### 3.2140 $$\int \frac{(a+b x+c x^2)^3}{(d+e x)^5} \, dx$$

Optimal. Leaf size=251 $\frac{(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^7 (d+e x)}-\frac{3 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{2 e^7 (d+e x)^2}+\frac{3 c \log (d+e x) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7 (d+e x)^3}-\frac{\left (a e^2-b d e+c d^2\right )^3}{4 e^7 (d+e x)^4}-\frac{c^2 x (5 c d-3 b e)}{e^6}+\frac{c^3 x^2}{2 e^5}$

[Out]

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

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Rubi [A]  time = 0.274186, antiderivative size = 251, 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} $\frac{(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^7 (d+e x)}-\frac{3 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{2 e^7 (d+e x)^2}+\frac{3 c \log (d+e x) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7 (d+e x)^3}-\frac{\left (a e^2-b d e+c d^2\right )^3}{4 e^7 (d+e x)^4}-\frac{c^2 x (5 c d-3 b e)}{e^6}+\frac{c^3 x^2}{2 e^5}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x + c*x^2)^3/(d + e*x)^5,x]

[Out]

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

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

Mathematica [A]  time = 0.175801, size = 402, normalized size = 1.6 $\frac{-c e^2 \left (a^2 e^2 \left (d^2+4 d e x+6 e^2 x^2\right )+6 a b e \left (4 d^2 e x+d^3+6 d e^2 x^2+4 e^3 x^3\right )+b^2 (-d) \left (88 d^2 e x+25 d^3+108 d e^2 x^2+48 e^3 x^3\right )\right )-e^3 \left (a^2 b e^2 (d+4 e x)+a^3 e^3+a b^2 e \left (d^2+4 d e x+6 e^2 x^2\right )+b^3 \left (4 d^2 e x+d^3+6 d e^2 x^2+4 e^3 x^3\right )\right )+12 c (d+e x)^4 \log (d+e x) \left (c e (a e-5 b d)+b^2 e^2+5 c^2 d^2\right )+c^2 e \left (a d e \left (88 d^2 e x+25 d^3+108 d e^2 x^2+48 e^3 x^3\right )-b \left (252 d^3 e^2 x^2+48 d^2 e^3 x^3+248 d^4 e x+77 d^5-48 d e^4 x^4-12 e^5 x^5\right )\right )+c^3 \left (132 d^4 e^2 x^2-32 d^3 e^3 x^3-68 d^2 e^4 x^4+168 d^5 e x+57 d^6-12 d e^5 x^5+2 e^6 x^6\right )}{4 e^7 (d+e x)^4}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + b*x + c*x^2)^3/(d + e*x)^5,x]

[Out]

(c^3*(57*d^6 + 168*d^5*e*x + 132*d^4*e^2*x^2 - 32*d^3*e^3*x^3 - 68*d^2*e^4*x^4 - 12*d*e^5*x^5 + 2*e^6*x^6) - e
^3*(a^3*e^3 + a^2*b*e^2*(d + 4*e*x) + a*b^2*e*(d^2 + 4*d*e*x + 6*e^2*x^2) + b^3*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2
+ 4*e^3*x^3)) - c*e^2*(a^2*e^2*(d^2 + 4*d*e*x + 6*e^2*x^2) + 6*a*b*e*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x
^3) - b^2*d*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 + 48*e^3*x^3)) + c^2*e*(a*d*e*(25*d^3 + 88*d^2*e*x + 108*d*e^
2*x^2 + 48*e^3*x^3) - b*(77*d^5 + 248*d^4*e*x + 252*d^3*e^2*x^2 + 48*d^2*e^3*x^3 - 48*d*e^4*x^4 - 12*e^5*x^5))
+ 12*c*(5*c^2*d^2 + b^2*e^2 + c*e*(-5*b*d + a*e))*(d + e*x)^4*Log[d + e*x])/(4*e^7*(d + e*x)^4)

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Maple [B]  time = 0.051, size = 678, normalized size = 2.7 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^3/(e*x+d)^5,x)

[Out]

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

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Maxima [A]  time = 1.04103, size = 595, normalized size = 2.37 \begin{align*} \frac{57 \, c^{3} d^{6} - 77 \, b c^{2} d^{5} e - a^{2} b d e^{5} - a^{3} e^{6} + 25 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} -{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 4 \,{\left (20 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} + 12 \,{\left (b^{2} c + a c^{2}\right )} d e^{5} -{\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 6 \,{\left (35 \, c^{3} d^{4} e^{2} - 50 \, b c^{2} d^{3} e^{3} + 18 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} -{\left (b^{3} + 6 \, a b c\right )} d e^{5} -{\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 4 \,{\left (47 \, c^{3} d^{5} e - 65 \, b c^{2} d^{4} e^{2} - a^{2} b e^{6} + 22 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} -{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x}{4 \,{\left (e^{11} x^{4} + 4 \, d e^{10} x^{3} + 6 \, d^{2} e^{9} x^{2} + 4 \, d^{3} e^{8} x + d^{4} e^{7}\right )}} + \frac{c^{3} e x^{2} - 2 \,{\left (5 \, c^{3} d - 3 \, b c^{2} e\right )} x}{2 \, e^{6}} + \frac{3 \,{\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e +{\left (b^{2} c + a c^{2}\right )} e^{2}\right )} \log \left (e x + d\right )}{e^{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)^5,x, algorithm="maxima")

[Out]

1/4*(57*c^3*d^6 - 77*b*c^2*d^5*e - a^2*b*d*e^5 - a^3*e^6 + 25*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^
3 - (a*b^2 + a^2*c)*d^2*e^4 + 4*(20*c^3*d^3*e^3 - 30*b*c^2*d^2*e^4 + 12*(b^2*c + a*c^2)*d*e^5 - (b^3 + 6*a*b*c
)*e^6)*x^3 + 6*(35*c^3*d^4*e^2 - 50*b*c^2*d^3*e^3 + 18*(b^2*c + a*c^2)*d^2*e^4 - (b^3 + 6*a*b*c)*d*e^5 - (a*b^
2 + a^2*c)*e^6)*x^2 + 4*(47*c^3*d^5*e - 65*b*c^2*d^4*e^2 - a^2*b*e^6 + 22*(b^2*c + a*c^2)*d^3*e^3 - (b^3 + 6*a
*b*c)*d^2*e^4 - (a*b^2 + a^2*c)*d*e^5)*x)/(e^11*x^4 + 4*d*e^10*x^3 + 6*d^2*e^9*x^2 + 4*d^3*e^8*x + d^4*e^7) +
1/2*(c^3*e*x^2 - 2*(5*c^3*d - 3*b*c^2*e)*x)/e^6 + 3*(5*c^3*d^2 - 5*b*c^2*d*e + (b^2*c + a*c^2)*e^2)*log(e*x +
d)/e^7

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Fricas [B]  time = 1.98217, size = 1308, normalized size = 5.21 \begin{align*} \frac{2 \, c^{3} e^{6} x^{6} + 57 \, c^{3} d^{6} - 77 \, b c^{2} d^{5} e - a^{2} b d e^{5} - a^{3} e^{6} + 25 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} -{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} - 12 \,{\left (c^{3} d e^{5} - b c^{2} e^{6}\right )} x^{5} - 4 \,{\left (17 \, c^{3} d^{2} e^{4} - 12 \, b c^{2} d e^{5}\right )} x^{4} - 4 \,{\left (8 \, c^{3} d^{3} e^{3} + 12 \, b c^{2} d^{2} e^{4} - 12 \,{\left (b^{2} c + a c^{2}\right )} d e^{5} +{\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 6 \,{\left (22 \, c^{3} d^{4} e^{2} - 42 \, b c^{2} d^{3} e^{3} + 18 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} -{\left (b^{3} + 6 \, a b c\right )} d e^{5} -{\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 4 \,{\left (42 \, c^{3} d^{5} e - 62 \, b c^{2} d^{4} e^{2} - a^{2} b e^{6} + 22 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} -{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x + 12 \,{\left (5 \, c^{3} d^{6} - 5 \, b c^{2} d^{5} e +{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} +{\left (5 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} +{\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} + 4 \,{\left (5 \, c^{3} d^{3} e^{3} - 5 \, b c^{2} d^{2} e^{4} +{\left (b^{2} c + a c^{2}\right )} d e^{5}\right )} x^{3} + 6 \,{\left (5 \, c^{3} d^{4} e^{2} - 5 \, b c^{2} d^{3} e^{3} +{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4}\right )} x^{2} + 4 \,{\left (5 \, c^{3} d^{5} e - 5 \, b c^{2} d^{4} e^{2} +{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3}\right )} x\right )} \log \left (e x + d\right )}{4 \,{\left (e^{11} x^{4} + 4 \, d e^{10} x^{3} + 6 \, d^{2} e^{9} x^{2} + 4 \, d^{3} e^{8} x + d^{4} 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)^5,x, algorithm="fricas")

[Out]

1/4*(2*c^3*e^6*x^6 + 57*c^3*d^6 - 77*b*c^2*d^5*e - a^2*b*d*e^5 - a^3*e^6 + 25*(b^2*c + a*c^2)*d^4*e^2 - (b^3 +
6*a*b*c)*d^3*e^3 - (a*b^2 + a^2*c)*d^2*e^4 - 12*(c^3*d*e^5 - b*c^2*e^6)*x^5 - 4*(17*c^3*d^2*e^4 - 12*b*c^2*d*
e^5)*x^4 - 4*(8*c^3*d^3*e^3 + 12*b*c^2*d^2*e^4 - 12*(b^2*c + a*c^2)*d*e^5 + (b^3 + 6*a*b*c)*e^6)*x^3 + 6*(22*c
^3*d^4*e^2 - 42*b*c^2*d^3*e^3 + 18*(b^2*c + a*c^2)*d^2*e^4 - (b^3 + 6*a*b*c)*d*e^5 - (a*b^2 + a^2*c)*e^6)*x^2
+ 4*(42*c^3*d^5*e - 62*b*c^2*d^4*e^2 - a^2*b*e^6 + 22*(b^2*c + a*c^2)*d^3*e^3 - (b^3 + 6*a*b*c)*d^2*e^4 - (a*b
^2 + a^2*c)*d*e^5)*x + 12*(5*c^3*d^6 - 5*b*c^2*d^5*e + (b^2*c + a*c^2)*d^4*e^2 + (5*c^3*d^2*e^4 - 5*b*c^2*d*e^
5 + (b^2*c + a*c^2)*e^6)*x^4 + 4*(5*c^3*d^3*e^3 - 5*b*c^2*d^2*e^4 + (b^2*c + a*c^2)*d*e^5)*x^3 + 6*(5*c^3*d^4*
e^2 - 5*b*c^2*d^3*e^3 + (b^2*c + a*c^2)*d^2*e^4)*x^2 + 4*(5*c^3*d^5*e - 5*b*c^2*d^4*e^2 + (b^2*c + a*c^2)*d^3*
e^3)*x)*log(e*x + d))/(e^11*x^4 + 4*d*e^10*x^3 + 6*d^2*e^9*x^2 + 4*d^3*e^8*x + d^4*e^7)

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Sympy [B]  time = 141.101, size = 518, normalized size = 2.06 \begin{align*} \frac{c^{3} x^{2}}{2 e^{5}} + \frac{3 c \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) \log{\left (d + e x \right )}}{e^{7}} - \frac{a^{3} e^{6} + a^{2} b d e^{5} + a^{2} c d^{2} e^{4} + a b^{2} d^{2} e^{4} + 6 a b c d^{3} e^{3} - 25 a c^{2} d^{4} e^{2} + b^{3} d^{3} e^{3} - 25 b^{2} c d^{4} e^{2} + 77 b c^{2} d^{5} e - 57 c^{3} d^{6} + x^{3} \left (24 a b c e^{6} - 48 a c^{2} d e^{5} + 4 b^{3} e^{6} - 48 b^{2} c d e^{5} + 120 b c^{2} d^{2} e^{4} - 80 c^{3} d^{3} e^{3}\right ) + x^{2} \left (6 a^{2} c e^{6} + 6 a b^{2} e^{6} + 36 a b c d e^{5} - 108 a c^{2} d^{2} e^{4} + 6 b^{3} d e^{5} - 108 b^{2} c d^{2} e^{4} + 300 b c^{2} d^{3} e^{3} - 210 c^{3} d^{4} e^{2}\right ) + x \left (4 a^{2} b e^{6} + 4 a^{2} c d e^{5} + 4 a b^{2} d e^{5} + 24 a b c d^{2} e^{4} - 88 a c^{2} d^{3} e^{3} + 4 b^{3} d^{2} e^{4} - 88 b^{2} c d^{3} e^{3} + 260 b c^{2} d^{4} e^{2} - 188 c^{3} d^{5} e\right )}{4 d^{4} e^{7} + 16 d^{3} e^{8} x + 24 d^{2} e^{9} x^{2} + 16 d e^{10} x^{3} + 4 e^{11} x^{4}} + \frac{x \left (3 b c^{2} e - 5 c^{3} d\right )}{e^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**3/(e*x+d)**5,x)

[Out]

c**3*x**2/(2*e**5) + 3*c*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2)*log(d + e*x)/e**7 - (a**3*e**6 + a**
2*b*d*e**5 + a**2*c*d**2*e**4 + a*b**2*d**2*e**4 + 6*a*b*c*d**3*e**3 - 25*a*c**2*d**4*e**2 + b**3*d**3*e**3 -
25*b**2*c*d**4*e**2 + 77*b*c**2*d**5*e - 57*c**3*d**6 + x**3*(24*a*b*c*e**6 - 48*a*c**2*d*e**5 + 4*b**3*e**6 -
48*b**2*c*d*e**5 + 120*b*c**2*d**2*e**4 - 80*c**3*d**3*e**3) + x**2*(6*a**2*c*e**6 + 6*a*b**2*e**6 + 36*a*b*c
*d*e**5 - 108*a*c**2*d**2*e**4 + 6*b**3*d*e**5 - 108*b**2*c*d**2*e**4 + 300*b*c**2*d**3*e**3 - 210*c**3*d**4*e
**2) + x*(4*a**2*b*e**6 + 4*a**2*c*d*e**5 + 4*a*b**2*d*e**5 + 24*a*b*c*d**2*e**4 - 88*a*c**2*d**3*e**3 + 4*b**
3*d**2*e**4 - 88*b**2*c*d**3*e**3 + 260*b*c**2*d**4*e**2 - 188*c**3*d**5*e))/(4*d**4*e**7 + 16*d**3*e**8*x + 2
4*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*e**11*x**4) + x*(3*b*c**2*e - 5*c**3*d)/e**6

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Giac [B]  time = 1.16593, size = 927, normalized size = 3.69 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^3/(e*x+d)^5,x, algorithm="giac")

[Out]

1/2*(c^3 - 6*(2*c^3*d*e - b*c^2*e^2)*e^(-1)/(x*e + d))*(x*e + d)^2*e^(-7) - 3*(5*c^3*d^2 - 5*b*c^2*d*e + b^2*c
*e^2 + a*c^2*e^2)*e^(-7)*log(abs(x*e + d)*e^(-1)/(x*e + d)^2) + 1/4*(80*c^3*d^3*e^29/(x*e + d) - 30*c^3*d^4*e^
29/(x*e + d)^2 + 8*c^3*d^5*e^29/(x*e + d)^3 - c^3*d^6*e^29/(x*e + d)^4 - 120*b*c^2*d^2*e^30/(x*e + d) + 60*b*c
^2*d^3*e^30/(x*e + d)^2 - 20*b*c^2*d^4*e^30/(x*e + d)^3 + 3*b*c^2*d^5*e^30/(x*e + d)^4 + 48*b^2*c*d*e^31/(x*e
+ d) + 48*a*c^2*d*e^31/(x*e + d) - 36*b^2*c*d^2*e^31/(x*e + d)^2 - 36*a*c^2*d^2*e^31/(x*e + d)^2 + 16*b^2*c*d^
3*e^31/(x*e + d)^3 + 16*a*c^2*d^3*e^31/(x*e + d)^3 - 3*b^2*c*d^4*e^31/(x*e + d)^4 - 3*a*c^2*d^4*e^31/(x*e + d)
^4 - 4*b^3*e^32/(x*e + d) - 24*a*b*c*e^32/(x*e + d) + 6*b^3*d*e^32/(x*e + d)^2 + 36*a*b*c*d*e^32/(x*e + d)^2 -
4*b^3*d^2*e^32/(x*e + d)^3 - 24*a*b*c*d^2*e^32/(x*e + d)^3 + b^3*d^3*e^32/(x*e + d)^4 + 6*a*b*c*d^3*e^32/(x*e
+ d)^4 - 6*a*b^2*e^33/(x*e + d)^2 - 6*a^2*c*e^33/(x*e + d)^2 + 8*a*b^2*d*e^33/(x*e + d)^3 + 8*a^2*c*d*e^33/(x
*e + d)^3 - 3*a*b^2*d^2*e^33/(x*e + d)^4 - 3*a^2*c*d^2*e^33/(x*e + d)^4 - 4*a^2*b*e^34/(x*e + d)^3 + 3*a^2*b*d
*e^34/(x*e + d)^4 - a^3*e^35/(x*e + d)^4)*e^(-36)