\,d^4\,e^6+210\,B\,a^4\,b^6\,d^6\,e^4+30\,A\,a^4\,b^6\,d^5\,e^5-\frac {17424\,B\,a^3\,b^7\,d^7\,e^3}{7}+120\,A\,a^3\,b^7\,d^6\,e^4+\frac {38961\,B\,a^2\,b^8\,d^8\,e^2}{7}-\frac {6534\,A\,a^2\,b^8\,d^7\,e^3}{7}-\frac {33490\,B\,a\,b^9\,d^9\,e}{7}+\frac {8658\,A\,a\,b^9\,d^8\,e^2}{7}+\frac {30371\,B\,b^{10}\,d^{10}}{21}-\frac {3349\,A\,b^{10}\,d^9\,e}{7}\right )+x^5\,\left (70\,B\,a^6\,b^4\,e^{10}+252\,B\,a^5\,b^5\,d\,e^9+84\,A\,a^5\,b^5\,e^{10}+1470\,B\,a^4\,b^6\,d^2\,e^8+210\,A\,a^4\,b^6\,d\,e^9-12320\,B\,a^3\,b^7\,d^3\,e^7+840\,A\,a^3\,b^7\,d^2\,e^8+24570\,B\,a^2\,b^8\,d^4\,e^6-4620\,A\,a^2\,b^8\,d^3\,e^7-19740\,B\,a\,b^9\,d^5\,e^5+5460\,A\,a\,b^9\,d^4\,e^6+5698\,B\,b^{10}\,d^6\,e^4-1974\,A\,b^{10}\,d^5\,e^5\right )+x^2\,\left (\frac {5\,B\,a^9\,b\,e^{10}}{3}+\frac {9\,B\,a^8\,b^2\,d\,e^9}{2}+\frac {15\,A\,a^8\,b^2\,e^{10}}{2}+12\,B\,a^7\,b^3\,d^2\,e^8+12\,A\,a^7\,b^3\,d\,e^9+35\,B\,a^6\,b^4\,d^3\,e^7+21\,A\,a^6\,b^4\,d^2\,e^8+126\,B\,a^5\,b^5\,d^4\,e^6+42\,A\,a^5\,b^5\,d^3\,e^7+735\,B\,a^4\,b^6\,d^5\,e^5+105\,A\,a^4\,b^6\,d^4\,e^6-8232\,B\,a^3\,b^7\,d^6\,e^4+420\,A\,a^3\,b^7\,d^5\,e^5+18063\,B\,a^2\,b^8\,d^7\,e^3-3087\,A\,a^2\,b^8\,d^6\,e^4-15345\,B\,a\,b^9\,d^8\,e^2+4014\,A\,a\,b^9\,d^7\,e^3+\frac {27599\,B\,b^{10}\,d^9\,e}{6}-\frac {3069\,A\,b^{10}\,d^8\,e^2}{2}\right )}{d^8\,e^{11}+8\,d^7\,e^{12}\,x+28\,d^6\,e^{13}\,x^2+56\,d^5\,e^{14}\,x^3+70\,d^4\,e^{15}\,x^4+56\,d^3\,e^{16}\,x^5+28\,d^2\,e^{17}\,x^6+8\,d\,e^{18}\,x^7+e^{19}\,x^8}-x\,\left (\frac {9\,d\,\left (\frac {A\,b^{10}+10\,B\,a\,b^9}{e^9}-\frac {9\,B\,b^{10}\,d}{e^{10}}\right )}{e}-\frac {5\,a\,b^8\,\left (2\,A\,b+9\,B\,a\right )}{e^9}+\frac {36\,B\,b^{10}\,d^2}{e^{11}}\right )+\frac {\ln \left (d+e\,x\right )\,\left (120\,B\,a^3\,b^7\,e^3-405\,B\,a^2\,b^8\,d\,e^2+45\,A\,a^2\,b^8\,e^3+450\,B\,a\,b^9\,d^2\,e-90\,A\,a\,b^9\,d\,e^2-165\,B\,b^{10}\,d^3+45\,A\,b^{10}\,d^2\,e\right )}{e^{12}}+\frac {B\,b^{10}\,x^3}{3\,e^9} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^10)/(d + e*x)^9,x)

[Out]

x^2*((A*b^10 + 10*B*a*b^9)/(2*e^9) - (9*B*b^10*d)/(2*e^10)) - (x^7*(120*A*a^3*b^7*e^10 + 210*B*a^4*b^6*e^10 -

120*A*b^10*d^3*e^7 + 330*B*b^10*d^4*e^6 + 360*A*a*b^9*d^2*e^8 - 360*A*a^2*b^8*d*e^9 - 1200*B*a*b^9*d^3*e^7 - 9

60*B*a^3*b^7*d*e^9 + 1620*B*a^2*b^8*d^2*e^8) + x^4*((105*A*a^6*b^4*e^10)/2 + 30*B*a^7*b^3*e^10 - (5985*A*b^10*

d^6*e^4)/2 + (17545*B*b^10*d^7*e^3)/2 + 8085*A*a*b^9*d^5*e^5 + 105*A*a^5*b^5*d*e^9 - 29925*B*a*b^9*d^6*e^4 + (

175*B*a^6*b^4*d*e^9)/2 - (13125*A*a^2*b^8*d^4*e^6)/2 + 1050*A*a^3*b^7*d^3*e^7 + (525*A*a^4*b^6*d^2*e^8)/2 + (7

2765*B*a^2*b^8*d^5*e^5)/2 - 17500*B*a^3*b^7*d^4*e^6 + (3675*B*a^4*b^6*d^3*e^7)/2 + 315*B*a^5*b^5*d^2*e^8) + x^

6*(105*A*a^4*b^6*e^10 + 126*B*a^5*b^5*e^10 - 735*A*b^10*d^4*e^6 + 2079*B*b^10*d^5*e^5 + 2100*A*a*b^9*d^3*e^7 +

420*A*a^3*b^7*d*e^9 - 7350*B*a*b^9*d^4*e^6 + 735*B*a^4*b^6*d*e^9 - 1890*A*a^2*b^8*d^2*e^8 + 9450*B*a^2*b^8*d^

3*e^7 - 5040*B*a^3*b^7*d^2*e^8) + x^3*(24*A*a^7*b^3*e^10 + 9*B*a^8*b^2*e^10 - 2754*A*b^10*d^7*e^3 + 8173*B*b^1

0*d^8*e^2 + 7308*A*a*b^9*d^6*e^4 + 42*A*a^6*b^4*d*e^9 - 27540*B*a*b^9*d^7*e^3 + 24*B*a^7*b^3*d*e^9 - 5754*A*a^

2*b^8*d^5*e^5 + 840*A*a^3*b^7*d^4*e^6 + 210*A*a^4*b^6*d^3*e^7 + 84*A*a^5*b^5*d^2*e^8 + 32886*B*a^2*b^8*d^6*e^4

- 15344*B*a^3*b^7*d^5*e^5 + 1470*B*a^4*b^6*d^4*e^6 + 252*B*a^5*b^5*d^3*e^7 + 70*B*a^6*b^4*d^2*e^8) + (21*A*a^

10*e^11 + 32891*B*b^10*d^11 - 10803*A*b^10*d^10*e + 3*B*a^10*d*e^10 + 27654*A*a*b^9*d^9*e^2 + 10*B*a^9*b*d^2*e

^9 - 20547*A*a^2*b^8*d^8*e^3 + 2520*A*a^3*b^7*d^7*e^4 + 630*A*a^4*b^6*d^6*e^5 + 252*A*a^5*b^5*d^5*e^6 + 126*A*

a^6*b^4*d^4*e^7 + 72*A*a^7*b^3*d^3*e^8 + 45*A*a^8*b^2*d^2*e^9 + 124443*B*a^2*b^8*d^9*e^2 - 54792*B*a^3*b^7*d^8

*e^3 + 4410*B*a^4*b^6*d^7*e^4 + 756*B*a^5*b^5*d^6*e^5 + 210*B*a^6*b^4*d^5*e^6 + 72*B*a^7*b^3*d^4*e^7 + 27*B*a^

8*b^2*d^3*e^8 + 30*A*a^9*b*d*e^10 - 108030*B*a*b^9*d^10*e)/(168*e) + x*((B*a^10*e^10)/7 + (30371*B*b^10*d^10)/

21 + (10*A*a^9*b*e^10)/7 - (3349*A*b^10*d^9*e)/7 + (8658*A*a*b^9*d^8*e^2)/7 + (15*A*a^8*b^2*d*e^9)/7 - (6534*A

*a^2*b^8*d^7*e^3)/7 + 120*A*a^3*b^7*d^6*e^4 + 30*A*a^4*b^6*d^5*e^5 + 12*A*a^5*b^5*d^4*e^6 + 6*A*a^6*b^4*d^3*e^

7 + (24*A*a^7*b^3*d^2*e^8)/7 + (38961*B*a^2*b^8*d^8*e^2)/7 - (17424*B*a^3*b^7*d^7*e^3)/7 + 210*B*a^4*b^6*d^6*e

^4 + 36*B*a^5*b^5*d^5*e^5 + 10*B*a^6*b^4*d^4*e^6 + (24*B*a^7*b^3*d^3*e^7)/7 + (9*B*a^8*b^2*d^2*e^8)/7 - (33490

*B*a*b^9*d^9*e)/7 + (10*B*a^9*b*d*e^9)/21) + x^5*(84*A*a^5*b^5*e^10 + 70*B*a^6*b^4*e^10 - 1974*A*b^10*d^5*e^5

+ 5698*B*b^10*d^6*e^4 + 5460*A*a*b^9*d^4*e^6 + 210*A*a^4*b^6*d*e^9 - 19740*B*a*b^9*d^5*e^5 + 252*B*a^5*b^5*d*e

^9 - 4620*A*a^2*b^8*d^3*e^7 + 840*A*a^3*b^7*d^2*e^8 + 24570*B*a^2*b^8*d^4*e^6 - 12320*B*a^3*b^7*d^3*e^7 + 1470

*B*a^4*b^6*d^2*e^8) + x^2*((5*B*a^9*b*e^10)/3 + (27599*B*b^10*d^9*e)/6 + (15*A*a^8*b^2*e^10)/2 - (3069*A*b^10*

d^8*e^2)/2 + 4014*A*a*b^9*d^7*e^3 + 12*A*a^7*b^3*d*e^9 - 15345*B*a*b^9*d^8*e^2 + (9*B*a^8*b^2*d*e^9)/2 - 3087*

A*a^2*b^8*d^6*e^4 + 420*A*a^3*b^7*d^5*e^5 + 105*A*a^4*b^6*d^4*e^6 + 42*A*a^5*b^5*d^3*e^7 + 21*A*a^6*b^4*d^2*e^

8 + 18063*B*a^2*b^8*d^7*e^3 - 8232*B*a^3*b^7*d^6*e^4 + 735*B*a^4*b^6*d^5*e^5 + 126*B*a^5*b^5*d^4*e^6 + 35*B*a^

6*b^4*d^3*e^7 + 12*B*a^7*b^3*d^2*e^8))/(d^8*e^11 + e^19*x^8 + 8*d^7*e^12*x + 8*d*e^18*x^7 + 28*d^6*e^13*x^2 +

56*d^5*e^14*x^3 + 70*d^4*e^15*x^4 + 56*d^3*e^16*x^5 + 28*d^2*e^17*x^6) - x*((9*d*((A*b^10 + 10*B*a*b^9)/e^9 -

(9*B*b^10*d)/e^10))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^9 + (36*B*b^10*d^2)/e^11) + (log(d + e*x)*(45*A*b^10*d^2*e

- 165*B*b^10*d^3 + 45*A*a^2*b^8*e^3 + 120*B*a^3*b^7*e^3 - 405*B*a^2*b^8*d*e^2 - 90*A*a*b^9*d*e^2 + 450*B*a*b^

9*d^2*e))/e^12 + (B*b^10*x^3)/(3*e^9)

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