3.22.44 \(\int \frac {(a+b x+c x^2)^3}{(d+e x)^9} \, dx\) [2144]

Optimal. Leaf size=269 \[ -\frac {\left (c d^2-b d e+a e^2\right )^3}{8 e^7 (d+e x)^8}+\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{7 e^7 (d+e x)^7}-\frac {\left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{2 e^7 (d+e x)^6}+\frac {(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )}{5 e^7 (d+e x)^5}-\frac {3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{4 e^7 (d+e x)^4}+\frac {c^2 (2 c d-b e)}{e^7 (d+e x)^3}-\frac {c^3}{2 e^7 (d+e x)^2} \]

[Out]

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

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Rubi [A]
time = 0.14, antiderivative size = 269, 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 )}{4 e^7 (d+e x)^4}+\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 )}{5 e^7 (d+e x)^5}-\frac {\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 )}{2 e^7 (d+e x)^6}+\frac {3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{7 e^7 (d+e x)^7}-\frac {\left (a e^2-b d e+c d^2\right )^3}{8 e^7 (d+e x)^8}+\frac {c^2 (2 c d-b e)}{e^7 (d+e x)^3}-\frac {c^3}{2 e^7 (d+e x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^3/(d + e*x)^9,x]

[Out]

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

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

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Mathematica [A]
time = 0.10, size = 375, normalized size = 1.39 \begin {gather*} -\frac {5 c^3 \left (d^6+8 d^5 e x+28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+56 d e^5 x^5+28 e^6 x^6\right )+e^3 \left (35 a^3 e^3+15 a^2 b e^2 (d+8 e x)+5 a b^2 e \left (d^2+8 d e x+28 e^2 x^2\right )+b^3 \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )\right )+c e^2 \left (5 a^2 e^2 \left (d^2+8 d e x+28 e^2 x^2\right )+6 a b e \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+3 b^2 \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )\right )+c^2 e \left (3 a e \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )+5 b \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )\right )}{280 e^7 (d+e x)^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^3/(d + e*x)^9,x]

[Out]

-1/280*(5*c^3*(d^6 + 8*d^5*e*x + 28*d^4*e^2*x^2 + 56*d^3*e^3*x^3 + 70*d^2*e^4*x^4 + 56*d*e^5*x^5 + 28*e^6*x^6)
+ e^3*(35*a^3*e^3 + 15*a^2*b*e^2*(d + 8*e*x) + 5*a*b^2*e*(d^2 + 8*d*e*x + 28*e^2*x^2) + b^3*(d^3 + 8*d^2*e*x
+ 28*d*e^2*x^2 + 56*e^3*x^3)) + c*e^2*(5*a^2*e^2*(d^2 + 8*d*e*x + 28*e^2*x^2) + 6*a*b*e*(d^3 + 8*d^2*e*x + 28*
d*e^2*x^2 + 56*e^3*x^3) + 3*b^2*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4)) + c^2*e*(3*a*e
*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4) + 5*b*(d^5 + 8*d^4*e*x + 28*d^3*e^2*x^2 + 56*d
^2*e^3*x^3 + 70*d*e^4*x^4 + 56*e^5*x^5)))/(e^7*(d + e*x)^8)

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Maple [A]
time = 0.73, size = 461, normalized size = 1.71

method	result	size
risch	\(\frac {-\frac {c^{3} x^{6}}{2 e}-\frac {c^{2} \left (b e +c d \right ) x^{5}}{e^{2}}-\frac {c \left (3 a c \,e^{2}+3 b^{2} e^{2}+5 b c d e +5 c^{2} d^{2}\right ) x^{4}}{4 e^{3}}-\frac {\left (6 a b c \,e^{3}+3 d \,e^{2} c^{2} a +b^{3} e^{3}+3 b^{2} d \,e^{2} c +5 b \,c^{2} d^{2} e +5 c^{3} d^{3}\right ) x^{3}}{5 e^{4}}-\frac {\left (5 e^{4} a^{2} c +5 a \,b^{2} e^{4}+6 a b c d \,e^{3}+3 d^{2} e^{2} c^{2} a +b^{3} d \,e^{3}+3 b^{2} c \,d^{2} e^{2}+5 d^{3} e b \,c^{2}+5 d^{4} c^{3}\right ) x^{2}}{10 e^{5}}-\frac {\left (15 a^{2} b \,e^{5}+5 d \,e^{4} a^{2} c +5 a \,b^{2} d \,e^{4}+6 a b c \,d^{2} e^{3}+3 d^{3} e^{2} c^{2} a +b^{3} d^{2} e^{3}+3 b^{2} c \,d^{3} e^{2}+5 b \,c^{2} d^{4} e +5 d^{5} c^{3}\right ) x}{35 e^{6}}-\frac {35 e^{6} a^{3}+15 a^{2} b d \,e^{5}+5 e^{4} d^{2} a^{2} c +5 a \,b^{2} d^{2} e^{4}+6 a b c \,d^{3} e^{3}+3 d^{4} e^{2} c^{2} a +b^{3} d^{3} e^{3}+3 b^{2} c \,d^{4} e^{2}+5 b \,c^{2} d^{5} e +5 d^{6} c^{3}}{280 e^{7}}}{\left (e x +d \right )^{8}}\)	\(436\)
norman	\(\frac {-\frac {c^{3} x^{6}}{2 e}-\frac {\left (e^{2} b \,c^{2}+d e \,c^{3}\right ) x^{5}}{e^{3}}-\frac {\left (3 a \,c^{2} e^{3}+3 b^{2} c \,e^{3}+5 d \,e^{2} b \,c^{2}+5 d^{2} e \,c^{3}\right ) x^{4}}{4 e^{4}}-\frac {\left (6 a b c \,e^{4}+3 d \,e^{3} c^{2} a +b^{3} e^{4}+3 b^{2} d \,e^{3} c +5 d^{2} e^{2} b \,c^{2}+5 d^{3} e \,c^{3}\right ) x^{3}}{5 e^{5}}-\frac {\left (5 a^{2} c \,e^{5}+5 a \,b^{2} e^{5}+6 a b c d \,e^{4}+3 e^{3} c^{2} d^{2} a +b^{3} d \,e^{4}+3 b^{2} c \,d^{2} e^{3}+5 b \,c^{2} d^{3} e^{2}+5 d^{4} e \,c^{3}\right ) x^{2}}{10 e^{6}}-\frac {\left (15 a^{2} b \,e^{6}+5 d \,e^{5} a^{2} c +5 a \,b^{2} d \,e^{5}+6 a b c \,d^{2} e^{4}+3 d^{3} e^{3} c^{2} a +b^{3} d^{2} e^{4}+3 b^{2} c \,d^{3} e^{3}+5 b \,c^{2} d^{4} e^{2}+5 d^{5} e \,c^{3}\right ) x}{35 e^{7}}-\frac {35 a^{3} e^{7}+15 a^{2} b d \,e^{6}+5 a^{2} c \,d^{2} e^{5}+5 a \,b^{2} d^{2} e^{5}+6 a b c \,d^{3} e^{4}+3 a \,c^{2} d^{4} e^{3}+b^{3} d^{3} e^{4}+3 b^{2} c \,d^{4} e^{3}+5 b \,c^{2} d^{5} e^{2}+5 c^{3} d^{6} e}{280 e^{8}}}{\left (e x +d \right )^{8}}\)	\(460\)
default	\(-\frac {3 e^{4} a^{2} c +3 a \,b^{2} e^{4}-18 a b c d \,e^{3}+18 d^{2} e^{2} c^{2} a -3 b^{3} d \,e^{3}+18 b^{2} c \,d^{2} e^{2}-30 d^{3} e b \,c^{2}+15 d^{4} c^{3}}{6 e^{7} \left (e x +d \right )^{6}}-\frac {c^{2} \left (b e -2 c d \right )}{e^{7} \left (e x +d \right )^{3}}-\frac {e^{6} a^{3}-3 a^{2} b d \,e^{5}+3 e^{4} d^{2} a^{2} c +3 a \,b^{2} d^{2} e^{4}-6 a b c \,d^{3} e^{3}+3 d^{4} e^{2} c^{2} a -b^{3} d^{3} e^{3}+3 b^{2} c \,d^{4} e^{2}-3 b \,c^{2} d^{5} e +d^{6} c^{3}}{8 e^{7} \left (e x +d \right )^{8}}-\frac {3 a^{2} b \,e^{5}-6 d \,e^{4} a^{2} c -6 a \,b^{2} d \,e^{4}+18 a b c \,d^{2} e^{3}-12 d^{3} e^{2} c^{2} a +3 b^{3} d^{2} e^{3}-12 b^{2} c \,d^{3} e^{2}+15 b \,c^{2} d^{4} e -6 d^{5} c^{3}}{7 e^{7} \left (e x +d \right )^{7}}-\frac {c^{3}}{2 e^{7} \left (e x +d \right )^{2}}-\frac {6 a b c \,e^{3}-12 d \,e^{2} c^{2} a +b^{3} e^{3}-12 b^{2} d \,e^{2} c +30 b \,c^{2} d^{2} e -20 c^{3} d^{3}}{5 e^{7} \left (e x +d \right )^{5}}-\frac {3 c \left (a c \,e^{2}+b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right )}{4 e^{7} \left (e x +d \right )^{4}}\)	\(461\)
gosper	\(-\frac {140 c^{3} e^{6} x^{6}+280 b \,c^{2} e^{6} x^{5}+280 c^{3} d \,e^{5} x^{5}+210 a \,c^{2} e^{6} x^{4}+210 b^{2} c \,e^{6} x^{4}+350 b \,c^{2} d \,e^{5} x^{4}+350 c^{3} d^{2} e^{4} x^{4}+336 a b c \,e^{6} x^{3}+168 a \,c^{2} d \,e^{5} x^{3}+56 b^{3} e^{6} x^{3}+168 b^{2} c d \,e^{5} x^{3}+280 b \,c^{2} d^{2} e^{4} x^{3}+280 c^{3} d^{3} e^{3} x^{3}+140 a^{2} c \,e^{6} x^{2}+140 a \,b^{2} e^{6} x^{2}+168 a b c d \,e^{5} x^{2}+84 a \,c^{2} d^{2} e^{4} x^{2}+28 b^{3} d \,e^{5} x^{2}+84 b^{2} c \,d^{2} e^{4} x^{2}+140 b \,c^{2} d^{3} e^{3} x^{2}+140 c^{3} d^{4} e^{2} x^{2}+120 a^{2} b \,e^{6} x +40 a^{2} c d \,e^{5} x +40 a \,b^{2} d \,e^{5} x +48 a b c \,d^{2} e^{4} x +24 a \,c^{2} d^{3} e^{3} x +8 b^{3} d^{2} e^{4} x +24 b^{2} c \,d^{3} e^{3} x +40 b \,c^{2} d^{4} e^{2} x +40 c^{3} d^{5} e x +35 e^{6} a^{3}+15 a^{2} b d \,e^{5}+5 e^{4} d^{2} a^{2} c +5 a \,b^{2} d^{2} e^{4}+6 a b c \,d^{3} e^{3}+3 d^{4} e^{2} c^{2} a +b^{3} d^{3} e^{3}+3 b^{2} c \,d^{4} e^{2}+5 b \,c^{2} d^{5} e +5 d^{6} c^{3}}{280 e^{7} \left (e x +d \right )^{8}}\)	\(494\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^3/(e*x+d)^9,x,method=_RETURNVERBOSE)

[Out]

-1/6*(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)^6-c^2/e^7*(b*e-2*c*d)/(e*x+d)^3-1/8*(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)^8-1/7*(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)^7-1/2*c^3/e^7/(e*x+d)^2-1/5/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+3
0*b*c^2*d^2*e-20*c^3*d^3)/(e*x+d)^5-3/4/e^7*c*(a*c*e^2+b^2*e^2-5*b*c*d*e+5*c^2*d^2)/(e*x+d)^4

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Maxima [A]
time = 0.30, size = 474, normalized size = 1.76 \begin {gather*} -\frac {140 \, c^{3} x^{6} e^{6} + 5 \, c^{3} d^{6} + 5 \, b c^{2} d^{5} e + 280 \, {\left (c^{3} d e^{5} + b c^{2} e^{6}\right )} x^{5} + 3 \, {\left (b^{2} c e^{2} + a c^{2} e^{2}\right )} d^{4} + 70 \, {\left (5 \, c^{3} d^{2} e^{4} + 5 \, b c^{2} d e^{5} + 3 \, b^{2} c e^{6} + 3 \, a c^{2} e^{6}\right )} x^{4} + 15 \, a^{2} b d e^{5} + {\left (b^{3} e^{3} + 6 \, a b c e^{3}\right )} d^{3} + 56 \, {\left (5 \, c^{3} d^{3} e^{3} + 5 \, b c^{2} d^{2} e^{4} + b^{3} e^{6} + 6 \, a b c e^{6} + 3 \, {\left (b^{2} c e^{5} + a c^{2} e^{5}\right )} d\right )} x^{3} + 35 \, a^{3} e^{6} + 5 \, {\left (a b^{2} e^{4} + a^{2} c e^{4}\right )} d^{2} + 28 \, {\left (5 \, c^{3} d^{4} e^{2} + 5 \, b c^{2} d^{3} e^{3} + 5 \, a b^{2} e^{6} + 5 \, a^{2} c e^{6} + 3 \, {\left (b^{2} c e^{4} + a c^{2} e^{4}\right )} d^{2} + {\left (b^{3} e^{5} + 6 \, a b c e^{5}\right )} d\right )} x^{2} + 8 \, {\left (5 \, c^{3} d^{5} e + 5 \, b c^{2} d^{4} e^{2} + 3 \, {\left (b^{2} c e^{3} + a c^{2} e^{3}\right )} d^{3} + 15 \, a^{2} b e^{6} + {\left (b^{3} e^{4} + 6 \, a b c e^{4}\right )} d^{2} + 5 \, {\left (a b^{2} e^{5} + a^{2} c e^{5}\right )} d\right )} x}{280 \, {\left (x^{8} e^{15} + 8 \, d x^{7} e^{14} + 28 \, d^{2} x^{6} e^{13} + 56 \, d^{3} x^{5} e^{12} + 70 \, d^{4} x^{4} e^{11} + 56 \, d^{5} x^{3} e^{10} + 28 \, d^{6} x^{2} e^{9} + 8 \, d^{7} x e^{8} + d^{8} 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)^9,x, algorithm="maxima")

[Out]

-1/280*(140*c^3*x^6*e^6 + 5*c^3*d^6 + 5*b*c^2*d^5*e + 280*(c^3*d*e^5 + b*c^2*e^6)*x^5 + 3*(b^2*c*e^2 + a*c^2*e
^2)*d^4 + 70*(5*c^3*d^2*e^4 + 5*b*c^2*d*e^5 + 3*b^2*c*e^6 + 3*a*c^2*e^6)*x^4 + 15*a^2*b*d*e^5 + (b^3*e^3 + 6*a
*b*c*e^3)*d^3 + 56*(5*c^3*d^3*e^3 + 5*b*c^2*d^2*e^4 + b^3*e^6 + 6*a*b*c*e^6 + 3*(b^2*c*e^5 + a*c^2*e^5)*d)*x^3
+ 35*a^3*e^6 + 5*(a*b^2*e^4 + a^2*c*e^4)*d^2 + 28*(5*c^3*d^4*e^2 + 5*b*c^2*d^3*e^3 + 5*a*b^2*e^6 + 5*a^2*c*e^
6 + 3*(b^2*c*e^4 + a*c^2*e^4)*d^2 + (b^3*e^5 + 6*a*b*c*e^5)*d)*x^2 + 8*(5*c^3*d^5*e + 5*b*c^2*d^4*e^2 + 3*(b^2
*c*e^3 + a*c^2*e^3)*d^3 + 15*a^2*b*e^6 + (b^3*e^4 + 6*a*b*c*e^4)*d^2 + 5*(a*b^2*e^5 + a^2*c*e^5)*d)*x)/(x^8*e^
15 + 8*d*x^7*e^14 + 28*d^2*x^6*e^13 + 56*d^3*x^5*e^12 + 70*d^4*x^4*e^11 + 56*d^5*x^3*e^10 + 28*d^6*x^2*e^9 + 8
*d^7*x*e^8 + d^8*e^7)

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Fricas [A]
time = 3.63, size = 445, normalized size = 1.65 \begin {gather*} -\frac {5 \, c^{3} d^{6} + {\left (140 \, c^{3} x^{6} + 280 \, b c^{2} x^{5} + 210 \, {\left (b^{2} c + a c^{2}\right )} x^{4} + 120 \, a^{2} b x + 56 \, {\left (b^{3} + 6 \, a b c\right )} x^{3} + 35 \, a^{3} + 140 \, {\left (a b^{2} + a^{2} c\right )} x^{2}\right )} e^{6} + {\left (280 \, c^{3} d x^{5} + 350 \, b c^{2} d x^{4} + 168 \, {\left (b^{2} c + a c^{2}\right )} d x^{3} + 15 \, a^{2} b d + 28 \, {\left (b^{3} + 6 \, a b c\right )} d x^{2} + 40 \, {\left (a b^{2} + a^{2} c\right )} d x\right )} e^{5} + {\left (350 \, c^{3} d^{2} x^{4} + 280 \, b c^{2} d^{2} x^{3} + 84 \, {\left (b^{2} c + a c^{2}\right )} d^{2} x^{2} + 8 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} x + 5 \, {\left (a b^{2} + a^{2} c\right )} d^{2}\right )} e^{4} + {\left (280 \, c^{3} d^{3} x^{3} + 140 \, b c^{2} d^{3} x^{2} + 24 \, {\left (b^{2} c + a c^{2}\right )} d^{3} x + {\left (b^{3} + 6 \, a b c\right )} d^{3}\right )} e^{3} + {\left (140 \, c^{3} d^{4} x^{2} + 40 \, b c^{2} d^{4} x + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{4}\right )} e^{2} + 5 \, {\left (8 \, c^{3} d^{5} x + b c^{2} d^{5}\right )} e}{280 \, {\left (x^{8} e^{15} + 8 \, d x^{7} e^{14} + 28 \, d^{2} x^{6} e^{13} + 56 \, d^{3} x^{5} e^{12} + 70 \, d^{4} x^{4} e^{11} + 56 \, d^{5} x^{3} e^{10} + 28 \, d^{6} x^{2} e^{9} + 8 \, d^{7} x e^{8} + d^{8} 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)^9,x, algorithm="fricas")

[Out]

-1/280*(5*c^3*d^6 + (140*c^3*x^6 + 280*b*c^2*x^5 + 210*(b^2*c + a*c^2)*x^4 + 120*a^2*b*x + 56*(b^3 + 6*a*b*c)*
x^3 + 35*a^3 + 140*(a*b^2 + a^2*c)*x^2)*e^6 + (280*c^3*d*x^5 + 350*b*c^2*d*x^4 + 168*(b^2*c + a*c^2)*d*x^3 + 1
5*a^2*b*d + 28*(b^3 + 6*a*b*c)*d*x^2 + 40*(a*b^2 + a^2*c)*d*x)*e^5 + (350*c^3*d^2*x^4 + 280*b*c^2*d^2*x^3 + 84
*(b^2*c + a*c^2)*d^2*x^2 + 8*(b^3 + 6*a*b*c)*d^2*x + 5*(a*b^2 + a^2*c)*d^2)*e^4 + (280*c^3*d^3*x^3 + 140*b*c^2
*d^3*x^2 + 24*(b^2*c + a*c^2)*d^3*x + (b^3 + 6*a*b*c)*d^3)*e^3 + (140*c^3*d^4*x^2 + 40*b*c^2*d^4*x + 3*(b^2*c
+ a*c^2)*d^4)*e^2 + 5*(8*c^3*d^5*x + b*c^2*d^5)*e)/(x^8*e^15 + 8*d*x^7*e^14 + 28*d^2*x^6*e^13 + 56*d^3*x^5*e^1
2 + 70*d^4*x^4*e^11 + 56*d^5*x^3*e^10 + 28*d^6*x^2*e^9 + 8*d^7*x*e^8 + d^8*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)**9,x)

[Out]

Timed out

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Giac [A]
time = 1.25, size = 458, normalized size = 1.70 \begin {gather*} -\frac {{\left (140 \, c^{3} x^{6} e^{6} + 280 \, c^{3} d x^{5} e^{5} + 350 \, c^{3} d^{2} x^{4} e^{4} + 280 \, c^{3} d^{3} x^{3} e^{3} + 140 \, c^{3} d^{4} x^{2} e^{2} + 40 \, c^{3} d^{5} x e + 5 \, c^{3} d^{6} + 280 \, b c^{2} x^{5} e^{6} + 350 \, b c^{2} d x^{4} e^{5} + 280 \, b c^{2} d^{2} x^{3} e^{4} + 140 \, b c^{2} d^{3} x^{2} e^{3} + 40 \, b c^{2} d^{4} x e^{2} + 5 \, b c^{2} d^{5} e + 210 \, b^{2} c x^{4} e^{6} + 210 \, a c^{2} x^{4} e^{6} + 168 \, b^{2} c d x^{3} e^{5} + 168 \, a c^{2} d x^{3} e^{5} + 84 \, b^{2} c d^{2} x^{2} e^{4} + 84 \, a c^{2} d^{2} x^{2} e^{4} + 24 \, b^{2} c d^{3} x e^{3} + 24 \, a c^{2} d^{3} x e^{3} + 3 \, b^{2} c d^{4} e^{2} + 3 \, a c^{2} d^{4} e^{2} + 56 \, b^{3} x^{3} e^{6} + 336 \, a b c x^{3} e^{6} + 28 \, b^{3} d x^{2} e^{5} + 168 \, a b c d x^{2} e^{5} + 8 \, b^{3} d^{2} x e^{4} + 48 \, a b c d^{2} x e^{4} + b^{3} d^{3} e^{3} + 6 \, a b c d^{3} e^{3} + 140 \, a b^{2} x^{2} e^{6} + 140 \, a^{2} c x^{2} e^{6} + 40 \, a b^{2} d x e^{5} + 40 \, a^{2} c d x e^{5} + 5 \, a b^{2} d^{2} e^{4} + 5 \, a^{2} c d^{2} e^{4} + 120 \, a^{2} b x e^{6} + 15 \, a^{2} b d e^{5} + 35 \, a^{3} e^{6}\right )} e^{\left (-7\right )}}{280 \, {\left (x e + d\right )}^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^3/(e*x+d)^9,x, algorithm="giac")

[Out]

-1/280*(140*c^3*x^6*e^6 + 280*c^3*d*x^5*e^5 + 350*c^3*d^2*x^4*e^4 + 280*c^3*d^3*x^3*e^3 + 140*c^3*d^4*x^2*e^2
+ 40*c^3*d^5*x*e + 5*c^3*d^6 + 280*b*c^2*x^5*e^6 + 350*b*c^2*d*x^4*e^5 + 280*b*c^2*d^2*x^3*e^4 + 140*b*c^2*d^3
*x^2*e^3 + 40*b*c^2*d^4*x*e^2 + 5*b*c^2*d^5*e + 210*b^2*c*x^4*e^6 + 210*a*c^2*x^4*e^6 + 168*b^2*c*d*x^3*e^5 +
168*a*c^2*d*x^3*e^5 + 84*b^2*c*d^2*x^2*e^4 + 84*a*c^2*d^2*x^2*e^4 + 24*b^2*c*d^3*x*e^3 + 24*a*c^2*d^3*x*e^3 +
3*b^2*c*d^4*e^2 + 3*a*c^2*d^4*e^2 + 56*b^3*x^3*e^6 + 336*a*b*c*x^3*e^6 + 28*b^3*d*x^2*e^5 + 168*a*b*c*d*x^2*e^
5 + 8*b^3*d^2*x*e^4 + 48*a*b*c*d^2*x*e^4 + b^3*d^3*e^3 + 6*a*b*c*d^3*e^3 + 140*a*b^2*x^2*e^6 + 140*a^2*c*x^2*e
^6 + 40*a*b^2*d*x*e^5 + 40*a^2*c*d*x*e^5 + 5*a*b^2*d^2*e^4 + 5*a^2*c*d^2*e^4 + 120*a^2*b*x*e^6 + 15*a^2*b*d*e^
5 + 35*a^3*e^6)*e^(-7)/(x*e + d)^8

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Mupad [B]
time = 0.16, size = 512, normalized size = 1.90 \begin {gather*} -\frac {\frac {35\,a^3\,e^6+15\,a^2\,b\,d\,e^5+5\,a^2\,c\,d^2\,e^4+5\,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+5\,b\,c^2\,d^5\,e+5\,c^3\,d^6}{280\,e^7}+\frac {x^3\,\left (b^3\,e^3+3\,b^2\,c\,d\,e^2+5\,b\,c^2\,d^2\,e+6\,a\,b\,c\,e^3+5\,c^3\,d^3+3\,a\,c^2\,d\,e^2\right )}{5\,e^4}+\frac {x^2\,\left (5\,a^2\,c\,e^4+5\,a\,b^2\,e^4+6\,a\,b\,c\,d\,e^3+3\,a\,c^2\,d^2\,e^2+b^3\,d\,e^3+3\,b^2\,c\,d^2\,e^2+5\,b\,c^2\,d^3\,e+5\,c^3\,d^4\right )}{10\,e^5}+\frac {c^3\,x^6}{2\,e}+\frac {x\,\left (15\,a^2\,b\,e^5+5\,a^2\,c\,d\,e^4+5\,a\,b^2\,d\,e^4+6\,a\,b\,c\,d^2\,e^3+3\,a\,c^2\,d^3\,e^2+b^3\,d^2\,e^3+3\,b^2\,c\,d^3\,e^2+5\,b\,c^2\,d^4\,e+5\,c^3\,d^5\right )}{35\,e^6}+\frac {c\,x^4\,\left (3\,b^2\,e^2+5\,b\,c\,d\,e+5\,c^2\,d^2+3\,a\,c\,e^2\right )}{4\,e^3}+\frac {c^2\,x^5\,\left (b\,e+c\,d\right )}{e^2}}{d^8+8\,d^7\,e\,x+28\,d^6\,e^2\,x^2+56\,d^5\,e^3\,x^3+70\,d^4\,e^4\,x^4+56\,d^3\,e^5\,x^5+28\,d^2\,e^6\,x^6+8\,d\,e^7\,x^7+e^8\,x^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x + c*x^2)^3/(d + e*x)^9,x)

[Out]

-((35*a^3*e^6 + 5*c^3*d^6 + b^3*d^3*e^3 + 5*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 5*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^
2 + 15*a^2*b*d*e^5 + 5*b*c^2*d^5*e + 6*a*b*c*d^3*e^3)/(280*e^7) + (x^3*(b^3*e^3 + 5*c^3*d^3 + 6*a*b*c*e^3 + 3*
a*c^2*d*e^2 + 5*b*c^2*d^2*e + 3*b^2*c*d*e^2))/(5*e^4) + (x^2*(5*c^3*d^4 + 5*a*b^2*e^4 + 5*a^2*c*e^4 + b^3*d*e^
3 + 3*a*c^2*d^2*e^2 + 3*b^2*c*d^2*e^2 + 5*b*c^2*d^3*e + 6*a*b*c*d*e^3))/(10*e^5) + (c^3*x^6)/(2*e) + (x*(5*c^3
*d^5 + 15*a^2*b*e^5 + b^3*d^2*e^3 + 3*a*c^2*d^3*e^2 + 3*b^2*c*d^3*e^2 + 5*a*b^2*d*e^4 + 5*a^2*c*d*e^4 + 5*b*c^
2*d^4*e + 6*a*b*c*d^2*e^3))/(35*e^6) + (c*x^4*(3*b^2*e^2 + 5*c^2*d^2 + 3*a*c*e^2 + 5*b*c*d*e))/(4*e^3) + (c^2*
x^5*(b*e + c*d))/e^2)/(d^8 + e^8*x^8 + 8*d*e^7*x^7 + 28*d^6*e^2*x^2 + 56*d^5*e^3*x^3 + 70*d^4*e^4*x^4 + 56*d^3
*e^5*x^5 + 28*d^2*e^6*x^6 + 8*d^7*e*x)

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