3.2145 \(\int \frac{(a+b x+c x^2)^3}{(d+e x)^{10}} \, dx\)

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

[Out]

-(c*d^2 - b*d*e + a*e^2)^3/(9*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)

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Rubi [A]  time = 0.209361, antiderivative size = 272, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {698} \[ -\frac{3 c \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^7 (d+e x)^5}+\frac{(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{6 e^7 (d+e x)^6}-\frac{3 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^7 (d+e x)^7}+\frac{3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{8 e^7 (d+e x)^8}-\frac{\left (a e^2-b d e+c d^2\right )^3}{9 e^7 (d+e x)^9}+\frac{3 c^2 (2 c d-b e)}{4 e^7 (d+e x)^4}-\frac{c^3}{3 e^7 (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c*d^2 - b*d*e + a*e^2)^3/(9*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 698

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*}

Mathematica [A]  time = 0.151566, size = 378, normalized size = 1.39 \[ -\frac{6 c e^2 \left (5 a^2 e^2 \left (d^2+9 d e x+36 e^2 x^2\right )+5 a b e \left (9 d^2 e x+d^3+36 d e^2 x^2+84 e^3 x^3\right )+2 b^2 \left (36 d^2 e^2 x^2+9 d^3 e x+d^4+84 d e^3 x^3+126 e^4 x^4\right )\right )+5 e^3 \left (21 a^2 b e^2 (d+9 e x)+56 a^3 e^3+6 a b^2 e \left (d^2+9 d e x+36 e^2 x^2\right )+b^3 \left (9 d^2 e x+d^3+36 d e^2 x^2+84 e^3 x^3\right )\right )+3 c^2 e \left (4 a e \left (36 d^2 e^2 x^2+9 d^3 e x+d^4+84 d e^3 x^3+126 e^4 x^4\right )+5 b \left (36 d^3 e^2 x^2+84 d^2 e^3 x^3+9 d^4 e x+d^5+126 d e^4 x^4+126 e^5 x^5\right )\right )+10 c^3 \left (36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+9 d^5 e x+d^6+126 d e^5 x^5+84 e^6 x^6\right )}{2520 e^7 (d+e x)^9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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)))/(2520*e^7*(d + e*x)^9)

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Maple [A]  time = 0.052, size = 461, normalized size = 1.7 \begin{align*} -{\frac{3\,{c}^{2} \left ( be-2\,cd \right ) }{4\,{e}^{7} \left ( ex+d \right ) ^{4}}}-{\frac{{c}^{3}}{3\,{e}^{7} \left ( ex+d \right ) ^{3}}}-{\frac{3\,b{a}^{2}{e}^{5}-6\,{a}^{2}cd{e}^{4}-6\,a{b}^{2}d{e}^{4}+18\,{d}^{2}abc{e}^{3}-12\,a{c}^{2}{d}^{3}{e}^{2}+3\,{b}^{3}{d}^{2}{e}^{3}-12\,{d}^{3}{b}^{2}c{e}^{2}+15\,{d}^{4}b{c}^{2}e-6\,{c}^{3}{d}^{5}}{8\,{e}^{7} \left ( ex+d \right ) ^{8}}}-{\frac{{a}^{3}{e}^{6}-3\,b{a}^{2}d{e}^{5}+3\,{a}^{2}c{d}^{2}{e}^{4}+3\,a{b}^{2}{d}^{2}{e}^{4}-6\,{d}^{3}abc{e}^{3}+3\,a{c}^{2}{d}^{4}{e}^{2}-{b}^{3}{d}^{3}{e}^{3}+3\,{d}^{4}{b}^{2}c{e}^{2}-3\,b{c}^{2}{d}^{5}e+{c}^{3}{d}^{6}}{9\,{e}^{7} \left ( ex+d \right ) ^{9}}}-{\frac{6\,abc{e}^{3}-12\,{c}^{2}ad{e}^{2}+{b}^{3}{e}^{3}-12\,{b}^{2}cd{e}^{2}+30\,b{c}^{2}{d}^{2}e-20\,{c}^{3}{d}^{3}}{6\,{e}^{7} \left ( ex+d \right ) ^{6}}}-{\frac{3\,c \left ( ac{e}^{2}+{b}^{2}{e}^{2}-5\,bcde+5\,{c}^{2}{d}^{2} \right ) }{5\,{e}^{7} \left ( ex+d \right ) ^{5}}}-{\frac{3\,{a}^{2}c{e}^{4}+3\,{b}^{2}a{e}^{4}-18\,cabd{e}^{3}+18\,a{c}^{2}{d}^{2}{e}^{2}-3\,{b}^{3}d{e}^{3}+18\,c{b}^{2}{d}^{2}{e}^{2}-30\,b{c}^{2}{d}^{3}e+15\,{c}^{3}{d}^{4}}{7\,{e}^{7} \left ( ex+d \right ) ^{7}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.09346, size = 674, normalized size = 2.48 \begin{align*} -\frac{840 \, c^{3} e^{6} x^{6} + 10 \, c^{3} d^{6} + 15 \, b c^{2} d^{5} e + 105 \, a^{2} b d e^{5} + 280 \, a^{3} e^{6} + 12 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} + 5 \,{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 30 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 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 \,{\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} + 84 \,{\left (10 \, c^{3} d^{3} e^{3} + 15 \, b c^{2} d^{2} e^{4} + 12 \,{\left (b^{2} c + a c^{2}\right )} d e^{5} + 5 \,{\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 36 \,{\left (10 \, c^{3} d^{4} e^{2} + 15 \, b c^{2} d^{3} e^{3} + 12 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} + 5 \,{\left (b^{3} + 6 \, a b c\right )} d e^{5} + 30 \,{\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 9 \,{\left (10 \, c^{3} d^{5} e + 15 \, b c^{2} d^{4} e^{2} + 105 \, a^{2} b e^{6} + 12 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} + 5 \,{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 30 \,{\left (a b^{2} + a^{2} c\right )} d e^{5}\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 )}} \end{align*}

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*e^6*x^6 + 10*c^3*d^6 + 15*b*c^2*d^5*e + 105*a^2*b*d*e^5 + 280*a^3*e^6 + 12*(b^2*c + a*c^2)*d^
4*e^2 + 5*(b^3 + 6*a*b*c)*d^3*e^3 + 30*(a*b^2 + a^2*c)*d^2*e^4 + 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 + a*c^2)*e^6)*x^4 + 84*(10*c^3*d^3*e^3 + 15*b*c^2*d^2*e^4 + 12*(b^2*
c + a*c^2)*d*e^5 + 5*(b^3 + 6*a*b*c)*e^6)*x^3 + 36*(10*c^3*d^4*e^2 + 15*b*c^2*d^3*e^3 + 12*(b^2*c + a*c^2)*d^2
*e^4 + 5*(b^3 + 6*a*b*c)*d*e^5 + 30*(a*b^2 + a^2*c)*e^6)*x^2 + 9*(10*c^3*d^5*e + 15*b*c^2*d^4*e^2 + 105*a^2*b*
e^6 + 12*(b^2*c + a*c^2)*d^3*e^3 + 5*(b^3 + 6*a*b*c)*d^2*e^4 + 30*(a*b^2 + a^2*c)*d*e^5)*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)

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Fricas [A]  time = 1.97955, size = 1083, normalized size = 3.98 \begin{align*} -\frac{840 \, c^{3} e^{6} x^{6} + 10 \, c^{3} d^{6} + 15 \, b c^{2} d^{5} e + 105 \, a^{2} b d e^{5} + 280 \, a^{3} e^{6} + 12 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} + 5 \,{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 30 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 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 \,{\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} + 84 \,{\left (10 \, c^{3} d^{3} e^{3} + 15 \, b c^{2} d^{2} e^{4} + 12 \,{\left (b^{2} c + a c^{2}\right )} d e^{5} + 5 \,{\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 36 \,{\left (10 \, c^{3} d^{4} e^{2} + 15 \, b c^{2} d^{3} e^{3} + 12 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} + 5 \,{\left (b^{3} + 6 \, a b c\right )} d e^{5} + 30 \,{\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 9 \,{\left (10 \, c^{3} d^{5} e + 15 \, b c^{2} d^{4} e^{2} + 105 \, a^{2} b e^{6} + 12 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} + 5 \,{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 30 \,{\left (a b^{2} + a^{2} c\right )} d e^{5}\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 )}} \end{align*}

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*(840*c^3*e^6*x^6 + 10*c^3*d^6 + 15*b*c^2*d^5*e + 105*a^2*b*d*e^5 + 280*a^3*e^6 + 12*(b^2*c + a*c^2)*d^
4*e^2 + 5*(b^3 + 6*a*b*c)*d^3*e^3 + 30*(a*b^2 + a^2*c)*d^2*e^4 + 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 + a*c^2)*e^6)*x^4 + 84*(10*c^3*d^3*e^3 + 15*b*c^2*d^2*e^4 + 12*(b^2*
c + a*c^2)*d*e^5 + 5*(b^3 + 6*a*b*c)*e^6)*x^3 + 36*(10*c^3*d^4*e^2 + 15*b*c^2*d^3*e^3 + 12*(b^2*c + a*c^2)*d^2
*e^4 + 5*(b^3 + 6*a*b*c)*d*e^5 + 30*(a*b^2 + a^2*c)*e^6)*x^2 + 9*(10*c^3*d^5*e + 15*b*c^2*d^4*e^2 + 105*a^2*b*
e^6 + 12*(b^2*c + a*c^2)*d^3*e^3 + 5*(b^3 + 6*a*b*c)*d^2*e^4 + 30*(a*b^2 + a^2*c)*d*e^5)*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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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 = 1.1227, size = 620, normalized size = 2.28 \begin{align*} -\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{align*}

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