∫ (a+bx)10(A+Bx)___________

3.11.96 \(\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^8} \, dx\) [1096]

Optimal. Leaf size=444 \[ -\frac {15 b^7 (b d-a e)^2 (11 b B d-3 A b e-8 a B e) x}{e^{11}}+\frac {(b d-a e)^{10} (B d-A e)}{7 e^{12} (d+e x)^7}-\frac {(b d-a e)^9 (11 b B d-10 A b e-a B e)}{6 e^{12} (d+e x)^6}+\frac {b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{e^{12} (d+e x)^5}-\frac {15 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e)}{4 e^{12} (d+e x)^4}+\frac {10 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e)}{e^{12} (d+e x)^3}-\frac {21 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e)}{e^{12} (d+e x)^2}+\frac {42 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e)}{e^{12} (d+e x)}+\frac {5 b^8 (b d-a e) (11 b B d-2 A b e-9 a B e) (d+e x)^2}{2 e^{12}}-\frac {b^9 (11 b B d-A b e-10 a B e) (d+e x)^3}{3 e^{12}}+\frac {b^{10} B (d+e x)^4}{4 e^{12}}+\frac {30 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e) \log (d+e x)}{e^{12}} \]

[Out]

-15*b^7*(-a*e+b*d)^2*(-3*A*b*e-8*B*a*e+11*B*b*d)*x/e^11+1/7*(-a*e+b*d)^10*(-A*e+B*d)/e^12/(e*x+d)^7-1/6*(-a*e+

b*d)^9*(-10*A*b*e-B*a*e+11*B*b*d)/e^12/(e*x+d)^6+b*(-a*e+b*d)^8*(-9*A*b*e-2*B*a*e+11*B*b*d)/e^12/(e*x+d)^5-15/

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

e^12/(e*x+d)^3-21*b^4*(-a*e+b*d)^5*(-6*A*b*e-5*B*a*e+11*B*b*d)/e^12/(e*x+d)^2+42*b^5*(-a*e+b*d)^4*(-5*A*b*e-6*

B*a*e+11*B*b*d)/e^12/(e*x+d)+5/2*b^8*(-a*e+b*d)*(-2*A*b*e-9*B*a*e+11*B*b*d)*(e*x+d)^2/e^12-1/3*b^9*(-A*b*e-10*

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

x+d)/e^12

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Rubi [A]
time = 0.70, antiderivative size = 444, 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)^3 (-10 a B e-A b e+11 b B d)}{3 e^{12}}+\frac {5 b^8 (d+e x)^2 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{2 e^{12}}-\frac {15 b^7 x (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{e^{11}}+\frac {30 b^6 (b d-a e)^3 \log (d+e x) (-7 a B e-4 A b e+11 b B d)}{e^{12}}+\frac {42 b^5 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{e^{12} (d+e x)}-\frac {21 b^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{e^{12} (d+e x)^2}+\frac {10 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{e^{12} (d+e x)^3}-\frac {15 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{4 e^{12} (d+e x)^4}+\frac {b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{e^{12} (d+e x)^5}-\frac {(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{6 e^{12} (d+e x)^6}+\frac {(b d-a e)^{10} (B d-A e)}{7 e^{12} (d+e x)^7}+\frac {b^{10} B (d+e x)^4}{4 e^{12}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Int[((a + b*x)^10*(A + B*x))/(d + e*x)^8,x]

[Out]

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

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

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

+ (10*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b*e - 4*a*B*e))/(e^12*(d + e*x)^3) - (21*b^4*(b*d - a*e)^5*(11*b*B*d -

6*A*b*e - 5*a*B*e))/(e^12*(d + e*x)^2) + (42*b^5*(b*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*B*e))/(e^12*(d + e*x

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

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

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Mathematica [A]
time = 0.36, size = 450, normalized size = 1.01 \begin {gather*} \frac {84 b^7 e \left (120 a^3 B e^3+40 a b^2 d e (9 B d-2 A e)+45 a^2 b e^2 (-8 B d+A e)+12 b^3 d^2 (-10 B d+3 A e)\right ) x-42 b^8 e^2 \left (-45 a^2 B e^2-10 a b e (-8 B d+A e)+4 b^2 d (-9 B d+2 A e)\right ) x^2+28 b^9 e^3 (-8 b B d+A b e+10 a B e) x^3+21 b^{10} B e^4 x^4+\frac {12 (b d-a e)^{10} (B d-A e)}{(d+e x)^7}-\frac {14 (b d-a e)^9 (11 b B d-10 A b e-a B e)}{(d+e x)^6}+\frac {84 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{(d+e x)^5}-\frac {315 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e)}{(d+e x)^4}+\frac {840 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e)}{(d+e x)^3}-\frac {1764 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e)}{(d+e x)^2}+\frac {3528 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e)}{d+e x}+2520 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e) \log (d+e x)}{84 e^{12}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^8,x]

[Out]

(84*b^7*e*(120*a^3*B*e^3 + 40*a*b^2*d*e*(9*B*d - 2*A*e) + 45*a^2*b*e^2*(-8*B*d + A*e) + 12*b^3*d^2*(-10*B*d +

3*A*e))*x - 42*b^8*e^2*(-45*a^2*B*e^2 - 10*a*b*e*(-8*B*d + A*e) + 4*b^2*d*(-9*B*d + 2*A*e))*x^2 + 28*b^9*e^3*(

-8*b*B*d + A*b*e + 10*a*B*e)*x^3 + 21*b^10*B*e^4*x^4 + (12*(b*d - a*e)^10*(B*d - A*e))/(d + e*x)^7 - (14*(b*d

- a*e)^9*(11*b*B*d - 10*A*b*e - a*B*e))/(d + e*x)^6 + (84*b*(b*d - a*e)^8*(11*b*B*d - 9*A*b*e - 2*a*B*e))/(d +

e*x)^5 - (315*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b*e - 3*a*B*e))/(d + e*x)^4 + (840*b^3*(b*d - a*e)^6*(11*b*B*

d - 7*A*b*e - 4*a*B*e))/(d + e*x)^3 - (1764*b^4*(b*d - a*e)^5*(11*b*B*d - 6*A*b*e - 5*a*B*e))/(d + e*x)^2 + (3

528*b^5*(b*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*B*e))/(d + e*x) + 2520*b^6*(b*d - a*e)^3*(11*b*B*d - 4*A*b*e -

7*a*B*e)*Log[d + e*x])/(84*e^12)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1952\) vs. \(2(432)=864\).
time = 0.10, size = 1953, normalized size = 4.40


method	result	size

norman	\(\text {Expression too large to display}\)	\(1925\)
default	\(\text {Expression too large to display}\)	\(1953\)
risch	\(\text {Expression too large to display}\)	\(1999\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10*(B*x+A)/(e*x+d)^8,x,method=_RETURNVERBOSE)

[Out]

b^7/e^11*(1/4*b^3*B*x^4*e^3+1/3*A*b^3*e^3*x^3+10/3*B*a*b^2*e^3*x^3-8/3*B*b^3*d*e^2*x^3+5*A*a*b^2*e^3*x^2-4*A*b

^3*d*e^2*x^2+45/2*B*a^2*b*e^3*x^2-40*B*a*b^2*d*e^2*x^2+18*B*b^3*d^2*e*x^2+45*A*a^2*b*e^3*x-80*A*a*b^2*d*e^2*x+

36*A*b^3*d^2*e*x+120*B*a^3*e^3*x-360*B*a^2*b*d*e^2*x+360*B*a*b^2*d^2*e*x-120*B*b^3*d^3*x)-21*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)^2-15/4*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+2

80*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)^4-42*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)-1/7*(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)^7-1/6/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)^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+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*x+d)^5-10*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)^3+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)*ln(e*x+d)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1912 vs. \(2 (462) = 924\).
time = 0.63, size = 1912, normalized size = 4.31 \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)^8,x, algorithm="maxima")

[Out]

30*(11*B*b^10*d^4 + 7*B*a^4*b^6*e^4 + 4*A*a^3*b^7*e^4 - 4*(10*B*a*b^9*e + A*b^10*e)*d^3 + 6*(9*B*a^2*b^8*e^2 +

2*A*a*b^9*e^2)*d^2 - 4*(8*B*a^3*b^7*e^3 + 3*A*a^2*b^8*e^3)*d)*e^(-12)*log(x*e + d) + 1/12*(3*B*b^10*x^4*e^3 -

4*(8*B*b^10*d*e^2 - 10*B*a*b^9*e^3 - A*b^10*e^3)*x^3 + 6*(36*B*b^10*d^2*e + 45*B*a^2*b^8*e^3 + 10*A*a*b^9*e^3

- 8*(10*B*a*b^9*e^2 + A*b^10*e^2)*d)*x^2 - 12*(120*B*b^10*d^3 - 120*B*a^3*b^7*e^3 - 45*A*a^2*b^8*e^3 - 36*(10

*B*a*b^9*e + A*b^10*e)*d^2 + 40*(9*B*a^2*b^8*e^2 + 2*A*a*b^9*e^2)*d)*x)*e^(-11) + 1/84*(25961*B*b^10*d^11 - 12

*A*a^10*e^11 - 11044*(10*B*a*b^9*e + A*b^10*e)*d^10 + 20094*(9*B*a^2*b^8*e^2 + 2*A*a*b^9*e^2)*d^9 - 17316*(8*B

*a^3*b^7*e^3 + 3*A*a^2*b^8*e^3)*d^8 + 6534*(7*B*a^4*b^6*e^4 + 4*A*a^3*b^7*e^4)*d^7 - 504*(6*B*a^5*b^5*e^5 + 5*

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

0*e^7)*d^4 + 10*(9*B*a^2*b^8*e^8 + 2*A*a*b^9*e^8)*d^3 - 10*(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 - 84*(5*B*a^6*b^4*e^6 + 6*A*a^5*b^5*e^6)*d^5 + 1764*(121*B*b^10*d^6*e^5

- 5*B*a^6*b^4*e^11 - 6*A*a^5*b^5*e^11 - 54*(10*B*a*b^9*e^6 + A*b^10*e^6)*d^5 + 105*(9*B*a^2*b^8*e^7 + 2*A*a*b^

9*e^7)*d^4 - 100*(8*B*a^3*b^7*e^8 + 3*A*a^2*b^8*e^8)*d^3 + 45*(7*B*a^4*b^6*e^9 + 4*A*a^3*b^7*e^9)*d^2 - 6*(6*B

*a^5*b^5*e^10 + 5*A*a^4*b^6*e^10)*d)*x^5 - 24*(4*B*a^7*b^3*e^7 + 7*A*a^6*b^4*e^7)*d^4 + 420*(1177*B*b^10*d^7*e

^4 - 8*B*a^7*b^3*e^11 - 14*A*a^6*b^4*e^11 - 518*(10*B*a*b^9*e^5 + A*b^10*e^5)*d^6 + 987*(9*B*a^2*b^8*e^6 + 2*A

*a*b^9*e^6)*d^5 - 910*(8*B*a^3*b^7*e^7 + 3*A*a^2*b^8*e^7)*d^4 + 385*(7*B*a^4*b^6*e^8 + 4*A*a^3*b^7*e^8)*d^3 -

42*(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 + 105*(5863*B*b^10*d^8*e^3 - 9*B*a^8*b^2*e^11 - 24*A*a^7*b^3*e^11 - 2552*(10*B*a*b^9

*e^4 + A*b^10*e^4)*d^7 + 4788*(9*B*a^2*b^8*e^5 + 2*A*a*b^9*e^5)*d^6 - 4312*(8*B*a^3*b^7*e^6 + 3*A*a^2*b^8*e^6)

*d^5 + 1750*(7*B*a^4*b^6*e^7 + 4*A*a^3*b^7*e^7)*d^4 - 168*(6*B*a^5*b^5*e^8 + 5*A*a^4*b^6*e^8)*d^3 - 28*(5*B*a^

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

*b^2*e^9)*d^2 + 21*(20669*B*b^10*d^9*e^2 - 8*B*a^9*b*e^11 - 36*A*a^8*b^2*e^11 - 8916*(10*B*a*b^9*e^3 + A*b^10*

e^3)*d^8 + 16524*(9*B*a^2*b^8*e^4 + 2*A*a*b^9*e^4)*d^7 - 14616*(8*B*a^3*b^7*e^5 + 3*A*a^2*b^8*e^5)*d^6 + 5754*

(7*B*a^4*b^6*e^6 + 4*A*a^3*b^7*e^6)*d^5 - 504*(6*B*a^5*b^5*e^7 + 5*A*a^4*b^6*e^7)*d^4 - 84*(5*B*a^6*b^4*e^8 +

6*A*a^5*b^5*e^8)*d^3 - 24*(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 - 2*(B*a^10*e^10 + 10*A*a^9*b*e^10)*d + 7*(23441*B*b^10*d^10*e - 2*B*a^10*e^11 - 20*A*a^9*b*e^11 - 10036*

(10*B*a*b^9*e^2 + A*b^10*e^2)*d^9 + 18414*(9*B*a^2*b^8*e^3 + 2*A*a*b^9*e^3)*d^8 - 16056*(8*B*a^3*b^7*e^4 + 3*A

*a^2*b^8*e^4)*d^7 + 6174*(7*B*a^4*b^6*e^5 + 4*A*a^3*b^7*e^5)*d^6 - 504*(6*B*a^5*b^5*e^6 + 5*A*a^4*b^6*e^6)*d^5

- 84*(5*B*a^6*b^4*e^7 + 6*A*a^5*b^5*e^7)*d^4 - 24*(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 - 4*(2*B*a^9*b*e^10 + 9*A*a^8*b^2*e^10)*d)*x)/(x^7*e^19 + 7*d*x^6*e^18 + 21*d^2*x^5*e

^17 + 35*d^3*x^4*e^16 + 35*d^4*x^3*e^15 + 21*d^5*x^2*e^14 + 7*d^6*x*e^13 + d^7*e^12)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2707 vs. \(2 (462) = 924\).
time = 1.17, size = 2707, normalized size = 6.10 \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)^8,x, algorithm="fricas")

[Out]

1/84*(25961*B*b^10*d^11 + (21*B*b^10*x^11 - 12*A*a^10 + 28*(10*B*a*b^9 + A*b^10)*x^10 + 210*(9*B*a^2*b^8 + 2*A

*a*b^9)*x^9 + 1260*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 - 3528*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 - 1764*(5*B*a^6*b^4

+ 6*A*a^5*b^5)*x^5 - 840*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 - 315*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 - 84*(2*B*a^9*b

+ 9*A*a^8*b^2)*x^2 - 14*(B*a^10 + 10*A*a^9*b)*x)*e^11 - (77*B*b^10*d*x^10 + 140*(10*B*a*b^9 + A*b^10)*d*x^9 +

1890*(9*B*a^2*b^8 + 2*A*a*b^9)*d*x^8 - 8820*(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 + 10584*(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 + 840*(4*B*a^7*b

^3 + 7*A*a^6*b^4)*d*x^3 + 189*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*x^2 + 28*(2*B*a^9*b + 9*A*a^8*b^2)*d*x + 2*(B*a^10

+ 10*A*a^9*b)*d)*e^10 + (385*B*b^10*d^2*x^9 + 1260*(10*B*a*b^9 + A*b^10)*d^2*x^8 - 19110*(9*B*a^2*b^8 + 2*A*a

*b^9)*d^2*x^7 - 8820*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*x^6 + 79380*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*x^5 - 17640*(

6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*x^4 - 2940*(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 - 63*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*x - 4*(2*B*a^9*b + 9*A*a^8*b^2)*d^2)*e^9 - (3465*B*b^10*d^3*x^8

- 15092*(10*B*a*b^9 + A*b^10)*d^3*x^7 + 27930*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*x^6 + 132300*(8*B*a^3*b^7 + 3*A*a

^2*b^8)*d^3*x^5 - 161700*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*x^4 + 17640*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*x^3 + 176

4*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*x^2 + 168*(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 - (45913*B*b^10*d^4*x^7 - 35084*(10*B*a*b^9 + A*b^10)*d^4*x^6 - 74970*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*x

^5 + 338100*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*x^4 - 183750*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*x^3 + 10584*(6*B*a^5*

b^5 + 5*A*a^4*b^6)*d^4*x^2 + 588*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*x + 24*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4)*e^7 -

7*(18193*B*b^10*d^5*x^6 + 84*(10*B*a*b^9 + A*b^10)*d^5*x^5 - 43050*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*x^4 + 60900*

(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*x^3 - 17262*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*x^2 + 504*(6*B*a^5*b^5 + 5*A*a^4*b

^6)*d^5*x + 12*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5)*e^6 - 7*(12999*B*b^10*d^6*x^5 + 16940*(10*B*a*b^9 + A*b^10)*d^

6*x^4 - 61950*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*x^3 + 42588*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*x^2 - 6174*(7*B*a^4*b^

6 + 4*A*a^3*b^7)*d^6*x + 72*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6)*e^5 + (171745*B*b^10*d^7*x^4 - 206780*(10*B*a*b^9

+ A*b^10)*d^7*x^3 + 323694*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*x^2 - 111132*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*x + 653

4*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7)*e^4 + (414295*B*b^10*d^8*x^3 - 166404*(10*B*a*b^9 + A*b^10)*d^8*x^2 + 12553

8*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*x - 17316*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8)*e^3 + (365001*B*b^10*d^9*x^2 - 6722

8*(10*B*a*b^9 + A*b^10)*d^9*x + 20094*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9)*e^2 + (154007*B*b^10*d^10*x - 11044*(10*B

*a*b^9 + A*b^10)*d^10)*e + 2520*(11*B*b^10*d^11 + (7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7*e^11 - (4*(8*B*a^3*b^7 + 3*A

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

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

^7)*d^3*x^4)*e^8 + (11*B*b^10*d^4*x^7 - 28*(10*B*a*b^9 + A*b^10)*d^4*x^6 + 126*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*x

^5 - 140*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*x^4 + 35*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*x^3)*e^7 + 7*(11*B*b^10*d^5*

x^6 - 12*(10*B*a*b^9 + A*b^10)*d^5*x^5 + 30*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*x^4 - 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)

*d^5*x^3 + 3*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*x^2)*e^6 + 7*(33*B*b^10*d^6*x^5 - 20*(10*B*a*b^9 + A*b^10)*d^6*x^

4 + 30*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*x^3 - 12*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*x^2 + (7*B*a^4*b^6 + 4*A*a^3*b^7

)*d^6*x)*e^5 + (385*B*b^10*d^7*x^4 - 140*(10*B*a*b^9 + A*b^10)*d^7*x^3 + 126*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*x^2

- 28*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*x + (7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7)*e^4 + (385*B*b^10*d^8*x^3 - 84*(10*

B*a*b^9 + A*b^10)*d^8*x^2 + 42*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*x - 4*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8)*e^3 + (231

*B*b^10*d^9*x^2 - 28*(10*B*a*b^9 + A*b^10)*d^9*x + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9)*e^2 + (77*B*b^10*d^10*x -

4*(10*B*a*b^9 + A*b^10)*d^10)*e)*log(x*e + d))/(x^7*e^19 + 7*d*x^6*e^18 + 21*d^2*x^5*e^17 + 35*d^3*x^4*e^16 +

35*d^4*x^3*e^15 + 21*d^5*x^2*e^14 + 7*d^6*x*e^13 + d^7*e^12)

________________________________________________________________________________________

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)**8,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1860 vs. \(2 (462) = 924\).
time = 1.60, size = 1860, normalized size = 4.19 \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)^8,x, algorithm="giac")

[Out]

30*(11*B*b^10*d^4 - 40*B*a*b^9*d^3*e - 4*A*b^10*d^3*e + 54*B*a^2*b^8*d^2*e^2 + 12*A*a*b^9*d^2*e^2 - 32*B*a^3*b

^7*d*e^3 - 12*A*a^2*b^8*d*e^3 + 7*B*a^4*b^6*e^4 + 4*A*a^3*b^7*e^4)*e^(-12)*log(abs(x*e + d)) + 1/12*(3*B*b^10*

x^4*e^24 - 32*B*b^10*d*x^3*e^23 + 216*B*b^10*d^2*x^2*e^22 - 1440*B*b^10*d^3*x*e^21 + 40*B*a*b^9*x^3*e^24 + 4*A

*b^10*x^3*e^24 - 480*B*a*b^9*d*x^2*e^23 - 48*A*b^10*d*x^2*e^23 + 4320*B*a*b^9*d^2*x*e^22 + 432*A*b^10*d^2*x*e^

22 + 270*B*a^2*b^8*x^2*e^24 + 60*A*a*b^9*x^2*e^24 - 4320*B*a^2*b^8*d*x*e^23 - 960*A*a*b^9*d*x*e^23 + 1440*B*a^

3*b^7*x*e^24 + 540*A*a^2*b^8*x*e^24)*e^(-32) + 1/84*(25961*B*b^10*d^11 - 110440*B*a*b^9*d^10*e - 11044*A*b^10*

d^10*e + 180846*B*a^2*b^8*d^9*e^2 + 40188*A*a*b^9*d^9*e^2 - 138528*B*a^3*b^7*d^8*e^3 - 51948*A*a^2*b^8*d^8*e^3

+ 45738*B*a^4*b^6*d^7*e^4 + 26136*A*a^3*b^7*d^7*e^4 - 3024*B*a^5*b^5*d^6*e^5 - 2520*A*a^4*b^6*d^6*e^5 - 420*B

*a^6*b^4*d^5*e^6 - 504*A*a^5*b^5*d^5*e^6 - 96*B*a^7*b^3*d^4*e^7 - 168*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 - 8*B*a^9*b*d^2*e^9 - 36*A*a^8*b^2*d^2*e^9 - 2*B*a^10*d*e^10 - 20*A*a^9*b*d*e^10 - 12*

A*a^10*e^11 + 3528*(11*B*b^10*d^5*e^6 - 50*B*a*b^9*d^4*e^7 - 5*A*b^10*d^4*e^7 + 90*B*a^2*b^8*d^3*e^8 + 20*A*a*

b^9*d^3*e^8 - 80*B*a^3*b^7*d^2*e^9 - 30*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 + 1764*(121*B*b^10*d^6*e^5 - 540*B*a*b^9*d^5*e^6 - 54*A*b^10*d^5*e^6 + 945*

B*a^2*b^8*d^4*e^7 + 210*A*a*b^9*d^4*e^7 - 800*B*a^3*b^7*d^3*e^8 - 300*A*a^2*b^8*d^3*e^8 + 315*B*a^4*b^6*d^2*e^

9 + 180*A*a^3*b^7*d^2*e^9 - 36*B*a^5*b^5*d*e^10 - 30*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*(1177*B*b^10*d^7*e^4 - 5180*B*a*b^9*d^6*e^5 - 518*A*b^10*d^6*e^5 + 8883*B*a^2*b^8*d^5*e^6 + 1974*A*a*

b^9*d^5*e^6 - 7280*B*a^3*b^7*d^4*e^7 - 2730*A*a^2*b^8*d^4*e^7 + 2695*B*a^4*b^6*d^3*e^8 + 1540*A*a^3*b^7*d^3*e^

8 - 252*B*a^5*b^5*d^2*e^9 - 210*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 - 8*B*a^7*b^3*e^

11 - 14*A*a^6*b^4*e^11)*x^4 + 105*(5863*B*b^10*d^8*e^3 - 25520*B*a*b^9*d^7*e^4 - 2552*A*b^10*d^7*e^4 + 43092*B

*a^2*b^8*d^6*e^5 + 9576*A*a*b^9*d^6*e^5 - 34496*B*a^3*b^7*d^5*e^6 - 12936*A*a^2*b^8*d^5*e^6 + 12250*B*a^4*b^6*

d^4*e^7 + 7000*A*a^3*b^7*d^4*e^7 - 1008*B*a^5*b^5*d^3*e^8 - 840*A*a^4*b^6*d^3*e^8 - 140*B*a^6*b^4*d^2*e^9 - 16

8*A*a^5*b^5*d^2*e^9 - 32*B*a^7*b^3*d*e^10 - 56*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 +

21*(20669*B*b^10*d^9*e^2 - 89160*B*a*b^9*d^8*e^3 - 8916*A*b^10*d^8*e^3 + 148716*B*a^2*b^8*d^7*e^4 + 33048*A*a*

b^9*d^7*e^4 - 116928*B*a^3*b^7*d^6*e^5 - 43848*A*a^2*b^8*d^6*e^5 + 40278*B*a^4*b^6*d^5*e^6 + 23016*A*a^3*b^7*d

^5*e^6 - 3024*B*a^5*b^5*d^4*e^7 - 2520*A*a^4*b^6*d^4*e^7 - 420*B*a^6*b^4*d^3*e^8 - 504*A*a^5*b^5*d^3*e^8 - 96*

B*a^7*b^3*d^2*e^9 - 168*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 - 8*B*a^9*b*e^11 - 36*A*

a^8*b^2*e^11)*x^2 + 7*(23441*B*b^10*d^10*e - 100360*B*a*b^9*d^9*e^2 - 10036*A*b^10*d^9*e^2 + 165726*B*a^2*b^8*

d^8*e^3 + 36828*A*a*b^9*d^8*e^3 - 128448*B*a^3*b^7*d^7*e^4 - 48168*A*a^2*b^8*d^7*e^4 + 43218*B*a^4*b^6*d^6*e^5

+ 24696*A*a^3*b^7*d^6*e^5 - 3024*B*a^5*b^5*d^5*e^6 - 2520*A*a^4*b^6*d^5*e^6 - 420*B*a^6*b^4*d^4*e^7 - 504*A*a

^5*b^5*d^4*e^7 - 96*B*a^7*b^3*d^3*e^8 - 168*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 -

8*B*a^9*b*d*e^10 - 36*A*a^8*b^2*d*e^10 - 2*B*a^10*e^11 - 20*A*a^9*b*e^11)*x)*e^(-12)/(x*e + d)^7

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Mupad [B]
time = 1.49, size = 2092, normalized size = 4.71 \begin {gather*} x^3\,\left (\frac {A\,b^{10}+10\,B\,a\,b^9}{3\,e^8}-\frac {8\,B\,b^{10}\,d}{3\,e^9}\right )-x\,\left (\frac {28\,d^2\,\left (\frac {A\,b^{10}+10\,B\,a\,b^9}{e^8}-\frac {8\,B\,b^{10}\,d}{e^9}\right )}{e^2}-\frac {8\,d\,\left (\frac {8\,d\,\left (\frac {A\,b^{10}+10\,B\,a\,b^9}{e^8}-\frac {8\,B\,b^{10}\,d}{e^9}\right )}{e}-\frac {5\,a\,b^8\,\left (2\,A\,b+9\,B\,a\right )}{e^8}+\frac {28\,B\,b^{10}\,d^2}{e^{10}}\right )}{e}-\frac {15\,a^2\,b^7\,\left (3\,A\,b+8\,B\,a\right )}{e^8}+\frac {56\,B\,b^{10}\,d^3}{e^{11}}\right )-x^2\,\left (\frac {4\,d\,\left (\frac {A\,b^{10}+10\,B\,a\,b^9}{e^8}-\frac {8\,B\,b^{10}\,d}{e^9}\right )}{e}-\frac {5\,a\,b^8\,\left (2\,A\,b+9\,B\,a\right )}{2\,e^8}+\frac {14\,B\,b^{10}\,d^2}{e^{10}}\right )-\frac {x^4\,\left (40\,B\,a^7\,b^3\,e^{10}+175\,B\,a^6\,b^4\,d\,e^9+70\,A\,a^6\,b^4\,e^{10}+1260\,B\,a^5\,b^5\,d^2\,e^8+210\,A\,a^5\,b^5\,d\,e^9-13475\,B\,a^4\,b^6\,d^3\,e^7+1050\,A\,a^4\,b^6\,d^2\,e^8+36400\,B\,a^3\,b^7\,d^4\,e^6-7700\,A\,a^3\,b^7\,d^3\,e^7-44415\,B\,a^2\,b^8\,d^5\,e^5+13650\,A\,a^2\,b^8\,d^4\,e^6+25900\,B\,a\,b^9\,d^6\,e^4-9870\,A\,a\,b^9\,d^5\,e^5-5885\,B\,b^{10}\,d^7\,e^3+2590\,A\,b^{10}\,d^6\,e^4\right )+x^6\,\left (252\,B\,a^5\,b^5\,e^{10}-1470\,B\,a^4\,b^6\,d\,e^9+210\,A\,a^4\,b^6\,e^{10}+3360\,B\,a^3\,b^7\,d^2\,e^8-840\,A\,a^3\,b^7\,d\,e^9-3780\,B\,a^2\,b^8\,d^3\,e^7+1260\,A\,a^2\,b^8\,d^2\,e^8+2100\,B\,a\,b^9\,d^4\,e^6-840\,A\,a\,b^9\,d^3\,e^7-462\,B\,b^{10}\,d^5\,e^5+210\,A\,b^{10}\,d^4\,e^6\right )+x^3\,\left (\frac {45\,B\,a^8\,b^2\,e^{10}}{4}+40\,B\,a^7\,b^3\,d\,e^9+30\,A\,a^7\,b^3\,e^{10}+175\,B\,a^6\,b^4\,d^2\,e^8+70\,A\,a^6\,b^4\,d\,e^9+1260\,B\,a^5\,b^5\,d^3\,e^7+210\,A\,a^5\,b^5\,d^2\,e^8-\frac {30625\,B\,a^4\,b^6\,d^4\,e^6}{2}+1050\,A\,a^4\,b^6\,d^3\,e^7+43120\,B\,a^3\,b^7\,d^5\,e^5-8750\,A\,a^3\,b^7\,d^4\,e^6-53865\,B\,a^2\,b^8\,d^6\,e^4+16170\,A\,a^2\,b^8\,d^5\,e^5+31900\,B\,a\,b^9\,d^7\,e^3-11970\,A\,a\,b^9\,d^6\,e^4-\frac {29315\,B\,b^{10}\,d^8\,e^2}{4}+3190\,A\,b^{10}\,d^7\,e^3\right )+\frac {2\,B\,a^{10}\,d\,e^{10}+12\,A\,a^{10}\,e^{11}+8\,B\,a^9\,b\,d^2\,e^9+20\,A\,a^9\,b\,d\,e^{10}+27\,B\,a^8\,b^2\,d^3\,e^8+36\,A\,a^8\,b^2\,d^2\,e^9+96\,B\,a^7\,b^3\,d^4\,e^7+72\,A\,a^7\,b^3\,d^3\,e^8+420\,B\,a^6\,b^4\,d^5\,e^6+168\,A\,a^6\,b^4\,d^4\,e^7+3024\,B\,a^5\,b^5\,d^6\,e^5+504\,A\,a^5\,b^5\,d^5\,e^6-45738\,B\,a^4\,b^6\,d^7\,e^4+2520\,A\,a^4\,b^6\,d^6\,e^5+138528\,B\,a^3\,b^7\,d^8\,e^3-26136\,A\,a^3\,b^7\,d^7\,e^4-180846\,B\,a^2\,b^8\,d^9\,e^2+51948\,A\,a^2\,b^8\,d^8\,e^3+110440\,B\,a\,b^9\,d^{10}\,e-40188\,A\,a\,b^9\,d^9\,e^2-25961\,B\,b^{10}\,d^{11}+11044\,A\,b^{10}\,d^{10}\,e}{84\,e}+x\,\left (\frac {B\,a^{10}\,e^{10}}{6}+\frac {2\,B\,a^9\,b\,d\,e^9}{3}+\frac {5\,A\,a^9\,b\,e^{10}}{3}+\frac {9\,B\,a^8\,b^2\,d^2\,e^8}{4}+3\,A\,a^8\,b^2\,d\,e^9+8\,B\,a^7\,b^3\,d^3\,e^7+6\,A\,a^7\,b^3\,d^2\,e^8+35\,B\,a^6\,b^4\,d^4\,e^6+14\,A\,a^6\,b^4\,d^3\,e^7+252\,B\,a^5\,b^5\,d^5\,e^5+42\,A\,a^5\,b^5\,d^4\,e^6-\frac {7203\,B\,a^4\,b^6\,d^6\,e^4}{2}+210\,A\,a^4\,b^6\,d^5\,e^5+10704\,B\,a^3\,b^7\,d^7\,e^3-2058\,A\,a^3\,b^7\,d^6\,e^4-\frac {27621\,B\,a^2\,b^8\,d^8\,e^2}{2}+4014\,A\,a^2\,b^8\,d^7\,e^3+\frac {25090\,B\,a\,b^9\,d^9\,e}{3}-3069\,A\,a\,b^9\,d^8\,e^2-\frac {23441\,B\,b^{10}\,d^{10}}{12}+\frac {2509\,A\,b^{10}\,d^9\,e}{3}\right )+x^5\,\left (105\,B\,a^6\,b^4\,e^{10}+756\,B\,a^5\,b^5\,d\,e^9+126\,A\,a^5\,b^5\,e^{10}-6615\,B\,a^4\,b^6\,d^2\,e^8+630\,A\,a^4\,b^6\,d\,e^9+16800\,B\,a^3\,b^7\,d^3\,e^7-3780\,A\,a^3\,b^7\,d^2\,e^8-19845\,B\,a^2\,b^8\,d^4\,e^6+6300\,A\,a^2\,b^8\,d^3\,e^7+11340\,B\,a\,b^9\,d^5\,e^5-4410\,A\,a\,b^9\,d^4\,e^6-2541\,B\,b^{10}\,d^6\,e^4+1134\,A\,b^{10}\,d^5\,e^5\right )+x^2\,\left (2\,B\,a^9\,b\,e^{10}+\frac {27\,B\,a^8\,b^2\,d\,e^9}{4}+9\,A\,a^8\,b^2\,e^{10}+24\,B\,a^7\,b^3\,d^2\,e^8+18\,A\,a^7\,b^3\,d\,e^9+105\,B\,a^6\,b^4\,d^3\,e^7+42\,A\,a^6\,b^4\,d^2\,e^8+756\,B\,a^5\,b^5\,d^4\,e^6+126\,A\,a^5\,b^5\,d^3\,e^7-\frac {20139\,B\,a^4\,b^6\,d^5\,e^5}{2}+630\,A\,a^4\,b^6\,d^4\,e^6+29232\,B\,a^3\,b^7\,d^6\,e^4-5754\,A\,a^3\,b^7\,d^5\,e^5-37179\,B\,a^2\,b^8\,d^7\,e^3+10962\,A\,a^2\,b^8\,d^6\,e^4+22290\,B\,a\,b^9\,d^8\,e^2-8262\,A\,a\,b^9\,d^7\,e^3-\frac {20669\,B\,b^{10}\,d^9\,e}{4}+2229\,A\,b^{10}\,d^8\,e^2\right )}{d^7\,e^{11}+7\,d^6\,e^{12}\,x+21\,d^5\,e^{13}\,x^2+35\,d^4\,e^{14}\,x^3+35\,d^3\,e^{15}\,x^4+21\,d^2\,e^{16}\,x^5+7\,d\,e^{17}\,x^6+e^{18}\,x^7}+\frac {\ln \left (d+e\,x\right )\,\left (210\,B\,a^4\,b^6\,e^4-960\,B\,a^3\,b^7\,d\,e^3+120\,A\,a^3\,b^7\,e^4+1620\,B\,a^2\,b^8\,d^2\,e^2-360\,A\,a^2\,b^8\,d\,e^3-1200\,B\,a\,b^9\,d^3\,e+360\,A\,a\,b^9\,d^2\,e^2+330\,B\,b^{10}\,d^4-120\,A\,b^{10}\,d^3\,e\right )}{e^{12}}+\frac {B\,b^{10}\,x^4}{4\,e^8} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^10)/(d + e*x)^8,x)

[Out]

x^3*((A*b^10 + 10*B*a*b^9)/(3*e^8) - (8*B*b^10*d)/(3*e^9)) - x*((28*d^2*((A*b^10 + 10*B*a*b^9)/e^8 - (8*B*b^10

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

(28*B*b^10*d^2)/e^10))/e - (15*a^2*b^7*(3*A*b + 8*B*a))/e^8 + (56*B*b^10*d^3)/e^11) - x^2*((4*d*((A*b^10 + 10

*B*a*b^9)/e^8 - (8*B*b^10*d)/e^9))/e - (5*a*b^8*(2*A*b + 9*B*a))/(2*e^8) + (14*B*b^10*d^2)/e^10) - (x^4*(70*A*

a^6*b^4*e^10 + 40*B*a^7*b^3*e^10 + 2590*A*b^10*d^6*e^4 - 5885*B*b^10*d^7*e^3 - 9870*A*a*b^9*d^5*e^5 + 210*A*a^

5*b^5*d*e^9 + 25900*B*a*b^9*d^6*e^4 + 175*B*a^6*b^4*d*e^9 + 13650*A*a^2*b^8*d^4*e^6 - 7700*A*a^3*b^7*d^3*e^7 +

1050*A*a^4*b^6*d^2*e^8 - 44415*B*a^2*b^8*d^5*e^5 + 36400*B*a^3*b^7*d^4*e^6 - 13475*B*a^4*b^6*d^3*e^7 + 1260*B

*a^5*b^5*d^2*e^8) + x^6*(210*A*a^4*b^6*e^10 + 252*B*a^5*b^5*e^10 + 210*A*b^10*d^4*e^6 - 462*B*b^10*d^5*e^5 - 8

40*A*a*b^9*d^3*e^7 - 840*A*a^3*b^7*d*e^9 + 2100*B*a*b^9*d^4*e^6 - 1470*B*a^4*b^6*d*e^9 + 1260*A*a^2*b^8*d^2*e^

8 - 3780*B*a^2*b^8*d^3*e^7 + 3360*B*a^3*b^7*d^2*e^8) + x^3*(30*A*a^7*b^3*e^10 + (45*B*a^8*b^2*e^10)/4 + 3190*A

*b^10*d^7*e^3 - (29315*B*b^10*d^8*e^2)/4 - 11970*A*a*b^9*d^6*e^4 + 70*A*a^6*b^4*d*e^9 + 31900*B*a*b^9*d^7*e^3

+ 40*B*a^7*b^3*d*e^9 + 16170*A*a^2*b^8*d^5*e^5 - 8750*A*a^3*b^7*d^4*e^6 + 1050*A*a^4*b^6*d^3*e^7 + 210*A*a^5*b

^5*d^2*e^8 - 53865*B*a^2*b^8*d^6*e^4 + 43120*B*a^3*b^7*d^5*e^5 - (30625*B*a^4*b^6*d^4*e^6)/2 + 1260*B*a^5*b^5*

d^3*e^7 + 175*B*a^6*b^4*d^2*e^8) + (12*A*a^10*e^11 - 25961*B*b^10*d^11 + 11044*A*b^10*d^10*e + 2*B*a^10*d*e^10

- 40188*A*a*b^9*d^9*e^2 + 8*B*a^9*b*d^2*e^9 + 51948*A*a^2*b^8*d^8*e^3 - 26136*A*a^3*b^7*d^7*e^4 + 2520*A*a^4*

b^6*d^6*e^5 + 504*A*a^5*b^5*d^5*e^6 + 168*A*a^6*b^4*d^4*e^7 + 72*A*a^7*b^3*d^3*e^8 + 36*A*a^8*b^2*d^2*e^9 - 18

0846*B*a^2*b^8*d^9*e^2 + 138528*B*a^3*b^7*d^8*e^3 - 45738*B*a^4*b^6*d^7*e^4 + 3024*B*a^5*b^5*d^6*e^5 + 420*B*a

^6*b^4*d^5*e^6 + 96*B*a^7*b^3*d^4*e^7 + 27*B*a^8*b^2*d^3*e^8 + 20*A*a^9*b*d*e^10 + 110440*B*a*b^9*d^10*e)/(84*

e) + x*((B*a^10*e^10)/6 - (23441*B*b^10*d^10)/12 + (5*A*a^9*b*e^10)/3 + (2509*A*b^10*d^9*e)/3 - 3069*A*a*b^9*d

^8*e^2 + 3*A*a^8*b^2*d*e^9 + 4014*A*a^2*b^8*d^7*e^3 - 2058*A*a^3*b^7*d^6*e^4 + 210*A*a^4*b^6*d^5*e^5 + 42*A*a^

5*b^5*d^4*e^6 + 14*A*a^6*b^4*d^3*e^7 + 6*A*a^7*b^3*d^2*e^8 - (27621*B*a^2*b^8*d^8*e^2)/2 + 10704*B*a^3*b^7*d^7

*e^3 - (7203*B*a^4*b^6*d^6*e^4)/2 + 252*B*a^5*b^5*d^5*e^5 + 35*B*a^6*b^4*d^4*e^6 + 8*B*a^7*b^3*d^3*e^7 + (9*B*

a^8*b^2*d^2*e^8)/4 + (25090*B*a*b^9*d^9*e)/3 + (2*B*a^9*b*d*e^9)/3) + x^5*(126*A*a^5*b^5*e^10 + 105*B*a^6*b^4*

e^10 + 1134*A*b^10*d^5*e^5 - 2541*B*b^10*d^6*e^4 - 4410*A*a*b^9*d^4*e^6 + 630*A*a^4*b^6*d*e^9 + 11340*B*a*b^9*

d^5*e^5 + 756*B*a^5*b^5*d*e^9 + 6300*A*a^2*b^8*d^3*e^7 - 3780*A*a^3*b^7*d^2*e^8 - 19845*B*a^2*b^8*d^4*e^6 + 16

800*B*a^3*b^7*d^3*e^7 - 6615*B*a^4*b^6*d^2*e^8) + x^2*(2*B*a^9*b*e^10 - (20669*B*b^10*d^9*e)/4 + 9*A*a^8*b^2*e

^10 + 2229*A*b^10*d^8*e^2 - 8262*A*a*b^9*d^7*e^3 + 18*A*a^7*b^3*d*e^9 + 22290*B*a*b^9*d^8*e^2 + (27*B*a^8*b^2*

d*e^9)/4 + 10962*A*a^2*b^8*d^6*e^4 - 5754*A*a^3*b^7*d^5*e^5 + 630*A*a^4*b^6*d^4*e^6 + 126*A*a^5*b^5*d^3*e^7 +

42*A*a^6*b^4*d^2*e^8 - 37179*B*a^2*b^8*d^7*e^3 + 29232*B*a^3*b^7*d^6*e^4 - (20139*B*a^4*b^6*d^5*e^5)/2 + 756*B

*a^5*b^5*d^4*e^6 + 105*B*a^6*b^4*d^3*e^7 + 24*B*a^7*b^3*d^2*e^8))/(d^7*e^11 + e^18*x^7 + 7*d^6*e^12*x + 7*d*e^

17*x^6 + 21*d^5*e^13*x^2 + 35*d^4*e^14*x^3 + 35*d^3*e^15*x^4 + 21*d^2*e^16*x^5) + (log(d + e*x)*(330*B*b^10*d^

4 - 120*A*b^10*d^3*e + 120*A*a^3*b^7*e^4 + 210*B*a^4*b^6*e^4 + 360*A*a*b^9*d^2*e^2 - 360*A*a^2*b^8*d*e^3 - 960

*B*a^3*b^7*d*e^3 + 1620*B*a^2*b^8*d^2*e^2 - 1200*B*a*b^9*d^3*e))/e^12 + (B*b^10*x^4)/(4*e^8)

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