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

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

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Rubi [A]  time = 1.32, 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 = {77} \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 {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 {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 {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 77

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 = 0.53, size = 686, normalized size = 1.55 \begin {gather*} \frac {-84 b^8 e^5 x^5 \left (-9 a^2 B e^2-2 a b e (A e-5 B d)+b^2 d (A e-3 B d)\right )+105 b^7 e^4 x^4 \left (24 a^3 B e^3+9 a^2 b e^2 (A e-5 B d)+10 a b^2 d e (3 B d-A e)+b^3 d^2 (3 A e-7 B d)\right )-140 b^6 e^3 x^3 \left (-42 a^4 B e^4-24 a^3 b e^3 (A e-5 B d)-45 a^2 b^2 d e^2 (3 B d-A e)+10 a b^3 d^2 e (7 B d-3 A e)+7 b^4 d^3 (A e-2 B d)\right )+42 b^5 e^2 x^2 \left (252 a^5 B e^5+210 a^4 b e^4 (A e-5 B d)+600 a^3 b^2 d e^3 (3 B d-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)-14 b^5 d^4 (9 B d-5 A e)\right )-84 b^4 e x \left (-210 a^6 B e^6-252 a^5 b e^5 (A e-5 B d)-1050 a^4 b^2 d e^4 (3 B d-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)+140 a b^5 d^4 e (9 B d-5 A e)-42 b^6 d^5 (5 B d-3 A e)\right )+14 b^9 e^6 x^6 (10 a B e+A b e-5 b B d)-2520 b^3 (b d-a e)^6 \log (d+e x) (-4 a B e-7 A b e+11 b B d)-\frac {1260 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{d+e x}+\frac {210 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{(d+e x)^2}-\frac {28 (b d-a e)^9 (-a B e-10 A b e+11 b B d)}{(d+e x)^3}+\frac {21 (b d-a e)^{10} (B d-A e)}{(d+e x)^4}+12 b^{10} B e^7 x^7}{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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.41, size = 2808, normalized size = 6.32

result too large to display

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*(12*B*b^10*e^11*x^11 - 11837*B*b^10*d^11 - 21*A*a^10*e^11 + 8449*(10*B*a*b^9 + A*b^10)*d^10*e - 28875*(9*
B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 55965*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 - 66990*(7*B*a^4*b^6 + 4*A*a^3*b^7)
*d^7*e^4 + 50274*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 - 22638*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 5250*(4*B*a
^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 - 315*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 - 35*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9
 - 7*(B*a^10 + 10*A*a^9*b)*d*e^10 - 2*(11*B*b^10*d*e^10 - 7*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 4*(11*B*b^10*d^
2*e^9 - 7*(10*B*a*b^9 + A*b^10)*d*e^10 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 - 9*(11*B*b^10*d^3*e^8 - 7*(10
*B*a*b^9 + A*b^10)*d^2*e^9 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 - 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 +
24*(11*B*b^10*d^4*e^7 - 7*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 - 35*(8*B*a^3*b
^7 + 3*A*a^2*b^8)*d*e^10 + 35*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 - 84*(11*B*b^10*d^5*e^6 - 7*(10*B*a*b^9 +
A*b^10)*d^4*e^7 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 - 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 35*(7*B*a^4*
b^6 + 4*A*a^3*b^7)*d*e^10 - 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 504*(11*B*b^10*d^6*e^5 - 7*(10*B*a*b^9
+ A*b^10)*d^5*e^6 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 - 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 35*(7*B*a^
4*b^6 + 4*A*a^3*b^7)*d^2*e^9 - 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 7*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5
 + 7*(6559*B*b^10*d^7*e^4 - 4043*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 11625*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 - 182
55*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 16680*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 - 8568*(6*B*a^5*b^5 + 5*A*a
^4*b^6)*d^2*e^9 + 2016*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10)*x^4 + 28*(2599*B*b^10*d^8*e^3 - 1523*(10*B*a*b^9 +
A*b^10)*d^7*e^4 + 4065*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 - 5655*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 4080*(7*
B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 - 1008*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 - 504*(5*B*a^6*b^4 + 6*A*a^5*b^5)*
d^2*e^9 + 360*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 - 45*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 + 42*(619*B*b^10*d
^9*e^2 - 263*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 285*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 + 645*(8*B*a^3*b^7 + 3*A*a^
2*b^8)*d^6*e^5 - 2220*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 + 2772*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 - 1764*(5
*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 + 540*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9 - 45*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d
*e^10 - 5*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 - 28*(701*B*b^10*d^10*e - 577*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 22
35*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 - 4845*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 6420*(7*B*a^4*b^6 + 4*A*a^3*
b^7)*d^6*e^5 - 5292*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 2604*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 - 660*(4*B*
a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 + 45*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 + 5*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 +
 (B*a^10 + 10*A*a^9*b)*e^11)*x - 2520*(11*B*b^10*d^11 - 7*(10*B*a*b^9 + A*b^10)*d^10*e + 21*(9*B*a^2*b^8 + 2*A
*a*b^9)*d^9*e^2 - 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 35*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 21*(6*B*a^
5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 7*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - (4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + (
11*B*b^10*d^7*e^4 - 7*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 - 35*(8*B*a^3*b^7 +
 3*A*a^2*b^8)*d^4*e^7 + 35*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 - 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9 + 7*(5
*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 - (4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 4*(11*B*b^10*d^8*e^3 - 7*(10*B*a*b^
9 + A*b^10)*d^7*e^4 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 - 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 35*(7*B*
a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 - 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + 7*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^
9 - (4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10)*x^3 + 6*(11*B*b^10*d^9*e^2 - 7*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 21*(9*B
*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 - 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 35*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^
6 - 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 + 7*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 - (4*B*a^7*b^3 + 7*A*a^6*b^
4)*d^2*e^9)*x^2 + 4*(11*B*b^10*d^10*e - 7*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3
 - 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 35*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 - 21*(6*B*a^5*b^5 + 5*A*a^4
*b^6)*d^5*e^6 + 7*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 - (4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8)*x)*log(e*x + d))/
(e^16*x^4 + 4*d*e^15*x^3 + 6*d^2*e^14*x^2 + 4*d^3*e^13*x + d^4*e^12)

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giac [B]  time = 1.47, size = 2513, normalized size = 5.66

result too large to display

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)

________________________________________________________________________________________

maple [B]  time = 0.04, size = 2673, normalized size = 6.02

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3/e^2/(e*x+d)^3*B*a^10-1/4/e/(e*x+d)^4*a^10*A+1/7*b^10/e^5*B*x^7+1/6*b^10/e^5*A*x^6-11/3/e^12/(e*x+d)^3*b^1
0*B*d^10-1/4/e^11/(e*x+d)^4*A*b^10*d^10+1/4/e^2/(e*x+d)^4*B*d*a^10+1/4/e^12/(e*x+d)^4*b^10*B*d^11+210*b^4/e^5*
ln(e*x+d)*A*a^6+252*b^5/e^5*A*x*a^5-126*b^10/e^10*A*x*d^5+3150*b^8/e^9*B*x*a^2*d^4-4200*b^7/e^8*B*x*a^3*d^3+31
50*b^6/e^7*B*x*a^4*d^2+700*b^9/e^9*A*x*a*d^4-1260*b^5/e^6*B*x*a^5*d+1800*b^7/e^7*A*x*a^3*d^2-1575*b^8/e^8*A*x*
a^2*d^3-1050*b^6/e^6*A*x*a^4*d-300*b^7/e^6*A*x^2*a^3*d+675/2*b^8/e^7*A*x^2*a^2*d^2-175*b^9/e^8*A*x^2*a*d^3-525
*b^6/e^6*B*x^2*a^4*d+900*b^7/e^7*B*x^2*a^3*d^2-1575/2*b^8/e^8*B*x^2*a^2*d^3+350*b^9/e^9*B*x^2*a*d^4-10*b^9/e^6
*B*x^5*a*d-25/2*b^9/e^6*A*x^4*a*d-225/4*b^8/e^6*B*x^4*a^2*d+75/2*b^9/e^7*B*x^4*a*d^2-75*b^8/e^6*A*x^3*a^2*d+50
*b^9/e^7*A*x^3*a*d^2-200*b^7/e^6*B*x^3*a^3*d+225*b^8/e^7*B*x^3*a^2*d^2-350/3*b^9/e^8*B*x^3*a*d^3-1260*b^9/e^10
*B*x*a*d^5+840*b^4/e^5/(e*x+d)*A*a^6*d-2520*b^5/e^6/(e*x+d)*A*a^5*d^2+4200*b^6/e^7/(e*x+d)*A*a^4*d^3-4200*b^7/
e^8/(e*x+d)*A*a^3*d^4+2520*b^8/e^9/(e*x+d)*A*a^2*d^5-840*b^9/e^10/(e*x+d)*A*a*d^6+480*b^3/e^5/(e*x+d)*B*a^7*d-
2100*b^4/e^6/(e*x+d)*B*a^6*d^2+5040*b^5/e^7/(e*x+d)*B*a^5*d^3-7350*b^6/e^8/(e*x+d)*B*a^4*d^4+6720*b^7/e^9/(e*x
+d)*B*a^3*d^5-3780*b^8/e^10/(e*x+d)*B*a^2*d^6+1200*b^9/e^11/(e*x+d)*B*a*d^7+30/e^3/(e*x+d)^3*A*a^8*b^2*d-120/e
^4/(e*x+d)^3*A*a^7*b^3*d^2+280/e^5/(e*x+d)^3*A*a^6*b^4*d^3-420/e^6/(e*x+d)^3*A*a^5*b^5*d^4+420/e^7/(e*x+d)^3*A
*a^4*b^6*d^5-280/e^8/(e*x+d)^3*A*a^3*b^7*d^6+120/e^9/(e*x+d)^3*A*a^2*b^8*d^7-30/e^10/(e*x+d)^3*A*a*b^9*d^8+20/
3/e^3/(e*x+d)^3*B*a^9*b*d-45/e^4/(e*x+d)^3*B*a^8*b^2*d^2+160/e^5/(e*x+d)^3*B*a^7*b^3*d^3-350/e^6/(e*x+d)^3*B*a
^6*b^4*d^4+504/e^7/(e*x+d)^3*B*a^5*b^5*d^5-490/e^8/(e*x+d)^3*B*a^4*b^6*d^6+320/e^9/(e*x+d)^3*B*a^3*b^7*d^7-135
/e^10/(e*x+d)^3*B*a^2*b^8*d^8+100/3/e^11/(e*x+d)^3*B*a*b^9*d^9+5/2/e^2/(e*x+d)^4*A*d*a^9*b-45/4/e^3/(e*x+d)^4*
A*d^2*a^8*b^2+30/e^4/(e*x+d)^4*A*d^3*a^7*b^3-105/2/e^5/(e*x+d)^4*A*d^4*a^6*b^4+63/e^6/(e*x+d)^4*A*a^5*b^5*d^5-
105/2/e^7/(e*x+d)^4*A*a^4*b^6*d^6+30/e^8/(e*x+d)^4*A*a^3*b^7*d^7-45/4/e^9/(e*x+d)^4*A*a^2*b^8*d^8+5/2/e^10/(e*
x+d)^4*A*a*b^9*d^9-5/2/e^3/(e*x+d)^4*B*d^2*a^9*b+45/4/e^4/(e*x+d)^4*B*d^3*a^8*b^2-30/e^5/(e*x+d)^4*B*d^4*a^7*b
^3+105/2/e^6/(e*x+d)^4*B*a^6*b^4*d^5-63/e^7/(e*x+d)^4*B*a^5*b^5*d^6+105/2/e^8/(e*x+d)^4*B*a^4*b^6*d^7-30/e^9/(
e*x+d)^4*B*a^3*b^7*d^8+45/4/e^10/(e*x+d)^4*B*a^2*b^8*d^9-5/2/e^11/(e*x+d)^4*B*a*b^9*d^10-1260*b^5/e^6*ln(e*x+d
)*A*a^5*d+3150*b^6/e^7*ln(e*x+d)*A*a^4*d^2-4200*b^7/e^8*ln(e*x+d)*A*a^3*d^3+3150*b^8/e^9*ln(e*x+d)*A*a^2*d^4-1
260*b^9/e^10*ln(e*x+d)*A*a*d^5-1050*b^4/e^6*ln(e*x+d)*B*a^6*d+3780*b^5/e^7*ln(e*x+d)*B*a^5*d^2-7350*b^6/e^8*ln
(e*x+d)*B*a^4*d^3+8400*b^7/e^9*ln(e*x+d)*B*a^3*d^4-5670*b^8/e^10*ln(e*x+d)*B*a^2*d^5+2100*b^9/e^11*ln(e*x+d)*B
*a*d^6+180*b^3/e^4/(e*x+d)^2*A*a^7*d-630*b^4/e^5/(e*x+d)^2*A*a^6*d^2+1260*b^5/e^6/(e*x+d)^2*A*a^5*d^3-1575*b^6
/e^7/(e*x+d)^2*A*a^4*d^4+1260*b^7/e^8/(e*x+d)^2*A*a^3*d^5-630*b^8/e^9/(e*x+d)^2*A*a^2*d^6+180*b^9/e^10/(e*x+d)
^2*A*a*d^7+135/2*b^2/e^4/(e*x+d)^2*B*a^8*d-360*b^3/e^5/(e*x+d)^2*B*a^7*d^2+1050*b^4/e^6/(e*x+d)^2*B*a^6*d^3-18
90*b^5/e^7/(e*x+d)^2*B*a^5*d^4+2205*b^6/e^8/(e*x+d)^2*B*a^4*d^5-1680*b^7/e^9/(e*x+d)^2*B*a^3*d^6+810*b^8/e^10/
(e*x+d)^2*B*a^2*d^7-225*b^9/e^11/(e*x+d)^2*B*a*d^8+126*b^5/e^5*B*x^2*a^5-63*b^10/e^10*B*x^2*d^5+105*b^6/e^5*A*
x^2*a^4+35*b^10/e^9*A*x^2*d^4+210*b^10/e^11*ln(e*x+d)*A*d^6+120*b^3/e^5*ln(e*x+d)*B*a^7-330*b^10/e^12*ln(e*x+d
)*B*d^7-45/2*b^2/e^3/(e*x+d)^2*A*a^8-45/2*b^10/e^11/(e*x+d)^2*A*d^8-5*b/e^3/(e*x+d)^2*B*a^9+55/2*b^10/e^12/(e*
x+d)^2*B*d^9+70*b^6/e^5*B*x^3*a^4+70/3*b^10/e^9*B*x^3*d^4-35/3*b^10/e^8*A*x^3*d^3+40*b^7/e^5*A*x^3*a^3+30*b^7/
e^5*B*x^4*a^3-35/4*b^10/e^8*B*x^4*d^3+15/4*b^10/e^7*A*x^4*d^2+9*b^8/e^5*B*x^5*a^2+3*b^10/e^7*B*x^5*d^2+45/4*b^
8/e^5*A*x^4*a^2-5/6*b^10/e^6*B*x^6*d+2*b^9/e^5*A*x^5*a-b^10/e^6*A*x^5*d+5/3*b^9/e^5*B*x^6*a-120*b^3/e^4/(e*x+d
)*A*a^7+120*b^10/e^11/(e*x+d)*A*d^7-45*b^2/e^4/(e*x+d)*B*a^8-165*b^10/e^12/(e*x+d)*B*d^8-10/3/e^2/(e*x+d)^3*A*
a^9*b+10/3/e^11/(e*x+d)^3*A*b^10*d^9+210*b^4/e^5*B*x*a^6+210*b^10/e^11*B*x*d^6

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maxima [B]  time = 1.32, size = 1846, normalized size = 4.16

result too large to display

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]

-1/12*(1691*B*b^10*d^11 + 3*A*a^10*e^11 - 1207*(10*B*a*b^9 + A*b^10)*d^10*e + 4125*(9*B*a^2*b^8 + 2*A*a*b^9)*d
^9*e^2 - 7995*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 9570*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 7182*(6*B*a^5*b
^5 + 5*A*a^4*b^6)*d^6*e^5 + 3234*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - 750*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7
 + 45*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 + 5*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 + (B*a^10 + 10*A*a^9*b)*d*e^10
 + 180*(11*B*b^10*d^8*e^3 - 8*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 - 56*(8*B*a
^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 70*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 - 56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^
8 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 - 8*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 + (3*B*a^8*b^2 + 8*A*a^7*b^3
)*e^11)*x^3 + 30*(187*B*b^10*d^9*e^2 - 135*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 468*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e
^4 - 924*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 1134*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 - 882*(6*B*a^5*b^5 + 5
*A*a^4*b^6)*d^4*e^7 + 420*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 - 108*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9 + 9*(3
*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 + (2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 4*(1331*B*b^10*d^10*e - 955*(10*B*a*b
^9 + A*b^10)*d^9*e^2 + 3285*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 - 6420*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 777
0*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 - 5922*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 2730*(5*B*a^6*b^4 + 6*A*a^5
*b^5)*d^4*e^7 - 660*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 + 45*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 + 5*(2*B*a^9*
b + 9*A*a^8*b^2)*d*e^10 + (B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^16*x^4 + 4*d*e^15*x^3 + 6*d^2*e^14*x^2 + 4*d^3*e^1
3*x + d^4*e^12) + 1/84*(12*B*b^10*e^6*x^7 - 14*(5*B*b^10*d*e^5 - (10*B*a*b^9 + A*b^10)*e^6)*x^6 + 84*(3*B*b^10
*d^2*e^4 - (10*B*a*b^9 + A*b^10)*d*e^5 + (9*B*a^2*b^8 + 2*A*a*b^9)*e^6)*x^5 - 105*(7*B*b^10*d^3*e^3 - 3*(10*B*
a*b^9 + A*b^10)*d^2*e^4 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^5 - 3*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^6)*x^4 + 140*(14
*B*b^10*d^4*e^2 - 7*(10*B*a*b^9 + A*b^10)*d^3*e^3 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^4 - 15*(8*B*a^3*b^7 + 3
*A*a^2*b^8)*d*e^5 + 6*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^6)*x^3 - 42*(126*B*b^10*d^5*e - 70*(10*B*a*b^9 + A*b^10)*d
^4*e^2 + 175*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^3 - 225*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^4 + 150*(7*B*a^4*b^6 +
4*A*a^3*b^7)*d*e^5 - 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^6)*x^2 + 84*(210*B*b^10*d^6 - 126*(10*B*a*b^9 + A*b^10)*
d^5*e + 350*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^2 - 525*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^3 + 450*(7*B*a^4*b^6 + 4
*A*a^3*b^7)*d^2*e^4 - 210*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^5 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^6)*x)/e^11 - 30
*(11*B*b^10*d^7 - 7*(10*B*a*b^9 + A*b^10)*d^6*e + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^2 - 35*(8*B*a^3*b^7 + 3*A
*a^2*b^8)*d^4*e^3 + 35*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^4 - 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^5 + 7*(5*B*a
^6*b^4 + 6*A*a^5*b^5)*d*e^6 - (4*B*a^7*b^3 + 7*A*a^6*b^4)*e^7)*log(e*x + d)/e^12

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mupad [B]  time = 1.45, size = 3655, normalized size = 8.23

result too large to display

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^3*e^7 - 1080*B*a^7*b^3*d^2*e^8))/(d^4*e^11 + e^15*x^4 + 4*d^3*e
^12*x + 4*d*e^14*x^3 + 6*d^2*e^13*x^2) - x^4*((5*d^2*((A*b^10 + 10*B*a*b^9)/e^5 - (5*B*b^10*d)/e^6))/(2*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))/(4*e) - (15*a^2*b^7*(3*A*b + 8*B*a))/(4*e^5) + (5*B*b^10*d^3)/(2*e^8)) + x^6*((A*b^10 + 10*B*a*b^9)/(6
*e^5) - (5*B*b^10*d)/(6*e^6)) + x^3*((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))/(3*e) - (10*d^3*((A*b^10 + 10*B*a*b^9)/e^5 - (5
*B*b^10*d)/e^6))/(3*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))/(3*e^2) + (10*a^3*b^6*(4*A*b + 7*B*a))/e^5 - (5*B*b^10*d^4)/(3*e^9)) + (l
og(d + e*x)*(210*A*b^10*d^6*e - 330*B*b^10*d^7 + 210*A*a^6*b^4*e^7 + 120*B*a^7*b^3*e^7 - 1260*A*a*b^9*d^5*e^2
- 1260*A*a^5*b^5*d*e^6 - 1050*B*a^6*b^4*d*e^6 + 3150*A*a^2*b^8*d^4*e^3 - 4200*A*a^3*b^7*d^3*e^4 + 3150*A*a^4*b
^6*d^2*e^5 - 5670*B*a^2*b^8*d^5*e^2 + 8400*B*a^3*b^7*d^4*e^3 - 7350*B*a^4*b^6*d^3*e^4 + 3780*B*a^5*b^5*d^2*e^5
 + 2100*B*a*b^9*d^6*e))/e^12 + (B*b^10*x^7)/(7*e^5)

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sympy [F(-1)]  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

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