\(\int \frac {(b x+c x^2)^3}{(d+e x)^{10}} \, dx\) [258]

Optimal result

Integrand size = 19, antiderivative size = 234 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^{10}} \, dx=-\frac {d^3 (c d-b e)^3}{9 e^7 (d+e x)^9}+\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{8 e^7 (d+e x)^8}-\frac {3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{7 e^7 (d+e x)^7}+\frac {(2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right )}{6 e^7 (d+e x)^6}-\frac {3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{5 e^7 (d+e x)^5}+\frac {3 c^2 (2 c d-b e)}{4 e^7 (d+e x)^4}-\frac {c^3}{3 e^7 (d+e x)^3} \]

[Out]

-1/9*d^3*(-b*e+c*d)^3/e^7/(e*x+d)^9+3/8*d^2*(-b*e+c*d)^2*(-b*e+2*c*d)/e^7/(e*x+d)^8-3/7*d*(-b*e+c*d)*(b^2*e^2-
5*b*c*d*e+5*c^2*d^2)/e^7/(e*x+d)^7+1/6*(-b*e+2*c*d)*(b^2*e^2-10*b*c*d*e+10*c^2*d^2)/e^7/(e*x+d)^6-3/5*c*(b^2*e
^2-5*b*c*d*e+5*c^2*d^2)/e^7/(e*x+d)^5+3/4*c^2*(-b*e+2*c*d)/e^7/(e*x+d)^4-1/3*c^3/e^7/(e*x+d)^3

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {712} \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^{10}} \, dx=-\frac {3 c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{5 e^7 (d+e x)^5}+\frac {(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{6 e^7 (d+e x)^6}-\frac {3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{7 e^7 (d+e x)^7}+\frac {3 c^2 (2 c d-b e)}{4 e^7 (d+e x)^4}-\frac {d^3 (c d-b e)^3}{9 e^7 (d+e x)^9}+\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{8 e^7 (d+e x)^8}-\frac {c^3}{3 e^7 (d+e x)^3} \]

[In]

Int[(b*x + c*x^2)^3/(d + e*x)^10,x]

[Out]

-1/9*(d^3*(c*d - b*e)^3)/(e^7*(d + e*x)^9) + (3*d^2*(c*d - b*e)^2*(2*c*d - b*e))/(8*e^7*(d + e*x)^8) - (3*d*(c
*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2))/(7*e^7*(d + e*x)^7) + ((2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + b
^2*e^2))/(6*e^7*(d + e*x)^6) - (3*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2))/(5*e^7*(d + e*x)^5) + (3*c^2*(2*c*d - b
*e))/(4*e^7*(d + e*x)^4) - c^3/(3*e^7*(d + e*x)^3)

Rule 712

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.95 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^{10}} \, dx=-\frac {5 b^3 e^3 \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )+12 b^2 c e^2 \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )+15 b c^2 e \left (d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right )+10 c^3 \left (d^6+9 d^5 e x+36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+126 d e^5 x^5+84 e^6 x^6\right )}{2520 e^7 (d+e x)^9} \]

[In]

Integrate[(b*x + c*x^2)^3/(d + e*x)^10,x]

[Out]

-1/2520*(5*b^3*e^3*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3) + 12*b^2*c*e^2*(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) + 15*b*c^2*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) + 10*c^3*(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))/(e^7*(d + e*x)^9)

Maple [A] (verified)

Time = 2.03 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.09

method	result	size
risch	\(\frac {-\frac {c^{3} x^{6}}{3 e}-\frac {c^{2} \left (3 b e +2 c d \right ) x^{5}}{4 e^{2}}-\frac {c \left (12 b^{2} e^{2}+15 b c d e +10 c^{2} d^{2}\right ) x^{4}}{20 e^{3}}-\frac {\left (5 b^{3} e^{3}+12 b^{2} d \,e^{2} c +15 b \,c^{2} d^{2} e +10 c^{3} d^{3}\right ) x^{3}}{30 e^{4}}-\frac {d \left (5 b^{3} e^{3}+12 b^{2} d \,e^{2} c +15 b \,c^{2} d^{2} e +10 c^{3} d^{3}\right ) x^{2}}{70 e^{5}}-\frac {d^{2} \left (5 b^{3} e^{3}+12 b^{2} d \,e^{2} c +15 b \,c^{2} d^{2} e +10 c^{3} d^{3}\right ) x}{280 e^{6}}-\frac {d^{3} \left (5 b^{3} e^{3}+12 b^{2} d \,e^{2} c +15 b \,c^{2} d^{2} e +10 c^{3} d^{3}\right )}{2520 e^{7}}}{\left (e x +d \right )^{9}}\)	\(255\)
default	\(-\frac {b^{3} e^{3}-12 b^{2} d \,e^{2} c +30 b \,c^{2} d^{2} e -20 c^{3} d^{3}}{6 e^{7} \left (e x +d \right )^{6}}+\frac {d^{3} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}{9 e^{7} \left (e x +d \right )^{9}}-\frac {c^{3}}{3 e^{7} \left (e x +d \right )^{3}}-\frac {3 c^{2} \left (b e -2 c d \right )}{4 e^{7} \left (e x +d \right )^{4}}+\frac {3 d \left (b^{3} e^{3}-6 b^{2} d \,e^{2} c +10 b \,c^{2} d^{2} e -5 c^{3} d^{3}\right )}{7 e^{7} \left (e x +d \right )^{7}}-\frac {3 c \left (b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right )}{5 e^{7} \left (e x +d \right )^{5}}-\frac {3 d^{2} \left (b^{3} e^{3}-4 b^{2} d \,e^{2} c +5 b \,c^{2} d^{2} e -2 c^{3} d^{3}\right )}{8 e^{7} \left (e x +d \right )^{8}}\)	\(274\)
gosper	\(-\frac {840 x^{6} c^{3} e^{6}+1890 x^{5} b \,c^{2} e^{6}+1260 x^{5} c^{3} d \,e^{5}+1512 x^{4} b^{2} c \,e^{6}+1890 x^{4} b \,c^{2} d \,e^{5}+1260 x^{4} c^{3} d^{2} e^{4}+420 x^{3} b^{3} e^{6}+1008 x^{3} b^{2} c d \,e^{5}+1260 x^{3} b \,c^{2} d^{2} e^{4}+840 x^{3} c^{3} d^{3} e^{3}+180 x^{2} b^{3} d \,e^{5}+432 x^{2} b^{2} c \,d^{2} e^{4}+540 x^{2} b \,c^{2} d^{3} e^{3}+360 x^{2} c^{3} d^{4} e^{2}+45 x \,b^{3} d^{2} e^{4}+108 x \,b^{2} c \,d^{3} e^{3}+135 x b \,c^{2} d^{4} e^{2}+90 x \,c^{3} d^{5} e +5 b^{3} d^{3} e^{3}+12 b^{2} c \,d^{4} e^{2}+15 b \,c^{2} d^{5} e +10 c^{3} d^{6}}{2520 e^{7} \left (e x +d \right )^{9}}\)	\(286\)
norman	\(\frac {-\frac {c^{3} x^{6}}{3 e}-\frac {\left (3 e^{3} b \,c^{2}+2 d \,e^{2} c^{3}\right ) x^{5}}{4 e^{4}}-\frac {\left (12 e^{4} b^{2} c +15 d \,e^{3} b \,c^{2}+10 d^{2} e^{2} c^{3}\right ) x^{4}}{20 e^{5}}-\frac {\left (5 b^{3} e^{5}+12 b^{2} c d \,e^{4}+15 b \,c^{2} d^{2} e^{3}+10 c^{3} d^{3} e^{2}\right ) x^{3}}{30 e^{6}}-\frac {d \left (5 b^{3} e^{5}+12 b^{2} c d \,e^{4}+15 b \,c^{2} d^{2} e^{3}+10 c^{3} d^{3} e^{2}\right ) x^{2}}{70 e^{7}}-\frac {d^{2} \left (5 b^{3} e^{5}+12 b^{2} c d \,e^{4}+15 b \,c^{2} d^{2} e^{3}+10 c^{3} d^{3} e^{2}\right ) x}{280 e^{8}}-\frac {d^{3} \left (5 b^{3} e^{5}+12 b^{2} c d \,e^{4}+15 b \,c^{2} d^{2} e^{3}+10 c^{3} d^{3} e^{2}\right )}{2520 e^{9}}}{\left (e x +d \right )^{9}}\)	\(289\)
parallelrisch	\(\frac {-840 c^{3} x^{6} e^{8}-1890 b \,c^{2} e^{8} x^{5}-1260 c^{3} d \,e^{7} x^{5}-1512 b^{2} c \,e^{8} x^{4}-1890 b \,c^{2} d \,e^{7} x^{4}-1260 c^{3} d^{2} e^{6} x^{4}-420 b^{3} e^{8} x^{3}-1008 b^{2} c d \,e^{7} x^{3}-1260 b \,c^{2} d^{2} e^{6} x^{3}-840 c^{3} d^{3} e^{5} x^{3}-180 b^{3} d \,e^{7} x^{2}-432 b^{2} c \,d^{2} e^{6} x^{2}-540 b \,c^{2} d^{3} e^{5} x^{2}-360 c^{3} d^{4} e^{4} x^{2}-45 b^{3} d^{2} e^{6} x -108 b^{2} c \,d^{3} e^{5} x -135 b \,c^{2} d^{4} e^{4} x -90 c^{3} d^{5} e^{3} x -5 b^{3} d^{3} e^{5}-12 b^{2} c \,d^{4} e^{4}-15 b \,c^{2} d^{5} e^{3}-10 c^{3} d^{6} e^{2}}{2520 e^{9} \left (e x +d \right )^{9}}\)	\(293\)

[In]

int((c*x^2+b*x)^3/(e*x+d)^10,x,method=_RETURNVERBOSE)

[Out]

(-1/3*c^3*x^6/e-1/4*c^2/e^2*(3*b*e+2*c*d)*x^5-1/20/e^3*c*(12*b^2*e^2+15*b*c*d*e+10*c^2*d^2)*x^4-1/30/e^4*(5*b^
3*e^3+12*b^2*c*d*e^2+15*b*c^2*d^2*e+10*c^3*d^3)*x^3-1/70*d/e^5*(5*b^3*e^3+12*b^2*c*d*e^2+15*b*c^2*d^2*e+10*c^3
*d^3)*x^2-1/280*d^2/e^6*(5*b^3*e^3+12*b^2*c*d*e^2+15*b*c^2*d^2*e+10*c^3*d^3)*x-1/2520*d^3/e^7*(5*b^3*e^3+12*b^
2*c*d*e^2+15*b*c^2*d^2*e+10*c^3*d^3))/(e*x+d)^9

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.54 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^{10}} \, dx=-\frac {840 \, c^{3} e^{6} x^{6} + 10 \, c^{3} d^{6} + 15 \, b c^{2} d^{5} e + 12 \, b^{2} c d^{4} e^{2} + 5 \, b^{3} d^{3} e^{3} + 630 \, {\left (2 \, c^{3} d e^{5} + 3 \, b c^{2} e^{6}\right )} x^{5} + 126 \, {\left (10 \, c^{3} d^{2} e^{4} + 15 \, b c^{2} d e^{5} + 12 \, b^{2} c e^{6}\right )} x^{4} + 84 \, {\left (10 \, c^{3} d^{3} e^{3} + 15 \, b c^{2} d^{2} e^{4} + 12 \, b^{2} c d e^{5} + 5 \, b^{3} e^{6}\right )} x^{3} + 36 \, {\left (10 \, c^{3} d^{4} e^{2} + 15 \, b c^{2} d^{3} e^{3} + 12 \, b^{2} c d^{2} e^{4} + 5 \, b^{3} d e^{5}\right )} x^{2} + 9 \, {\left (10 \, c^{3} d^{5} e + 15 \, b c^{2} d^{4} e^{2} + 12 \, b^{2} c d^{3} e^{3} + 5 \, b^{3} d^{2} e^{4}\right )} x}{2520 \, {\left (e^{16} x^{9} + 9 \, d e^{15} x^{8} + 36 \, d^{2} e^{14} x^{7} + 84 \, d^{3} e^{13} x^{6} + 126 \, d^{4} e^{12} x^{5} + 126 \, d^{5} e^{11} x^{4} + 84 \, d^{6} e^{10} x^{3} + 36 \, d^{7} e^{9} x^{2} + 9 \, d^{8} e^{8} x + d^{9} e^{7}\right )}} \]

[In]

integrate((c*x^2+b*x)^3/(e*x+d)^10,x, algorithm="fricas")

[Out]

-1/2520*(840*c^3*e^6*x^6 + 10*c^3*d^6 + 15*b*c^2*d^5*e + 12*b^2*c*d^4*e^2 + 5*b^3*d^3*e^3 + 630*(2*c^3*d*e^5 +
3*b*c^2*e^6)*x^5 + 126*(10*c^3*d^2*e^4 + 15*b*c^2*d*e^5 + 12*b^2*c*e^6)*x^4 + 84*(10*c^3*d^3*e^3 + 15*b*c^2*d
^2*e^4 + 12*b^2*c*d*e^5 + 5*b^3*e^6)*x^3 + 36*(10*c^3*d^4*e^2 + 15*b*c^2*d^3*e^3 + 12*b^2*c*d^2*e^4 + 5*b^3*d*
e^5)*x^2 + 9*(10*c^3*d^5*e + 15*b*c^2*d^4*e^2 + 12*b^2*c*d^3*e^3 + 5*b^3*d^2*e^4)*x)/(e^16*x^9 + 9*d*e^15*x^8
+ 36*d^2*e^14*x^7 + 84*d^3*e^13*x^6 + 126*d^4*e^12*x^5 + 126*d^5*e^11*x^4 + 84*d^6*e^10*x^3 + 36*d^7*e^9*x^2 +
9*d^8*e^8*x + d^9*e^7)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^{10}} \, dx=\text {Timed out} \]

[In]

integrate((c*x**2+b*x)**3/(e*x+d)**10,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.54 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^{10}} \, dx=-\frac {840 \, c^{3} e^{6} x^{6} + 10 \, c^{3} d^{6} + 15 \, b c^{2} d^{5} e + 12 \, b^{2} c d^{4} e^{2} + 5 \, b^{3} d^{3} e^{3} + 630 \, {\left (2 \, c^{3} d e^{5} + 3 \, b c^{2} e^{6}\right )} x^{5} + 126 \, {\left (10 \, c^{3} d^{2} e^{4} + 15 \, b c^{2} d e^{5} + 12 \, b^{2} c e^{6}\right )} x^{4} + 84 \, {\left (10 \, c^{3} d^{3} e^{3} + 15 \, b c^{2} d^{2} e^{4} + 12 \, b^{2} c d e^{5} + 5 \, b^{3} e^{6}\right )} x^{3} + 36 \, {\left (10 \, c^{3} d^{4} e^{2} + 15 \, b c^{2} d^{3} e^{3} + 12 \, b^{2} c d^{2} e^{4} + 5 \, b^{3} d e^{5}\right )} x^{2} + 9 \, {\left (10 \, c^{3} d^{5} e + 15 \, b c^{2} d^{4} e^{2} + 12 \, b^{2} c d^{3} e^{3} + 5 \, b^{3} d^{2} e^{4}\right )} x}{2520 \, {\left (e^{16} x^{9} + 9 \, d e^{15} x^{8} + 36 \, d^{2} e^{14} x^{7} + 84 \, d^{3} e^{13} x^{6} + 126 \, d^{4} e^{12} x^{5} + 126 \, d^{5} e^{11} x^{4} + 84 \, d^{6} e^{10} x^{3} + 36 \, d^{7} e^{9} x^{2} + 9 \, d^{8} e^{8} x + d^{9} e^{7}\right )}} \]

[In]

integrate((c*x^2+b*x)^3/(e*x+d)^10,x, algorithm="maxima")

[Out]

-1/2520*(840*c^3*e^6*x^6 + 10*c^3*d^6 + 15*b*c^2*d^5*e + 12*b^2*c*d^4*e^2 + 5*b^3*d^3*e^3 + 630*(2*c^3*d*e^5 +
3*b*c^2*e^6)*x^5 + 126*(10*c^3*d^2*e^4 + 15*b*c^2*d*e^5 + 12*b^2*c*e^6)*x^4 + 84*(10*c^3*d^3*e^3 + 15*b*c^2*d
^2*e^4 + 12*b^2*c*d*e^5 + 5*b^3*e^6)*x^3 + 36*(10*c^3*d^4*e^2 + 15*b*c^2*d^3*e^3 + 12*b^2*c*d^2*e^4 + 5*b^3*d*
e^5)*x^2 + 9*(10*c^3*d^5*e + 15*b*c^2*d^4*e^2 + 12*b^2*c*d^3*e^3 + 5*b^3*d^2*e^4)*x)/(e^16*x^9 + 9*d*e^15*x^8
+ 36*d^2*e^14*x^7 + 84*d^3*e^13*x^6 + 126*d^4*e^12*x^5 + 126*d^5*e^11*x^4 + 84*d^6*e^10*x^3 + 36*d^7*e^9*x^2 +
9*d^8*e^8*x + d^9*e^7)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.22 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^{10}} \, dx=-\frac {840 \, c^{3} e^{6} x^{6} + 1260 \, c^{3} d e^{5} x^{5} + 1890 \, b c^{2} e^{6} x^{5} + 1260 \, c^{3} d^{2} e^{4} x^{4} + 1890 \, b c^{2} d e^{5} x^{4} + 1512 \, b^{2} c e^{6} x^{4} + 840 \, c^{3} d^{3} e^{3} x^{3} + 1260 \, b c^{2} d^{2} e^{4} x^{3} + 1008 \, b^{2} c d e^{5} x^{3} + 420 \, b^{3} e^{6} x^{3} + 360 \, c^{3} d^{4} e^{2} x^{2} + 540 \, b c^{2} d^{3} e^{3} x^{2} + 432 \, b^{2} c d^{2} e^{4} x^{2} + 180 \, b^{3} d e^{5} x^{2} + 90 \, c^{3} d^{5} e x + 135 \, b c^{2} d^{4} e^{2} x + 108 \, b^{2} c d^{3} e^{3} x + 45 \, b^{3} d^{2} e^{4} x + 10 \, c^{3} d^{6} + 15 \, b c^{2} d^{5} e + 12 \, b^{2} c d^{4} e^{2} + 5 \, b^{3} d^{3} e^{3}}{2520 \, {\left (e x + d\right )}^{9} e^{7}} \]

[In]

integrate((c*x^2+b*x)^3/(e*x+d)^10,x, algorithm="giac")

[Out]

-1/2520*(840*c^3*e^6*x^6 + 1260*c^3*d*e^5*x^5 + 1890*b*c^2*e^6*x^5 + 1260*c^3*d^2*e^4*x^4 + 1890*b*c^2*d*e^5*x
^4 + 1512*b^2*c*e^6*x^4 + 840*c^3*d^3*e^3*x^3 + 1260*b*c^2*d^2*e^4*x^3 + 1008*b^2*c*d*e^5*x^3 + 420*b^3*e^6*x^
3 + 360*c^3*d^4*e^2*x^2 + 540*b*c^2*d^3*e^3*x^2 + 432*b^2*c*d^2*e^4*x^2 + 180*b^3*d*e^5*x^2 + 90*c^3*d^5*e*x +
135*b*c^2*d^4*e^2*x + 108*b^2*c*d^3*e^3*x + 45*b^3*d^2*e^4*x + 10*c^3*d^6 + 15*b*c^2*d^5*e + 12*b^2*c*d^4*e^2
+ 5*b^3*d^3*e^3)/((e*x + d)^9*e^7)

Mupad [B] (verification not implemented)

Time = 9.66 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.47 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^{10}} \, dx=-\frac {\frac {d^3\,\left (5\,b^3\,e^3+12\,b^2\,c\,d\,e^2+15\,b\,c^2\,d^2\,e+10\,c^3\,d^3\right )}{2520\,e^7}+\frac {x^3\,\left (5\,b^3\,e^3+12\,b^2\,c\,d\,e^2+15\,b\,c^2\,d^2\,e+10\,c^3\,d^3\right )}{30\,e^4}+\frac {c^3\,x^6}{3\,e}+\frac {c^2\,x^5\,\left (3\,b\,e+2\,c\,d\right )}{4\,e^2}+\frac {c\,x^4\,\left (12\,b^2\,e^2+15\,b\,c\,d\,e+10\,c^2\,d^2\right )}{20\,e^3}+\frac {d\,x^2\,\left (5\,b^3\,e^3+12\,b^2\,c\,d\,e^2+15\,b\,c^2\,d^2\,e+10\,c^3\,d^3\right )}{70\,e^5}+\frac {d^2\,x\,\left (5\,b^3\,e^3+12\,b^2\,c\,d\,e^2+15\,b\,c^2\,d^2\,e+10\,c^3\,d^3\right )}{280\,e^6}}{d^9+9\,d^8\,e\,x+36\,d^7\,e^2\,x^2+84\,d^6\,e^3\,x^3+126\,d^5\,e^4\,x^4+126\,d^4\,e^5\,x^5+84\,d^3\,e^6\,x^6+36\,d^2\,e^7\,x^7+9\,d\,e^8\,x^8+e^9\,x^9} \]

[In]

int((b*x + c*x^2)^3/(d + e*x)^10,x)

[Out]

-((d^3*(5*b^3*e^3 + 10*c^3*d^3 + 15*b*c^2*d^2*e + 12*b^2*c*d*e^2))/(2520*e^7) + (x^3*(5*b^3*e^3 + 10*c^3*d^3 +
15*b*c^2*d^2*e + 12*b^2*c*d*e^2))/(30*e^4) + (c^3*x^6)/(3*e) + (c^2*x^5*(3*b*e + 2*c*d))/(4*e^2) + (c*x^4*(12
*b^2*e^2 + 10*c^2*d^2 + 15*b*c*d*e))/(20*e^3) + (d*x^2*(5*b^3*e^3 + 10*c^3*d^3 + 15*b*c^2*d^2*e + 12*b^2*c*d*e
^2))/(70*e^5) + (d^2*x*(5*b^3*e^3 + 10*c^3*d^3 + 15*b*c^2*d^2*e + 12*b^2*c*d*e^2))/(280*e^6))/(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)