3.22.45 \(\int \frac {(a+b x+c x^2)^3}{(d+e x)^{10}} \, dx\) [2145]

Optimal. Leaf size=272 \[ -\frac {\left (c d^2-b d e+a e^2\right )^3}{9 e^7 (d+e x)^9}+\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{8 e^7 (d+e x)^8}-\frac {3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{7 e^7 (d+e x)^7}+\frac {(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )}{6 e^7 (d+e x)^6}-\frac {3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{5 e^7 (d+e x)^5}+\frac {3 c^2 (2 c d-b e)}{4 e^7 (d+e x)^4}-\frac {c^3}{3 e^7 (d+e x)^3} \]

[Out]

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

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Rubi [A]
time = 0.14, antiderivative size = 272, 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 = {712} \begin {gather*} -\frac {3 c \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^7 (d+e x)^5}+\frac {(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{6 e^7 (d+e x)^6}-\frac {3 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^7 (d+e x)^7}+\frac {3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{8 e^7 (d+e x)^8}-\frac {\left (a e^2-b d e+c d^2\right )^3}{9 e^7 (d+e x)^9}+\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 {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/9*(c*d^2 - b*d*e + a*e^2)^3/(e^7*(d + e*x)^9) + (3*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2)/(8*e^7*(d + e*x
)^8) - (3*(c*d^2 - b*d*e + a*e^2)*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e)))/(7*e^7*(d + e*x)^7) + ((2*c*d - b
*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e)))/(6*e^7*(d + e*x)^6) - (3*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b
*d - a*e)))/(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*} \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^{10}} \, dx &=\int \left (\frac {\left (c d^2-b d e+a e^2\right )^3}{e^6 (d+e x)^{10}}+\frac {3 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2}{e^6 (d+e x)^9}+\frac {3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2-5 b c d e+b^2 e^2+a c e^2\right )}{e^6 (d+e x)^8}+\frac {(2 c d-b e) \left (-10 c^2 d^2-b^2 e^2+2 c e (5 b d-3 a e)\right )}{e^6 (d+e x)^7}+\frac {3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\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 {\left (c d^2-b d e+a e^2\right )^3}{9 e^7 (d+e x)^9}+\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{8 e^7 (d+e x)^8}-\frac {3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{7 e^7 (d+e x)^7}+\frac {(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )}{6 e^7 (d+e x)^6}-\frac {3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\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*}

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Mathematica [A]
time = 0.10, size = 378, normalized size = 1.39 \begin {gather*} -\frac {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 )+5 e^3 \left (56 a^3 e^3+21 a^2 b e^2 (d+9 e x)+6 a b^2 e \left (d^2+9 d e x+36 e^2 x^2\right )+b^3 \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )\right )+6 c e^2 \left (5 a^2 e^2 \left (d^2+9 d e x+36 e^2 x^2\right )+5 a b e \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )+2 b^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 )\right )+3 c^2 e \left (4 a e \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 )+5 b \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 )\right )}{2520 e^7 (d+e x)^9} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2520*(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) + 5*e^3*(56*a^3*e^3 + 21*a^2*b*e^2*(d + 9*e*x) + 6*a*b^2*e*(d^2 + 9*d*e*x + 36*e^2*x^2) + b^3*(d^3 + 9*d^
2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3)) + 6*c*e^2*(5*a^2*e^2*(d^2 + 9*d*e*x + 36*e^2*x^2) + 5*a*b*e*(d^3 + 9*d^2*e
*x + 36*d*e^2*x^2 + 84*e^3*x^3) + 2*b^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)) + 3*c
^2*e*(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)))/(e^7*(d + e*x)^9)

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Maple [A]
time = 0.82, size = 461, normalized size = 1.69

method	result	size
risch	\(\frac {-\frac {c^{3} x^{6}}{3 e}-\frac {c^{2} \left (3 b e +2 c d \right ) x^{5}}{4 e^{2}}-\frac {c \left (12 a c \,e^{2}+12 b^{2} e^{2}+15 b c d e +10 c^{2} d^{2}\right ) x^{4}}{20 e^{3}}-\frac {\left (30 a b c \,e^{3}+12 d \,e^{2} c^{2} a +5 b^{3} e^{3}+12 b^{2} d \,e^{2} c +15 b \,c^{2} d^{2} e +10 c^{3} d^{3}\right ) x^{3}}{30 e^{4}}-\frac {\left (30 e^{4} a^{2} c +30 a \,b^{2} e^{4}+30 a b c d \,e^{3}+12 d^{2} e^{2} c^{2} a +5 b^{3} d \,e^{3}+12 b^{2} c \,d^{2} e^{2}+15 d^{3} e b \,c^{2}+10 d^{4} c^{3}\right ) x^{2}}{70 e^{5}}-\frac {\left (105 a^{2} b \,e^{5}+30 d \,e^{4} a^{2} c +30 a \,b^{2} d \,e^{4}+30 a b c \,d^{2} e^{3}+12 d^{3} e^{2} c^{2} a +5 b^{3} d^{2} e^{3}+12 b^{2} c \,d^{3} e^{2}+15 b \,c^{2} d^{4} e +10 d^{5} c^{3}\right ) x}{280 e^{6}}-\frac {280 e^{6} a^{3}+105 a^{2} b d \,e^{5}+30 e^{4} d^{2} a^{2} c +30 a \,b^{2} d^{2} e^{4}+30 a b c \,d^{3} e^{3}+12 d^{4} e^{2} c^{2} a +5 b^{3} d^{3} e^{3}+12 b^{2} c \,d^{4} e^{2}+15 b \,c^{2} d^{5} e +10 d^{6} c^{3}}{2520 e^{7}}}{\left (e x +d \right )^{9}}\)	\(442\)
default	\(-\frac {6 a b c \,e^{3}-12 d \,e^{2} c^{2} a +b^{3} e^{3}-12 b^{2} d \,e^{2} c +30 b \,c^{2} d^{2} e -20 c^{3} d^{3}}{6 e^{7} \left (e x +d \right )^{6}}-\frac {e^{6} a^{3}-3 a^{2} b d \,e^{5}+3 e^{4} d^{2} a^{2} c +3 a \,b^{2} d^{2} e^{4}-6 a b c \,d^{3} e^{3}+3 d^{4} e^{2} c^{2} a -b^{3} d^{3} e^{3}+3 b^{2} c \,d^{4} e^{2}-3 b \,c^{2} d^{5} e +d^{6} c^{3}}{9 e^{7} \left (e x +d \right )^{9}}-\frac {c^{3}}{3 e^{7} \left (e x +d \right )^{3}}-\frac {3 a^{2} b \,e^{5}-6 d \,e^{4} a^{2} c -6 a \,b^{2} d \,e^{4}+18 a b c \,d^{2} e^{3}-12 d^{3} e^{2} c^{2} a +3 b^{3} d^{2} e^{3}-12 b^{2} c \,d^{3} e^{2}+15 b \,c^{2} d^{4} e -6 d^{5} c^{3}}{8 e^{7} \left (e x +d \right )^{8}}-\frac {3 e^{4} a^{2} c +3 a \,b^{2} e^{4}-18 a b c d \,e^{3}+18 d^{2} e^{2} c^{2} a -3 b^{3} d \,e^{3}+18 b^{2} c \,d^{2} e^{2}-30 d^{3} e b \,c^{2}+15 d^{4} c^{3}}{7 e^{7} \left (e x +d \right )^{7}}-\frac {3 c \left (a c \,e^{2}+b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right )}{5 e^{7} \left (e x +d \right )^{5}}-\frac {3 c^{2} \left (b e -2 c d \right )}{4 e^{7} \left (e x +d \right )^{4}}\)	\(461\)
norman	\(\frac {-\frac {c^{3} x^{6}}{3 e}-\frac {\left (3 e^{3} b \,c^{2}+2 d \,e^{2} c^{3}\right ) x^{5}}{4 e^{4}}-\frac {\left (12 a \,c^{2} e^{4}+12 b^{2} c \,e^{4}+15 d \,e^{3} b \,c^{2}+10 d^{2} e^{2} c^{3}\right ) x^{4}}{20 e^{5}}-\frac {\left (30 a b c \,e^{5}+12 a \,c^{2} d \,e^{4}+5 b^{3} e^{5}+12 b^{2} c d \,e^{4}+15 b \,c^{2} d^{2} e^{3}+10 e^{2} c^{3} d^{3}\right ) x^{3}}{30 e^{6}}-\frac {\left (30 e^{6} a^{2} c +30 a \,b^{2} e^{6}+30 a b c d \,e^{5}+12 e^{4} d^{2} c^{2} a +5 b^{3} d \,e^{5}+12 b^{2} c \,d^{2} e^{4}+15 b \,c^{2} d^{3} e^{3}+10 d^{4} e^{2} c^{3}\right ) x^{2}}{70 e^{7}}-\frac {\left (105 a^{2} b \,e^{7}+30 a^{2} c d \,e^{6}+30 a \,b^{2} d \,e^{6}+30 a b c \,d^{2} e^{5}+12 a \,c^{2} d^{3} e^{4}+5 b^{3} d^{2} e^{5}+12 b^{2} c \,d^{3} e^{4}+15 b \,c^{2} d^{4} e^{3}+10 c^{3} d^{5} e^{2}\right ) x}{280 e^{8}}-\frac {280 a^{3} e^{8}+105 a^{2} b d \,e^{7}+30 a^{2} c \,d^{2} e^{6}+30 a \,b^{2} d^{2} e^{6}+30 a b c \,d^{3} e^{5}+12 a \,c^{2} d^{4} e^{4}+5 b^{3} d^{3} e^{5}+12 b^{2} c \,d^{4} e^{4}+15 b \,c^{2} d^{5} e^{3}+10 c^{3} d^{6} e^{2}}{2520 e^{9}}}{\left (e x +d \right )^{9}}\)	\(478\)
gosper	\(-\frac {840 c^{3} e^{6} x^{6}+1890 b \,c^{2} e^{6} x^{5}+1260 c^{3} d \,e^{5} x^{5}+1512 a \,c^{2} e^{6} x^{4}+1512 b^{2} c \,e^{6} x^{4}+1890 b \,c^{2} d \,e^{5} x^{4}+1260 c^{3} d^{2} e^{4} x^{4}+2520 a b c \,e^{6} x^{3}+1008 a \,c^{2} d \,e^{5} x^{3}+420 b^{3} e^{6} x^{3}+1008 b^{2} c d \,e^{5} x^{3}+1260 b \,c^{2} d^{2} e^{4} x^{3}+840 c^{3} d^{3} e^{3} x^{3}+1080 a^{2} c \,e^{6} x^{2}+1080 a \,b^{2} e^{6} x^{2}+1080 a b c d \,e^{5} x^{2}+432 a \,c^{2} d^{2} e^{4} x^{2}+180 b^{3} d \,e^{5} x^{2}+432 b^{2} c \,d^{2} e^{4} x^{2}+540 b \,c^{2} d^{3} e^{3} x^{2}+360 c^{3} d^{4} e^{2} x^{2}+945 a^{2} b \,e^{6} x +270 a^{2} c d \,e^{5} x +270 a \,b^{2} d \,e^{5} x +270 a b c \,d^{2} e^{4} x +108 a \,c^{2} d^{3} e^{3} x +45 b^{3} d^{2} e^{4} x +108 b^{2} c \,d^{3} e^{3} x +135 b \,c^{2} d^{4} e^{2} x +90 c^{3} d^{5} e x +280 e^{6} a^{3}+105 a^{2} b d \,e^{5}+30 e^{4} d^{2} a^{2} c +30 a \,b^{2} d^{2} e^{4}+30 a b c \,d^{3} e^{3}+12 d^{4} e^{2} c^{2} a +5 b^{3} d^{3} e^{3}+12 b^{2} c \,d^{4} e^{2}+15 b \,c^{2} d^{5} e +10 d^{6} c^{3}}{2520 e^{7} \left (e x +d \right )^{9}}\)	\(495\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6/e^7*(6*a*b*c*e^3-12*a*c^2*d*e^2+b^3*e^3-12*b^2*c*d*e^2+30*b*c^2*d^2*e-20*c^3*d^3)/(e*x+d)^6-1/9*(a^3*e^6-
3*a^2*b*d*e^5+3*a^2*c*d^2*e^4+3*a*b^2*d^2*e^4-6*a*b*c*d^3*e^3+3*a*c^2*d^4*e^2-b^3*d^3*e^3+3*b^2*c*d^4*e^2-3*b*
c^2*d^5*e+c^3*d^6)/e^7/(e*x+d)^9-1/3*c^3/e^7/(e*x+d)^3-1/8*(3*a^2*b*e^5-6*a^2*c*d*e^4-6*a*b^2*d*e^4+18*a*b*c*d
^2*e^3-12*a*c^2*d^3*e^2+3*b^3*d^2*e^3-12*b^2*c*d^3*e^2+15*b*c^2*d^4*e-6*c^3*d^5)/e^7/(e*x+d)^8-1/7*(3*a^2*c*e^
4+3*a*b^2*e^4-18*a*b*c*d*e^3+18*a*c^2*d^2*e^2-3*b^3*d*e^3+18*b^2*c*d^2*e^2-30*b*c^2*d^3*e+15*c^3*d^4)/e^7/(e*x
+d)^7-3/5/e^7*c*(a*c*e^2+b^2*e^2-5*b*c*d*e+5*c^2*d^2)/(e*x+d)^5-3/4*c^2/e^7*(b*e-2*c*d)/(e*x+d)^4

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Maxima [A]
time = 0.30, size = 490, normalized size = 1.80 \begin {gather*} -\frac {840 \, c^{3} x^{6} e^{6} + 10 \, c^{3} d^{6} + 15 \, b c^{2} d^{5} e + 630 \, {\left (2 \, c^{3} d e^{5} + 3 \, b c^{2} e^{6}\right )} x^{5} + 12 \, {\left (b^{2} c e^{2} + a c^{2} e^{2}\right )} d^{4} + 126 \, {\left (10 \, c^{3} d^{2} e^{4} + 15 \, b c^{2} d e^{5} + 12 \, b^{2} c e^{6} + 12 \, a c^{2} e^{6}\right )} x^{4} + 105 \, a^{2} b d e^{5} + 5 \, {\left (b^{3} e^{3} + 6 \, a b c e^{3}\right )} d^{3} + 84 \, {\left (10 \, c^{3} d^{3} e^{3} + 15 \, b c^{2} d^{2} e^{4} + 5 \, b^{3} e^{6} + 30 \, a b c e^{6} + 12 \, {\left (b^{2} c e^{5} + a c^{2} e^{5}\right )} d\right )} x^{3} + 280 \, a^{3} e^{6} + 30 \, {\left (a b^{2} e^{4} + a^{2} c e^{4}\right )} d^{2} + 36 \, {\left (10 \, c^{3} d^{4} e^{2} + 15 \, b c^{2} d^{3} e^{3} + 30 \, a b^{2} e^{6} + 30 \, a^{2} c e^{6} + 12 \, {\left (b^{2} c e^{4} + a c^{2} e^{4}\right )} d^{2} + 5 \, {\left (b^{3} e^{5} + 6 \, a b c e^{5}\right )} d\right )} x^{2} + 9 \, {\left (10 \, c^{3} d^{5} e + 15 \, b c^{2} d^{4} e^{2} + 12 \, {\left (b^{2} c e^{3} + a c^{2} e^{3}\right )} d^{3} + 105 \, a^{2} b e^{6} + 5 \, {\left (b^{3} e^{4} + 6 \, a b c e^{4}\right )} d^{2} + 30 \, {\left (a b^{2} e^{5} + a^{2} c e^{5}\right )} d\right )} x}{2520 \, {\left (x^{9} e^{16} + 9 \, d x^{8} e^{15} + 36 \, d^{2} x^{7} e^{14} + 84 \, d^{3} x^{6} e^{13} + 126 \, d^{4} x^{5} e^{12} + 126 \, d^{5} x^{4} e^{11} + 84 \, d^{6} x^{3} e^{10} + 36 \, d^{7} x^{2} e^{9} + 9 \, d^{8} x e^{8} + d^{9} e^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2520*(840*c^3*x^6*e^6 + 10*c^3*d^6 + 15*b*c^2*d^5*e + 630*(2*c^3*d*e^5 + 3*b*c^2*e^6)*x^5 + 12*(b^2*c*e^2 +
a*c^2*e^2)*d^4 + 126*(10*c^3*d^2*e^4 + 15*b*c^2*d*e^5 + 12*b^2*c*e^6 + 12*a*c^2*e^6)*x^4 + 105*a^2*b*d*e^5 +
5*(b^3*e^3 + 6*a*b*c*e^3)*d^3 + 84*(10*c^3*d^3*e^3 + 15*b*c^2*d^2*e^4 + 5*b^3*e^6 + 30*a*b*c*e^6 + 12*(b^2*c*e
^5 + a*c^2*e^5)*d)*x^3 + 280*a^3*e^6 + 30*(a*b^2*e^4 + a^2*c*e^4)*d^2 + 36*(10*c^3*d^4*e^2 + 15*b*c^2*d^3*e^3
+ 30*a*b^2*e^6 + 30*a^2*c*e^6 + 12*(b^2*c*e^4 + a*c^2*e^4)*d^2 + 5*(b^3*e^5 + 6*a*b*c*e^5)*d)*x^2 + 9*(10*c^3*
d^5*e + 15*b*c^2*d^4*e^2 + 12*(b^2*c*e^3 + a*c^2*e^3)*d^3 + 105*a^2*b*e^6 + 5*(b^3*e^4 + 6*a*b*c*e^4)*d^2 + 30
*(a*b^2*e^5 + a^2*c*e^5)*d)*x)/(x^9*e^16 + 9*d*x^8*e^15 + 36*d^2*x^7*e^14 + 84*d^3*x^6*e^13 + 126*d^4*x^5*e^12
+ 126*d^5*x^4*e^11 + 84*d^6*x^3*e^10 + 36*d^7*x^2*e^9 + 9*d^8*x*e^8 + d^9*e^7)

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Fricas [A]
time = 3.50, size = 459, normalized size = 1.69 \begin {gather*} -\frac {10 \, c^{3} d^{6} + {\left (840 \, c^{3} x^{6} + 1890 \, b c^{2} x^{5} + 1512 \, {\left (b^{2} c + a c^{2}\right )} x^{4} + 945 \, a^{2} b x + 420 \, {\left (b^{3} + 6 \, a b c\right )} x^{3} + 280 \, a^{3} + 1080 \, {\left (a b^{2} + a^{2} c\right )} x^{2}\right )} e^{6} + 3 \, {\left (420 \, c^{3} d x^{5} + 630 \, b c^{2} d x^{4} + 336 \, {\left (b^{2} c + a c^{2}\right )} d x^{3} + 35 \, a^{2} b d + 60 \, {\left (b^{3} + 6 \, a b c\right )} d x^{2} + 90 \, {\left (a b^{2} + a^{2} c\right )} d x\right )} e^{5} + 3 \, {\left (420 \, c^{3} d^{2} x^{4} + 420 \, b c^{2} d^{2} x^{3} + 144 \, {\left (b^{2} c + a c^{2}\right )} d^{2} x^{2} + 15 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} x + 10 \, {\left (a b^{2} + a^{2} c\right )} d^{2}\right )} e^{4} + {\left (840 \, c^{3} d^{3} x^{3} + 540 \, b c^{2} d^{3} x^{2} + 108 \, {\left (b^{2} c + a c^{2}\right )} d^{3} x + 5 \, {\left (b^{3} + 6 \, a b c\right )} d^{3}\right )} e^{3} + 3 \, {\left (120 \, c^{3} d^{4} x^{2} + 45 \, b c^{2} d^{4} x + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{4}\right )} e^{2} + 15 \, {\left (6 \, c^{3} d^{5} x + b c^{2} d^{5}\right )} e}{2520 \, {\left (x^{9} e^{16} + 9 \, d x^{8} e^{15} + 36 \, d^{2} x^{7} e^{14} + 84 \, d^{3} x^{6} e^{13} + 126 \, d^{4} x^{5} e^{12} + 126 \, d^{5} x^{4} e^{11} + 84 \, d^{6} x^{3} e^{10} + 36 \, d^{7} x^{2} e^{9} + 9 \, d^{8} x e^{8} + d^{9} e^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2520*(10*c^3*d^6 + (840*c^3*x^6 + 1890*b*c^2*x^5 + 1512*(b^2*c + a*c^2)*x^4 + 945*a^2*b*x + 420*(b^3 + 6*a*
b*c)*x^3 + 280*a^3 + 1080*(a*b^2 + a^2*c)*x^2)*e^6 + 3*(420*c^3*d*x^5 + 630*b*c^2*d*x^4 + 336*(b^2*c + a*c^2)*
d*x^3 + 35*a^2*b*d + 60*(b^3 + 6*a*b*c)*d*x^2 + 90*(a*b^2 + a^2*c)*d*x)*e^5 + 3*(420*c^3*d^2*x^4 + 420*b*c^2*d
^2*x^3 + 144*(b^2*c + a*c^2)*d^2*x^2 + 15*(b^3 + 6*a*b*c)*d^2*x + 10*(a*b^2 + a^2*c)*d^2)*e^4 + (840*c^3*d^3*x
^3 + 540*b*c^2*d^3*x^2 + 108*(b^2*c + a*c^2)*d^3*x + 5*(b^3 + 6*a*b*c)*d^3)*e^3 + 3*(120*c^3*d^4*x^2 + 45*b*c^
2*d^4*x + 4*(b^2*c + a*c^2)*d^4)*e^2 + 15*(6*c^3*d^5*x + b*c^2*d^5)*e)/(x^9*e^16 + 9*d*x^8*e^15 + 36*d^2*x^7*e
^14 + 84*d^3*x^6*e^13 + 126*d^4*x^5*e^12 + 126*d^5*x^4*e^11 + 84*d^6*x^3*e^10 + 36*d^7*x^2*e^9 + 9*d^8*x*e^8 +
d^9*e^7)

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Sympy [F(-1)] Timed out
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((c*x**2+b*x+a)**3/(e*x+d)**10,x)

[Out]

Timed out

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Giac [A]
time = 1.10, size = 459, normalized size = 1.69 \begin {gather*} -\frac {{\left (840 \, c^{3} x^{6} e^{6} + 1260 \, c^{3} d x^{5} e^{5} + 1260 \, c^{3} d^{2} x^{4} e^{4} + 840 \, c^{3} d^{3} x^{3} e^{3} + 360 \, c^{3} d^{4} x^{2} e^{2} + 90 \, c^{3} d^{5} x e + 10 \, c^{3} d^{6} + 1890 \, b c^{2} x^{5} e^{6} + 1890 \, b c^{2} d x^{4} e^{5} + 1260 \, b c^{2} d^{2} x^{3} e^{4} + 540 \, b c^{2} d^{3} x^{2} e^{3} + 135 \, b c^{2} d^{4} x e^{2} + 15 \, b c^{2} d^{5} e + 1512 \, b^{2} c x^{4} e^{6} + 1512 \, a c^{2} x^{4} e^{6} + 1008 \, b^{2} c d x^{3} e^{5} + 1008 \, a c^{2} d x^{3} e^{5} + 432 \, b^{2} c d^{2} x^{2} e^{4} + 432 \, a c^{2} d^{2} x^{2} e^{4} + 108 \, b^{2} c d^{3} x e^{3} + 108 \, a c^{2} d^{3} x e^{3} + 12 \, b^{2} c d^{4} e^{2} + 12 \, a c^{2} d^{4} e^{2} + 420 \, b^{3} x^{3} e^{6} + 2520 \, a b c x^{3} e^{6} + 180 \, b^{3} d x^{2} e^{5} + 1080 \, a b c d x^{2} e^{5} + 45 \, b^{3} d^{2} x e^{4} + 270 \, a b c d^{2} x e^{4} + 5 \, b^{3} d^{3} e^{3} + 30 \, a b c d^{3} e^{3} + 1080 \, a b^{2} x^{2} e^{6} + 1080 \, a^{2} c x^{2} e^{6} + 270 \, a b^{2} d x e^{5} + 270 \, a^{2} c d x e^{5} + 30 \, a b^{2} d^{2} e^{4} + 30 \, a^{2} c d^{2} e^{4} + 945 \, a^{2} b x e^{6} + 105 \, a^{2} b d e^{5} + 280 \, a^{3} e^{6}\right )} e^{\left (-7\right )}}{2520 \, {\left (x e + d\right )}^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2520*(840*c^3*x^6*e^6 + 1260*c^3*d*x^5*e^5 + 1260*c^3*d^2*x^4*e^4 + 840*c^3*d^3*x^3*e^3 + 360*c^3*d^4*x^2*e
^2 + 90*c^3*d^5*x*e + 10*c^3*d^6 + 1890*b*c^2*x^5*e^6 + 1890*b*c^2*d*x^4*e^5 + 1260*b*c^2*d^2*x^3*e^4 + 540*b*
c^2*d^3*x^2*e^3 + 135*b*c^2*d^4*x*e^2 + 15*b*c^2*d^5*e + 1512*b^2*c*x^4*e^6 + 1512*a*c^2*x^4*e^6 + 1008*b^2*c*
d*x^3*e^5 + 1008*a*c^2*d*x^3*e^5 + 432*b^2*c*d^2*x^2*e^4 + 432*a*c^2*d^2*x^2*e^4 + 108*b^2*c*d^3*x*e^3 + 108*a
*c^2*d^3*x*e^3 + 12*b^2*c*d^4*e^2 + 12*a*c^2*d^4*e^2 + 420*b^3*x^3*e^6 + 2520*a*b*c*x^3*e^6 + 180*b^3*d*x^2*e^
5 + 1080*a*b*c*d*x^2*e^5 + 45*b^3*d^2*x*e^4 + 270*a*b*c*d^2*x*e^4 + 5*b^3*d^3*e^3 + 30*a*b*c*d^3*e^3 + 1080*a*
b^2*x^2*e^6 + 1080*a^2*c*x^2*e^6 + 270*a*b^2*d*x*e^5 + 270*a^2*c*d*x*e^5 + 30*a*b^2*d^2*e^4 + 30*a^2*c*d^2*e^4
+ 945*a^2*b*x*e^6 + 105*a^2*b*d*e^5 + 280*a^3*e^6)*e^(-7)/(x*e + d)^9

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Mupad [B]
time = 0.76, size = 530, normalized size = 1.95 \begin {gather*} -\frac {\frac {280\,a^3\,e^6+105\,a^2\,b\,d\,e^5+30\,a^2\,c\,d^2\,e^4+30\,a\,b^2\,d^2\,e^4+30\,a\,b\,c\,d^3\,e^3+12\,a\,c^2\,d^4\,e^2+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}+\frac {x^3\,\left (5\,b^3\,e^3+12\,b^2\,c\,d\,e^2+15\,b\,c^2\,d^2\,e+30\,a\,b\,c\,e^3+10\,c^3\,d^3+12\,a\,c^2\,d\,e^2\right )}{30\,e^4}+\frac {x^2\,\left (30\,a^2\,c\,e^4+30\,a\,b^2\,e^4+30\,a\,b\,c\,d\,e^3+12\,a\,c^2\,d^2\,e^2+5\,b^3\,d\,e^3+12\,b^2\,c\,d^2\,e^2+15\,b\,c^2\,d^3\,e+10\,c^3\,d^4\right )}{70\,e^5}+\frac {c^3\,x^6}{3\,e}+\frac {x\,\left (105\,a^2\,b\,e^5+30\,a^2\,c\,d\,e^4+30\,a\,b^2\,d\,e^4+30\,a\,b\,c\,d^2\,e^3+12\,a\,c^2\,d^3\,e^2+5\,b^3\,d^2\,e^3+12\,b^2\,c\,d^3\,e^2+15\,b\,c^2\,d^4\,e+10\,c^3\,d^5\right )}{280\,e^6}+\frac {c\,x^4\,\left (12\,b^2\,e^2+15\,b\,c\,d\,e+10\,c^2\,d^2+12\,a\,c\,e^2\right )}{20\,e^3}+\frac {c^2\,x^5\,\left (3\,b\,e+2\,c\,d\right )}{4\,e^2}}{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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((280*a^3*e^6 + 10*c^3*d^6 + 5*b^3*d^3*e^3 + 30*a*b^2*d^2*e^4 + 12*a*c^2*d^4*e^2 + 30*a^2*c*d^2*e^4 + 12*b^2*
c*d^4*e^2 + 105*a^2*b*d*e^5 + 15*b*c^2*d^5*e + 30*a*b*c*d^3*e^3)/(2520*e^7) + (x^3*(5*b^3*e^3 + 10*c^3*d^3 + 3
0*a*b*c*e^3 + 12*a*c^2*d*e^2 + 15*b*c^2*d^2*e + 12*b^2*c*d*e^2))/(30*e^4) + (x^2*(10*c^3*d^4 + 30*a*b^2*e^4 +
30*a^2*c*e^4 + 5*b^3*d*e^3 + 12*a*c^2*d^2*e^2 + 12*b^2*c*d^2*e^2 + 15*b*c^2*d^3*e + 30*a*b*c*d*e^3))/(70*e^5)
+ (c^3*x^6)/(3*e) + (x*(10*c^3*d^5 + 105*a^2*b*e^5 + 5*b^3*d^2*e^3 + 12*a*c^2*d^3*e^2 + 12*b^2*c*d^3*e^2 + 30*
a*b^2*d*e^4 + 30*a^2*c*d*e^4 + 15*b*c^2*d^4*e + 30*a*b*c*d^2*e^3))/(280*e^6) + (c*x^4*(12*b^2*e^2 + 10*c^2*d^2
+ 12*a*c*e^2 + 15*b*c*d*e))/(20*e^3) + (c^2*x^5*(3*b*e + 2*c*d))/(4*e^2))/(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)

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