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

Optimal. Leaf size=441 \[ -\frac {b^9 x (-10 a B e-A b e+10 b B d)}{e^{11}}+\frac {5 b^8 (b d-a e) \log (d+e x) (-9 a B e-2 A b e+11 b B d)}{e^{12}}+\frac {15 b^7 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{e^{12} (d+e x)}-\frac {15 b^6 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{e^{12} (d+e x)^2}+\frac {14 b^5 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{e^{12} (d+e x)^3}-\frac {21 b^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{2 e^{12} (d+e x)^4}+\frac {6 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{e^{12} (d+e x)^5}-\frac {5 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{2 e^{12} (d+e x)^6}+\frac {5 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{7 e^{12} (d+e x)^7}-\frac {(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{8 e^{12} (d+e x)^8}+\frac {(b d-a e)^{10} (B d-A e)}{9 e^{12} (d+e x)^9}+\frac {b^{10} B x^2}{2 e^{10}} \]

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Rubi [A]  time = 0.89, antiderivative size = 441, 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 x (-10 a B e-A b e+10 b B d)}{e^{11}}+\frac {15 b^7 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{e^{12} (d+e x)}-\frac {15 b^6 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{e^{12} (d+e x)^2}+\frac {14 b^5 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{e^{12} (d+e x)^3}-\frac {21 b^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{2 e^{12} (d+e x)^4}+\frac {6 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{e^{12} (d+e x)^5}-\frac {5 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)^6}+\frac {5 b^8 (b d-a e) \log (d+e x) (-9 a B e-2 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)}{7 e^{12} (d+e x)^7}-\frac {(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{8 e^{12} (d+e x)^8}+\frac {(b d-a e)^{10} (B d-A e)}{9 e^{12} (d+e x)^9}+\frac {b^{10} B x^2}{2 e^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [B]  time = 1.11, size = 1460, normalized size = 3.31 \begin {gather*} -\frac {-\left (\left (B \left (42131 d^{11}+351459 e x d^{10}+1281096 e^2 x^2 d^9+2656584 e^3 x^3 d^8+3402756 e^4 x^4 d^7+2704212 e^5 x^5 d^6+1220688 e^6 x^6 d^5+190512 e^7 x^7 d^4-77112 e^8 x^8 d^3-36288 e^9 x^9 d^2-2772 e^{10} x^{10} d+252 e^{11} x^{11}\right )-2 A e \left (4861 d^{10}+41229 e x d^9+153576 e^2 x^2 d^8+328104 e^3 x^3 d^7+439236 e^4 x^4 d^6+375732 e^5 x^5 d^5+197568 e^6 x^6 d^4+54432 e^7 x^7 d^3+2268 e^8 x^8 d^2-2268 e^9 x^9 d-252 e^{10} x^{10}\right )\right ) b^{10}\right )-2 a e \left (A d e \left (7129 d^8+61641 e x d^7+235224 e^2 x^2 d^6+518616 e^3 x^3 d^5+725004 e^4 x^4 d^4+661500 e^5 x^5 d^3+388080 e^6 x^6 d^2+136080 e^7 x^7 d+22680 e^8 x^8\right )-10 B \left (4861 d^{10}+41229 e x d^9+153576 e^2 x^2 d^8+328104 e^3 x^3 d^7+439236 e^4 x^4 d^6+375732 e^5 x^5 d^5+197568 e^6 x^6 d^4+54432 e^7 x^7 d^3+2268 e^8 x^8 d^2-2268 e^9 x^9 d-252 e^{10} x^{10}\right )\right ) b^9-9 a^2 e^2 \left (B d \left (7129 d^8+61641 e x d^7+235224 e^2 x^2 d^6+518616 e^3 x^3 d^5+725004 e^4 x^4 d^4+661500 e^5 x^5 d^3+388080 e^6 x^6 d^2+136080 e^7 x^7 d+22680 e^8 x^8\right )-280 A e \left (d^8+9 e x d^7+36 e^2 x^2 d^6+84 e^3 x^3 d^5+126 e^4 x^4 d^4+126 e^5 x^5 d^3+84 e^6 x^6 d^2+36 e^7 x^7 d+9 e^8 x^8\right )\right ) b^8-2520 (b d-a e) (11 b B d-2 A b e-9 a B e) (d+e x)^9 \log (d+e x) b^8+840 a^3 e^3 \left (A e \left (d^7+9 e x d^6+36 e^2 x^2 d^5+84 e^3 x^3 d^4+126 e^4 x^4 d^3+126 e^5 x^5 d^2+84 e^6 x^6 d+36 e^7 x^7\right )+8 B \left (d^8+9 e x d^7+36 e^2 x^2 d^6+84 e^3 x^3 d^5+126 e^4 x^4 d^4+126 e^5 x^5 d^3+84 e^6 x^6 d^2+36 e^7 x^7 d+9 e^8 x^8\right )\right ) b^7+210 a^4 e^4 \left (2 A e \left (d^6+9 e x d^5+36 e^2 x^2 d^4+84 e^3 x^3 d^3+126 e^4 x^4 d^2+126 e^5 x^5 d+84 e^6 x^6\right )+7 B \left (d^7+9 e x d^6+36 e^2 x^2 d^5+84 e^3 x^3 d^4+126 e^4 x^4 d^3+126 e^5 x^5 d^2+84 e^6 x^6 d+36 e^7 x^7\right )\right ) b^6+252 a^5 e^5 \left (A e \left (d^5+9 e x d^4+36 e^2 x^2 d^3+84 e^3 x^3 d^2+126 e^4 x^4 d+126 e^5 x^5\right )+2 B \left (d^6+9 e x d^5+36 e^2 x^2 d^4+84 e^3 x^3 d^3+126 e^4 x^4 d^2+126 e^5 x^5 d+84 e^6 x^6\right )\right ) b^5+42 a^6 e^6 \left (4 A e \left (d^4+9 e x d^3+36 e^2 x^2 d^2+84 e^3 x^3 d+126 e^4 x^4\right )+5 B \left (d^5+9 e x d^4+36 e^2 x^2 d^3+84 e^3 x^3 d^2+126 e^4 x^4 d+126 e^5 x^5\right )\right ) b^4+24 a^7 e^7 \left (5 A e \left (d^3+9 e x d^2+36 e^2 x^2 d+84 e^3 x^3\right )+4 B \left (d^4+9 e x d^3+36 e^2 x^2 d^2+84 e^3 x^3 d+126 e^4 x^4\right )\right ) b^3+45 a^8 e^8 \left (2 A e \left (d^2+9 e x d+36 e^2 x^2\right )+B \left (d^3+9 e x d^2+36 e^2 x^2 d+84 e^3 x^3\right )\right ) b^2+10 a^9 e^9 \left (7 A e (d+9 e x)+2 B \left (d^2+9 e x d+36 e^2 x^2\right )\right ) b+7 a^{10} e^{10} (8 A e+B (d+9 e x))}{504 e^{12} (d+e x)^9} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/504*(7*a^10*e^10*(8*A*e + B*(d + 9*e*x)) + 10*a^9*b*e^9*(7*A*e*(d + 9*e*x) + 2*B*(d^2 + 9*d*e*x + 36*e^2*x^
2)) + 45*a^8*b^2*e^8*(2*A*e*(d^2 + 9*d*e*x + 36*e^2*x^2) + B*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3)) +
24*a^7*b^3*e^7*(5*A*e*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3) + 4*B*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 +
84*d*e^3*x^3 + 126*e^4*x^4)) + 42*a^6*b^4*e^6*(4*A*e*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^
4*x^4) + 5*B*(d^5 + 9*d^4*e*x + 36*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 126*d*e^4*x^4 + 126*e^5*x^5)) + 252*a^5*b^5*
e^5*(A*e*(d^5 + 9*d^4*e*x + 36*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 126*d*e^4*x^4 + 126*e^5*x^5) + 2*B*(d^6 + 9*d^5*
e*x + 36*d^4*e^2*x^2 + 84*d^3*e^3*x^3 + 126*d^2*e^4*x^4 + 126*d*e^5*x^5 + 84*e^6*x^6)) + 210*a^4*b^6*e^4*(2*A*
e*(d^6 + 9*d^5*e*x + 36*d^4*e^2*x^2 + 84*d^3*e^3*x^3 + 126*d^2*e^4*x^4 + 126*d*e^5*x^5 + 84*e^6*x^6) + 7*B*(d^
7 + 9*d^6*e*x + 36*d^5*e^2*x^2 + 84*d^4*e^3*x^3 + 126*d^3*e^4*x^4 + 126*d^2*e^5*x^5 + 84*d*e^6*x^6 + 36*e^7*x^
7)) + 840*a^3*b^7*e^3*(A*e*(d^7 + 9*d^6*e*x + 36*d^5*e^2*x^2 + 84*d^4*e^3*x^3 + 126*d^3*e^4*x^4 + 126*d^2*e^5*
x^5 + 84*d*e^6*x^6 + 36*e^7*x^7) + 8*B*(d^8 + 9*d^7*e*x + 36*d^6*e^2*x^2 + 84*d^5*e^3*x^3 + 126*d^4*e^4*x^4 +
126*d^3*e^5*x^5 + 84*d^2*e^6*x^6 + 36*d*e^7*x^7 + 9*e^8*x^8)) - 9*a^2*b^8*e^2*(-280*A*e*(d^8 + 9*d^7*e*x + 36*
d^6*e^2*x^2 + 84*d^5*e^3*x^3 + 126*d^4*e^4*x^4 + 126*d^3*e^5*x^5 + 84*d^2*e^6*x^6 + 36*d*e^7*x^7 + 9*e^8*x^8)
+ B*d*(7129*d^8 + 61641*d^7*e*x + 235224*d^6*e^2*x^2 + 518616*d^5*e^3*x^3 + 725004*d^4*e^4*x^4 + 661500*d^3*e^
5*x^5 + 388080*d^2*e^6*x^6 + 136080*d*e^7*x^7 + 22680*e^8*x^8)) - 2*a*b^9*e*(A*d*e*(7129*d^8 + 61641*d^7*e*x +
 235224*d^6*e^2*x^2 + 518616*d^5*e^3*x^3 + 725004*d^4*e^4*x^4 + 661500*d^3*e^5*x^5 + 388080*d^2*e^6*x^6 + 1360
80*d*e^7*x^7 + 22680*e^8*x^8) - 10*B*(4861*d^10 + 41229*d^9*e*x + 153576*d^8*e^2*x^2 + 328104*d^7*e^3*x^3 + 43
9236*d^6*e^4*x^4 + 375732*d^5*e^5*x^5 + 197568*d^4*e^6*x^6 + 54432*d^3*e^7*x^7 + 2268*d^2*e^8*x^8 - 2268*d*e^9
*x^9 - 252*e^10*x^10)) - b^10*(-2*A*e*(4861*d^10 + 41229*d^9*e*x + 153576*d^8*e^2*x^2 + 328104*d^7*e^3*x^3 + 4
39236*d^6*e^4*x^4 + 375732*d^5*e^5*x^5 + 197568*d^4*e^6*x^6 + 54432*d^3*e^7*x^7 + 2268*d^2*e^8*x^8 - 2268*d*e^
9*x^9 - 252*e^10*x^10) + B*(42131*d^11 + 351459*d^10*e*x + 1281096*d^9*e^2*x^2 + 2656584*d^8*e^3*x^3 + 3402756
*d^7*e^4*x^4 + 2704212*d^6*e^5*x^5 + 1220688*d^5*e^6*x^6 + 190512*d^4*e^7*x^7 - 77112*d^3*e^8*x^8 - 36288*d^2*
e^9*x^9 - 2772*d*e^10*x^10 + 252*e^11*x^11)) - 2520*b^8*(b*d - a*e)*(11*b*B*d - 2*A*b*e - 9*a*B*e)*(d + e*x)^9
*Log[d + e*x])/(e^12*(d + e*x)^9)

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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)^{10}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.29, size = 2501, normalized size = 5.67

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)^10,x, algorithm="fricas")

[Out]

1/504*(252*B*b^10*e^11*x^11 + 42131*B*b^10*d^11 - 56*A*a^10*e^11 - 9722*(10*B*a*b^9 + A*b^10)*d^10*e + 7129*(9
*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 - 840*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 - 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^
7*e^4 - 84*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 - 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - 24*(4*B*a^7*b^3 + 7*
A*a^6*b^4)*d^4*e^7 - 15*(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 - 7*(B*a^10
 + 10*A*a^9*b)*d*e^10 - 252*(11*B*b^10*d*e^10 - 2*(10*B*a*b^9 + A*b^10)*e^11)*x^10 - 4536*(8*B*b^10*d^2*e^9 -
(10*B*a*b^9 + A*b^10)*d*e^10)*x^9 - 1512*(51*B*b^10*d^3*e^8 + 3*(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 + 5*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 1512*(126*B*b^10*d^4*e^7 - 72*(10*B*a*b^9 +
A*b^10)*d^3*e^8 + 90*(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 - 5*(7*B*a^4*b^
6 + 4*A*a^3*b^7)*e^11)*x^7 + 3528*(346*B*b^10*d^5*e^6 - 112*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 110*(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 - 5*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 - 2*(6*B*a
^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 5292*(511*B*b^10*d^6*e^5 - 142*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 125*(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 - 5*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 - 2*
(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 + 756*(4501*B*b^10*d^7*e^4 - 1162*(
10*B*a*b^9 + A*b^10)*d^6*e^5 + 959*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 - 140*(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 - 14*(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*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 252*(10542*B*b^10*d^8*e^3 - 2604*(10*B*a*b^9 + A*b^10)
*d^7*e^4 + 2058*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 - 280*(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 - 28*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 - 14*(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 - 5*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 + 36*(35586*B*b^10*d^9*e^2 - 8532*
(10*B*a*b^9 + A*b^10)*d^8*e^3 + 6534*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 - 840*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e
^5 - 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 - 84*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 - 42*(5*B*a^6*b^4 + 6*A*
a^5*b^5)*d^3*e^8 - 24*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9 - 15*(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 + 9*(39051*B*b^10*d^10*e - 9162*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 6849*(9*B*a^2*b^8
 + 2*A*a*b^9)*d^8*e^3 - 840*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 - 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 - 84
*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 - 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 - 24*(4*B*a^7*b^3 + 7*A*a^6*b^4)
*d^3*e^8 - 15*(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 - 7*(B*a^10 + 10*A*a^9
*b)*e^11)*x + 2520*(11*B*b^10*d^11 - 2*(10*B*a*b^9 + A*b^10)*d^10*e + (9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + (11*
B*b^10*d^2*e^9 - 2*(10*B*a*b^9 + A*b^10)*d*e^10 + (9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 9*(11*B*b^10*d^3*e^8 -
 2*(10*B*a*b^9 + A*b^10)*d^2*e^9 + (9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10)*x^8 + 36*(11*B*b^10*d^4*e^7 - 2*(10*B*a*b
^9 + A*b^10)*d^3*e^8 + (9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9)*x^7 + 84*(11*B*b^10*d^5*e^6 - 2*(10*B*a*b^9 + A*b^10
)*d^4*e^7 + (9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8)*x^6 + 126*(11*B*b^10*d^6*e^5 - 2*(10*B*a*b^9 + A*b^10)*d^5*e^6
+ (9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7)*x^5 + 126*(11*B*b^10*d^7*e^4 - 2*(10*B*a*b^9 + A*b^10)*d^6*e^5 + (9*B*a^2
*b^8 + 2*A*a*b^9)*d^5*e^6)*x^4 + 84*(11*B*b^10*d^8*e^3 - 2*(10*B*a*b^9 + A*b^10)*d^7*e^4 + (9*B*a^2*b^8 + 2*A*
a*b^9)*d^6*e^5)*x^3 + 36*(11*B*b^10*d^9*e^2 - 2*(10*B*a*b^9 + A*b^10)*d^8*e^3 + (9*B*a^2*b^8 + 2*A*a*b^9)*d^7*
e^4)*x^2 + 9*(11*B*b^10*d^10*e - 2*(10*B*a*b^9 + A*b^10)*d^9*e^2 + (9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3)*x)*log(e
*x + d))/(e^21*x^9 + 9*d*e^20*x^8 + 36*d^2*e^19*x^7 + 84*d^3*e^18*x^6 + 126*d^4*e^17*x^5 + 126*d^5*e^16*x^4 +
84*d^6*e^15*x^3 + 36*d^7*e^14*x^2 + 9*d^8*e^13*x + d^9*e^12)

________________________________________________________________________________________

giac [B]  time = 1.34, size = 1843, normalized size = 4.18

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)^10,x, algorithm="giac")

[Out]

5*(11*B*b^10*d^2 - 20*B*a*b^9*d*e - 2*A*b^10*d*e + 9*B*a^2*b^8*e^2 + 2*A*a*b^9*e^2)*e^(-12)*log(abs(x*e + d))
+ 1/2*(B*b^10*x^2*e^10 - 20*B*b^10*d*x*e^9 + 20*B*a*b^9*x*e^10 + 2*A*b^10*x*e^10)*e^(-20) + 1/504*(42131*B*b^1
0*d^11 - 97220*B*a*b^9*d^10*e - 9722*A*b^10*d^10*e + 64161*B*a^2*b^8*d^9*e^2 + 14258*A*a*b^9*d^9*e^2 - 6720*B*
a^3*b^7*d^8*e^3 - 2520*A*a^2*b^8*d^8*e^3 - 1470*B*a^4*b^6*d^7*e^4 - 840*A*a^3*b^7*d^7*e^4 - 504*B*a^5*b^5*d^6*
e^5 - 420*A*a^4*b^6*d^6*e^5 - 210*B*a^6*b^4*d^5*e^6 - 252*A*a^5*b^5*d^5*e^6 - 96*B*a^7*b^3*d^4*e^7 - 168*A*a^6
*b^4*d^4*e^7 - 45*B*a^8*b^2*d^3*e^8 - 120*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 - 7*B*
a^10*d*e^10 - 70*A*a^9*b*d*e^10 - 56*A*a^10*e^11 + 7560*(11*B*b^10*d^3*e^8 - 30*B*a*b^9*d^2*e^9 - 3*A*b^10*d^2
*e^9 + 27*B*a^2*b^8*d*e^10 + 6*A*a*b^9*d*e^10 - 8*B*a^3*b^7*e^11 - 3*A*a^2*b^8*e^11)*x^8 + 7560*(77*B*b^10*d^4
*e^7 - 200*B*a*b^9*d^3*e^8 - 20*A*b^10*d^3*e^8 + 162*B*a^2*b^8*d^2*e^9 + 36*A*a*b^9*d^2*e^9 - 32*B*a^3*b^7*d*e
^10 - 12*A*a^2*b^8*d*e^10 - 7*B*a^4*b^6*e^11 - 4*A*a^3*b^7*e^11)*x^7 + 3528*(517*B*b^10*d^5*e^6 - 1300*B*a*b^9
*d^4*e^7 - 130*A*b^10*d^4*e^7 + 990*B*a^2*b^8*d^3*e^8 + 220*A*a*b^9*d^3*e^8 - 160*B*a^3*b^7*d^2*e^9 - 60*A*a^2
*b^8*d^2*e^9 - 35*B*a^4*b^6*d*e^10 - 20*A*a^3*b^7*d*e^10 - 12*B*a^5*b^5*e^11 - 10*A*a^4*b^6*e^11)*x^6 + 5292*(
627*B*b^10*d^6*e^5 - 1540*B*a*b^9*d^5*e^6 - 154*A*b^10*d^5*e^6 + 1125*B*a^2*b^8*d^4*e^7 + 250*A*a*b^9*d^4*e^7
- 160*B*a^3*b^7*d^3*e^8 - 60*A*a^2*b^8*d^3*e^8 - 35*B*a^4*b^6*d^2*e^9 - 20*A*a^3*b^7*d^2*e^9 - 12*B*a^5*b^5*d*
e^10 - 10*A*a^4*b^6*d*e^10 - 5*B*a^6*b^4*e^11 - 6*A*a^5*b^5*e^11)*x^5 + 756*(5049*B*b^10*d^7*e^4 - 12180*B*a*b
^9*d^6*e^5 - 1218*A*b^10*d^6*e^5 + 8631*B*a^2*b^8*d^5*e^6 + 1918*A*a*b^9*d^5*e^6 - 1120*B*a^3*b^7*d^4*e^7 - 42
0*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 - 84*B*a^5*b^5*d^2*e^9 - 70*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 - 16*B*a^7*b^3*e^11 - 28*A*a^6*b^4*e^11)*x^4 + 252*(11253*B*b^
10*d^8*e^3 - 26760*B*a*b^9*d^7*e^4 - 2676*A*b^10*d^7*e^4 + 18522*B*a^2*b^8*d^6*e^5 + 4116*A*a*b^9*d^6*e^5 - 22
40*B*a^3*b^7*d^5*e^6 - 840*A*a^2*b^8*d^5*e^6 - 490*B*a^4*b^6*d^4*e^7 - 280*A*a^3*b^7*d^4*e^7 - 168*B*a^5*b^5*d
^3*e^8 - 140*A*a^4*b^6*d^3*e^8 - 70*B*a^6*b^4*d^2*e^9 - 84*A*a^5*b^5*d^2*e^9 - 32*B*a^7*b^3*d*e^10 - 56*A*a^6*
b^4*d*e^10 - 15*B*a^8*b^2*e^11 - 40*A*a^7*b^3*e^11)*x^3 + 36*(36839*B*b^10*d^9*e^2 - 86580*B*a*b^9*d^8*e^3 - 8
658*A*b^10*d^8*e^3 + 58806*B*a^2*b^8*d^7*e^4 + 13068*A*a*b^9*d^7*e^4 - 6720*B*a^3*b^7*d^6*e^5 - 2520*A*a^2*b^8
*d^6*e^5 - 1470*B*a^4*b^6*d^5*e^6 - 840*A*a^3*b^7*d^5*e^6 - 504*B*a^5*b^5*d^4*e^7 - 420*A*a^4*b^6*d^4*e^7 - 21
0*B*a^6*b^4*d^3*e^8 - 252*A*a^5*b^5*d^3*e^8 - 96*B*a^7*b^3*d^2*e^9 - 168*A*a^6*b^4*d^2*e^9 - 45*B*a^8*b^2*d*e^
10 - 120*A*a^7*b^3*d*e^10 - 20*B*a^9*b*e^11 - 90*A*a^8*b^2*e^11)*x^2 + 9*(39611*B*b^10*d^10*e - 92180*B*a*b^9*
d^9*e^2 - 9218*A*b^10*d^9*e^2 + 61641*B*a^2*b^8*d^8*e^3 + 13698*A*a*b^9*d^8*e^3 - 6720*B*a^3*b^7*d^7*e^4 - 252
0*A*a^2*b^8*d^7*e^4 - 1470*B*a^4*b^6*d^6*e^5 - 840*A*a^3*b^7*d^6*e^5 - 504*B*a^5*b^5*d^5*e^6 - 420*A*a^4*b^6*d
^5*e^6 - 210*B*a^6*b^4*d^4*e^7 - 252*A*a^5*b^5*d^4*e^7 - 96*B*a^7*b^3*d^3*e^8 - 168*A*a^6*b^4*d^3*e^8 - 45*B*a
^8*b^2*d^2*e^9 - 120*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 - 7*B*a^10*e^11 - 70*A*a^9*b*
e^11)*x)*e^(-12)/(x*e + d)^9

________________________________________________________________________________________

maple [B]  time = 0.03, size = 2882, normalized size = 6.54 \begin {gather*} \text {output 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)^10,x)

[Out]

300*b^4/e^6/(e*x+d)^7*B*a^6*d^3-540*b^5/e^7/(e*x+d)^7*B*a^5*d^4+630*b^6/e^8/(e*x+d)^7*B*a^4*d^5-480*b^7/e^9/(e
*x+d)^7*B*a^3*d^6+1620/7*b^8/e^10/(e*x+d)^7*B*a^2*d^7-450/7*b^9/e^11/(e*x+d)^7*B*a*d^8+1134*b^8/e^10/(e*x+d)^5
*B*a^2*d^5-420*b^9/e^11/(e*x+d)^5*B*a*d^6-100*b^9/e^11*ln(e*x+d)*B*a*d+180*b^8/e^9/(e*x+d)^2*A*a^2*d+1/9/e^2/(
e*x+d)^9*B*d*a^10+1/9/e^12/(e*x+d)^9*b^10*B*d^11-20*b^3/e^4/(e*x+d)^6*A*a^7+20*b^10/e^11/(e*x+d)^6*A*d^7-15/2*
b^2/e^4/(e*x+d)^6*B*a^8-55/2*b^10/e^12/(e*x+d)^6*B*d^8+10*b^9/e^10*B*x*a-10*b^10/e^11*B*x*d-10/7*b/e^3/(e*x+d)
^7*B*a^9+55/7*b^10/e^12/(e*x+d)^7*B*d^9-42*b^4/e^5/(e*x+d)^5*A*a^6-42*b^10/e^11/(e*x+d)^5*A*d^6-24*b^3/e^5/(e*
x+d)^5*B*a^7+66*b^10/e^12/(e*x+d)^5*B*d^7+10*b^9/e^10*ln(e*x+d)*A*a-10*b^10/e^11*ln(e*x+d)*A*d+45*b^8/e^10*ln(
e*x+d)*B*a^2+55*b^10/e^12*ln(e*x+d)*B*d^2-60*b^7/e^8/(e*x+d)^2*A*a^3+60*b^10/e^11/(e*x+d)^2*A*d^3-105*b^6/e^8/
(e*x+d)^2*B*a^4-165*b^10/e^12/(e*x+d)^2*B*d^4+b^10/e^10*A*x-1/8/e^2/(e*x+d)^8*B*a^10-1/9/e/(e*x+d)^9*a^10*A-45
*b^8/e^9/(e*x+d)*A*a^2-45*b^10/e^11/(e*x+d)*A*d^2-120*b^7/e^9/(e*x+d)*B*a^3+165*b^10/e^12/(e*x+d)*B*d^3-70*b^6
/e^7/(e*x+d)^3*A*a^4-70*b^10/e^11/(e*x+d)^3*A*d^4-84*b^5/e^7/(e*x+d)^3*B*a^5+154*b^10/e^12/(e*x+d)^3*B*d^5-63*
b^5/e^6/(e*x+d)^4*A*a^5+63*b^10/e^11/(e*x+d)^4*A*d^5-105/2*b^4/e^6/(e*x+d)^4*B*a^6-231/2*b^10/e^12/(e*x+d)^4*B
*d^6-5/4/e^2/(e*x+d)^8*A*a^9*b+5/4/e^11/(e*x+d)^8*A*b^10*d^9-11/8/e^12/(e*x+d)^8*b^10*B*d^10+5/2/e^3/(e*x+d)^8
*B*a^9*b*d+1/2*b^10*B*x^2/e^10+45/e^9/(e*x+d)^8*A*a^2*b^8*d^7-45/4/e^10/(e*x+d)^8*A*a*b^9*d^8-1/9/e^11/(e*x+d)
^9*A*b^10*d^10-180*b^9/e^10/(e*x+d)^2*A*a*d^2+480*b^7/e^9/(e*x+d)^2*B*a^3*d-810*b^8/e^10/(e*x+d)^2*B*a^2*d^2+6
00*b^9/e^11/(e*x+d)^2*B*a*d^3+45/4/e^3/(e*x+d)^8*A*a^8*b^2*d-45/e^4/(e*x+d)^8*A*a^7*b^3*d^2+105/e^5/(e*x+d)^8*
A*a^6*b^4*d^3-315/2/e^6/(e*x+d)^8*A*a^5*b^5*d^4+315/2/e^7/(e*x+d)^8*A*a^4*b^6*d^5-105/e^8/(e*x+d)^8*A*a^3*b^7*
d^6+1120*b^7/e^9/(e*x+d)^6*B*a^3*d^5-630*b^8/e^10/(e*x+d)^6*B*a^2*d^6+200*b^9/e^11/(e*x+d)^6*B*a*d^7+360/7*b^3
/e^4/(e*x+d)^7*A*a^7*d-180*b^4/e^5/(e*x+d)^7*A*a^6*d^2+360*b^5/e^6/(e*x+d)^7*A*a^5*d^3-450*b^6/e^7/(e*x+d)^7*A
*a^4*d^4+360*b^7/e^8/(e*x+d)^7*A*a^3*d^5-180*b^8/e^9/(e*x+d)^7*A*a^2*d^6+360/7*b^9/e^10/(e*x+d)^7*A*a*d^7+135/
7*b^2/e^4/(e*x+d)^7*B*a^8*d-720/7*b^3/e^5/(e*x+d)^7*B*a^7*d^2+10/9/e^2/(e*x+d)^9*A*d*a^9*b-5/e^3/(e*x+d)^9*A*d
^2*a^8*b^2+40/3/e^4/(e*x+d)^9*A*d^3*a^7*b^3-70/3/e^5/(e*x+d)^9*A*d^4*a^6*b^4+28/e^6/(e*x+d)^9*A*d^5*a^5*b^5-70
/3/e^7/(e*x+d)^9*A*d^6*a^4*b^6+40/3/e^8/(e*x+d)^9*A*d^7*a^3*b^7-5/e^9/(e*x+d)^9*A*d^8*a^2*b^8+10/9/e^10/(e*x+d
)^9*A*d^9*a*b^9-10/9/e^3/(e*x+d)^9*B*d^2*a^9*b+5/e^4/(e*x+d)^9*B*d^3*a^8*b^2-40/3/e^5/(e*x+d)^9*B*d^4*a^7*b^3+
70/3/e^6/(e*x+d)^9*B*d^5*a^6*b^4-28/e^7/(e*x+d)^9*B*d^6*a^5*b^5+70/3/e^8/(e*x+d)^9*B*d^7*a^4*b^6-40/3/e^9/(e*x
+d)^9*B*d^8*a^3*b^7+5/e^10/(e*x+d)^9*B*d^9*a^2*b^8-10/9/e^11/(e*x+d)^9*B*a*b^9*d^10+140*b^4/e^5/(e*x+d)^6*A*a^
6*d-420*b^5/e^6/(e*x+d)^6*A*a^5*d^2+700*b^6/e^7/(e*x+d)^6*A*a^4*d^3-700*b^7/e^8/(e*x+d)^6*A*a^3*d^4+420*b^8/e^
9/(e*x+d)^6*A*a^2*d^5-140*b^9/e^10/(e*x+d)^6*A*a*d^6+80*b^3/e^5/(e*x+d)^6*B*a^7*d-350*b^4/e^6/(e*x+d)^6*B*a^6*
d^2+840*b^5/e^7/(e*x+d)^6*B*a^5*d^3-1225*b^6/e^8/(e*x+d)^6*B*a^4*d^4-450*b^9/e^11/(e*x+d)*B*a*d^2+280*b^7/e^8/
(e*x+d)^3*A*a^3*d+90*b^9/e^10/(e*x+d)*A*a*d+405*b^8/e^10/(e*x+d)*B*a^2*d-420*b^8/e^9/(e*x+d)^3*A*a^2*d^2+280*b
^9/e^10/(e*x+d)^3*A*a*d^3+490*b^6/e^8/(e*x+d)^3*B*a^4*d-1120*b^7/e^9/(e*x+d)^3*B*a^3*d^2+1260*b^8/e^10/(e*x+d)
^3*B*a^2*d^3-700*b^9/e^11/(e*x+d)^3*B*a*d^4+315*b^6/e^7/(e*x+d)^4*A*a^4*d-630*b^7/e^8/(e*x+d)^4*A*a^3*d^2+630*
b^8/e^9/(e*x+d)^4*A*a^2*d^3-315*b^9/e^10/(e*x+d)^4*A*a*d^4+378*b^5/e^7/(e*x+d)^4*B*a^5*d-2205/2*b^6/e^8/(e*x+d
)^4*B*a^4*d^2+1680*b^7/e^9/(e*x+d)^4*B*a^3*d^3-2835/2*b^8/e^10/(e*x+d)^4*B*a^2*d^4+630*b^9/e^11/(e*x+d)^4*B*a*
d^5+252*b^5/e^6/(e*x+d)^5*A*a^5*d-630*b^6/e^7/(e*x+d)^5*A*a^4*d^2+840*b^7/e^8/(e*x+d)^5*A*a^3*d^3-630*b^8/e^9/
(e*x+d)^5*A*a^2*d^4+252*b^9/e^10/(e*x+d)^5*A*a*d^5+210*b^4/e^6/(e*x+d)^5*B*a^6*d-756*b^5/e^7/(e*x+d)^5*B*a^5*d
^2+1470*b^6/e^8/(e*x+d)^5*B*a^4*d^3-1680*b^7/e^9/(e*x+d)^5*B*a^3*d^4-45/7*b^2/e^3/(e*x+d)^7*A*a^8-45/7*b^10/e^
11/(e*x+d)^7*A*d^8-135/8/e^4/(e*x+d)^8*B*a^8*b^2*d^2+60/e^5/(e*x+d)^8*B*a^7*b^3*d^3-525/4/e^6/(e*x+d)^8*B*a^6*
b^4*d^4+189/e^7/(e*x+d)^8*B*a^5*b^5*d^5-735/4/e^8/(e*x+d)^8*B*a^4*b^6*d^6+120/e^9/(e*x+d)^8*B*a^3*b^7*d^7-405/
8/e^10/(e*x+d)^8*B*a^2*b^8*d^8+25/2/e^11/(e*x+d)^8*B*a*b^9*d^9

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maxima [B]  time = 1.76, size = 1904, normalized size = 4.32

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)^10,x, algorithm="maxima")

[Out]

1/504*(42131*B*b^10*d^11 - 56*A*a^10*e^11 - 9722*(10*B*a*b^9 + A*b^10)*d^10*e + 7129*(9*B*a^2*b^8 + 2*A*a*b^9)
*d^9*e^2 - 840*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 - 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 84*(6*B*a^5*b^5
 + 5*A*a^4*b^6)*d^6*e^5 - 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - 24*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 - 15
*(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 - 7*(B*a^10 + 10*A*a^9*b)*d*e^10 +
 7560*(11*B*b^10*d^3*e^8 - 3*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 3*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 - (8*B*a^3*b^7
 + 3*A*a^2*b^8)*e^11)*x^8 + 7560*(77*B*b^10*d^4*e^7 - 20*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 18*(9*B*a^2*b^8 + 2*A
*a*b^9)*d^2*e^9 - 4*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 - (7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 3528*(517*B*b
^10*d^5*e^6 - 130*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 110*(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 - 5*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 - 2*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 5292*(62
7*B*b^10*d^6*e^5 - 154*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 125*(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 - 5*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 - 2*(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 + 756*(5049*B*b^10*d^7*e^4 - 1218*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 959*(9*B*a^
2*b^8 + 2*A*a*b^9)*d^5*e^6 - 140*(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
- 14*(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*(4*B*a^7*b^3 + 7*A*a^6*b^4
)*e^11)*x^4 + 252*(11253*B*b^10*d^8*e^3 - 2676*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 2058*(9*B*a^2*b^8 + 2*A*a*b^9)*
d^6*e^5 - 280*(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 - 28*(6*B*a^5*b^5 +
 5*A*a^4*b^6)*d^3*e^8 - 14*(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 - 5*(3*B
*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 + 36*(36839*B*b^10*d^9*e^2 - 8658*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 6534*(9*B*
a^2*b^8 + 2*A*a*b^9)*d^7*e^4 - 840*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 - 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e
^6 - 84*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 - 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 - 24*(4*B*a^7*b^3 + 7*A*a
^6*b^4)*d^2*e^9 - 15*(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 + 9*(39611*B*
b^10*d^10*e - 9218*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 6849*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 - 840*(8*B*a^3*b^7 +
 3*A*a^2*b^8)*d^7*e^4 - 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 - 84*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 - 42*
(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 - 24*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 - 15*(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 - 7*(B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^21*x^9 + 9*d*e^20*x^8 + 36
*d^2*e^19*x^7 + 84*d^3*e^18*x^6 + 126*d^4*e^17*x^5 + 126*d^5*e^16*x^4 + 84*d^6*e^15*x^3 + 36*d^7*e^14*x^2 + 9*
d^8*e^13*x + d^9*e^12) + 1/2*(B*b^10*e*x^2 - 2*(10*B*b^10*d - (10*B*a*b^9 + A*b^10)*e)*x)/e^11 + 5*(11*B*b^10*
d^2 - 2*(10*B*a*b^9 + A*b^10)*d*e + (9*B*a^2*b^8 + 2*A*a*b^9)*e^2)*log(e*x + d)/e^12

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mupad [B]  time = 1.53, size = 2048, normalized size = 4.64

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)^10,x)

[Out]

x*((A*b^10 + 10*B*a*b^9)/e^10 - (10*B*b^10*d)/e^11) - (x^7*(60*A*a^3*b^7*e^10 + 105*B*a^4*b^6*e^10 + 300*A*b^1
0*d^3*e^7 - 1155*B*b^10*d^4*e^6 - 540*A*a*b^9*d^2*e^8 + 180*A*a^2*b^8*d*e^9 + 3000*B*a*b^9*d^3*e^7 + 480*B*a^3
*b^7*d*e^9 - 2430*B*a^2*b^8*d^2*e^8) + x^4*(42*A*a^6*b^4*e^10 + 24*B*a^7*b^3*e^10 + 1827*A*b^10*d^6*e^4 - (151
47*B*b^10*d^7*e^3)/2 - 2877*A*a*b^9*d^5*e^5 + 63*A*a^5*b^5*d*e^9 + 18270*B*a*b^9*d^6*e^4 + (105*B*a^6*b^4*d*e^
9)/2 + 630*A*a^2*b^8*d^4*e^6 + 210*A*a^3*b^7*d^3*e^7 + 105*A*a^4*b^6*d^2*e^8 - (25893*B*a^2*b^8*d^5*e^5)/2 + 1
680*B*a^3*b^7*d^4*e^6 + (735*B*a^4*b^6*d^3*e^7)/2 + 126*B*a^5*b^5*d^2*e^8) + x^6*(70*A*a^4*b^6*e^10 + 84*B*a^5
*b^5*e^10 + 910*A*b^10*d^4*e^6 - 3619*B*b^10*d^5*e^5 - 1540*A*a*b^9*d^3*e^7 + 140*A*a^3*b^7*d*e^9 + 9100*B*a*b
^9*d^4*e^6 + 245*B*a^4*b^6*d*e^9 + 420*A*a^2*b^8*d^2*e^8 - 6930*B*a^2*b^8*d^3*e^7 + 1120*B*a^3*b^7*d^2*e^8) +
x^3*(20*A*a^7*b^3*e^10 + (15*B*a^8*b^2*e^10)/2 + 1338*A*b^10*d^7*e^3 - (11253*B*b^10*d^8*e^2)/2 - 2058*A*a*b^9
*d^6*e^4 + 28*A*a^6*b^4*d*e^9 + 13380*B*a*b^9*d^7*e^3 + 16*B*a^7*b^3*d*e^9 + 420*A*a^2*b^8*d^5*e^5 + 140*A*a^3
*b^7*d^4*e^6 + 70*A*a^4*b^6*d^3*e^7 + 42*A*a^5*b^5*d^2*e^8 - 9261*B*a^2*b^8*d^6*e^4 + 1120*B*a^3*b^7*d^5*e^5 +
 245*B*a^4*b^6*d^4*e^6 + 84*B*a^5*b^5*d^3*e^7 + 35*B*a^6*b^4*d^2*e^8) + (56*A*a^10*e^11 - 42131*B*b^10*d^11 +
9722*A*b^10*d^10*e + 7*B*a^10*d*e^10 - 14258*A*a*b^9*d^9*e^2 + 20*B*a^9*b*d^2*e^9 + 2520*A*a^2*b^8*d^8*e^3 + 8
40*A*a^3*b^7*d^7*e^4 + 420*A*a^4*b^6*d^6*e^5 + 252*A*a^5*b^5*d^5*e^6 + 168*A*a^6*b^4*d^4*e^7 + 120*A*a^7*b^3*d
^3*e^8 + 90*A*a^8*b^2*d^2*e^9 - 64161*B*a^2*b^8*d^9*e^2 + 6720*B*a^3*b^7*d^8*e^3 + 1470*B*a^4*b^6*d^7*e^4 + 50
4*B*a^5*b^5*d^6*e^5 + 210*B*a^6*b^4*d^5*e^6 + 96*B*a^7*b^3*d^4*e^7 + 45*B*a^8*b^2*d^3*e^8 + 70*A*a^9*b*d*e^10
+ 97220*B*a*b^9*d^10*e)/(504*e) + x*((B*a^10*e^10)/8 - (39611*B*b^10*d^10)/56 + (5*A*a^9*b*e^10)/4 + (4609*A*b
^10*d^9*e)/28 - (6849*A*a*b^9*d^8*e^2)/28 + (45*A*a^8*b^2*d*e^9)/28 + 45*A*a^2*b^8*d^7*e^3 + 15*A*a^3*b^7*d^6*
e^4 + (15*A*a^4*b^6*d^5*e^5)/2 + (9*A*a^5*b^5*d^4*e^6)/2 + 3*A*a^6*b^4*d^3*e^7 + (15*A*a^7*b^3*d^2*e^8)/7 - (6
1641*B*a^2*b^8*d^8*e^2)/56 + 120*B*a^3*b^7*d^7*e^3 + (105*B*a^4*b^6*d^6*e^4)/4 + 9*B*a^5*b^5*d^5*e^5 + (15*B*a
^6*b^4*d^4*e^6)/4 + (12*B*a^7*b^3*d^3*e^7)/7 + (45*B*a^8*b^2*d^2*e^8)/56 + (23045*B*a*b^9*d^9*e)/14 + (5*B*a^9
*b*d*e^9)/14) + x^8*(45*A*a^2*b^8*e^10 + 120*B*a^3*b^7*e^10 + 45*A*b^10*d^2*e^8 - 165*B*b^10*d^3*e^7 + 450*B*a
*b^9*d^2*e^8 - 405*B*a^2*b^8*d*e^9 - 90*A*a*b^9*d*e^9) + x^5*(63*A*a^5*b^5*e^10 + (105*B*a^6*b^4*e^10)/2 + 161
7*A*b^10*d^5*e^5 - (13167*B*b^10*d^6*e^4)/2 - 2625*A*a*b^9*d^4*e^6 + 105*A*a^4*b^6*d*e^9 + 16170*B*a*b^9*d^5*e
^5 + 126*B*a^5*b^5*d*e^9 + 630*A*a^2*b^8*d^3*e^7 + 210*A*a^3*b^7*d^2*e^8 - (23625*B*a^2*b^8*d^4*e^6)/2 + 1680*
B*a^3*b^7*d^3*e^7 + (735*B*a^4*b^6*d^2*e^8)/2) + x^2*((10*B*a^9*b*e^10)/7 - (36839*B*b^10*d^9*e)/14 + (45*A*a^
8*b^2*e^10)/7 + (4329*A*b^10*d^8*e^2)/7 - (6534*A*a*b^9*d^7*e^3)/7 + (60*A*a^7*b^3*d*e^9)/7 + (43290*B*a*b^9*d
^8*e^2)/7 + (45*B*a^8*b^2*d*e^9)/14 + 180*A*a^2*b^8*d^6*e^4 + 60*A*a^3*b^7*d^5*e^5 + 30*A*a^4*b^6*d^4*e^6 + 18
*A*a^5*b^5*d^3*e^7 + 12*A*a^6*b^4*d^2*e^8 - (29403*B*a^2*b^8*d^7*e^3)/7 + 480*B*a^3*b^7*d^6*e^4 + 105*B*a^4*b^
6*d^5*e^5 + 36*B*a^5*b^5*d^4*e^6 + 15*B*a^6*b^4*d^3*e^7 + (48*B*a^7*b^3*d^2*e^8)/7))/(d^9*e^11 + e^20*x^9 + 9*
d^8*e^12*x + 9*d*e^19*x^8 + 36*d^7*e^13*x^2 + 84*d^6*e^14*x^3 + 126*d^5*e^15*x^4 + 126*d^4*e^16*x^5 + 84*d^3*e
^17*x^6 + 36*d^2*e^18*x^7) + (log(d + e*x)*(55*B*b^10*d^2 - 10*A*b^10*d*e + 10*A*a*b^9*e^2 + 45*B*a^2*b^8*e^2
- 100*B*a*b^9*d*e))/e^12 + (B*b^10*x^2)/(2*e^10)

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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)**10,x)

[Out]

Timed out

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