3.22.43 \(\int \frac {(a+b x+c x^2)^3}{(d+e x)^8} \, dx\) [2143]

Optimal. Leaf size=268 \[ -\frac {\left (c d^2-b d e+a e^2\right )^3}{7 e^7 (d+e x)^7}+\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{2 e^7 (d+e x)^6}-\frac {3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{5 e^7 (d+e x)^5}+\frac {(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )}{4 e^7 (d+e x)^4}-\frac {c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{e^7 (d+e x)^3}+\frac {3 c^2 (2 c d-b e)}{2 e^7 (d+e x)^2}-\frac {c^3}{e^7 (d+e x)} \]

[Out]

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

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Rubi [A]
time = 0.14, antiderivative size = 268, 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 {c \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7 (d+e x)^3}+\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 )}{4 e^7 (d+e x)^4}-\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 )}{5 e^7 (d+e x)^5}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{2 e^7 (d+e x)^6}-\frac {\left (a e^2-b d e+c d^2\right )^3}{7 e^7 (d+e x)^7}+\frac {3 c^2 (2 c d-b e)}{2 e^7 (d+e x)^2}-\frac {c^3}{e^7 (d+e x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^3/(d + e*x)^8,x]

[Out]

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

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

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Mathematica [A]
time = 0.09, size = 377, normalized size = 1.41 \begin {gather*} -\frac {20 c^3 \left (d^6+7 d^5 e x+21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+21 d e^5 x^5+7 e^6 x^6\right )+e^3 \left (20 a^3 e^3+10 a^2 b e^2 (d+7 e x)+4 a b^2 e \left (d^2+7 d e x+21 e^2 x^2\right )+b^3 \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )\right )+2 c e^2 \left (2 a^2 e^2 \left (d^2+7 d e x+21 e^2 x^2\right )+3 a b e \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+2 b^2 \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )\right )+2 c^2 e \left (2 a e \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )+5 b \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )\right )}{140 e^7 (d+e x)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^3/(d + e*x)^8,x]

[Out]

-1/140*(20*c^3*(d^6 + 7*d^5*e*x + 21*d^4*e^2*x^2 + 35*d^3*e^3*x^3 + 35*d^2*e^4*x^4 + 21*d*e^5*x^5 + 7*e^6*x^6)
+ e^3*(20*a^3*e^3 + 10*a^2*b*e^2*(d + 7*e*x) + 4*a*b^2*e*(d^2 + 7*d*e*x + 21*e^2*x^2) + b^3*(d^3 + 7*d^2*e*x
+ 21*d*e^2*x^2 + 35*e^3*x^3)) + 2*c*e^2*(2*a^2*e^2*(d^2 + 7*d*e*x + 21*e^2*x^2) + 3*a*b*e*(d^3 + 7*d^2*e*x + 2
1*d*e^2*x^2 + 35*e^3*x^3) + 2*b^2*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4)) + 2*c^2*e*(2
*a*e*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4) + 5*b*(d^5 + 7*d^4*e*x + 21*d^3*e^2*x^2 +
35*d^2*e^3*x^3 + 35*d*e^4*x^4 + 21*e^5*x^5)))/(e^7*(d + e*x)^7)

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Maple [A]
time = 0.74, size = 461, normalized size = 1.72

method	result	size
risch	\(\frac {-\frac {c^{3} x^{6}}{e}-\frac {3 c^{2} \left (b e +2 c d \right ) x^{5}}{2 e^{2}}-\frac {c \left (2 a c \,e^{2}+2 b^{2} e^{2}+5 b c d e +10 c^{2} d^{2}\right ) x^{4}}{2 e^{3}}-\frac {\left (6 a b c \,e^{3}+4 d \,e^{2} c^{2} a +b^{3} e^{3}+4 b^{2} d \,e^{2} c +10 b \,c^{2} d^{2} e +20 c^{3} d^{3}\right ) x^{3}}{4 e^{4}}-\frac {3 \left (4 e^{4} a^{2} c +4 a \,b^{2} e^{4}+6 a b c d \,e^{3}+4 d^{2} e^{2} c^{2} a +b^{3} d \,e^{3}+4 b^{2} c \,d^{2} e^{2}+10 d^{3} e b \,c^{2}+20 d^{4} c^{3}\right ) x^{2}}{20 e^{5}}-\frac {\left (10 a^{2} b \,e^{5}+4 d \,e^{4} a^{2} c +4 a \,b^{2} d \,e^{4}+6 a b c \,d^{2} e^{3}+4 d^{3} e^{2} c^{2} a +b^{3} d^{2} e^{3}+4 b^{2} c \,d^{3} e^{2}+10 b \,c^{2} d^{4} e +20 d^{5} c^{3}\right ) x}{20 e^{6}}-\frac {20 e^{6} a^{3}+10 a^{2} b d \,e^{5}+4 e^{4} d^{2} a^{2} c +4 a \,b^{2} d^{2} e^{4}+6 a b c \,d^{3} e^{3}+4 d^{4} e^{2} c^{2} a +b^{3} d^{3} e^{3}+4 b^{2} c \,d^{4} e^{2}+10 b \,c^{2} d^{5} e +20 d^{6} c^{3}}{140 e^{7}}}{\left (e x +d \right )^{7}}\)	\(437\)
norman	\(\frac {-\frac {c^{3} x^{6}}{e}-\frac {3 \left (b \,c^{2} e +2 c^{3} d \right ) x^{5}}{2 e^{2}}-\frac {\left (2 e^{2} c^{2} a +2 b^{2} e^{2} c +5 d e b \,c^{2}+10 c^{3} d^{2}\right ) x^{4}}{2 e^{3}}-\frac {\left (6 a b c \,e^{3}+4 d \,e^{2} c^{2} a +b^{3} e^{3}+4 b^{2} d \,e^{2} c +10 b \,c^{2} d^{2} e +20 c^{3} d^{3}\right ) x^{3}}{4 e^{4}}-\frac {3 \left (4 e^{4} a^{2} c +4 a \,b^{2} e^{4}+6 a b c d \,e^{3}+4 d^{2} e^{2} c^{2} a +b^{3} d \,e^{3}+4 b^{2} c \,d^{2} e^{2}+10 d^{3} e b \,c^{2}+20 d^{4} c^{3}\right ) x^{2}}{20 e^{5}}-\frac {\left (10 a^{2} b \,e^{5}+4 d \,e^{4} a^{2} c +4 a \,b^{2} d \,e^{4}+6 a b c \,d^{2} e^{3}+4 d^{3} e^{2} c^{2} a +b^{3} d^{2} e^{3}+4 b^{2} c \,d^{3} e^{2}+10 b \,c^{2} d^{4} e +20 d^{5} c^{3}\right ) x}{20 e^{6}}-\frac {20 e^{6} a^{3}+10 a^{2} b d \,e^{5}+4 e^{4} d^{2} a^{2} c +4 a \,b^{2} d^{2} e^{4}+6 a b c \,d^{3} e^{3}+4 d^{4} e^{2} c^{2} a +b^{3} d^{3} e^{3}+4 b^{2} c \,d^{4} e^{2}+10 b \,c^{2} d^{5} e +20 d^{6} c^{3}}{140 e^{7}}}{\left (e x +d \right )^{7}}\)	\(443\)
default	\(-\frac {3 a^{2} b \,e^{5}-6 d \,e^{4} a^{2} c -6 a \,b^{2} d \,e^{4}+18 a b c \,d^{2} e^{3}-12 d^{3} e^{2} c^{2} a +3 b^{3} d^{2} e^{3}-12 b^{2} c \,d^{3} e^{2}+15 b \,c^{2} d^{4} e -6 d^{5} c^{3}}{6 e^{7} \left (e x +d \right )^{6}}-\frac {c \left (a c \,e^{2}+b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right )}{e^{7} \left (e x +d \right )^{3}}-\frac {c^{3}}{e^{7} \left (e x +d \right )}-\frac {e^{6} a^{3}-3 a^{2} b d \,e^{5}+3 e^{4} d^{2} a^{2} c +3 a \,b^{2} d^{2} e^{4}-6 a b c \,d^{3} e^{3}+3 d^{4} e^{2} c^{2} a -b^{3} d^{3} e^{3}+3 b^{2} c \,d^{4} e^{2}-3 b \,c^{2} d^{5} e +d^{6} c^{3}}{7 e^{7} \left (e x +d \right )^{7}}-\frac {3 c^{2} \left (b e -2 c d \right )}{2 e^{7} \left (e x +d \right )^{2}}-\frac {3 e^{4} a^{2} c +3 a \,b^{2} e^{4}-18 a b c d \,e^{3}+18 d^{2} e^{2} c^{2} a -3 b^{3} d \,e^{3}+18 b^{2} c \,d^{2} e^{2}-30 d^{3} e b \,c^{2}+15 d^{4} c^{3}}{5 e^{7} \left (e x +d \right )^{5}}-\frac {6 a b c \,e^{3}-12 d \,e^{2} c^{2} a +b^{3} e^{3}-12 b^{2} d \,e^{2} c +30 b \,c^{2} d^{2} e -20 c^{3} d^{3}}{4 e^{7} \left (e x +d \right )^{4}}\)	\(461\)
gosper	\(-\frac {140 c^{3} e^{6} x^{6}+210 b \,c^{2} e^{6} x^{5}+420 c^{3} d \,e^{5} x^{5}+140 a \,c^{2} e^{6} x^{4}+140 b^{2} c \,e^{6} x^{4}+350 b \,c^{2} d \,e^{5} x^{4}+700 c^{3} d^{2} e^{4} x^{4}+210 a b c \,e^{6} x^{3}+140 a \,c^{2} d \,e^{5} x^{3}+35 b^{3} e^{6} x^{3}+140 b^{2} c d \,e^{5} x^{3}+350 b \,c^{2} d^{2} e^{4} x^{3}+700 c^{3} d^{3} e^{3} x^{3}+84 a^{2} c \,e^{6} x^{2}+84 a \,b^{2} e^{6} x^{2}+126 a b c d \,e^{5} x^{2}+84 a \,c^{2} d^{2} e^{4} x^{2}+21 b^{3} d \,e^{5} x^{2}+84 b^{2} c \,d^{2} e^{4} x^{2}+210 b \,c^{2} d^{3} e^{3} x^{2}+420 c^{3} d^{4} e^{2} x^{2}+70 a^{2} b \,e^{6} x +28 a^{2} c d \,e^{5} x +28 a \,b^{2} d \,e^{5} x +42 a b c \,d^{2} e^{4} x +28 a \,c^{2} d^{3} e^{3} x +7 b^{3} d^{2} e^{4} x +28 b^{2} c \,d^{3} e^{3} x +70 b \,c^{2} d^{4} e^{2} x +140 c^{3} d^{5} e x +20 e^{6} a^{3}+10 a^{2} b d \,e^{5}+4 e^{4} d^{2} a^{2} c +4 a \,b^{2} d^{2} e^{4}+6 a b c \,d^{3} e^{3}+4 d^{4} e^{2} c^{2} a +b^{3} d^{3} e^{3}+4 b^{2} c \,d^{4} e^{2}+10 b \,c^{2} d^{5} e +20 d^{6} c^{3}}{140 \left (e x +d \right )^{7} e^{7}}\)	\(494\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^3/(e*x+d)^8,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 465, normalized size = 1.74 \begin {gather*} -\frac {140 \, c^{3} x^{6} e^{6} + 20 \, c^{3} d^{6} + 10 \, b c^{2} d^{5} e + 210 \, {\left (2 \, c^{3} d e^{5} + b c^{2} e^{6}\right )} x^{5} + 4 \, {\left (b^{2} c e^{2} + a c^{2} e^{2}\right )} d^{4} + 70 \, {\left (10 \, c^{3} d^{2} e^{4} + 5 \, b c^{2} d e^{5} + 2 \, b^{2} c e^{6} + 2 \, a c^{2} e^{6}\right )} x^{4} + 10 \, a^{2} b d e^{5} + {\left (b^{3} e^{3} + 6 \, a b c e^{3}\right )} d^{3} + 35 \, {\left (20 \, c^{3} d^{3} e^{3} + 10 \, b c^{2} d^{2} e^{4} + b^{3} e^{6} + 6 \, a b c e^{6} + 4 \, {\left (b^{2} c e^{5} + a c^{2} e^{5}\right )} d\right )} x^{3} + 20 \, a^{3} e^{6} + 4 \, {\left (a b^{2} e^{4} + a^{2} c e^{4}\right )} d^{2} + 21 \, {\left (20 \, c^{3} d^{4} e^{2} + 10 \, b c^{2} d^{3} e^{3} + 4 \, a b^{2} e^{6} + 4 \, a^{2} c e^{6} + 4 \, {\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} + 7 \, {\left (20 \, c^{3} d^{5} e + 10 \, b c^{2} d^{4} e^{2} + 4 \, {\left (b^{2} c e^{3} + a c^{2} e^{3}\right )} d^{3} + 10 \, a^{2} b e^{6} + {\left (b^{3} e^{4} + 6 \, a b c e^{4}\right )} d^{2} + 4 \, {\left (a b^{2} e^{5} + a^{2} c e^{5}\right )} d\right )} x}{140 \, {\left (x^{7} e^{14} + 7 \, d x^{6} e^{13} + 21 \, d^{2} x^{5} e^{12} + 35 \, d^{3} x^{4} e^{11} + 35 \, d^{4} x^{3} e^{10} + 21 \, d^{5} x^{2} e^{9} + 7 \, d^{6} x e^{8} + d^{7} 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)^8,x, algorithm="maxima")

[Out]

-1/140*(140*c^3*x^6*e^6 + 20*c^3*d^6 + 10*b*c^2*d^5*e + 210*(2*c^3*d*e^5 + b*c^2*e^6)*x^5 + 4*(b^2*c*e^2 + a*c
^2*e^2)*d^4 + 70*(10*c^3*d^2*e^4 + 5*b*c^2*d*e^5 + 2*b^2*c*e^6 + 2*a*c^2*e^6)*x^4 + 10*a^2*b*d*e^5 + (b^3*e^3
+ 6*a*b*c*e^3)*d^3 + 35*(20*c^3*d^3*e^3 + 10*b*c^2*d^2*e^4 + b^3*e^6 + 6*a*b*c*e^6 + 4*(b^2*c*e^5 + a*c^2*e^5)
*d)*x^3 + 20*a^3*e^6 + 4*(a*b^2*e^4 + a^2*c*e^4)*d^2 + 21*(20*c^3*d^4*e^2 + 10*b*c^2*d^3*e^3 + 4*a*b^2*e^6 + 4
*a^2*c*e^6 + 4*(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 + 7*(20*c^3*d^5*e + 10*b*c^2*d^4*e
^2 + 4*(b^2*c*e^3 + a*c^2*e^3)*d^3 + 10*a^2*b*e^6 + (b^3*e^4 + 6*a*b*c*e^4)*d^2 + 4*(a*b^2*e^5 + a^2*c*e^5)*d)
*x)/(x^7*e^14 + 7*d*x^6*e^13 + 21*d^2*x^5*e^12 + 35*d^3*x^4*e^11 + 35*d^4*x^3*e^10 + 21*d^5*x^2*e^9 + 7*d^6*x*
e^8 + d^7*e^7)

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Fricas [A]
time = 2.68, size = 436, normalized size = 1.63 \begin {gather*} -\frac {20 \, c^{3} d^{6} + {\left (140 \, c^{3} x^{6} + 210 \, b c^{2} x^{5} + 140 \, {\left (b^{2} c + a c^{2}\right )} x^{4} + 70 \, a^{2} b x + 35 \, {\left (b^{3} + 6 \, a b c\right )} x^{3} + 20 \, a^{3} + 84 \, {\left (a b^{2} + a^{2} c\right )} x^{2}\right )} e^{6} + {\left (420 \, c^{3} d x^{5} + 350 \, b c^{2} d x^{4} + 140 \, {\left (b^{2} c + a c^{2}\right )} d x^{3} + 10 \, a^{2} b d + 21 \, {\left (b^{3} + 6 \, a b c\right )} d x^{2} + 28 \, {\left (a b^{2} + a^{2} c\right )} d x\right )} e^{5} + {\left (700 \, c^{3} d^{2} x^{4} + 350 \, b c^{2} d^{2} x^{3} + 84 \, {\left (b^{2} c + a c^{2}\right )} d^{2} x^{2} + 7 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} x + 4 \, {\left (a b^{2} + a^{2} c\right )} d^{2}\right )} e^{4} + {\left (700 \, c^{3} d^{3} x^{3} + 210 \, b c^{2} d^{3} x^{2} + 28 \, {\left (b^{2} c + a c^{2}\right )} d^{3} x + {\left (b^{3} + 6 \, a b c\right )} d^{3}\right )} e^{3} + 2 \, {\left (210 \, c^{3} d^{4} x^{2} + 35 \, b c^{2} d^{4} x + 2 \, {\left (b^{2} c + a c^{2}\right )} d^{4}\right )} e^{2} + 10 \, {\left (14 \, c^{3} d^{5} x + b c^{2} d^{5}\right )} e}{140 \, {\left (x^{7} e^{14} + 7 \, d x^{6} e^{13} + 21 \, d^{2} x^{5} e^{12} + 35 \, d^{3} x^{4} e^{11} + 35 \, d^{4} x^{3} e^{10} + 21 \, d^{5} x^{2} e^{9} + 7 \, d^{6} x e^{8} + d^{7} 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)^8,x, algorithm="fricas")

[Out]

-1/140*(20*c^3*d^6 + (140*c^3*x^6 + 210*b*c^2*x^5 + 140*(b^2*c + a*c^2)*x^4 + 70*a^2*b*x + 35*(b^3 + 6*a*b*c)*
x^3 + 20*a^3 + 84*(a*b^2 + a^2*c)*x^2)*e^6 + (420*c^3*d*x^5 + 350*b*c^2*d*x^4 + 140*(b^2*c + a*c^2)*d*x^3 + 10
*a^2*b*d + 21*(b^3 + 6*a*b*c)*d*x^2 + 28*(a*b^2 + a^2*c)*d*x)*e^5 + (700*c^3*d^2*x^4 + 350*b*c^2*d^2*x^3 + 84*
(b^2*c + a*c^2)*d^2*x^2 + 7*(b^3 + 6*a*b*c)*d^2*x + 4*(a*b^2 + a^2*c)*d^2)*e^4 + (700*c^3*d^3*x^3 + 210*b*c^2*
d^3*x^2 + 28*(b^2*c + a*c^2)*d^3*x + (b^3 + 6*a*b*c)*d^3)*e^3 + 2*(210*c^3*d^4*x^2 + 35*b*c^2*d^4*x + 2*(b^2*c
+ a*c^2)*d^4)*e^2 + 10*(14*c^3*d^5*x + b*c^2*d^5)*e)/(x^7*e^14 + 7*d*x^6*e^13 + 21*d^2*x^5*e^12 + 35*d^3*x^4*
e^11 + 35*d^4*x^3*e^10 + 21*d^5*x^2*e^9 + 7*d^6*x*e^8 + d^7*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)**8,x)

[Out]

Timed out

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Giac [A]
time = 1.10, size = 458, normalized size = 1.71 \begin {gather*} -\frac {{\left (140 \, c^{3} x^{6} e^{6} + 420 \, c^{3} d x^{5} e^{5} + 700 \, c^{3} d^{2} x^{4} e^{4} + 700 \, c^{3} d^{3} x^{3} e^{3} + 420 \, c^{3} d^{4} x^{2} e^{2} + 140 \, c^{3} d^{5} x e + 20 \, c^{3} d^{6} + 210 \, b c^{2} x^{5} e^{6} + 350 \, b c^{2} d x^{4} e^{5} + 350 \, b c^{2} d^{2} x^{3} e^{4} + 210 \, b c^{2} d^{3} x^{2} e^{3} + 70 \, b c^{2} d^{4} x e^{2} + 10 \, b c^{2} d^{5} e + 140 \, b^{2} c x^{4} e^{6} + 140 \, a c^{2} x^{4} e^{6} + 140 \, b^{2} c d x^{3} e^{5} + 140 \, 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} + 28 \, b^{2} c d^{3} x e^{3} + 28 \, a c^{2} d^{3} x e^{3} + 4 \, b^{2} c d^{4} e^{2} + 4 \, a c^{2} d^{4} e^{2} + 35 \, b^{3} x^{3} e^{6} + 210 \, a b c x^{3} e^{6} + 21 \, b^{3} d x^{2} e^{5} + 126 \, a b c d x^{2} e^{5} + 7 \, b^{3} d^{2} x e^{4} + 42 \, a b c d^{2} x e^{4} + b^{3} d^{3} e^{3} + 6 \, a b c d^{3} e^{3} + 84 \, a b^{2} x^{2} e^{6} + 84 \, a^{2} c x^{2} e^{6} + 28 \, a b^{2} d x e^{5} + 28 \, a^{2} c d x e^{5} + 4 \, a b^{2} d^{2} e^{4} + 4 \, a^{2} c d^{2} e^{4} + 70 \, a^{2} b x e^{6} + 10 \, a^{2} b d e^{5} + 20 \, a^{3} e^{6}\right )} e^{\left (-7\right )}}{140 \, {\left (x e + d\right )}^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^3/(e*x+d)^8,x, algorithm="giac")

[Out]

-1/140*(140*c^3*x^6*e^6 + 420*c^3*d*x^5*e^5 + 700*c^3*d^2*x^4*e^4 + 700*c^3*d^3*x^3*e^3 + 420*c^3*d^4*x^2*e^2
+ 140*c^3*d^5*x*e + 20*c^3*d^6 + 210*b*c^2*x^5*e^6 + 350*b*c^2*d*x^4*e^5 + 350*b*c^2*d^2*x^3*e^4 + 210*b*c^2*d
^3*x^2*e^3 + 70*b*c^2*d^4*x*e^2 + 10*b*c^2*d^5*e + 140*b^2*c*x^4*e^6 + 140*a*c^2*x^4*e^6 + 140*b^2*c*d*x^3*e^5
+ 140*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 + 28*b^2*c*d^3*x*e^3 + 28*a*c^2*d^3*x*e^3
+ 4*b^2*c*d^4*e^2 + 4*a*c^2*d^4*e^2 + 35*b^3*x^3*e^6 + 210*a*b*c*x^3*e^6 + 21*b^3*d*x^2*e^5 + 126*a*b*c*d*x^2
*e^5 + 7*b^3*d^2*x*e^4 + 42*a*b*c*d^2*x*e^4 + b^3*d^3*e^3 + 6*a*b*c*d^3*e^3 + 84*a*b^2*x^2*e^6 + 84*a^2*c*x^2*
e^6 + 28*a*b^2*d*x*e^5 + 28*a^2*c*d*x*e^5 + 4*a*b^2*d^2*e^4 + 4*a^2*c*d^2*e^4 + 70*a^2*b*x*e^6 + 10*a^2*b*d*e^
5 + 20*a^3*e^6)*e^(-7)/(x*e + d)^7

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Mupad [B]
time = 0.17, size = 502, normalized size = 1.87 \begin {gather*} -\frac {\frac {20\,a^3\,e^6+10\,a^2\,b\,d\,e^5+4\,a^2\,c\,d^2\,e^4+4\,a\,b^2\,d^2\,e^4+6\,a\,b\,c\,d^3\,e^3+4\,a\,c^2\,d^4\,e^2+b^3\,d^3\,e^3+4\,b^2\,c\,d^4\,e^2+10\,b\,c^2\,d^5\,e+20\,c^3\,d^6}{140\,e^7}+\frac {x^3\,\left (b^3\,e^3+4\,b^2\,c\,d\,e^2+10\,b\,c^2\,d^2\,e+6\,a\,b\,c\,e^3+20\,c^3\,d^3+4\,a\,c^2\,d\,e^2\right )}{4\,e^4}+\frac {3\,x^2\,\left (4\,a^2\,c\,e^4+4\,a\,b^2\,e^4+6\,a\,b\,c\,d\,e^3+4\,a\,c^2\,d^2\,e^2+b^3\,d\,e^3+4\,b^2\,c\,d^2\,e^2+10\,b\,c^2\,d^3\,e+20\,c^3\,d^4\right )}{20\,e^5}+\frac {c^3\,x^6}{e}+\frac {x\,\left (10\,a^2\,b\,e^5+4\,a^2\,c\,d\,e^4+4\,a\,b^2\,d\,e^4+6\,a\,b\,c\,d^2\,e^3+4\,a\,c^2\,d^3\,e^2+b^3\,d^2\,e^3+4\,b^2\,c\,d^3\,e^2+10\,b\,c^2\,d^4\,e+20\,c^3\,d^5\right )}{20\,e^6}+\frac {c\,x^4\,\left (2\,b^2\,e^2+5\,b\,c\,d\,e+10\,c^2\,d^2+2\,a\,c\,e^2\right )}{2\,e^3}+\frac {3\,c^2\,x^5\,\left (b\,e+2\,c\,d\right )}{2\,e^2}}{d^7+7\,d^6\,e\,x+21\,d^5\,e^2\,x^2+35\,d^4\,e^3\,x^3+35\,d^3\,e^4\,x^4+21\,d^2\,e^5\,x^5+7\,d\,e^6\,x^6+e^7\,x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x + c*x^2)^3/(d + e*x)^8,x)

[Out]

-((20*a^3*e^6 + 20*c^3*d^6 + b^3*d^3*e^3 + 4*a*b^2*d^2*e^4 + 4*a*c^2*d^4*e^2 + 4*a^2*c*d^2*e^4 + 4*b^2*c*d^4*e
^2 + 10*a^2*b*d*e^5 + 10*b*c^2*d^5*e + 6*a*b*c*d^3*e^3)/(140*e^7) + (x^3*(b^3*e^3 + 20*c^3*d^3 + 6*a*b*c*e^3 +
4*a*c^2*d*e^2 + 10*b*c^2*d^2*e + 4*b^2*c*d*e^2))/(4*e^4) + (3*x^2*(20*c^3*d^4 + 4*a*b^2*e^4 + 4*a^2*c*e^4 + b
^3*d*e^3 + 4*a*c^2*d^2*e^2 + 4*b^2*c*d^2*e^2 + 10*b*c^2*d^3*e + 6*a*b*c*d*e^3))/(20*e^5) + (c^3*x^6)/e + (x*(2
0*c^3*d^5 + 10*a^2*b*e^5 + b^3*d^2*e^3 + 4*a*c^2*d^3*e^2 + 4*b^2*c*d^3*e^2 + 4*a*b^2*d*e^4 + 4*a^2*c*d*e^4 + 1
0*b*c^2*d^4*e + 6*a*b*c*d^2*e^3))/(20*e^6) + (c*x^4*(2*b^2*e^2 + 10*c^2*d^2 + 2*a*c*e^2 + 5*b*c*d*e))/(2*e^3)
+ (3*c^2*x^5*(b*e + 2*c*d))/(2*e^2))/(d^7 + e^7*x^7 + 7*d*e^6*x^6 + 21*d^5*e^2*x^2 + 35*d^4*e^3*x^3 + 35*d^3*e
^4*x^4 + 21*d^2*e^5*x^5 + 7*d^6*e*x)

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