### 3.2143 $$\int \frac{(a+b x+c x^2)^3}{(d+e x)^8} \, dx$$

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

[Out]

-(c*d^2 - b*d*e + a*e^2)^3/(7*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))

________________________________________________________________________________________

Rubi [A]  time = 0.207406, antiderivative size = 268, 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} $-\frac{c \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7 (d+e x)^3}+\frac{(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{4 e^7 (d+e x)^4}-\frac{3 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^7 (d+e x)^5}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{2 e^7 (d+e x)^6}-\frac{\left (a e^2-b d e+c d^2\right )^3}{7 e^7 (d+e x)^7}+\frac{3 c^2 (2 c d-b e)}{2 e^7 (d+e x)^2}-\frac{c^3}{e^7 (d+e x)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(c*d^2 - b*d*e + a*e^2)^3/(7*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 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)^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*}

Mathematica [A]  time = 0.142781, size = 377, normalized size = 1.41 $-\frac{2 c e^2 \left (2 a^2 e^2 \left (d^2+7 d e x+21 e^2 x^2\right )+3 a b e \left (7 d^2 e x+d^3+21 d e^2 x^2+35 e^3 x^3\right )+2 b^2 \left (21 d^2 e^2 x^2+7 d^3 e x+d^4+35 d e^3 x^3+35 e^4 x^4\right )\right )+e^3 \left (10 a^2 b e^2 (d+7 e x)+20 a^3 e^3+4 a b^2 e \left (d^2+7 d e x+21 e^2 x^2\right )+b^3 \left (7 d^2 e x+d^3+21 d e^2 x^2+35 e^3 x^3\right )\right )+2 c^2 e \left (2 a e \left (21 d^2 e^2 x^2+7 d^3 e x+d^4+35 d e^3 x^3+35 e^4 x^4\right )+5 b \left (21 d^3 e^2 x^2+35 d^2 e^3 x^3+7 d^4 e x+d^5+35 d e^4 x^4+21 e^5 x^5\right )\right )+20 c^3 \left (21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+7 d^5 e x+d^6+21 d e^5 x^5+7 e^6 x^6\right )}{140 e^7 (d+e x)^7}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(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 + 21*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)))/(140*e^7*(d + e*x)^7)

________________________________________________________________________________________

Maple [A]  time = 0.049, size = 461, normalized size = 1.7 \begin{align*} -{\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}}{4\,{e}^{7} \left ( ex+d \right ) ^{4}}}-{\frac{c \left ( ac{e}^{2}+{b}^{2}{e}^{2}-5\,bcde+5\,{c}^{2}{d}^{2} \right ) }{{e}^{7} \left ( ex+d \right ) ^{3}}}-{\frac{3\,{c}^{2} \left ( be-2\,cd \right ) }{2\,{e}^{7} \left ( ex+d \right ) ^{2}}}-{\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}}{6\,{e}^{7} \left ( ex+d \right ) ^{6}}}-{\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}}{5\,{e}^{7} \left ( ex+d \right ) ^{5}}}-{\frac{{c}^{3}}{{e}^{7} \left ( ex+d \right ) }}-{\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}}{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)^8,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.18849, size = 637, normalized size = 2.38 \begin{align*} -\frac{140 \, c^{3} e^{6} x^{6} + 20 \, c^{3} d^{6} + 10 \, b c^{2} d^{5} e + 10 \, a^{2} b d e^{5} + 20 \, a^{3} e^{6} + 4 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} +{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 4 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 210 \,{\left (2 \, c^{3} d e^{5} + b c^{2} e^{6}\right )} x^{5} + 70 \,{\left (10 \, c^{3} d^{2} e^{4} + 5 \, b c^{2} d e^{5} + 2 \,{\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} + 35 \,{\left (20 \, c^{3} d^{3} e^{3} + 10 \, b c^{2} d^{2} e^{4} + 4 \,{\left (b^{2} c + a c^{2}\right )} d e^{5} +{\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 21 \,{\left (20 \, c^{3} d^{4} e^{2} + 10 \, b c^{2} d^{3} e^{3} + 4 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} +{\left (b^{3} + 6 \, a b c\right )} d e^{5} + 4 \,{\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 7 \,{\left (20 \, c^{3} d^{5} e + 10 \, b c^{2} d^{4} e^{2} + 10 \, a^{2} b e^{6} + 4 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} +{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 4 \,{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x}{140 \,{\left (e^{14} x^{7} + 7 \, d e^{13} x^{6} + 21 \, d^{2} e^{12} x^{5} + 35 \, d^{3} e^{11} x^{4} + 35 \, d^{4} e^{10} x^{3} + 21 \, d^{5} e^{9} x^{2} + 7 \, d^{6} e^{8} x + d^{7} 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)^8,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.14285, size = 999, normalized size = 3.73 \begin{align*} -\frac{140 \, c^{3} e^{6} x^{6} + 20 \, c^{3} d^{6} + 10 \, b c^{2} d^{5} e + 10 \, a^{2} b d e^{5} + 20 \, a^{3} e^{6} + 4 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} +{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 4 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 210 \,{\left (2 \, c^{3} d e^{5} + b c^{2} e^{6}\right )} x^{5} + 70 \,{\left (10 \, c^{3} d^{2} e^{4} + 5 \, b c^{2} d e^{5} + 2 \,{\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} + 35 \,{\left (20 \, c^{3} d^{3} e^{3} + 10 \, b c^{2} d^{2} e^{4} + 4 \,{\left (b^{2} c + a c^{2}\right )} d e^{5} +{\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 21 \,{\left (20 \, c^{3} d^{4} e^{2} + 10 \, b c^{2} d^{3} e^{3} + 4 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} +{\left (b^{3} + 6 \, a b c\right )} d e^{5} + 4 \,{\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 7 \,{\left (20 \, c^{3} d^{5} e + 10 \, b c^{2} d^{4} e^{2} + 10 \, a^{2} b e^{6} + 4 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} +{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 4 \,{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x}{140 \,{\left (e^{14} x^{7} + 7 \, d e^{13} x^{6} + 21 \, d^{2} e^{12} x^{5} + 35 \, d^{3} e^{11} x^{4} + 35 \, d^{4} e^{10} x^{3} + 21 \, d^{5} e^{9} x^{2} + 7 \, d^{6} e^{8} x + d^{7} 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)^8,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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)**8,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.09907, size = 618, normalized size = 2.31 \begin{align*} -\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{align*}

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