\(\int \frac {(a+b x)^3 (A+B x)}{(d+e x)^{10}} \, dx\) [1050]

Optimal result

Integrand size = 20, antiderivative size = 163 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^{10}} \, dx=-\frac {(b d-a e)^3 (B d-A e)}{9 e^5 (d+e x)^9}+\frac {(b d-a e)^2 (4 b B d-3 A b e-a B e)}{8 e^5 (d+e x)^8}-\frac {3 b (b d-a e) (2 b B d-A b e-a B e)}{7 e^5 (d+e x)^7}+\frac {b^2 (4 b B d-A b e-3 a B e)}{6 e^5 (d+e x)^6}-\frac {b^3 B}{5 e^5 (d+e x)^5} \]

[Out]

-1/9*(-a*e+b*d)^3*(-A*e+B*d)/e^5/(e*x+d)^9+1/8*(-a*e+b*d)^2*(-3*A*b*e-B*a*e+4*B*b*d)/e^5/(e*x+d)^8-3/7*b*(-a*e
+b*d)*(-A*b*e-B*a*e+2*B*b*d)/e^5/(e*x+d)^7+1/6*b^2*(-A*b*e-3*B*a*e+4*B*b*d)/e^5/(e*x+d)^6-1/5*b^3*B/e^5/(e*x+d
)^5

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78} \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^{10}} \, dx=\frac {b^2 (-3 a B e-A b e+4 b B d)}{6 e^5 (d+e x)^6}-\frac {3 b (b d-a e) (-a B e-A b e+2 b B d)}{7 e^5 (d+e x)^7}+\frac {(b d-a e)^2 (-a B e-3 A b e+4 b B d)}{8 e^5 (d+e x)^8}-\frac {(b d-a e)^3 (B d-A e)}{9 e^5 (d+e x)^9}-\frac {b^3 B}{5 e^5 (d+e x)^5} \]

[In]

Int[((a + b*x)^3*(A + B*x))/(d + e*x)^10,x]

[Out]

-1/9*((b*d - a*e)^3*(B*d - A*e))/(e^5*(d + e*x)^9) + ((b*d - a*e)^2*(4*b*B*d - 3*A*b*e - a*B*e))/(8*e^5*(d + e
*x)^8) - (3*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e))/(7*e^5*(d + e*x)^7) + (b^2*(4*b*B*d - A*b*e - 3*a*B*e))/(
6*e^5*(d + e*x)^6) - (b^3*B)/(5*e^5*(d + e*x)^5)

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-b d+a e)^3 (-B d+A e)}{e^4 (d+e x)^{10}}+\frac {(-b d+a e)^2 (-4 b B d+3 A b e+a B e)}{e^4 (d+e x)^9}-\frac {3 b (b d-a e) (-2 b B d+A b e+a B e)}{e^4 (d+e x)^8}+\frac {b^2 (-4 b B d+A b e+3 a B e)}{e^4 (d+e x)^7}+\frac {b^3 B}{e^4 (d+e x)^6}\right ) \, dx \\ & = -\frac {(b d-a e)^3 (B d-A e)}{9 e^5 (d+e x)^9}+\frac {(b d-a e)^2 (4 b B d-3 A b e-a B e)}{8 e^5 (d+e x)^8}-\frac {3 b (b d-a e) (2 b B d-A b e-a B e)}{7 e^5 (d+e x)^7}+\frac {b^2 (4 b B d-A b e-3 a B e)}{6 e^5 (d+e x)^6}-\frac {b^3 B}{5 e^5 (d+e x)^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.31 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^{10}} \, dx=-\frac {35 a^3 e^3 (8 A e+B (d+9 e x))+15 a^2 b e^2 \left (7 A e (d+9 e x)+2 B \left (d^2+9 d e x+36 e^2 x^2\right )\right )+15 a b^2 e \left (2 A e \left (d^2+9 d e x+36 e^2 x^2\right )+B \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )\right )+b^3 \left (5 A e \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )+4 B \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )\right )}{2520 e^5 (d+e x)^9} \]

[In]

Integrate[((a + b*x)^3*(A + B*x))/(d + e*x)^10,x]

[Out]

-1/2520*(35*a^3*e^3*(8*A*e + B*(d + 9*e*x)) + 15*a^2*b*e^2*(7*A*e*(d + 9*e*x) + 2*B*(d^2 + 9*d*e*x + 36*e^2*x^
2)) + 15*a*b^2*e*(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)) + b^3*
(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)))/(e^5*(d + e*x)^9)

Maple [A] (verified)

Time = 0.69 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.66

method	result	size
risch	\(\frac {-\frac {b^{3} B \,x^{4}}{5 e}-\frac {b^{2} \left (5 A b e +15 B a e +4 B b d \right ) x^{3}}{30 e^{2}}-\frac {b \left (30 A a b \,e^{2}+5 A \,b^{2} d e +30 B \,a^{2} e^{2}+15 B a b d e +4 b^{2} B \,d^{2}\right ) x^{2}}{70 e^{3}}-\frac {\left (105 A \,a^{2} b \,e^{3}+30 A a \,b^{2} d \,e^{2}+5 A \,b^{3} d^{2} e +35 B \,a^{3} e^{3}+30 B \,a^{2} b d \,e^{2}+15 B a \,b^{2} d^{2} e +4 b^{3} B \,d^{3}\right ) x}{280 e^{4}}-\frac {280 a^{3} A \,e^{4}+105 A \,a^{2} b d \,e^{3}+30 A a \,b^{2} d^{2} e^{2}+5 A \,b^{3} d^{3} e +35 B \,a^{3} d \,e^{3}+30 B \,a^{2} b \,d^{2} e^{2}+15 B a \,b^{2} d^{3} e +4 b^{3} B \,d^{4}}{2520 e^{5}}}{\left (e x +d \right )^{9}}\)	\(270\)
default	\(-\frac {3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +B \,a^{3} e^{3}-6 B \,a^{2} b d \,e^{2}+9 B a \,b^{2} d^{2} e -4 b^{3} B \,d^{3}}{8 e^{5} \left (e x +d \right )^{8}}-\frac {3 b \left (A a b \,e^{2}-A \,b^{2} d e +B \,a^{2} e^{2}-3 B a b d e +2 b^{2} B \,d^{2}\right )}{7 e^{5} \left (e x +d \right )^{7}}-\frac {b^{3} B}{5 e^{5} \left (e x +d \right )^{5}}-\frac {a^{3} A \,e^{4}-3 A \,a^{2} b d \,e^{3}+3 A a \,b^{2} d^{2} e^{2}-A \,b^{3} d^{3} e -B \,a^{3} d \,e^{3}+3 B \,a^{2} b \,d^{2} e^{2}-3 B a \,b^{2} d^{3} e +b^{3} B \,d^{4}}{9 e^{5} \left (e x +d \right )^{9}}-\frac {b^{2} \left (A b e +3 B a e -4 B b d \right )}{6 e^{5} \left (e x +d \right )^{6}}\)	\(281\)
gosper	\(-\frac {504 B \,x^{4} b^{3} e^{4}+420 A \,x^{3} b^{3} e^{4}+1260 B \,x^{3} a \,b^{2} e^{4}+336 B \,x^{3} b^{3} d \,e^{3}+1080 A \,x^{2} a \,b^{2} e^{4}+180 A \,x^{2} b^{3} d \,e^{3}+1080 B \,x^{2} a^{2} b \,e^{4}+540 B \,x^{2} a \,b^{2} d \,e^{3}+144 B \,x^{2} b^{3} d^{2} e^{2}+945 A x \,a^{2} b \,e^{4}+270 A x a \,b^{2} d \,e^{3}+45 A x \,b^{3} d^{2} e^{2}+315 B x \,a^{3} e^{4}+270 B x \,a^{2} b d \,e^{3}+135 B x a \,b^{2} d^{2} e^{2}+36 B x \,b^{3} d^{3} e +280 a^{3} A \,e^{4}+105 A \,a^{2} b d \,e^{3}+30 A a \,b^{2} d^{2} e^{2}+5 A \,b^{3} d^{3} e +35 B \,a^{3} d \,e^{3}+30 B \,a^{2} b \,d^{2} e^{2}+15 B a \,b^{2} d^{3} e +4 b^{3} B \,d^{4}}{2520 e^{5} \left (e x +d \right )^{9}}\)	\(301\)
norman	\(\frac {-\frac {b^{3} B \,x^{4}}{5 e}-\frac {\left (5 A \,b^{3} e^{5}+15 B a \,b^{2} e^{5}+4 b^{3} B d \,e^{4}\right ) x^{3}}{30 e^{6}}-\frac {\left (30 a \,b^{2} A \,e^{6}+5 A \,b^{3} d \,e^{5}+30 a^{2} b B \,e^{6}+15 B a \,b^{2} d \,e^{5}+4 B \,b^{3} d^{2} e^{4}\right ) x^{2}}{70 e^{7}}-\frac {\left (105 A \,a^{2} b \,e^{7}+30 A a \,b^{2} d \,e^{6}+5 A \,b^{3} d^{2} e^{5}+35 B \,a^{3} e^{7}+30 B \,a^{2} b d \,e^{6}+15 B a \,b^{2} d^{2} e^{5}+4 B \,b^{3} d^{3} e^{4}\right ) x}{280 e^{8}}-\frac {280 a^{3} A \,e^{8}+105 A \,a^{2} b d \,e^{7}+30 A a \,b^{2} d^{2} e^{6}+5 A \,b^{3} d^{3} e^{5}+35 B \,a^{3} d \,e^{7}+30 B \,a^{2} b \,d^{2} e^{6}+15 B a \,b^{2} d^{3} e^{5}+4 B \,b^{3} d^{4} e^{4}}{2520 e^{9}}}{\left (e x +d \right )^{9}}\)	\(306\)
parallelrisch	\(-\frac {504 b^{3} B \,x^{4} e^{8}+420 A \,b^{3} e^{8} x^{3}+1260 B a \,b^{2} e^{8} x^{3}+336 B \,b^{3} d \,e^{7} x^{3}+1080 A a \,b^{2} e^{8} x^{2}+180 A \,b^{3} d \,e^{7} x^{2}+1080 B \,a^{2} b \,e^{8} x^{2}+540 B a \,b^{2} d \,e^{7} x^{2}+144 B \,b^{3} d^{2} e^{6} x^{2}+945 A \,a^{2} b \,e^{8} x +270 A a \,b^{2} d \,e^{7} x +45 A \,b^{3} d^{2} e^{6} x +315 B \,a^{3} e^{8} x +270 B \,a^{2} b d \,e^{7} x +135 B a \,b^{2} d^{2} e^{6} x +36 B \,b^{3} d^{3} e^{5} x +280 a^{3} A \,e^{8}+105 A \,a^{2} b d \,e^{7}+30 A a \,b^{2} d^{2} e^{6}+5 A \,b^{3} d^{3} e^{5}+35 B \,a^{3} d \,e^{7}+30 B \,a^{2} b \,d^{2} e^{6}+15 B a \,b^{2} d^{3} e^{5}+4 B \,b^{3} d^{4} e^{4}}{2520 e^{9} \left (e x +d \right )^{9}}\)	\(310\)

[In]

int((b*x+a)^3*(B*x+A)/(e*x+d)^10,x,method=_RETURNVERBOSE)

[Out]

(-1/5*b^3*B/e*x^4-1/30*b^2/e^2*(5*A*b*e+15*B*a*e+4*B*b*d)*x^3-1/70*b/e^3*(30*A*a*b*e^2+5*A*b^2*d*e+30*B*a^2*e^
2+15*B*a*b*d*e+4*B*b^2*d^2)*x^2-1/280/e^4*(105*A*a^2*b*e^3+30*A*a*b^2*d*e^2+5*A*b^3*d^2*e+35*B*a^3*e^3+30*B*a^
2*b*d*e^2+15*B*a*b^2*d^2*e+4*B*b^3*d^3)*x-1/2520/e^5*(280*A*a^3*e^4+105*A*a^2*b*d*e^3+30*A*a*b^2*d^2*e^2+5*A*b
^3*d^3*e+35*B*a^3*d*e^3+30*B*a^2*b*d^2*e^2+15*B*a*b^2*d^3*e+4*B*b^3*d^4))/(e*x+d)^9

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 354 vs. \(2 (153) = 306\).

Time = 0.23 (sec) , antiderivative size = 354, normalized size of antiderivative = 2.17 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^{10}} \, dx=-\frac {504 \, B b^{3} e^{4} x^{4} + 4 \, B b^{3} d^{4} + 280 \, A a^{3} e^{4} + 5 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 30 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 35 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 84 \, {\left (4 \, B b^{3} d e^{3} + 5 \, {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 36 \, {\left (4 \, B b^{3} d^{2} e^{2} + 5 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 30 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 9 \, {\left (4 \, B b^{3} d^{3} e + 5 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 30 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} + 35 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{2520 \, {\left (e^{14} x^{9} + 9 \, d e^{13} x^{8} + 36 \, d^{2} e^{12} x^{7} + 84 \, d^{3} e^{11} x^{6} + 126 \, d^{4} e^{10} x^{5} + 126 \, d^{5} e^{9} x^{4} + 84 \, d^{6} e^{8} x^{3} + 36 \, d^{7} e^{7} x^{2} + 9 \, d^{8} e^{6} x + d^{9} e^{5}\right )}} \]

[In]

integrate((b*x+a)^3*(B*x+A)/(e*x+d)^10,x, algorithm="fricas")

[Out]

-1/2520*(504*B*b^3*e^4*x^4 + 4*B*b^3*d^4 + 280*A*a^3*e^4 + 5*(3*B*a*b^2 + A*b^3)*d^3*e + 30*(B*a^2*b + A*a*b^2
)*d^2*e^2 + 35*(B*a^3 + 3*A*a^2*b)*d*e^3 + 84*(4*B*b^3*d*e^3 + 5*(3*B*a*b^2 + A*b^3)*e^4)*x^3 + 36*(4*B*b^3*d^
2*e^2 + 5*(3*B*a*b^2 + A*b^3)*d*e^3 + 30*(B*a^2*b + A*a*b^2)*e^4)*x^2 + 9*(4*B*b^3*d^3*e + 5*(3*B*a*b^2 + A*b^
3)*d^2*e^2 + 30*(B*a^2*b + A*a*b^2)*d*e^3 + 35*(B*a^3 + 3*A*a^2*b)*e^4)*x)/(e^14*x^9 + 9*d*e^13*x^8 + 36*d^2*e
^12*x^7 + 84*d^3*e^11*x^6 + 126*d^4*e^10*x^5 + 126*d^5*e^9*x^4 + 84*d^6*e^8*x^3 + 36*d^7*e^7*x^2 + 9*d^8*e^6*x
+ d^9*e^5)

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^{10}} \, dx=\text {Timed out} \]

[In]

integrate((b*x+a)**3*(B*x+A)/(e*x+d)**10,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 354 vs. \(2 (153) = 306\).

Time = 0.22 (sec) , antiderivative size = 354, normalized size of antiderivative = 2.17 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^{10}} \, dx=-\frac {504 \, B b^{3} e^{4} x^{4} + 4 \, B b^{3} d^{4} + 280 \, A a^{3} e^{4} + 5 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 30 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 35 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 84 \, {\left (4 \, B b^{3} d e^{3} + 5 \, {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 36 \, {\left (4 \, B b^{3} d^{2} e^{2} + 5 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 30 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 9 \, {\left (4 \, B b^{3} d^{3} e + 5 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 30 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} + 35 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{2520 \, {\left (e^{14} x^{9} + 9 \, d e^{13} x^{8} + 36 \, d^{2} e^{12} x^{7} + 84 \, d^{3} e^{11} x^{6} + 126 \, d^{4} e^{10} x^{5} + 126 \, d^{5} e^{9} x^{4} + 84 \, d^{6} e^{8} x^{3} + 36 \, d^{7} e^{7} x^{2} + 9 \, d^{8} e^{6} x + d^{9} e^{5}\right )}} \]

[In]

integrate((b*x+a)^3*(B*x+A)/(e*x+d)^10,x, algorithm="maxima")

[Out]

-1/2520*(504*B*b^3*e^4*x^4 + 4*B*b^3*d^4 + 280*A*a^3*e^4 + 5*(3*B*a*b^2 + A*b^3)*d^3*e + 30*(B*a^2*b + A*a*b^2
)*d^2*e^2 + 35*(B*a^3 + 3*A*a^2*b)*d*e^3 + 84*(4*B*b^3*d*e^3 + 5*(3*B*a*b^2 + A*b^3)*e^4)*x^3 + 36*(4*B*b^3*d^
2*e^2 + 5*(3*B*a*b^2 + A*b^3)*d*e^3 + 30*(B*a^2*b + A*a*b^2)*e^4)*x^2 + 9*(4*B*b^3*d^3*e + 5*(3*B*a*b^2 + A*b^
3)*d^2*e^2 + 30*(B*a^2*b + A*a*b^2)*d*e^3 + 35*(B*a^3 + 3*A*a^2*b)*e^4)*x)/(e^14*x^9 + 9*d*e^13*x^8 + 36*d^2*e
^12*x^7 + 84*d^3*e^11*x^6 + 126*d^4*e^10*x^5 + 126*d^5*e^9*x^4 + 84*d^6*e^8*x^3 + 36*d^7*e^7*x^2 + 9*d^8*e^6*x
+ d^9*e^5)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.84 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^{10}} \, dx=-\frac {504 \, B b^{3} e^{4} x^{4} + 336 \, B b^{3} d e^{3} x^{3} + 1260 \, B a b^{2} e^{4} x^{3} + 420 \, A b^{3} e^{4} x^{3} + 144 \, B b^{3} d^{2} e^{2} x^{2} + 540 \, B a b^{2} d e^{3} x^{2} + 180 \, A b^{3} d e^{3} x^{2} + 1080 \, B a^{2} b e^{4} x^{2} + 1080 \, A a b^{2} e^{4} x^{2} + 36 \, B b^{3} d^{3} e x + 135 \, B a b^{2} d^{2} e^{2} x + 45 \, A b^{3} d^{2} e^{2} x + 270 \, B a^{2} b d e^{3} x + 270 \, A a b^{2} d e^{3} x + 315 \, B a^{3} e^{4} x + 945 \, A a^{2} b e^{4} x + 4 \, B b^{3} d^{4} + 15 \, B a b^{2} d^{3} e + 5 \, A b^{3} d^{3} e + 30 \, B a^{2} b d^{2} e^{2} + 30 \, A a b^{2} d^{2} e^{2} + 35 \, B a^{3} d e^{3} + 105 \, A a^{2} b d e^{3} + 280 \, A a^{3} e^{4}}{2520 \, {\left (e x + d\right )}^{9} e^{5}} \]

[In]

integrate((b*x+a)^3*(B*x+A)/(e*x+d)^10,x, algorithm="giac")

[Out]

-1/2520*(504*B*b^3*e^4*x^4 + 336*B*b^3*d*e^3*x^3 + 1260*B*a*b^2*e^4*x^3 + 420*A*b^3*e^4*x^3 + 144*B*b^3*d^2*e^
2*x^2 + 540*B*a*b^2*d*e^3*x^2 + 180*A*b^3*d*e^3*x^2 + 1080*B*a^2*b*e^4*x^2 + 1080*A*a*b^2*e^4*x^2 + 36*B*b^3*d
^3*e*x + 135*B*a*b^2*d^2*e^2*x + 45*A*b^3*d^2*e^2*x + 270*B*a^2*b*d*e^3*x + 270*A*a*b^2*d*e^3*x + 315*B*a^3*e^
4*x + 945*A*a^2*b*e^4*x + 4*B*b^3*d^4 + 15*B*a*b^2*d^3*e + 5*A*b^3*d^3*e + 30*B*a^2*b*d^2*e^2 + 30*A*a*b^2*d^2
*e^2 + 35*B*a^3*d*e^3 + 105*A*a^2*b*d*e^3 + 280*A*a^3*e^4)/((e*x + d)^9*e^5)

Mupad [B] (verification not implemented)

Time = 1.30 (sec) , antiderivative size = 358, normalized size of antiderivative = 2.20 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^{10}} \, dx=-\frac {\frac {35\,B\,a^3\,d\,e^3+280\,A\,a^3\,e^4+30\,B\,a^2\,b\,d^2\,e^2+105\,A\,a^2\,b\,d\,e^3+15\,B\,a\,b^2\,d^3\,e+30\,A\,a\,b^2\,d^2\,e^2+4\,B\,b^3\,d^4+5\,A\,b^3\,d^3\,e}{2520\,e^5}+\frac {x\,\left (35\,B\,a^3\,e^3+30\,B\,a^2\,b\,d\,e^2+105\,A\,a^2\,b\,e^3+15\,B\,a\,b^2\,d^2\,e+30\,A\,a\,b^2\,d\,e^2+4\,B\,b^3\,d^3+5\,A\,b^3\,d^2\,e\right )}{280\,e^4}+\frac {b^2\,x^3\,\left (5\,A\,b\,e+15\,B\,a\,e+4\,B\,b\,d\right )}{30\,e^2}+\frac {b\,x^2\,\left (30\,B\,a^2\,e^2+15\,B\,a\,b\,d\,e+30\,A\,a\,b\,e^2+4\,B\,b^2\,d^2+5\,A\,b^2\,d\,e\right )}{70\,e^3}+\frac {B\,b^3\,x^4}{5\,e}}{d^9+9\,d^8\,e\,x+36\,d^7\,e^2\,x^2+84\,d^6\,e^3\,x^3+126\,d^5\,e^4\,x^4+126\,d^4\,e^5\,x^5+84\,d^3\,e^6\,x^6+36\,d^2\,e^7\,x^7+9\,d\,e^8\,x^8+e^9\,x^9} \]

[In]

int(((A + B*x)*(a + b*x)^3)/(d + e*x)^10,x)

[Out]

-((280*A*a^3*e^4 + 4*B*b^3*d^4 + 5*A*b^3*d^3*e + 35*B*a^3*d*e^3 + 30*A*a*b^2*d^2*e^2 + 30*B*a^2*b*d^2*e^2 + 10
5*A*a^2*b*d*e^3 + 15*B*a*b^2*d^3*e)/(2520*e^5) + (x*(35*B*a^3*e^3 + 4*B*b^3*d^3 + 105*A*a^2*b*e^3 + 5*A*b^3*d^
2*e + 30*A*a*b^2*d*e^2 + 15*B*a*b^2*d^2*e + 30*B*a^2*b*d*e^2))/(280*e^4) + (b^2*x^3*(5*A*b*e + 15*B*a*e + 4*B*
b*d))/(30*e^2) + (b*x^2*(30*B*a^2*e^2 + 4*B*b^2*d^2 + 30*A*a*b*e^2 + 5*A*b^2*d*e + 15*B*a*b*d*e))/(70*e^3) + (
B*b^3*x^4)/(5*e))/(d^9 + e^9*x^9 + 9*d*e^8*x^8 + 36*d^7*e^2*x^2 + 84*d^6*e^3*x^3 + 126*d^5*e^4*x^4 + 126*d^4*e
^5*x^5 + 84*d^3*e^6*x^6 + 36*d^2*e^7*x^7 + 9*d^8*e*x)