3.11.93 \(\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^5} \, dx\) [1093]

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

[Out]

42*b^4*(-a*e+b*d)^5*(-6*A*b*e-5*B*a*e+11*B*b*d)*x/e^11+1/4*(-a*e+b*d)^10*(-A*e+B*d)/e^12/(e*x+d)^4-1/3*(-a*e+b
*d)^9*(-10*A*b*e-B*a*e+11*B*b*d)/e^12/(e*x+d)^3+5/2*b*(-a*e+b*d)^8*(-9*A*b*e-2*B*a*e+11*B*b*d)/e^12/(e*x+d)^2-
15*b^2*(-a*e+b*d)^7*(-8*A*b*e-3*B*a*e+11*B*b*d)/e^12/(e*x+d)-21*b^5*(-a*e+b*d)^4*(-5*A*b*e-6*B*a*e+11*B*b*d)*(
e*x+d)^2/e^12+10*b^6*(-a*e+b*d)^3*(-4*A*b*e-7*B*a*e+11*B*b*d)*(e*x+d)^3/e^12-15/4*b^7*(-a*e+b*d)^2*(-3*A*b*e-8
*B*a*e+11*B*b*d)*(e*x+d)^4/e^12+b^8*(-a*e+b*d)*(-2*A*b*e-9*B*a*e+11*B*b*d)*(e*x+d)^5/e^12-1/6*b^9*(-A*b*e-10*B
*a*e+11*B*b*d)*(e*x+d)^6/e^12+1/7*b^10*B*(e*x+d)^7/e^12-30*b^3*(-a*e+b*d)^6*(-7*A*b*e-4*B*a*e+11*B*b*d)*ln(e*x
+d)/e^12

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Rubi [A]
time = 0.86, 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)^6 (-10 a B e-A b e+11 b B d)}{6 e^{12}}+\frac {b^8 (d+e x)^5 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{e^{12}}-\frac {15 b^7 (d+e x)^4 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{4 e^{12}}+\frac {10 b^6 (d+e x)^3 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{e^{12}}-\frac {21 b^5 (d+e x)^2 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{e^{12}}+\frac {42 b^4 x (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{e^{11}}-\frac {30 b^3 (b d-a e)^6 \log (d+e x) (-4 a B e-7 A b e+11 b B d)}{e^{12}}-\frac {15 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{e^{12} (d+e x)}+\frac {5 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{2 e^{12} (d+e x)^2}-\frac {(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{3 e^{12} (d+e x)^3}+\frac {(b d-a e)^{10} (B d-A e)}{4 e^{12} (d+e x)^4}+\frac {b^{10} B (d+e x)^7}{7 e^{12}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^10*(A + B*x))/(d + e*x)^5,x]

[Out]

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

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Mathematica [A]
time = 0.32, size = 686, normalized size = 1.55 \begin {gather*} \frac {-84 b^4 e \left (-210 a^6 B e^6+140 a b^5 d^4 e (9 B d-5 A e)-42 b^6 d^5 (5 B d-3 A e)+600 a^3 b^3 d^2 e^3 (7 B d-3 A e)-1575 a^2 b^4 d^3 e^2 (2 B d-A e)-1050 a^4 b^2 d e^4 (3 B d-A e)-252 a^5 b e^5 (-5 B d+A e)\right ) x+42 b^5 e^2 \left (252 a^5 B e^5-14 b^5 d^4 (9 B d-5 A e)-225 a^2 b^3 d^2 e^2 (7 B d-3 A e)+350 a b^4 d^3 e (2 B d-A e)+600 a^3 b^2 d e^3 (3 B d-A e)+210 a^4 b e^4 (-5 B d+A e)\right ) x^2-140 b^6 e^3 \left (-42 a^4 B e^4+10 a b^3 d^2 e (7 B d-3 A e)-45 a^2 b^2 d e^2 (3 B d-A e)-24 a^3 b e^3 (-5 B d+A e)+7 b^4 d^3 (-2 B d+A e)\right ) x^3+105 b^7 e^4 \left (24 a^3 B e^3+10 a b^2 d e (3 B d-A e)+9 a^2 b e^2 (-5 B d+A e)+b^3 d^2 (-7 B d+3 A e)\right ) x^4-84 b^8 e^5 \left (-9 a^2 B e^2-2 a b e (-5 B d+A e)+b^2 d (-3 B d+A e)\right ) x^5+14 b^9 e^6 (-5 b B d+A b e+10 a B e) x^6+12 b^{10} B e^7 x^7+\frac {21 (b d-a e)^{10} (B d-A e)}{(d+e x)^4}-\frac {28 (b d-a e)^9 (11 b B d-10 A b e-a B e)}{(d+e x)^3}+\frac {210 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{(d+e x)^2}-\frac {1260 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e)}{d+e x}-2520 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 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)^5,x]

[Out]

(-84*b^4*e*(-210*a^6*B*e^6 + 140*a*b^5*d^4*e*(9*B*d - 5*A*e) - 42*b^6*d^5*(5*B*d - 3*A*e) + 600*a^3*b^3*d^2*e^
3*(7*B*d - 3*A*e) - 1575*a^2*b^4*d^3*e^2*(2*B*d - A*e) - 1050*a^4*b^2*d*e^4*(3*B*d - A*e) - 252*a^5*b*e^5*(-5*
B*d + A*e))*x + 42*b^5*e^2*(252*a^5*B*e^5 - 14*b^5*d^4*(9*B*d - 5*A*e) - 225*a^2*b^3*d^2*e^2*(7*B*d - 3*A*e) +
350*a*b^4*d^3*e*(2*B*d - A*e) + 600*a^3*b^2*d*e^3*(3*B*d - A*e) + 210*a^4*b*e^4*(-5*B*d + A*e))*x^2 - 140*b^6
*e^3*(-42*a^4*B*e^4 + 10*a*b^3*d^2*e*(7*B*d - 3*A*e) - 45*a^2*b^2*d*e^2*(3*B*d - A*e) - 24*a^3*b*e^3*(-5*B*d +
A*e) + 7*b^4*d^3*(-2*B*d + A*e))*x^3 + 105*b^7*e^4*(24*a^3*B*e^3 + 10*a*b^2*d*e*(3*B*d - A*e) + 9*a^2*b*e^2*(
-5*B*d + A*e) + b^3*d^2*(-7*B*d + 3*A*e))*x^4 - 84*b^8*e^5*(-9*a^2*B*e^2 - 2*a*b*e*(-5*B*d + A*e) + b^2*d*(-3*
B*d + A*e))*x^5 + 14*b^9*e^6*(-5*b*B*d + A*b*e + 10*a*B*e)*x^6 + 12*b^10*B*e^7*x^7 + (21*(b*d - a*e)^10*(B*d -
A*e))/(d + e*x)^4 - (28*(b*d - a*e)^9*(11*b*B*d - 10*A*b*e - a*B*e))/(d + e*x)^3 + (210*b*(b*d - a*e)^8*(11*b
*B*d - 9*A*b*e - 2*a*B*e))/(d + e*x)^2 - (1260*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b*e - 3*a*B*e))/(d + e*x) - 2
520*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b*e - 4*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. \(2033\) vs. \(2(432)=864\).
time = 0.11, size = 2034, normalized size = 4.58

method	result	size
norman	\(\text {Expression too large to display}\)	\(1910\)
default	\(\text {Expression too large to display}\)	\(2034\)
risch	\(\text {Expression too large to display}\)	\(2127\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10*(B*x+A)/(e*x+d)^5,x,method=_RETURNVERBOSE)

[Out]

b^4/e^11*(5/3*B*a*b^5*e^6*x^6-200*B*a^3*b^3*d*e^5*x^3+225*B*a^2*b^4*d^2*e^4*x^3-350/3*B*a*b^5*d^3*e^3*x^3-300*
A*a^3*b^3*d*e^5*x^2+675/2*A*a^2*b^4*d^2*e^4*x^2-175*A*a*b^5*d^3*e^3*x^2-525*B*a^4*b^2*d*e^5*x^2+900*B*a^3*b^3*
d^2*e^4*x^2-1575/2*B*a^2*b^4*d^3*e^3*x^2+350*B*a*b^5*d^4*e^2*x^2-10*B*a*b^5*d*e^5*x^5-25/2*A*a*b^5*d*e^5*x^4-2
25/4*B*a^2*b^4*d*e^5*x^4+75/2*B*a*b^5*d^2*e^4*x^4-75*A*a^2*b^4*d*e^5*x^3+50*A*a*b^5*d^2*e^4*x^3+3150*B*a^2*b^4
*d^4*e^2*x-1260*B*a*b^5*d^5*e*x-1050*A*a^4*b^2*d*e^5*x+1800*A*a^3*b^3*d^2*e^4*x-1575*A*a^2*b^4*d^3*e^3*x+700*A
*a*b^5*d^4*e^2*x-1260*B*a^5*b*d*e^5*x+3150*B*a^4*b^2*d^2*e^4*x-4200*B*a^3*b^3*d^3*e^3*x+1/7*b^6*B*x^7*e^6+1/6*
A*b^6*e^6*x^6+210*B*a^6*e^6*x+210*B*b^6*d^6*x+3*B*b^6*d^2*e^4*x^5+45/4*A*a^2*b^4*e^6*x^4+15/4*A*b^6*d^2*e^4*x^
4+30*B*a^3*b^3*e^6*x^4-5/6*B*b^6*d*e^5*x^6+2*A*a*b^5*e^6*x^5-A*b^6*d*e^5*x^5+9*B*a^2*b^4*e^6*x^5+252*A*a^5*b*e
^6*x-126*A*b^6*d^5*e*x-35/4*B*b^6*d^3*e^3*x^4+40*A*a^3*b^3*e^6*x^3-35/3*A*b^6*d^3*e^3*x^3+70*B*a^4*b^2*e^6*x^3
+70/3*B*b^6*d^4*e^2*x^3+105*A*a^4*b^2*e^6*x^2+35*A*b^6*d^4*e^2*x^2+126*B*a^5*b*e^6*x^2-63*B*b^6*d^5*e*x^2)-5/2
*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^5-50
4*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)^2-1/4*(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)^4-15*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)-1/3/e^12*(10*A*a^9*b*e^10-9
0*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+84
0*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^4-9
60*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)^3+30*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)*ln(e*x+d)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1880 vs. \(2 (462) = 924\).
time = 0.52, size = 1880, normalized size = 4.23 \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)^5,x, algorithm="maxima")

[Out]

-30*(11*B*b^10*d^7 - 4*B*a^7*b^3*e^7 - 7*A*a^6*b^4*e^7 - 7*(10*B*a*b^9*e + A*b^10*e)*d^6 + 21*(9*B*a^2*b^8*e^2
+ 2*A*a*b^9*e^2)*d^5 - 35*(8*B*a^3*b^7*e^3 + 3*A*a^2*b^8*e^3)*d^4 + 35*(7*B*a^4*b^6*e^4 + 4*A*a^3*b^7*e^4)*d^
3 - 21*(6*B*a^5*b^5*e^5 + 5*A*a^4*b^6*e^5)*d^2 + 7*(5*B*a^6*b^4*e^6 + 6*A*a^5*b^5*e^6)*d)*e^(-12)*log(x*e + d)
+ 1/84*(12*B*b^10*x^7*e^6 - 14*(5*B*b^10*d*e^5 - 10*B*a*b^9*e^6 - A*b^10*e^6)*x^6 + 84*(3*B*b^10*d^2*e^4 + 9*
B*a^2*b^8*e^6 + 2*A*a*b^9*e^6 - (10*B*a*b^9*e^5 + A*b^10*e^5)*d)*x^5 - 105*(7*B*b^10*d^3*e^3 - 24*B*a^3*b^7*e^
6 - 9*A*a^2*b^8*e^6 - 3*(10*B*a*b^9*e^4 + A*b^10*e^4)*d^2 + 5*(9*B*a^2*b^8*e^5 + 2*A*a*b^9*e^5)*d)*x^4 + 140*(
14*B*b^10*d^4*e^2 + 42*B*a^4*b^6*e^6 + 24*A*a^3*b^7*e^6 - 7*(10*B*a*b^9*e^3 + A*b^10*e^3)*d^3 + 15*(9*B*a^2*b^
8*e^4 + 2*A*a*b^9*e^4)*d^2 - 15*(8*B*a^3*b^7*e^5 + 3*A*a^2*b^8*e^5)*d)*x^3 - 42*(126*B*b^10*d^5*e - 252*B*a^5*
b^5*e^6 - 210*A*a^4*b^6*e^6 - 70*(10*B*a*b^9*e^2 + A*b^10*e^2)*d^4 + 175*(9*B*a^2*b^8*e^3 + 2*A*a*b^9*e^3)*d^3
- 225*(8*B*a^3*b^7*e^4 + 3*A*a^2*b^8*e^4)*d^2 + 150*(7*B*a^4*b^6*e^5 + 4*A*a^3*b^7*e^5)*d)*x^2 + 84*(210*B*b^
10*d^6 + 210*B*a^6*b^4*e^6 + 252*A*a^5*b^5*e^6 - 126*(10*B*a*b^9*e + A*b^10*e)*d^5 + 350*(9*B*a^2*b^8*e^2 + 2*
A*a*b^9*e^2)*d^4 - 525*(8*B*a^3*b^7*e^3 + 3*A*a^2*b^8*e^3)*d^3 + 450*(7*B*a^4*b^6*e^4 + 4*A*a^3*b^7*e^4)*d^2 -
210*(6*B*a^5*b^5*e^5 + 5*A*a^4*b^6*e^5)*d)*x)*e^(-11) - 1/12*(1691*B*b^10*d^11 + 3*A*a^10*e^11 - 1207*(10*B*a
*b^9*e + A*b^10*e)*d^10 + 4125*(9*B*a^2*b^8*e^2 + 2*A*a*b^9*e^2)*d^9 - 7995*(8*B*a^3*b^7*e^3 + 3*A*a^2*b^8*e^3
)*d^8 + 9570*(7*B*a^4*b^6*e^4 + 4*A*a^3*b^7*e^4)*d^7 - 7182*(6*B*a^5*b^5*e^5 + 5*A*a^4*b^6*e^5)*d^6 + 3234*(5*
B*a^6*b^4*e^6 + 6*A*a^5*b^5*e^6)*d^5 - 750*(4*B*a^7*b^3*e^7 + 7*A*a^6*b^4*e^7)*d^4 + 45*(3*B*a^8*b^2*e^8 + 8*A
*a^7*b^3*e^8)*d^3 + 180*(11*B*b^10*d^8*e^3 + 3*B*a^8*b^2*e^11 + 8*A*a^7*b^3*e^11 - 8*(10*B*a*b^9*e^4 + A*b^10*
e^4)*d^7 + 28*(9*B*a^2*b^8*e^5 + 2*A*a*b^9*e^5)*d^6 - 56*(8*B*a^3*b^7*e^6 + 3*A*a^2*b^8*e^6)*d^5 + 70*(7*B*a^4
*b^6*e^7 + 4*A*a^3*b^7*e^7)*d^4 - 56*(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 + 5*(2*B*a^9*b*e^9 + 9*A*a^8*b^2*e^9)*d^2 + 30*(1
87*B*b^10*d^9*e^2 + 2*B*a^9*b*e^11 + 9*A*a^8*b^2*e^11 - 135*(10*B*a*b^9*e^3 + A*b^10*e^3)*d^8 + 468*(9*B*a^2*b
^8*e^4 + 2*A*a*b^9*e^4)*d^7 - 924*(8*B*a^3*b^7*e^5 + 3*A*a^2*b^8*e^5)*d^6 + 1134*(7*B*a^4*b^6*e^6 + 4*A*a^3*b^
7*e^6)*d^5 - 882*(6*B*a^5*b^5*e^7 + 5*A*a^4*b^6*e^7)*d^4 + 420*(5*B*a^6*b^4*e^8 + 6*A*a^5*b^5*e^8)*d^3 - 108*(
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 + (B*a^10*e^10 + 10*A*
a^9*b*e^10)*d + 4*(1331*B*b^10*d^10*e + B*a^10*e^11 + 10*A*a^9*b*e^11 - 955*(10*B*a*b^9*e^2 + A*b^10*e^2)*d^9
+ 3285*(9*B*a^2*b^8*e^3 + 2*A*a*b^9*e^3)*d^8 - 6420*(8*B*a^3*b^7*e^4 + 3*A*a^2*b^8*e^4)*d^7 + 7770*(7*B*a^4*b^
6*e^5 + 4*A*a^3*b^7*e^5)*d^6 - 5922*(6*B*a^5*b^5*e^6 + 5*A*a^4*b^6*e^6)*d^5 + 2730*(5*B*a^6*b^4*e^7 + 6*A*a^5*
b^5*e^7)*d^4 - 660*(4*B*a^7*b^3*e^8 + 7*A*a^6*b^4*e^8)*d^3 + 45*(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^4*e^16 + 4*d*x^3*e^15 + 6*d^2*x^2*e^14 + 4*d^3*x*e^13 + d^4*e^12)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2736 vs. \(2 (462) = 924\).
time = 1.16, size = 2736, normalized size = 6.16 \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)^5,x, algorithm="fricas")

[Out]

-1/84*(11837*B*b^10*d^11 - (12*B*b^10*x^11 - 21*A*a^10 + 14*(10*B*a*b^9 + A*b^10)*x^10 + 84*(9*B*a^2*b^8 + 2*A
*a*b^9)*x^9 + 315*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 840*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 1764*(6*B*a^5*b^5 +
5*A*a^4*b^6)*x^6 + 3528*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 - 1260*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 - 210*(2*B*a^9*
b + 9*A*a^8*b^2)*x^2 - 28*(B*a^10 + 10*A*a^9*b)*x)*e^11 + (22*B*b^10*d*x^10 + 28*(10*B*a*b^9 + A*b^10)*d*x^9 +
189*(9*B*a^2*b^8 + 2*A*a*b^9)*d*x^8 + 840*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*x^7 + 2940*(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 - 14112*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*x^4 - 10080*(4*B*a^7*b
^3 + 7*A*a^6*b^4)*d*x^3 + 1890*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*x^2 + 140*(2*B*a^9*b + 9*A*a^8*b^2)*d*x + 7*(B*a^
10 + 10*A*a^9*b)*d)*e^10 - (44*B*b^10*d^2*x^9 + 63*(10*B*a*b^9 + A*b^10)*d^2*x^8 + 504*(9*B*a^2*b^8 + 2*A*a*b^
9)*d^2*x^7 + 2940*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*x^6 + 17640*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*x^5 - 59976*(6*B
*a^5*b^5 + 5*A*a^4*b^6)*d^2*x^4 - 14112*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*x^3 + 22680*(4*B*a^7*b^3 + 7*A*a^6*b^4
)*d^2*x^2 - 1260*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*x - 35*(2*B*a^9*b + 9*A*a^8*b^2)*d^2)*e^9 + 3*(33*B*b^10*d^3*
x^8 + 56*(10*B*a*b^9 + A*b^10)*d^3*x^7 + 588*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*x^6 + 5880*(8*B*a^3*b^7 + 3*A*a^2*b
^8)*d^3*x^5 - 38920*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*x^4 + 9408*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*x^3 + 24696*(5*
B*a^6*b^4 + 6*A*a^5*b^5)*d^3*x^2 - 6160*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*x + 105*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^
3)*e^8 - 3*(88*B*b^10*d^4*x^7 + 196*(10*B*a*b^9 + A*b^10)*d^4*x^6 + 3528*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*x^5 - 4
2595*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*x^4 + 38080*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*x^3 + 38808*(6*B*a^5*b^5 + 5*
A*a^4*b^6)*d^4*x^2 - 24304*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*x + 1750*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4)*e^7 + 21*
(44*B*b^10*d^5*x^6 + 168*(10*B*a*b^9 + A*b^10)*d^5*x^5 - 3875*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*x^4 + 7540*(8*B*a^
3*b^7 + 3*A*a^2*b^8)*d^5*x^3 + 4440*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*x^2 - 7056*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5
*x + 1078*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5)*e^6 - 7*(792*B*b^10*d^6*x^5 - 4043*(10*B*a*b^9 + A*b^10)*d^6*x^4 +
16260*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*x^3 + 3870*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*x^2 - 25680*(7*B*a^4*b^6 + 4*A*
a^3*b^7)*d^6*x + 7182*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6)*e^5 - 7*(6559*B*b^10*d^7*x^4 - 6092*(10*B*a*b^9 + A*b^1
0)*d^7*x^3 + 1710*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*x^2 + 19380*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*x - 9570*(7*B*a^4*
b^6 + 4*A*a^3*b^7)*d^7)*e^4 - 7*(10396*B*b^10*d^8*x^3 - 1578*(10*B*a*b^9 + A*b^10)*d^8*x^2 - 8940*(9*B*a^2*b^8
+ 2*A*a*b^9)*d^8*x + 7995*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8)*e^3 - 7*(3714*B*b^10*d^9*x^2 + 2308*(10*B*a*b^9 +
A*b^10)*d^9*x - 4125*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9)*e^2 + 7*(2804*B*b^10*d^10*x - 1207*(10*B*a*b^9 + A*b^10)*d
^10)*e + 2520*(11*B*b^10*d^11 - (4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4*e^11 + (7*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*x^4 -
4*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*x^3)*e^10 - (21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*x^4 - 28*(5*B*a^6*b^4 + 6*A*a^
5*b^5)*d^2*x^3 + 6*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*x^2)*e^9 + (35*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*x^4 - 84*(6*
B*a^5*b^5 + 5*A*a^4*b^6)*d^3*x^3 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*x^2 - 4*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*
x)*e^8 - (35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*x^4 - 140*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*x^3 + 126*(6*B*a^5*b^5
+ 5*A*a^4*b^6)*d^4*x^2 - 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*x + (4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4)*e^7 + 7*(3*(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 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*
x^2 - 12*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*x + (5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5)*e^6 - 7*((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 + 30*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*x^2 - 20*(7*B*a^4*b^6 + 4*A
*a^3*b^7)*d^6*x + 3*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6)*e^5 + (11*B*b^10*d^7*x^4 - 28*(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 - 140*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*x + 35*(7*B*a^4*b^6 + 4*A*a^3
*b^7)*d^7)*e^4 + (44*B*b^10*d^8*x^3 - 42*(10*B*a*b^9 + A*b^10)*d^8*x^2 + 84*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*x -
35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8)*e^3 + (66*B*b^10*d^9*x^2 - 28*(10*B*a*b^9 + A*b^10)*d^9*x + 21*(9*B*a^2*b^
8 + 2*A*a*b^9)*d^9)*e^2 + (44*B*b^10*d^10*x - 7*(10*B*a*b^9 + A*b^10)*d^10)*e)*log(x*e + d))/(x^4*e^16 + 4*d*x
^3*e^15 + 6*d^2*x^2*e^14 + 4*d^3*x*e^13 + d^4*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)**5,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2513 vs. \(2 (462) = 924\).
time = 1.69, size = 2513, normalized size = 5.66 \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)^5,x, algorithm="giac")

[Out]

1/84*(12*B*b^10 - 14*(11*B*b^10*d*e - 10*B*a*b^9*e^2 - A*b^10*e^2)*e^(-1)/(x*e + d) + 84*(11*B*b^10*d^2*e^2 -
20*B*a*b^9*d*e^3 - 2*A*b^10*d*e^3 + 9*B*a^2*b^8*e^4 + 2*A*a*b^9*e^4)*e^(-2)/(x*e + d)^2 - 315*(11*B*b^10*d^3*e
^3 - 30*B*a*b^9*d^2*e^4 - 3*A*b^10*d^2*e^4 + 27*B*a^2*b^8*d*e^5 + 6*A*a*b^9*d*e^5 - 8*B*a^3*b^7*e^6 - 3*A*a^2*
b^8*e^6)*e^(-3)/(x*e + d)^3 + 840*(11*B*b^10*d^4*e^4 - 40*B*a*b^9*d^3*e^5 - 4*A*b^10*d^3*e^5 + 54*B*a^2*b^8*d^
2*e^6 + 12*A*a*b^9*d^2*e^6 - 32*B*a^3*b^7*d*e^7 - 12*A*a^2*b^8*d*e^7 + 7*B*a^4*b^6*e^8 + 4*A*a^3*b^7*e^8)*e^(-
4)/(x*e + d)^4 - 1764*(11*B*b^10*d^5*e^5 - 50*B*a*b^9*d^4*e^6 - 5*A*b^10*d^4*e^6 + 90*B*a^2*b^8*d^3*e^7 + 20*A
*a*b^9*d^3*e^7 - 80*B*a^3*b^7*d^2*e^8 - 30*A*a^2*b^8*d^2*e^8 + 35*B*a^4*b^6*d*e^9 + 20*A*a^3*b^7*d*e^9 - 6*B*a
^5*b^5*e^10 - 5*A*a^4*b^6*e^10)*e^(-5)/(x*e + d)^5 + 3528*(11*B*b^10*d^6*e^6 - 60*B*a*b^9*d^5*e^7 - 6*A*b^10*d
^5*e^7 + 135*B*a^2*b^8*d^4*e^8 + 30*A*a*b^9*d^4*e^8 - 160*B*a^3*b^7*d^3*e^9 - 60*A*a^2*b^8*d^3*e^9 + 105*B*a^4
*b^6*d^2*e^10 + 60*A*a^3*b^7*d^2*e^10 - 36*B*a^5*b^5*d*e^11 - 30*A*a^4*b^6*d*e^11 + 5*B*a^6*b^4*e^12 + 6*A*a^5
*b^5*e^12)*e^(-6)/(x*e + d)^6)*(x*e + d)^7*e^(-12) + 30*(11*B*b^10*d^7 - 70*B*a*b^9*d^6*e - 7*A*b^10*d^6*e + 1
89*B*a^2*b^8*d^5*e^2 + 42*A*a*b^9*d^5*e^2 - 280*B*a^3*b^7*d^4*e^3 - 105*A*a^2*b^8*d^4*e^3 + 245*B*a^4*b^6*d^3*
e^4 + 140*A*a^3*b^7*d^3*e^4 - 126*B*a^5*b^5*d^2*e^5 - 105*A*a^4*b^6*d^2*e^5 + 35*B*a^6*b^4*d*e^6 + 42*A*a^5*b^
5*d*e^6 - 4*B*a^7*b^3*e^7 - 7*A*a^6*b^4*e^7)*e^(-12)*log(abs(x*e + d)*e^(-1)/(x*e + d)^2) - 1/12*(1980*B*b^10*
d^8*e^64/(x*e + d) - 330*B*b^10*d^9*e^64/(x*e + d)^2 + 44*B*b^10*d^10*e^64/(x*e + d)^3 - 3*B*b^10*d^11*e^64/(x
*e + d)^4 - 14400*B*a*b^9*d^7*e^65/(x*e + d) - 1440*A*b^10*d^7*e^65/(x*e + d) + 2700*B*a*b^9*d^8*e^65/(x*e + d
)^2 + 270*A*b^10*d^8*e^65/(x*e + d)^2 - 400*B*a*b^9*d^9*e^65/(x*e + d)^3 - 40*A*b^10*d^9*e^65/(x*e + d)^3 + 30
*B*a*b^9*d^10*e^65/(x*e + d)^4 + 3*A*b^10*d^10*e^65/(x*e + d)^4 + 45360*B*a^2*b^8*d^6*e^66/(x*e + d) + 10080*A
*a*b^9*d^6*e^66/(x*e + d) - 9720*B*a^2*b^8*d^7*e^66/(x*e + d)^2 - 2160*A*a*b^9*d^7*e^66/(x*e + d)^2 + 1620*B*a
^2*b^8*d^8*e^66/(x*e + d)^3 + 360*A*a*b^9*d^8*e^66/(x*e + d)^3 - 135*B*a^2*b^8*d^9*e^66/(x*e + d)^4 - 30*A*a*b
^9*d^9*e^66/(x*e + d)^4 - 80640*B*a^3*b^7*d^5*e^67/(x*e + d) - 30240*A*a^2*b^8*d^5*e^67/(x*e + d) + 20160*B*a^
3*b^7*d^6*e^67/(x*e + d)^2 + 7560*A*a^2*b^8*d^6*e^67/(x*e + d)^2 - 3840*B*a^3*b^7*d^7*e^67/(x*e + d)^3 - 1440*
A*a^2*b^8*d^7*e^67/(x*e + d)^3 + 360*B*a^3*b^7*d^8*e^67/(x*e + d)^4 + 135*A*a^2*b^8*d^8*e^67/(x*e + d)^4 + 882
00*B*a^4*b^6*d^4*e^68/(x*e + d) + 50400*A*a^3*b^7*d^4*e^68/(x*e + d) - 26460*B*a^4*b^6*d^5*e^68/(x*e + d)^2 -
15120*A*a^3*b^7*d^5*e^68/(x*e + d)^2 + 5880*B*a^4*b^6*d^6*e^68/(x*e + d)^3 + 3360*A*a^3*b^7*d^6*e^68/(x*e + d)
^3 - 630*B*a^4*b^6*d^7*e^68/(x*e + d)^4 - 360*A*a^3*b^7*d^7*e^68/(x*e + d)^4 - 60480*B*a^5*b^5*d^3*e^69/(x*e +
d) - 50400*A*a^4*b^6*d^3*e^69/(x*e + d) + 22680*B*a^5*b^5*d^4*e^69/(x*e + d)^2 + 18900*A*a^4*b^6*d^4*e^69/(x*
e + d)^2 - 6048*B*a^5*b^5*d^5*e^69/(x*e + d)^3 - 5040*A*a^4*b^6*d^5*e^69/(x*e + d)^3 + 756*B*a^5*b^5*d^6*e^69/
(x*e + d)^4 + 630*A*a^4*b^6*d^6*e^69/(x*e + d)^4 + 25200*B*a^6*b^4*d^2*e^70/(x*e + d) + 30240*A*a^5*b^5*d^2*e^
70/(x*e + d) - 12600*B*a^6*b^4*d^3*e^70/(x*e + d)^2 - 15120*A*a^5*b^5*d^3*e^70/(x*e + d)^2 + 4200*B*a^6*b^4*d^
4*e^70/(x*e + d)^3 + 5040*A*a^5*b^5*d^4*e^70/(x*e + d)^3 - 630*B*a^6*b^4*d^5*e^70/(x*e + d)^4 - 756*A*a^5*b^5*
d^5*e^70/(x*e + d)^4 - 5760*B*a^7*b^3*d*e^71/(x*e + d) - 10080*A*a^6*b^4*d*e^71/(x*e + d) + 4320*B*a^7*b^3*d^2
*e^71/(x*e + d)^2 + 7560*A*a^6*b^4*d^2*e^71/(x*e + d)^2 - 1920*B*a^7*b^3*d^3*e^71/(x*e + d)^3 - 3360*A*a^6*b^4
*d^3*e^71/(x*e + d)^3 + 360*B*a^7*b^3*d^4*e^71/(x*e + d)^4 + 630*A*a^6*b^4*d^4*e^71/(x*e + d)^4 + 540*B*a^8*b^
2*e^72/(x*e + d) + 1440*A*a^7*b^3*e^72/(x*e + d) - 810*B*a^8*b^2*d*e^72/(x*e + d)^2 - 2160*A*a^7*b^3*d*e^72/(x
*e + d)^2 + 540*B*a^8*b^2*d^2*e^72/(x*e + d)^3 + 1440*A*a^7*b^3*d^2*e^72/(x*e + d)^3 - 135*B*a^8*b^2*d^3*e^72/
(x*e + d)^4 - 360*A*a^7*b^3*d^3*e^72/(x*e + d)^4 + 60*B*a^9*b*e^73/(x*e + d)^2 + 270*A*a^8*b^2*e^73/(x*e + d)^
2 - 80*B*a^9*b*d*e^73/(x*e + d)^3 - 360*A*a^8*b^2*d*e^73/(x*e + d)^3 + 30*B*a^9*b*d^2*e^73/(x*e + d)^4 + 135*A
*a^8*b^2*d^2*e^73/(x*e + d)^4 + 4*B*a^10*e^74/(x*e + d)^3 + 40*A*a^9*b*e^74/(x*e + d)^3 - 3*B*a^10*d*e^74/(x*e
+ d)^4 - 30*A*a^9*b*d*e^74/(x*e + d)^4 + 3*A*a^10*e^75/(x*e + d)^4)*e^(-76)

________________________________________________________________________________________

Mupad [B]
time = 1.45, size = 2500, normalized size = 5.63 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^10)/(d + e*x)^5,x)

[Out]

x*((10*d^3*((10*d^2*((A*b^10 + 10*B*a*b^9)/e^5 - (5*B*b^10*d)/e^6))/e^2 - (5*d*((5*d*((A*b^10 + 10*B*a*b^9)/e^
5 - (5*B*b^10*d)/e^6))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^5 + (10*B*b^10*d^2)/e^7))/e - (15*a^2*b^7*(3*A*b + 8*B*
a))/e^5 + (10*B*b^10*d^3)/e^8))/e^3 - (d^5*((A*b^10 + 10*B*a*b^9)/e^5 - (5*B*b^10*d)/e^6))/e^5 - (10*d^2*((5*d
*((10*d^2*((A*b^10 + 10*B*a*b^9)/e^5 - (5*B*b^10*d)/e^6))/e^2 - (5*d*((5*d*((A*b^10 + 10*B*a*b^9)/e^5 - (5*B*b
^10*d)/e^6))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^5 + (10*B*b^10*d^2)/e^7))/e - (15*a^2*b^7*(3*A*b + 8*B*a))/e^5 +
(10*B*b^10*d^3)/e^8))/e - (10*d^3*((A*b^10 + 10*B*a*b^9)/e^5 - (5*B*b^10*d)/e^6))/e^3 + (10*d^2*((5*d*((A*b^10
+ 10*B*a*b^9)/e^5 - (5*B*b^10*d)/e^6))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^5 + (10*B*b^10*d^2)/e^7))/e^2 + (30*a^
3*b^6*(4*A*b + 7*B*a))/e^5 - (5*B*b^10*d^4)/e^9))/e^2 + (5*d*((5*d*((5*d*((10*d^2*((A*b^10 + 10*B*a*b^9)/e^5 -
(5*B*b^10*d)/e^6))/e^2 - (5*d*((5*d*((A*b^10 + 10*B*a*b^9)/e^5 - (5*B*b^10*d)/e^6))/e - (5*a*b^8*(2*A*b + 9*B
*a))/e^5 + (10*B*b^10*d^2)/e^7))/e - (15*a^2*b^7*(3*A*b + 8*B*a))/e^5 + (10*B*b^10*d^3)/e^8))/e - (10*d^3*((A*
b^10 + 10*B*a*b^9)/e^5 - (5*B*b^10*d)/e^6))/e^3 + (10*d^2*((5*d*((A*b^10 + 10*B*a*b^9)/e^5 - (5*B*b^10*d)/e^6)
)/e - (5*a*b^8*(2*A*b + 9*B*a))/e^5 + (10*B*b^10*d^2)/e^7))/e^2 + (30*a^3*b^6*(4*A*b + 7*B*a))/e^5 - (5*B*b^10
*d^4)/e^9))/e - (10*d^2*((10*d^2*((A*b^10 + 10*B*a*b^9)/e^5 - (5*B*b^10*d)/e^6))/e^2 - (5*d*((5*d*((A*b^10 + 1
0*B*a*b^9)/e^5 - (5*B*b^10*d)/e^6))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^5 + (10*B*b^10*d^2)/e^7))/e - (15*a^2*b^7*
(3*A*b + 8*B*a))/e^5 + (10*B*b^10*d^3)/e^8))/e^2 + (5*d^4*((A*b^10 + 10*B*a*b^9)/e^5 - (5*B*b^10*d)/e^6))/e^4
- (10*d^3*((5*d*((A*b^10 + 10*B*a*b^9)/e^5 - (5*B*b^10*d)/e^6))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^5 + (10*B*b^10
*d^2)/e^7))/e^3 - (42*a^4*b^5*(5*A*b + 6*B*a))/e^5 + (B*b^10*d^5)/e^10))/e + (5*d^4*((5*d*((A*b^10 + 10*B*a*b^
9)/e^5 - (5*B*b^10*d)/e^6))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^5 + (10*B*b^10*d^2)/e^7))/e^4 + (42*a^5*b^4*(6*A*b
+ 5*B*a))/e^5) - x^5*((d*((A*b^10 + 10*B*a*b^9)/e^5 - (5*B*b^10*d)/e^6))/e - (a*b^8*(2*A*b + 9*B*a))/e^5 + (2
*B*b^10*d^2)/e^7) - x^2*((5*d*((5*d*((10*d^2*((A*b^10 + 10*B*a*b^9)/e^5 - (5*B*b^10*d)/e^6))/e^2 - (5*d*((5*d*
((A*b^10 + 10*B*a*b^9)/e^5 - (5*B*b^10*d)/e^6))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^5 + (10*B*b^10*d^2)/e^7))/e -
(15*a^2*b^7*(3*A*b + 8*B*a))/e^5 + (10*B*b^10*d^3)/e^8))/e - (10*d^3*((A*b^10 + 10*B*a*b^9)/e^5 - (5*B*b^10*d)
/e^6))/e^3 + (10*d^2*((5*d*((A*b^10 + 10*B*a*b^9)/e^5 - (5*B*b^10*d)/e^6))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^5 +
(10*B*b^10*d^2)/e^7))/e^2 + (30*a^3*b^6*(4*A*b + 7*B*a))/e^5 - (5*B*b^10*d^4)/e^9))/(2*e) - (5*d^2*((10*d^2*(
(A*b^10 + 10*B*a*b^9)/e^5 - (5*B*b^10*d)/e^6))/e^2 - (5*d*((5*d*((A*b^10 + 10*B*a*b^9)/e^5 - (5*B*b^10*d)/e^6)
)/e - (5*a*b^8*(2*A*b + 9*B*a))/e^5 + (10*B*b^10*d^2)/e^7))/e - (15*a^2*b^7*(3*A*b + 8*B*a))/e^5 + (10*B*b^10*
d^3)/e^8))/e^2 + (5*d^4*((A*b^10 + 10*B*a*b^9)/e^5 - (5*B*b^10*d)/e^6))/(2*e^4) - (5*d^3*((5*d*((A*b^10 + 10*B
*a*b^9)/e^5 - (5*B*b^10*d)/e^6))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^5 + (10*B*b^10*d^2)/e^7))/e^3 - (21*a^4*b^5*(
5*A*b + 6*B*a))/e^5 + (B*b^10*d^5)/(2*e^10)) - (x^3*(120*A*a^7*b^3*e^10 + 45*B*a^8*b^2*e^10 - 120*A*b^10*d^7*e
^3 + 165*B*b^10*d^8*e^2 + 840*A*a*b^9*d^6*e^4 - 840*A*a^6*b^4*d*e^9 - 1200*B*a*b^9*d^7*e^3 - 480*B*a^7*b^3*d*e
^9 - 2520*A*a^2*b^8*d^5*e^5 + 4200*A*a^3*b^7*d^4*e^6 - 4200*A*a^4*b^6*d^3*e^7 + 2520*A*a^5*b^5*d^2*e^8 + 3780*
B*a^2*b^8*d^6*e^4 - 6720*B*a^3*b^7*d^5*e^5 + 7350*B*a^4*b^6*d^4*e^6 - 5040*B*a^5*b^5*d^3*e^7 + 2100*B*a^6*b^4*
d^2*e^8) + (3*A*a^10*e^11 + 1691*B*b^10*d^11 - 1207*A*b^10*d^10*e + B*a^10*d*e^10 + 8250*A*a*b^9*d^9*e^2 + 10*
B*a^9*b*d^2*e^9 - 23985*A*a^2*b^8*d^8*e^3 + 38280*A*a^3*b^7*d^7*e^4 - 35910*A*a^4*b^6*d^6*e^5 + 19404*A*a^5*b^
5*d^5*e^6 - 5250*A*a^6*b^4*d^4*e^7 + 360*A*a^7*b^3*d^3*e^8 + 45*A*a^8*b^2*d^2*e^9 + 37125*B*a^2*b^8*d^9*e^2 -
63960*B*a^3*b^7*d^8*e^3 + 66990*B*a^4*b^6*d^7*e^4 - 43092*B*a^5*b^5*d^6*e^5 + 16170*B*a^6*b^4*d^5*e^6 - 3000*B
*a^7*b^3*d^4*e^7 + 135*B*a^8*b^2*d^3*e^8 + 10*A*a^9*b*d*e^10 - 12070*B*a*b^9*d^10*e)/(12*e) + x*((B*a^10*e^10)
/3 + (1331*B*b^10*d^10)/3 + (10*A*a^9*b*e^10)/3 - (955*A*b^10*d^9*e)/3 + 2190*A*a*b^9*d^8*e^2 + 15*A*a^8*b^2*d
*e^9 - 6420*A*a^2*b^8*d^7*e^3 + 10360*A*a^3*b^7*d^6*e^4 - 9870*A*a^4*b^6*d^5*e^5 + 5460*A*a^5*b^5*d^4*e^6 - 15
40*A*a^6*b^4*d^3*e^7 + 120*A*a^7*b^3*d^2*e^8 + 9855*B*a^2*b^8*d^8*e^2 - 17120*B*a^3*b^7*d^7*e^3 + 18130*B*a^4*
b^6*d^6*e^4 - 11844*B*a^5*b^5*d^5*e^5 + 4550*B*a^6*b^4*d^4*e^6 - 880*B*a^7*b^3*d^3*e^7 + 45*B*a^8*b^2*d^2*e^8
- (9550*B*a*b^9*d^9*e)/3 + (10*B*a^9*b*d*e^9)/3) + x^2*(5*B*a^9*b*e^10 + (935*B*b^10*d^9*e)/2 + (45*A*a^8*b^2*
e^10)/2 - (675*A*b^10*d^8*e^2)/2 + 2340*A*a*b^9*d^7*e^3 + 180*A*a^7*b^3*d*e^9 - 3375*B*a*b^9*d^8*e^2 + (135*B*
a^8*b^2*d*e^9)/2 - 6930*A*a^2*b^8*d^6*e^4 + 11340*A*a^3*b^7*d^5*e^5 - 11025*A*a^4*b^6*d^4*e^6 + 6300*A*a^5*b^5
*d^3*e^7 - 1890*A*a^6*b^4*d^2*e^8 + 10530*B*a^2*b^8*d^7*e^3 - 18480*B*a^3*b^7*d^6*e^4 + 19845*B*a^4*b^6*d^5*e^
5 - 13230*B*a^5*b^5*d^4*e^6 + 5250*B*a^6*b^4*d^...

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