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3.10.55 \(\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^6} \, dx\)

Optimal. Leaf size=447 \[ -\frac {b^9 (d+e x)^5 (-10 a B e-A b e+11 b B d)}{5 e^{12}}+\frac {5 b^8 (d+e x)^4 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{4 e^{12}}-\frac {5 b^7 (d+e x)^3 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{e^{12}}+\frac {15 b^6 (d+e x)^2 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{e^{12}}-\frac {42 b^5 x (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{e^{11}}+\frac {42 b^4 (b d-a e)^5 \log (d+e x) (-5 a B e-6 A b e+11 b B d)}{e^{12}}+\frac {30 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{e^{12} (d+e x)}-\frac {15 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{2 e^{12} (d+e x)^2}+\frac {5 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{3 e^{12} (d+e x)^3}-\frac {(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{4 e^{12} (d+e x)^4}+\frac {(b d-a e)^{10} (B d-A e)}{5 e^{12} (d+e x)^5}+\frac {b^{10} B (d+e x)^6}{6 e^{12}} \]

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Rubi [A]  time = 1.25, antiderivative size = 447, 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)^5 (-10 a B e-A b e+11 b B d)}{5 e^{12}}+\frac {5 b^8 (d+e x)^4 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{4 e^{12}}-\frac {5 b^7 (d+e x)^3 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{e^{12}}+\frac {15 b^6 (d+e x)^2 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{e^{12}}-\frac {42 b^5 x (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{e^{11}}+\frac {30 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{e^{12} (d+e x)}-\frac {15 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{2 e^{12} (d+e x)^2}+\frac {42 b^4 (b d-a e)^5 \log (d+e x) (-5 a B e-6 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)}{3 e^{12} (d+e x)^3}-\frac {(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{4 e^{12} (d+e x)^4}+\frac {(b d-a e)^{10} (B d-A e)}{5 e^{12} (d+e x)^5}+\frac {b^{10} B (d+e x)^6}{6 e^{12}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^10*(A + B*x))/(d + e*x)^6,x]

[Out]

(-42*b^5*(b*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*B*e)*x)/e^11 + ((b*d - a*e)^10*(B*d - A*e))/(5*e^12*(d + e*x)
^5) - ((b*d - a*e)^9*(11*b*B*d - 10*A*b*e - a*B*e))/(4*e^12*(d + e*x)^4) + (5*b*(b*d - a*e)^8*(11*b*B*d - 9*A*
b*e - 2*a*B*e))/(3*e^12*(d + e*x)^3) - (15*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b*e - 3*a*B*e))/(2*e^12*(d + e*x)
^2) + (30*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b*e - 4*a*B*e))/(e^12*(d + e*x)) + (15*b^6*(b*d - a*e)^3*(11*b*B*d
 - 4*A*b*e - 7*a*B*e)*(d + e*x)^2)/e^12 - (5*b^7*(b*d - a*e)^2*(11*b*B*d - 3*A*b*e - 8*a*B*e)*(d + e*x)^3)/e^1
2 + (5*b^8*(b*d - a*e)*(11*b*B*d - 2*A*b*e - 9*a*B*e)*(d + e*x)^4)/(4*e^12) - (b^9*(11*b*B*d - A*b*e - 10*a*B*
e)*(d + e*x)^5)/(5*e^12) + (b^10*B*(d + e*x)^6)/(6*e^12) + (42*b^4*(b*d - a*e)^5*(11*b*B*d - 6*A*b*e - 5*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)^6} \, dx &=\int \left (\frac {42 b^5 (b d-a e)^4 (-11 b B d+5 A b e+6 a B e)}{e^{11}}+\frac {(-b d+a e)^{10} (-B d+A e)}{e^{11} (d+e x)^6}+\frac {(-b d+a e)^9 (-11 b B d+10 A b e+a B e)}{e^{11} (d+e x)^5}+\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)^4}-\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)^3}+\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)^2}-\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)}-\frac {30 b^6 (b d-a e)^3 (-11 b B d+4 A b e+7 a B e) (d+e x)}{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)^2}{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)^3}{e^{11}}+\frac {b^9 (-11 b B d+A b e+10 a B e) (d+e x)^4}{e^{11}}+\frac {b^{10} B (d+e x)^5}{e^{11}}\right ) \, dx\\ &=-\frac {42 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e) x}{e^{11}}+\frac {(b d-a e)^{10} (B d-A e)}{5 e^{12} (d+e x)^5}-\frac {(b d-a e)^9 (11 b B d-10 A b e-a B e)}{4 e^{12} (d+e x)^4}+\frac {5 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{3 e^{12} (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)}{2 e^{12} (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^{12} (d+e x)}+\frac {15 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e) (d+e x)^2}{e^{12}}-\frac {5 b^7 (b d-a e)^2 (11 b B d-3 A b e-8 a B e) (d+e x)^3}{e^{12}}+\frac {5 b^8 (b d-a e) (11 b B d-2 A b e-9 a B e) (d+e x)^4}{4 e^{12}}-\frac {b^9 (11 b B d-A b e-10 a B e) (d+e x)^5}{5 e^{12}}+\frac {b^{10} B (d+e x)^6}{6 e^{12}}+\frac {42 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e) \log (d+e x)}{e^{12}}\\ \end {align*}

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Mathematica [A]  time = 0.58, size = 587, normalized size = 1.31 \begin {gather*} \frac {-15 b^8 e^4 x^4 \left (-45 a^2 B e^2-10 a b e (A e-6 B d)+3 b^2 d (2 A e-7 B d)\right )+20 b^7 e^3 x^3 \left (120 a^3 B e^3+45 a^2 b e^2 (A e-6 B d)+30 a b^2 d e (7 B d-2 A e)-7 b^3 d^2 (8 B d-3 A e)\right )-30 b^6 e^2 x^2 \left (-210 a^4 B e^4-120 a^3 b e^3 (A e-6 B d)-135 a^2 b^2 d e^2 (7 B d-2 A e)+70 a b^3 d^2 e (8 B d-3 A e)-14 b^4 d^3 (9 B d-4 A e)\right )+60 b^5 e x \left (252 a^5 B e^5+210 a^4 b e^4 (A e-6 B d)+360 a^3 b^2 d e^3 (7 B d-2 A e)-315 a^2 b^3 d^2 e^2 (8 B d-3 A e)+140 a b^4 d^3 e (9 B d-4 A e)-126 b^5 d^4 (2 B d-A e)\right )+12 b^9 e^5 x^5 (10 a B e+A b e-6 b B d)+2520 b^4 (b d-a e)^5 \log (d+e x) (-5 a B e-6 A b e+11 b B d)+\frac {1800 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{d+e x}-\frac {450 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{(d+e x)^2}+\frac {100 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{(d+e x)^3}-\frac {15 (b d-a e)^9 (-a B e-10 A b e+11 b B d)}{(d+e x)^4}+\frac {12 (b d-a e)^{10} (B d-A e)}{(d+e x)^5}+10 b^{10} B e^6 x^6}{60 e^{12}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^6,x]

[Out]

(60*b^5*e*(252*a^5*B*e^5 + 140*a*b^4*d^3*e*(9*B*d - 4*A*e) - 315*a^2*b^3*d^2*e^2*(8*B*d - 3*A*e) + 360*a^3*b^2
*d*e^3*(7*B*d - 2*A*e) - 126*b^5*d^4*(2*B*d - A*e) + 210*a^4*b*e^4*(-6*B*d + A*e))*x - 30*b^6*e^2*(-210*a^4*B*
e^4 - 14*b^4*d^3*(9*B*d - 4*A*e) + 70*a*b^3*d^2*e*(8*B*d - 3*A*e) - 135*a^2*b^2*d*e^2*(7*B*d - 2*A*e) - 120*a^
3*b*e^3*(-6*B*d + A*e))*x^2 + 20*b^7*e^3*(120*a^3*B*e^3 - 7*b^3*d^2*(8*B*d - 3*A*e) + 30*a*b^2*d*e*(7*B*d - 2*
A*e) + 45*a^2*b*e^2*(-6*B*d + A*e))*x^3 - 15*b^8*e^4*(-45*a^2*B*e^2 - 10*a*b*e*(-6*B*d + A*e) + 3*b^2*d*(-7*B*
d + 2*A*e))*x^4 + 12*b^9*e^5*(-6*b*B*d + A*b*e + 10*a*B*e)*x^5 + 10*b^10*B*e^6*x^6 + (12*(b*d - a*e)^10*(B*d -
 A*e))/(d + e*x)^5 - (15*(b*d - a*e)^9*(11*b*B*d - 10*A*b*e - a*B*e))/(d + e*x)^4 + (100*b*(b*d - a*e)^8*(11*b
*B*d - 9*A*b*e - 2*a*B*e))/(d + e*x)^3 - (450*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b*e - 3*a*B*e))/(d + e*x)^2 +
(1800*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b*e - 4*a*B*e))/(d + e*x) + 2520*b^4*(b*d - a*e)^5*(11*b*B*d - 6*A*b*e
 - 5*a*B*e)*Log[d + e*x])/(60*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)^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x)^10*(A + B*x))/(d + e*x)^6,x]

[Out]

IntegrateAlgebraic[((a + b*x)^10*(A + B*x))/(d + e*x)^6, x]

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fricas [B]  time = 1.32, size = 2851, normalized size = 6.38

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)^6,x, algorithm="fricas")

[Out]

1/60*(10*B*b^10*e^11*x^11 + 15797*B*b^10*d^11 - 12*A*a^10*e^11 - 9762*(10*B*a*b^9 + A*b^10)*d^10*e + 28185*(9*
B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 - 44580*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 41310*(7*B*a^4*b^6 + 4*A*a^3*b^7)
*d^7*e^4 - 21924*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 5754*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - 360*(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 - 10*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 -
3*(B*a^10 + 10*A*a^9*b)*d*e^10 - 2*(11*B*b^10*d*e^10 - 6*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 5*(11*B*b^10*d^2*e
^9 - 6*(10*B*a*b^9 + A*b^10)*d*e^10 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 - 15*(11*B*b^10*d^3*e^8 - 6*(10*B
*a*b^9 + A*b^10)*d^2*e^9 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 - 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 60
*(11*B*b^10*d^4*e^7 - 6*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 - 20*(8*B*a^3*b^7
 + 3*A*a^2*b^8)*d*e^10 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 - 420*(11*B*b^10*d^5*e^6 - 6*(10*B*a*b^9 + A
*b^10)*d^4*e^7 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 - 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 15*(7*B*a^4*b
^6 + 4*A*a^3*b^7)*d*e^10 - 6*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 - (47497*B*b^10*d^6*e^5 - 24762*(10*B*a*b^9
 + A*b^10)*d^5*e^6 + 58125*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 - 70500*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 450
00*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 - 12600*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10)*x^5 - 5*(19777*B*b^10*d^7*e
^4 - 9642*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 20325*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 - 20100*(8*B*a^3*b^7 + 3*A*a
^2*b^8)*d^4*e^7 + 7200*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 + 2520*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9 - 2520*(
5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 + 360*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 - 10*(5917*B*b^10*d^8*e^3 - 2082
*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 1425*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 + 5100*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5
*e^6 - 11700*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 + 10080*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 - 3780*(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
+ 10*(3323*B*b^10*d^9*e^2 - 2958*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 11175*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 - 219
00*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 24300*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 - 15120*(6*B*a^5*b^5 + 5*A*
a^4*b^6)*d^4*e^7 + 4620*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 - 360*(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 - 10*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 5*(10253*B*b^10*d^10*e - 6738*(10*B
*a*b^9 + A*b^10)*d^9*e^2 + 20625*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 - 34500*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4
 + 33750*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 - 18900*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 5250*(5*B*a^6*b^4 +
 6*A*a^5*b^5)*d^4*e^7 - 360*(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 - 10*
(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 - 3*(B*a^10 + 10*A*a^9*b)*e^11)*x + 2520*(11*B*b^10*d^11 - 6*(10*B*a*b^9 + A*
b^10)*d^10*e + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 - 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 15*(7*B*a^4*b^6
 + 4*A*a^3*b^7)*d^7*e^4 - 6*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + (5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + (11*B*
b^10*d^6*e^5 - 6*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 - 20*(8*B*a^3*b^7 + 3*A*
a^2*b^8)*d^3*e^8 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 - 6*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + (5*B*a^6*b^
4 + 6*A*a^5*b^5)*e^11)*x^5 + 5*(11*B*b^10*d^7*e^4 - 6*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 15*(9*B*a^2*b^8 + 2*A*a*
b^9)*d^5*e^6 - 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 - 6*(6*B*a^5*b^
5 + 5*A*a^4*b^6)*d^2*e^9 + (5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10)*x^4 + 10*(11*B*b^10*d^8*e^3 - 6*(10*B*a*b^9 + A
*b^10)*d^7*e^4 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 - 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 15*(7*B*a^4*b
^6 + 4*A*a^3*b^7)*d^4*e^7 - 6*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + (5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9)*x^3 +
 10*(11*B*b^10*d^9*e^2 - 6*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 - 20*(8*B*a^3*
b^7 + 3*A*a^2*b^8)*d^6*e^5 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 - 6*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 +
(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8)*x^2 + 5*(11*B*b^10*d^10*e - 6*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 15*(9*B*a^2
*b^8 + 2*A*a*b^9)*d^8*e^3 - 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 -
6*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + (5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7)*x)*log(e*x + d))/(e^17*x^5 + 5*d*
e^16*x^4 + 10*d^2*e^15*x^3 + 10*d^3*e^14*x^2 + 5*d^4*e^13*x + d^5*e^12)

________________________________________________________________________________________

giac [B]  time = 1.27, size = 1900, normalized size = 4.25

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)^6,x, algorithm="giac")

[Out]

42*(11*B*b^10*d^6 - 60*B*a*b^9*d^5*e - 6*A*b^10*d^5*e + 135*B*a^2*b^8*d^4*e^2 + 30*A*a*b^9*d^4*e^2 - 160*B*a^3
*b^7*d^3*e^3 - 60*A*a^2*b^8*d^3*e^3 + 105*B*a^4*b^6*d^2*e^4 + 60*A*a^3*b^7*d^2*e^4 - 36*B*a^5*b^5*d*e^5 - 30*A
*a^4*b^6*d*e^5 + 5*B*a^6*b^4*e^6 + 6*A*a^5*b^5*e^6)*e^(-12)*log(abs(x*e + d)) + 1/60*(10*B*b^10*x^6*e^30 - 72*
B*b^10*d*x^5*e^29 + 315*B*b^10*d^2*x^4*e^28 - 1120*B*b^10*d^3*x^3*e^27 + 3780*B*b^10*d^4*x^2*e^26 - 15120*B*b^
10*d^5*x*e^25 + 120*B*a*b^9*x^5*e^30 + 12*A*b^10*x^5*e^30 - 900*B*a*b^9*d*x^4*e^29 - 90*A*b^10*d*x^4*e^29 + 42
00*B*a*b^9*d^2*x^3*e^28 + 420*A*b^10*d^2*x^3*e^28 - 16800*B*a*b^9*d^3*x^2*e^27 - 1680*A*b^10*d^3*x^2*e^27 + 75
600*B*a*b^9*d^4*x*e^26 + 7560*A*b^10*d^4*x*e^26 + 675*B*a^2*b^8*x^4*e^30 + 150*A*a*b^9*x^4*e^30 - 5400*B*a^2*b
^8*d*x^3*e^29 - 1200*A*a*b^9*d*x^3*e^29 + 28350*B*a^2*b^8*d^2*x^2*e^28 + 6300*A*a*b^9*d^2*x^2*e^28 - 151200*B*
a^2*b^8*d^3*x*e^27 - 33600*A*a*b^9*d^3*x*e^27 + 2400*B*a^3*b^7*x^3*e^30 + 900*A*a^2*b^8*x^3*e^30 - 21600*B*a^3
*b^7*d*x^2*e^29 - 8100*A*a^2*b^8*d*x^2*e^29 + 151200*B*a^3*b^7*d^2*x*e^28 + 56700*A*a^2*b^8*d^2*x*e^28 + 6300*
B*a^4*b^6*x^2*e^30 + 3600*A*a^3*b^7*x^2*e^30 - 75600*B*a^4*b^6*d*x*e^29 - 43200*A*a^3*b^7*d*x*e^29 + 15120*B*a
^5*b^5*x*e^30 + 12600*A*a^4*b^6*x*e^30)*e^(-36) + 1/60*(15797*B*b^10*d^11 - 97620*B*a*b^9*d^10*e - 9762*A*b^10
*d^10*e + 253665*B*a^2*b^8*d^9*e^2 + 56370*A*a*b^9*d^9*e^2 - 356640*B*a^3*b^7*d^8*e^3 - 133740*A*a^2*b^8*d^8*e
^3 + 289170*B*a^4*b^6*d^7*e^4 + 165240*A*a^3*b^7*d^7*e^4 - 131544*B*a^5*b^5*d^6*e^5 - 109620*A*a^4*b^6*d^6*e^5
 + 28770*B*a^6*b^4*d^5*e^6 + 34524*A*a^5*b^5*d^5*e^6 - 1440*B*a^7*b^3*d^4*e^7 - 2520*A*a^6*b^4*d^4*e^7 - 135*B
*a^8*b^2*d^3*e^8 - 360*A*a^7*b^3*d^3*e^8 - 20*B*a^9*b*d^2*e^9 - 90*A*a^8*b^2*d^2*e^9 - 3*B*a^10*d*e^10 - 30*A*
a^9*b*d*e^10 - 12*A*a^10*e^11 + 1800*(11*B*b^10*d^7*e^4 - 70*B*a*b^9*d^6*e^5 - 7*A*b^10*d^6*e^5 + 189*B*a^2*b^
8*d^5*e^6 + 42*A*a*b^9*d^5*e^6 - 280*B*a^3*b^7*d^4*e^7 - 105*A*a^2*b^8*d^4*e^7 + 245*B*a^4*b^6*d^3*e^8 + 140*A
*a^3*b^7*d^3*e^8 - 126*B*a^5*b^5*d^2*e^9 - 105*A*a^4*b^6*d^2*e^9 + 35*B*a^6*b^4*d*e^10 + 42*A*a^5*b^5*d*e^10 -
 4*B*a^7*b^3*e^11 - 7*A*a^6*b^4*e^11)*x^4 + 450*(165*B*b^10*d^8*e^3 - 1040*B*a*b^9*d^7*e^4 - 104*A*b^10*d^7*e^
4 + 2772*B*a^2*b^8*d^6*e^5 + 616*A*a*b^9*d^6*e^5 - 4032*B*a^3*b^7*d^5*e^6 - 1512*A*a^2*b^8*d^5*e^6 + 3430*B*a^
4*b^6*d^4*e^7 + 1960*A*a^3*b^7*d^4*e^7 - 1680*B*a^5*b^5*d^3*e^8 - 1400*A*a^4*b^6*d^3*e^8 + 420*B*a^6*b^4*d^2*e
^9 + 504*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 - 3*B*a^8*b^2*e^11 - 8*A*a^7*b^3*e^11)*
x^3 + 50*(2101*B*b^10*d^9*e^2 - 13140*B*a*b^9*d^8*e^3 - 1314*A*b^10*d^8*e^3 + 34668*B*a^2*b^8*d^7*e^4 + 7704*A
*a*b^9*d^7*e^4 - 49728*B*a^3*b^7*d^6*e^5 - 18648*A*a^2*b^8*d^6*e^5 + 41454*B*a^4*b^6*d^5*e^6 + 23688*A*a^3*b^7
*d^5*e^6 - 19656*B*a^5*b^5*d^4*e^7 - 16380*A*a^4*b^6*d^4*e^7 + 4620*B*a^6*b^4*d^3*e^8 + 5544*A*a^5*b^5*d^3*e^8
 - 288*B*a^7*b^3*d^2*e^9 - 504*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 - 4*B*a^9*b*e^11
- 18*A*a^8*b^2*e^11)*x^2 + 5*(13277*B*b^10*d^10*e - 82500*B*a*b^9*d^9*e^2 - 8250*A*b^10*d^9*e^2 + 215865*B*a^2
*b^8*d^8*e^3 + 47970*A*a*b^9*d^8*e^3 - 306240*B*a^3*b^7*d^7*e^4 - 114840*A*a^2*b^8*d^7*e^4 + 251370*B*a^4*b^6*
d^6*e^5 + 143640*A*a^3*b^7*d^6*e^5 - 116424*B*a^5*b^5*d^5*e^6 - 97020*A*a^4*b^6*d^5*e^6 + 26250*B*a^6*b^4*d^4*
e^7 + 31500*A*a^5*b^5*d^4*e^7 - 1440*B*a^7*b^3*d^3*e^8 - 2520*A*a^6*b^4*d^3*e^8 - 135*B*a^8*b^2*d^2*e^9 - 360*
A*a^7*b^3*d^2*e^9 - 20*B*a^9*b*d*e^10 - 90*A*a^8*b^2*d*e^10 - 3*B*a^10*e^11 - 30*A*a^9*b*e^11)*x)*e^(-12)/(x*e
 + d)^5

________________________________________________________________________________________

maple [B]  time = 0.04, size = 2731, normalized size = 6.11 \begin {gather*} \text {Expression too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10*(B*x+A)/(e*x+d)^6,x)

[Out]

2520*b^7/e^8*B*a^3*d^2*x-2520*b^8/e^9*B*a^2*d^3*x+1260*b^9/e^10*B*a*d^4*x-15*b^9/e^7*B*x^4*a*d+1260*b^5/e^6/(e
*x+d)*A*a^5*d-3150*b^6/e^7/(e*x+d)*A*a^4*d^2+4200*b^7/e^8/(e*x+d)*A*a^3*d^3-3150*b^8/e^9/(e*x+d)*A*a^2*d^4+126
0*b^9/e^10/(e*x+d)*A*a*d^5+1050*b^4/e^6/(e*x+d)*B*a^6*d-3780*b^5/e^7/(e*x+d)*B*a^5*d^2+7350*b^6/e^8/(e*x+d)*B*
a^4*d^3-8400*b^7/e^9/(e*x+d)*B*a^3*d^4+5670*b^8/e^10/(e*x+d)*B*a^2*d^5-2100*b^9/e^11/(e*x+d)*B*a*d^6+120*b^3/e
^4/(e*x+d)^3*A*a^7*d-420*b^4/e^5/(e*x+d)^3*A*a^6*d^2+840*b^5/e^6/(e*x+d)^3*A*a^5*d^3-1050*b^6/e^7/(e*x+d)^3*A*
a^4*d^4+840*b^7/e^8/(e*x+d)^3*A*a^3*d^5-420*b^8/e^9/(e*x+d)^3*A*a^2*d^6+120*b^9/e^10/(e*x+d)^3*A*a*d^7+45*b^2/
e^4/(e*x+d)^3*B*a^8*d-240*b^3/e^5/(e*x+d)^3*B*a^7*d^2+700*b^4/e^6/(e*x+d)^3*B*a^6*d^3-1260*b^5/e^7/(e*x+d)^3*B
*a^5*d^4+1470*b^6/e^8/(e*x+d)^3*B*a^4*d^5-1120*b^7/e^9/(e*x+d)^3*B*a^3*d^6+540*b^8/e^10/(e*x+d)^3*B*a^2*d^7-15
0*b^9/e^11/(e*x+d)^3*B*a*d^8+45/2/e^3/(e*x+d)^4*A*a^8*b^2*d-90/e^4/(e*x+d)^4*A*a^7*b^3*d^2+210/e^5/(e*x+d)^4*A
*a^6*b^4*d^3-315/e^6/(e*x+d)^4*A*a^5*b^5*d^4+315/e^7/(e*x+d)^4*A*a^4*b^6*d^5-210/e^8/(e*x+d)^4*A*a^3*b^7*d^6-7
20*b^7/e^7*A*a^3*d*x+945/2*b^8/e^8*B*x^2*a^2*d^2-280*b^9/e^9*B*x^2*a*d^3+945*b^8/e^8*A*a^2*d^2*x-560*b^9/e^9*A
*a*d^3*x-1260*b^6/e^7*B*a^4*d*x-1/5/e/(e*x+d)^5*a^10*A+1/6*b^10/e^6*B*x^6+1/5*b^10/e^6*A*x^5-1/4/e^2/(e*x+d)^4
*B*a^10+90/e^9/(e*x+d)^4*A*a^2*b^8*d^7-45/2/e^10/(e*x+d)^4*A*a*b^9*d^8+5/e^3/(e*x+d)^4*B*a^9*b*d-135/4/e^4/(e*
x+d)^4*B*a^8*b^2*d^2+120/e^5/(e*x+d)^4*B*a^7*b^3*d^3-525/2/e^6/(e*x+d)^4*B*a^6*b^4*d^4+378/e^7/(e*x+d)^4*B*a^5
*b^5*d^5-735/2/e^8/(e*x+d)^4*B*a^4*b^6*d^6+240/e^9/(e*x+d)^4*B*a^3*b^7*d^7-405/4/e^10/(e*x+d)^4*B*a^2*b^8*d^8+
25/e^11/(e*x+d)^4*B*a*b^9*d^9+2/e^2/(e*x+d)^5*A*d*a^9*b-9/e^3/(e*x+d)^5*A*d^2*a^8*b^2+24/e^4/(e*x+d)^5*A*d^3*a
^7*b^3-42/e^5/(e*x+d)^5*A*d^4*a^6*b^4+252/5/e^6/(e*x+d)^5*A*d^5*a^5*b^5-42/e^7/(e*x+d)^5*A*a^4*b^6*d^6+24/e^8/
(e*x+d)^5*A*a^3*b^7*d^7-9/e^9/(e*x+d)^5*A*a^2*b^8*d^8+2/e^10/(e*x+d)^5*A*a*b^9*d^9-2/e^3/(e*x+d)^5*B*d^2*a^9*b
+9/e^4/(e*x+d)^5*B*d^3*a^8*b^2-24/e^5/(e*x+d)^5*B*d^4*a^7*b^3+42/e^6/(e*x+d)^5*B*d^5*a^6*b^4-252/5/e^7/(e*x+d)
^5*B*a^5*b^5*d^6+42/e^8/(e*x+d)^5*B*a^4*b^6*d^7-24/e^9/(e*x+d)^5*B*a^3*b^7*d^8+9/e^10/(e*x+d)^5*B*a^2*b^8*d^9-
2/e^11/(e*x+d)^5*B*a*b^9*d^10-1260*b^6/e^7*ln(e*x+d)*A*a^4*d+2520*b^7/e^8*ln(e*x+d)*A*a^3*d^2-2520*b^8/e^9*ln(
e*x+d)*A*a^2*d^3+1260*b^9/e^10*ln(e*x+d)*A*a*d^4-1512*b^5/e^7*ln(e*x+d)*B*a^5*d+4410*b^6/e^8*ln(e*x+d)*B*a^4*d
^2-6720*b^7/e^9*ln(e*x+d)*B*a^3*d^3+5670*b^8/e^10*ln(e*x+d)*B*a^2*d^4-2520*b^9/e^11*ln(e*x+d)*B*a*d^5+420*b^4/
e^5/(e*x+d)^2*A*a^6*d-1260*b^5/e^6/(e*x+d)^2*A*a^5*d^2+2100*b^6/e^7/(e*x+d)^2*A*a^4*d^3-2100*b^7/e^8/(e*x+d)^2
*A*a^3*d^4+1260*b^8/e^9/(e*x+d)^2*A*a^2*d^5-420*b^9/e^10/(e*x+d)^2*A*a*d^6+240*b^3/e^5/(e*x+d)^2*B*a^7*d-1050*
b^4/e^6/(e*x+d)^2*B*a^6*d^2+2520*b^5/e^7/(e*x+d)^2*B*a^5*d^3-3675*b^6/e^8/(e*x+d)^2*B*a^4*d^4+3360*b^7/e^9/(e*
x+d)^2*B*a^3*d^5-1890*b^8/e^10/(e*x+d)^2*B*a^2*d^6+600*b^9/e^11/(e*x+d)^2*B*a*d^7+70*b^9/e^8*B*x^3*a*d^2-135*b
^8/e^7*A*x^2*a^2*d+105*b^9/e^8*A*x^2*a*d^2-360*b^7/e^7*B*x^2*a^3*d-20*b^9/e^7*A*x^3*a*d-90*b^8/e^7*B*x^3*a^2*d
+252*b^5/e^6*B*a^5*x-252*b^10/e^11*B*d^5*x-28*b^10/e^9*A*x^2*d^3+105*b^6/e^6*B*x^2*a^4+63*b^10/e^10*B*x^2*d^4+
210*b^6/e^6*A*a^4*x+126*b^10/e^10*A*d^4*x-3/2*b^10/e^7*A*x^4*d+45/4*b^8/e^6*B*x^4*a^2+21/4*b^10/e^8*B*x^4*d^2-
210*b^4/e^5/(e*x+d)*A*a^6-210*b^10/e^11/(e*x+d)*A*d^6-120*b^3/e^5/(e*x+d)*B*a^7+330*b^10/e^12/(e*x+d)*B*d^7-15
*b^2/e^3/(e*x+d)^3*A*a^8-15*b^10/e^11/(e*x+d)^3*A*d^8-10/3*b/e^3/(e*x+d)^3*B*a^9+55/3*b^10/e^12/(e*x+d)^3*B*d^
9-5/2/e^2/(e*x+d)^4*A*a^9*b+5/2/e^11/(e*x+d)^4*A*b^10*d^9-11/4/e^12/(e*x+d)^4*b^10*B*d^10-1/5/e^11/(e*x+d)^5*A
*b^10*d^10+1/5/e^2/(e*x+d)^5*B*d*a^10+1/5/e^12/(e*x+d)^5*b^10*B*d^11+252*b^5/e^6*ln(e*x+d)*A*a^5-252*b^10/e^11
*ln(e*x+d)*A*d^5+210*b^4/e^6*ln(e*x+d)*B*a^6+462*b^10/e^12*ln(e*x+d)*B*d^6-60*b^3/e^4/(e*x+d)^2*A*a^7+60*b^10/
e^11/(e*x+d)^2*A*d^7-45/2*b^2/e^4/(e*x+d)^2*B*a^8-165/2*b^10/e^12/(e*x+d)^2*B*d^8+7*b^10/e^8*A*x^3*d^2+15*b^8/
e^6*A*x^3*a^2+40*b^7/e^6*B*x^3*a^3-56/3*b^10/e^9*B*x^3*d^3+60*b^7/e^6*A*x^2*a^3-6/5*b^10/e^7*B*x^5*d+5/2*b^9/e
^6*A*x^4*a+2*b^9/e^6*B*x^5*a

________________________________________________________________________________________

maxima [B]  time = 1.59, size = 1861, 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)^6,x, algorithm="maxima")

[Out]

1/60*(15797*B*b^10*d^11 - 12*A*a^10*e^11 - 9762*(10*B*a*b^9 + A*b^10)*d^10*e + 28185*(9*B*a^2*b^8 + 2*A*a*b^9)
*d^9*e^2 - 44580*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 41310*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 21924*(6*B*
a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 5754*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - 360*(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 - 10*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 - 3*(B*a^10 + 10*A*a^9*b
)*d*e^10 + 1800*(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 + 450*(165*B*b^10*d^
8*e^3 - 104*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 308*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 - 504*(8*B*a^3*b^7 + 3*A*a^2
*b^8)*d^5*e^6 + 490*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 - 280*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + 84*(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 +
50*(2101*B*b^10*d^9*e^2 - 1314*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 3852*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 - 6216*(
8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 5922*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 - 3276*(6*B*a^5*b^5 + 5*A*a^4*b^
6)*d^4*e^7 + 924*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 - 72*(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*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 5*(13277*B*b^10*d^10*e - 8250*(10*B*a*b^9 + A
*b^10)*d^9*e^2 + 23985*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 - 38280*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 35910*(
7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 - 19404*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 5250*(5*B*a^6*b^4 + 6*A*a^5*b
^5)*d^4*e^7 - 360*(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 - 10*(2*B*a^9*b
 + 9*A*a^8*b^2)*d*e^10 - 3*(B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^17*x^5 + 5*d*e^16*x^4 + 10*d^2*e^15*x^3 + 10*d^3*
e^14*x^2 + 5*d^4*e^13*x + d^5*e^12) + 1/60*(10*B*b^10*e^5*x^6 - 12*(6*B*b^10*d*e^4 - (10*B*a*b^9 + A*b^10)*e^5
)*x^5 + 15*(21*B*b^10*d^2*e^3 - 6*(10*B*a*b^9 + A*b^10)*d*e^4 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*e^5)*x^4 - 20*(56*
B*b^10*d^3*e^2 - 21*(10*B*a*b^9 + A*b^10)*d^2*e^3 + 30*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^4 - 15*(8*B*a^3*b^7 + 3*A
*a^2*b^8)*e^5)*x^3 + 30*(126*B*b^10*d^4*e - 56*(10*B*a*b^9 + A*b^10)*d^3*e^2 + 105*(9*B*a^2*b^8 + 2*A*a*b^9)*d
^2*e^3 - 90*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^4 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^5)*x^2 - 60*(252*B*b^10*d^5 -
 126*(10*B*a*b^9 + A*b^10)*d^4*e + 280*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^2 - 315*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2
*e^3 + 180*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^4 - 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^5)*x)/e^11 + 42*(11*B*b^10*d^6
 - 6*(10*B*a*b^9 + A*b^10)*d^5*e + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^2 - 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e
^3 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^4 - 6*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^5 + (5*B*a^6*b^4 + 6*A*a^5*b^5
)*e^6)*log(e*x + d)/e^12

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mupad [B]  time = 0.42, size = 2681, normalized size = 6.00

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)^6,x)

[Out]

x^5*((A*b^10 + 10*B*a*b^9)/(5*e^6) - (6*B*b^10*d)/(5*e^7)) - x^4*((3*d*((A*b^10 + 10*B*a*b^9)/e^6 - (6*B*b^10*
d)/e^7))/(2*e) - (5*a*b^8*(2*A*b + 9*B*a))/(4*e^6) + (15*B*b^10*d^2)/(4*e^8)) - (x^4*(210*A*a^6*b^4*e^10 + 120
*B*a^7*b^3*e^10 + 210*A*b^10*d^6*e^4 - 330*B*b^10*d^7*e^3 - 1260*A*a*b^9*d^5*e^5 - 1260*A*a^5*b^5*d*e^9 + 2100
*B*a*b^9*d^6*e^4 - 1050*B*a^6*b^4*d*e^9 + 3150*A*a^2*b^8*d^4*e^6 - 4200*A*a^3*b^7*d^3*e^7 + 3150*A*a^4*b^6*d^2
*e^8 - 5670*B*a^2*b^8*d^5*e^5 + 8400*B*a^3*b^7*d^4*e^6 - 7350*B*a^4*b^6*d^3*e^7 + 3780*B*a^5*b^5*d^2*e^8) + x^
3*(60*A*a^7*b^3*e^10 + (45*B*a^8*b^2*e^10)/2 + 780*A*b^10*d^7*e^3 - (2475*B*b^10*d^8*e^2)/2 - 4620*A*a*b^9*d^6
*e^4 + 420*A*a^6*b^4*d*e^9 + 7800*B*a*b^9*d^7*e^3 + 240*B*a^7*b^3*d*e^9 + 11340*A*a^2*b^8*d^5*e^5 - 14700*A*a^
3*b^7*d^4*e^6 + 10500*A*a^4*b^6*d^3*e^7 - 3780*A*a^5*b^5*d^2*e^8 - 20790*B*a^2*b^8*d^6*e^4 + 30240*B*a^3*b^7*d
^5*e^5 - 25725*B*a^4*b^6*d^4*e^6 + 12600*B*a^5*b^5*d^3*e^7 - 3150*B*a^6*b^4*d^2*e^8) + (12*A*a^10*e^11 - 15797
*B*b^10*d^11 + 9762*A*b^10*d^10*e + 3*B*a^10*d*e^10 - 56370*A*a*b^9*d^9*e^2 + 20*B*a^9*b*d^2*e^9 + 133740*A*a^
2*b^8*d^8*e^3 - 165240*A*a^3*b^7*d^7*e^4 + 109620*A*a^4*b^6*d^6*e^5 - 34524*A*a^5*b^5*d^5*e^6 + 2520*A*a^6*b^4
*d^4*e^7 + 360*A*a^7*b^3*d^3*e^8 + 90*A*a^8*b^2*d^2*e^9 - 253665*B*a^2*b^8*d^9*e^2 + 356640*B*a^3*b^7*d^8*e^3
- 289170*B*a^4*b^6*d^7*e^4 + 131544*B*a^5*b^5*d^6*e^5 - 28770*B*a^6*b^4*d^5*e^6 + 1440*B*a^7*b^3*d^4*e^7 + 135
*B*a^8*b^2*d^3*e^8 + 30*A*a^9*b*d*e^10 + 97620*B*a*b^9*d^10*e)/(60*e) + x*((B*a^10*e^10)/4 - (13277*B*b^10*d^1
0)/12 + (5*A*a^9*b*e^10)/2 + (1375*A*b^10*d^9*e)/2 - (7995*A*a*b^9*d^8*e^2)/2 + (15*A*a^8*b^2*d*e^9)/2 + 9570*
A*a^2*b^8*d^7*e^3 - 11970*A*a^3*b^7*d^6*e^4 + 8085*A*a^4*b^6*d^5*e^5 - 2625*A*a^5*b^5*d^4*e^6 + 210*A*a^6*b^4*
d^3*e^7 + 30*A*a^7*b^3*d^2*e^8 - (71955*B*a^2*b^8*d^8*e^2)/4 + 25520*B*a^3*b^7*d^7*e^3 - (41895*B*a^4*b^6*d^6*
e^4)/2 + 9702*B*a^5*b^5*d^5*e^5 - (4375*B*a^6*b^4*d^4*e^6)/2 + 120*B*a^7*b^3*d^3*e^7 + (45*B*a^8*b^2*d^2*e^8)/
4 + 6875*B*a*b^9*d^9*e + (5*B*a^9*b*d*e^9)/3) + x^2*((10*B*a^9*b*e^10)/3 - (10505*B*b^10*d^9*e)/6 + 15*A*a^8*b
^2*e^10 + 1095*A*b^10*d^8*e^2 - 6420*A*a*b^9*d^7*e^3 + 60*A*a^7*b^3*d*e^9 + 10950*B*a*b^9*d^8*e^2 + (45*B*a^8*
b^2*d*e^9)/2 + 15540*A*a^2*b^8*d^6*e^4 - 19740*A*a^3*b^7*d^5*e^5 + 13650*A*a^4*b^6*d^4*e^6 - 4620*A*a^5*b^5*d^
3*e^7 + 420*A*a^6*b^4*d^2*e^8 - 28890*B*a^2*b^8*d^7*e^3 + 41440*B*a^3*b^7*d^6*e^4 - 34545*B*a^4*b^6*d^5*e^5 +
16380*B*a^5*b^5*d^4*e^6 - 3850*B*a^6*b^4*d^3*e^7 + 240*B*a^7*b^3*d^2*e^8))/(d^5*e^11 + e^16*x^5 + 5*d^4*e^12*x
 + 5*d*e^15*x^4 + 10*d^3*e^13*x^2 + 10*d^2*e^14*x^3) - x^3*((5*d^2*((A*b^10 + 10*B*a*b^9)/e^6 - (6*B*b^10*d)/e
^7))/e^2 - (2*d*((6*d*((A*b^10 + 10*B*a*b^9)/e^6 - (6*B*b^10*d)/e^7))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^6 + (15*
B*b^10*d^2)/e^8))/e - (5*a^2*b^7*(3*A*b + 8*B*a))/e^6 + (20*B*b^10*d^3)/(3*e^9)) - x*((6*d*((6*d*((15*d^2*((A*
b^10 + 10*B*a*b^9)/e^6 - (6*B*b^10*d)/e^7))/e^2 - (6*d*((6*d*((A*b^10 + 10*B*a*b^9)/e^6 - (6*B*b^10*d)/e^7))/e
 - (5*a*b^8*(2*A*b + 9*B*a))/e^6 + (15*B*b^10*d^2)/e^8))/e - (15*a^2*b^7*(3*A*b + 8*B*a))/e^6 + (20*B*b^10*d^3
)/e^9))/e - (20*d^3*((A*b^10 + 10*B*a*b^9)/e^6 - (6*B*b^10*d)/e^7))/e^3 + (15*d^2*((6*d*((A*b^10 + 10*B*a*b^9)
/e^6 - (6*B*b^10*d)/e^7))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^6 + (15*B*b^10*d^2)/e^8))/e^2 + (30*a^3*b^6*(4*A*b +
 7*B*a))/e^6 - (15*B*b^10*d^4)/e^10))/e - (15*d^2*((15*d^2*((A*b^10 + 10*B*a*b^9)/e^6 - (6*B*b^10*d)/e^7))/e^2
 - (6*d*((6*d*((A*b^10 + 10*B*a*b^9)/e^6 - (6*B*b^10*d)/e^7))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^6 + (15*B*b^10*d
^2)/e^8))/e - (15*a^2*b^7*(3*A*b + 8*B*a))/e^6 + (20*B*b^10*d^3)/e^9))/e^2 + (15*d^4*((A*b^10 + 10*B*a*b^9)/e^
6 - (6*B*b^10*d)/e^7))/e^4 - (20*d^3*((6*d*((A*b^10 + 10*B*a*b^9)/e^6 - (6*B*b^10*d)/e^7))/e - (5*a*b^8*(2*A*b
 + 9*B*a))/e^6 + (15*B*b^10*d^2)/e^8))/e^3 - (42*a^4*b^5*(5*A*b + 6*B*a))/e^6 + (6*B*b^10*d^5)/e^11) + x^2*((3
*d*((15*d^2*((A*b^10 + 10*B*a*b^9)/e^6 - (6*B*b^10*d)/e^7))/e^2 - (6*d*((6*d*((A*b^10 + 10*B*a*b^9)/e^6 - (6*B
*b^10*d)/e^7))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^6 + (15*B*b^10*d^2)/e^8))/e - (15*a^2*b^7*(3*A*b + 8*B*a))/e^6
+ (20*B*b^10*d^3)/e^9))/e - (10*d^3*((A*b^10 + 10*B*a*b^9)/e^6 - (6*B*b^10*d)/e^7))/e^3 + (15*d^2*((6*d*((A*b^
10 + 10*B*a*b^9)/e^6 - (6*B*b^10*d)/e^7))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^6 + (15*B*b^10*d^2)/e^8))/(2*e^2) +
(15*a^3*b^6*(4*A*b + 7*B*a))/e^6 - (15*B*b^10*d^4)/(2*e^10)) + (log(d + e*x)*(462*B*b^10*d^6 - 252*A*b^10*d^5*
e + 252*A*a^5*b^5*e^6 + 210*B*a^6*b^4*e^6 + 1260*A*a*b^9*d^4*e^2 - 1260*A*a^4*b^6*d*e^5 - 1512*B*a^5*b^5*d*e^5
 - 2520*A*a^2*b^8*d^3*e^3 + 2520*A*a^3*b^7*d^2*e^4 + 5670*B*a^2*b^8*d^4*e^2 - 6720*B*a^3*b^7*d^3*e^3 + 4410*B*
a^4*b^6*d^2*e^4 - 2520*B*a*b^9*d^5*e))/e^12 + (B*b^10*x^6)/(6*e^6)

________________________________________________________________________________________

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)**6,x)

[Out]

Timed out

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