### 3.441 $$\int (d+e x)^m (b x+c x^2)^2 \, dx$$

Optimal. Leaf size=159 $\frac{\left (b^2 e^2-6 b c d e+6 c^2 d^2\right ) (d+e x)^{m+3}}{e^5 (m+3)}+\frac{d^2 (c d-b e)^2 (d+e x)^{m+1}}{e^5 (m+1)}-\frac{2 d (c d-b e) (2 c d-b e) (d+e x)^{m+2}}{e^5 (m+2)}-\frac{2 c (2 c d-b e) (d+e x)^{m+4}}{e^5 (m+4)}+\frac{c^2 (d+e x)^{m+5}}{e^5 (m+5)}$

[Out]

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

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Rubi [A]  time = 0.0825785, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.053, Rules used = {698} $\frac{\left (b^2 e^2-6 b c d e+6 c^2 d^2\right ) (d+e x)^{m+3}}{e^5 (m+3)}+\frac{d^2 (c d-b e)^2 (d+e x)^{m+1}}{e^5 (m+1)}-\frac{2 d (c d-b e) (2 c d-b e) (d+e x)^{m+2}}{e^5 (m+2)}-\frac{2 c (2 c d-b e) (d+e x)^{m+4}}{e^5 (m+4)}+\frac{c^2 (d+e x)^{m+5}}{e^5 (m+5)}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^m*(b*x + c*x^2)^2,x]

[Out]

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

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 (d+e x)^m \left (b x+c x^2\right )^2 \, dx &=\int \left (\frac{d^2 (c d-b e)^2 (d+e x)^m}{e^4}+\frac{2 d (c d-b e) (-2 c d+b e) (d+e x)^{1+m}}{e^4}+\frac{\left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^{2+m}}{e^4}-\frac{2 c (2 c d-b e) (d+e x)^{3+m}}{e^4}+\frac{c^2 (d+e x)^{4+m}}{e^4}\right ) \, dx\\ &=\frac{d^2 (c d-b e)^2 (d+e x)^{1+m}}{e^5 (1+m)}-\frac{2 d (c d-b e) (2 c d-b e) (d+e x)^{2+m}}{e^5 (2+m)}+\frac{\left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^{3+m}}{e^5 (3+m)}-\frac{2 c (2 c d-b e) (d+e x)^{4+m}}{e^5 (4+m)}+\frac{c^2 (d+e x)^{5+m}}{e^5 (5+m)}\\ \end{align*}

Mathematica [A]  time = 0.0946224, size = 138, normalized size = 0.87 $\frac{(d+e x)^{m+1} \left (\frac{(d+e x)^2 \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{m+3}+\frac{d^2 (c d-b e)^2}{m+1}-\frac{2 c (d+e x)^3 (2 c d-b e)}{m+4}-\frac{2 d (d+e x) (c d-b e) (2 c d-b e)}{m+2}+\frac{c^2 (d+e x)^4}{m+5}\right )}{e^5}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^m*(b*x + c*x^2)^2,x]

[Out]

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

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Maple [B]  time = 0.057, size = 547, normalized size = 3.4 \begin{align*}{\frac{ \left ( ex+d \right ) ^{1+m} \left ({c}^{2}{e}^{4}{m}^{4}{x}^{4}+2\,bc{e}^{4}{m}^{4}{x}^{3}+10\,{c}^{2}{e}^{4}{m}^{3}{x}^{4}+{b}^{2}{e}^{4}{m}^{4}{x}^{2}+22\,bc{e}^{4}{m}^{3}{x}^{3}-4\,{c}^{2}d{e}^{3}{m}^{3}{x}^{3}+35\,{c}^{2}{e}^{4}{m}^{2}{x}^{4}+12\,{b}^{2}{e}^{4}{m}^{3}{x}^{2}-6\,bcd{e}^{3}{m}^{3}{x}^{2}+82\,bc{e}^{4}{m}^{2}{x}^{3}-24\,{c}^{2}d{e}^{3}{m}^{2}{x}^{3}+50\,{c}^{2}{e}^{4}m{x}^{4}-2\,{b}^{2}d{e}^{3}{m}^{3}x+49\,{b}^{2}{e}^{4}{m}^{2}{x}^{2}-48\,bcd{e}^{3}{m}^{2}{x}^{2}+122\,bc{e}^{4}m{x}^{3}+12\,{c}^{2}{d}^{2}{e}^{2}{m}^{2}{x}^{2}-44\,{c}^{2}d{e}^{3}m{x}^{3}+24\,{c}^{2}{x}^{4}{e}^{4}-20\,{b}^{2}d{e}^{3}{m}^{2}x+78\,{b}^{2}{e}^{4}m{x}^{2}+12\,bc{d}^{2}{e}^{2}{m}^{2}x-102\,bcd{e}^{3}m{x}^{2}+60\,bc{e}^{4}{x}^{3}+36\,{c}^{2}{d}^{2}{e}^{2}m{x}^{2}-24\,{c}^{2}d{e}^{3}{x}^{3}+2\,{b}^{2}{d}^{2}{e}^{2}{m}^{2}-58\,{b}^{2}d{e}^{3}mx+40\,{b}^{2}{e}^{4}{x}^{2}+72\,bc{d}^{2}{e}^{2}mx-60\,bcd{e}^{3}{x}^{2}-24\,{c}^{2}{d}^{3}emx+24\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}+18\,{b}^{2}{d}^{2}{e}^{2}m-40\,{b}^{2}d{e}^{3}x-12\,bc{d}^{3}em+60\,bc{d}^{2}{e}^{2}x-24\,{c}^{2}{d}^{3}ex+40\,{b}^{2}{d}^{2}{e}^{2}-60\,bc{d}^{3}e+24\,{c}^{2}{d}^{4} \right ) }{{e}^{5} \left ({m}^{5}+15\,{m}^{4}+85\,{m}^{3}+225\,{m}^{2}+274\,m+120 \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^m*(c*x^2+b*x)^2,x)

[Out]

(e*x+d)^(1+m)*(c^2*e^4*m^4*x^4+2*b*c*e^4*m^4*x^3+10*c^2*e^4*m^3*x^4+b^2*e^4*m^4*x^2+22*b*c*e^4*m^3*x^3-4*c^2*d
*e^3*m^3*x^3+35*c^2*e^4*m^2*x^4+12*b^2*e^4*m^3*x^2-6*b*c*d*e^3*m^3*x^2+82*b*c*e^4*m^2*x^3-24*c^2*d*e^3*m^2*x^3
+50*c^2*e^4*m*x^4-2*b^2*d*e^3*m^3*x+49*b^2*e^4*m^2*x^2-48*b*c*d*e^3*m^2*x^2+122*b*c*e^4*m*x^3+12*c^2*d^2*e^2*m
^2*x^2-44*c^2*d*e^3*m*x^3+24*c^2*e^4*x^4-20*b^2*d*e^3*m^2*x+78*b^2*e^4*m*x^2+12*b*c*d^2*e^2*m^2*x-102*b*c*d*e^
3*m*x^2+60*b*c*e^4*x^3+36*c^2*d^2*e^2*m*x^2-24*c^2*d*e^3*x^3+2*b^2*d^2*e^2*m^2-58*b^2*d*e^3*m*x+40*b^2*e^4*x^2
+72*b*c*d^2*e^2*m*x-60*b*c*d*e^3*x^2-24*c^2*d^3*e*m*x+24*c^2*d^2*e^2*x^2+18*b^2*d^2*e^2*m-40*b^2*d*e^3*x-12*b*
c*d^3*e*m+60*b*c*d^2*e^2*x-24*c^2*d^3*e*x+40*b^2*d^2*e^2-60*b*c*d^3*e+24*c^2*d^4)/e^5/(m^5+15*m^4+85*m^3+225*m
^2+274*m+120)

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Maxima [A]  time = 1.25286, size = 429, normalized size = 2.7 \begin{align*} \frac{{\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} +{\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )}{\left (e x + d\right )}^{m} b^{2}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac{2 \,{\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{4} x^{4} +{\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d e^{3} x^{3} - 3 \,{\left (m^{2} + m\right )} d^{2} e^{2} x^{2} + 6 \, d^{3} e m x - 6 \, d^{4}\right )}{\left (e x + d\right )}^{m} b c}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} + \frac{{\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{5} x^{5} +{\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d e^{4} x^{4} - 4 \,{\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{2} e^{3} x^{3} + 12 \,{\left (m^{2} + m\right )} d^{3} e^{2} x^{2} - 24 \, d^{4} e m x + 24 \, d^{5}\right )}{\left (e x + d\right )}^{m} c^{2}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(c*x^2+b*x)^2,x, algorithm="maxima")

[Out]

((m^2 + 3*m + 2)*e^3*x^3 + (m^2 + m)*d*e^2*x^2 - 2*d^2*e*m*x + 2*d^3)*(e*x + d)^m*b^2/((m^3 + 6*m^2 + 11*m + 6
)*e^3) + 2*((m^3 + 6*m^2 + 11*m + 6)*e^4*x^4 + (m^3 + 3*m^2 + 2*m)*d*e^3*x^3 - 3*(m^2 + m)*d^2*e^2*x^2 + 6*d^3
*e*m*x - 6*d^4)*(e*x + d)^m*b*c/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^4) + ((m^4 + 10*m^3 + 35*m^2 + 50*m + 2
4)*e^5*x^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*d*e^4*x^4 - 4*(m^3 + 3*m^2 + 2*m)*d^2*e^3*x^3 + 12*(m^2 + m)*d^3*e^2
*x^2 - 24*d^4*e*m*x + 24*d^5)*(e*x + d)^m*c^2/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^5)

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Fricas [B]  time = 1.97357, size = 1220, normalized size = 7.67 \begin{align*} \frac{{\left (2 \, b^{2} d^{3} e^{2} m^{2} + 24 \, c^{2} d^{5} - 60 \, b c d^{4} e + 40 \, b^{2} d^{3} e^{2} +{\left (c^{2} e^{5} m^{4} + 10 \, c^{2} e^{5} m^{3} + 35 \, c^{2} e^{5} m^{2} + 50 \, c^{2} e^{5} m + 24 \, c^{2} e^{5}\right )} x^{5} +{\left (60 \, b c e^{5} +{\left (c^{2} d e^{4} + 2 \, b c e^{5}\right )} m^{4} + 2 \,{\left (3 \, c^{2} d e^{4} + 11 \, b c e^{5}\right )} m^{3} +{\left (11 \, c^{2} d e^{4} + 82 \, b c e^{5}\right )} m^{2} + 2 \,{\left (3 \, c^{2} d e^{4} + 61 \, b c e^{5}\right )} m\right )} x^{4} +{\left (40 \, b^{2} e^{5} +{\left (2 \, b c d e^{4} + b^{2} e^{5}\right )} m^{4} - 4 \,{\left (c^{2} d^{2} e^{3} - 4 \, b c d e^{4} - 3 \, b^{2} e^{5}\right )} m^{3} -{\left (12 \, c^{2} d^{2} e^{3} - 34 \, b c d e^{4} - 49 \, b^{2} e^{5}\right )} m^{2} - 2 \,{\left (4 \, c^{2} d^{2} e^{3} - 10 \, b c d e^{4} - 39 \, b^{2} e^{5}\right )} m\right )} x^{3} +{\left (b^{2} d e^{4} m^{4} - 2 \,{\left (3 \, b c d^{2} e^{3} - 5 \, b^{2} d e^{4}\right )} m^{3} +{\left (12 \, c^{2} d^{3} e^{2} - 36 \, b c d^{2} e^{3} + 29 \, b^{2} d e^{4}\right )} m^{2} + 2 \,{\left (6 \, c^{2} d^{3} e^{2} - 15 \, b c d^{2} e^{3} + 10 \, b^{2} d e^{4}\right )} m\right )} x^{2} - 6 \,{\left (2 \, b c d^{4} e - 3 \, b^{2} d^{3} e^{2}\right )} m - 2 \,{\left (b^{2} d^{2} e^{3} m^{3} - 3 \,{\left (2 \, b c d^{3} e^{2} - 3 \, b^{2} d^{2} e^{3}\right )} m^{2} + 2 \,{\left (6 \, c^{2} d^{4} e - 15 \, b c d^{3} e^{2} + 10 \, b^{2} d^{2} e^{3}\right )} m\right )} x\right )}{\left (e x + d\right )}^{m}}{e^{5} m^{5} + 15 \, e^{5} m^{4} + 85 \, e^{5} m^{3} + 225 \, e^{5} m^{2} + 274 \, e^{5} m + 120 \, e^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(c*x^2+b*x)^2,x, algorithm="fricas")

[Out]

(2*b^2*d^3*e^2*m^2 + 24*c^2*d^5 - 60*b*c*d^4*e + 40*b^2*d^3*e^2 + (c^2*e^5*m^4 + 10*c^2*e^5*m^3 + 35*c^2*e^5*m
^2 + 50*c^2*e^5*m + 24*c^2*e^5)*x^5 + (60*b*c*e^5 + (c^2*d*e^4 + 2*b*c*e^5)*m^4 + 2*(3*c^2*d*e^4 + 11*b*c*e^5)
*m^3 + (11*c^2*d*e^4 + 82*b*c*e^5)*m^2 + 2*(3*c^2*d*e^4 + 61*b*c*e^5)*m)*x^4 + (40*b^2*e^5 + (2*b*c*d*e^4 + b^
2*e^5)*m^4 - 4*(c^2*d^2*e^3 - 4*b*c*d*e^4 - 3*b^2*e^5)*m^3 - (12*c^2*d^2*e^3 - 34*b*c*d*e^4 - 49*b^2*e^5)*m^2
- 2*(4*c^2*d^2*e^3 - 10*b*c*d*e^4 - 39*b^2*e^5)*m)*x^3 + (b^2*d*e^4*m^4 - 2*(3*b*c*d^2*e^3 - 5*b^2*d*e^4)*m^3
+ (12*c^2*d^3*e^2 - 36*b*c*d^2*e^3 + 29*b^2*d*e^4)*m^2 + 2*(6*c^2*d^3*e^2 - 15*b*c*d^2*e^3 + 10*b^2*d*e^4)*m)*
x^2 - 6*(2*b*c*d^4*e - 3*b^2*d^3*e^2)*m - 2*(b^2*d^2*e^3*m^3 - 3*(2*b*c*d^3*e^2 - 3*b^2*d^2*e^3)*m^2 + 2*(6*c^
2*d^4*e - 15*b*c*d^3*e^2 + 10*b^2*d^2*e^3)*m)*x)*(e*x + d)^m/(e^5*m^5 + 15*e^5*m^4 + 85*e^5*m^3 + 225*e^5*m^2
+ 274*e^5*m + 120*e^5)

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Sympy [A]  time = 7.87121, size = 6205, normalized size = 39.03 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**m*(c*x**2+b*x)**2,x)

[Out]

Piecewise((d**m*(b**2*x**3/3 + b*c*x**4/2 + c**2*x**5/5), Eq(e, 0)), (4*b**2*d*e**5*x**3/(12*d**6*e**5 + 48*d*
*5*e**6*x + 72*d**4*e**7*x**2 + 48*d**3*e**8*x**3 + 12*d**2*e**9*x**4) + b**2*e**6*x**4/(12*d**6*e**5 + 48*d**
5*e**6*x + 72*d**4*e**7*x**2 + 48*d**3*e**8*x**3 + 12*d**2*e**9*x**4) + 6*b*c*d*e**5*x**4/(12*d**6*e**5 + 48*d
**5*e**6*x + 72*d**4*e**7*x**2 + 48*d**3*e**8*x**3 + 12*d**2*e**9*x**4) + 12*c**2*d**6*log(d/e + x)/(12*d**6*e
**5 + 48*d**5*e**6*x + 72*d**4*e**7*x**2 + 48*d**3*e**8*x**3 + 12*d**2*e**9*x**4) + 7*c**2*d**6/(12*d**6*e**5
+ 48*d**5*e**6*x + 72*d**4*e**7*x**2 + 48*d**3*e**8*x**3 + 12*d**2*e**9*x**4) + 48*c**2*d**5*e*x*log(d/e + x)/
(12*d**6*e**5 + 48*d**5*e**6*x + 72*d**4*e**7*x**2 + 48*d**3*e**8*x**3 + 12*d**2*e**9*x**4) + 16*c**2*d**5*e*x
/(12*d**6*e**5 + 48*d**5*e**6*x + 72*d**4*e**7*x**2 + 48*d**3*e**8*x**3 + 12*d**2*e**9*x**4) + 72*c**2*d**4*e*
*2*x**2*log(d/e + x)/(12*d**6*e**5 + 48*d**5*e**6*x + 72*d**4*e**7*x**2 + 48*d**3*e**8*x**3 + 12*d**2*e**9*x**
4) + 48*c**2*d**3*e**3*x**3*log(d/e + x)/(12*d**6*e**5 + 48*d**5*e**6*x + 72*d**4*e**7*x**2 + 48*d**3*e**8*x**
3 + 12*d**2*e**9*x**4) - 24*c**2*d**3*e**3*x**3/(12*d**6*e**5 + 48*d**5*e**6*x + 72*d**4*e**7*x**2 + 48*d**3*e
**8*x**3 + 12*d**2*e**9*x**4) + 12*c**2*d**2*e**4*x**4*log(d/e + x)/(12*d**6*e**5 + 48*d**5*e**6*x + 72*d**4*e
**7*x**2 + 48*d**3*e**8*x**3 + 12*d**2*e**9*x**4) - 18*c**2*d**2*e**4*x**4/(12*d**6*e**5 + 48*d**5*e**6*x + 72
*d**4*e**7*x**2 + 48*d**3*e**8*x**3 + 12*d**2*e**9*x**4), Eq(m, -5)), (b**2*e**5*x**3/(3*d**4*e**5 + 9*d**3*e*
*6*x + 9*d**2*e**7*x**2 + 3*d*e**8*x**3) + 6*b*c*d**4*e*log(d/e + x)/(3*d**4*e**5 + 9*d**3*e**6*x + 9*d**2*e**
7*x**2 + 3*d*e**8*x**3) + 5*b*c*d**4*e/(3*d**4*e**5 + 9*d**3*e**6*x + 9*d**2*e**7*x**2 + 3*d*e**8*x**3) + 18*b
*c*d**3*e**2*x*log(d/e + x)/(3*d**4*e**5 + 9*d**3*e**6*x + 9*d**2*e**7*x**2 + 3*d*e**8*x**3) + 9*b*c*d**3*e**2
*x/(3*d**4*e**5 + 9*d**3*e**6*x + 9*d**2*e**7*x**2 + 3*d*e**8*x**3) + 18*b*c*d**2*e**3*x**2*log(d/e + x)/(3*d*
*4*e**5 + 9*d**3*e**6*x + 9*d**2*e**7*x**2 + 3*d*e**8*x**3) + 6*b*c*d*e**4*x**3*log(d/e + x)/(3*d**4*e**5 + 9*
d**3*e**6*x + 9*d**2*e**7*x**2 + 3*d*e**8*x**3) - 6*b*c*d*e**4*x**3/(3*d**4*e**5 + 9*d**3*e**6*x + 9*d**2*e**7
*x**2 + 3*d*e**8*x**3) - 12*c**2*d**5*log(d/e + x)/(3*d**4*e**5 + 9*d**3*e**6*x + 9*d**2*e**7*x**2 + 3*d*e**8*
x**3) - 10*c**2*d**5/(3*d**4*e**5 + 9*d**3*e**6*x + 9*d**2*e**7*x**2 + 3*d*e**8*x**3) - 36*c**2*d**4*e*x*log(d
/e + x)/(3*d**4*e**5 + 9*d**3*e**6*x + 9*d**2*e**7*x**2 + 3*d*e**8*x**3) - 18*c**2*d**4*e*x/(3*d**4*e**5 + 9*d
**3*e**6*x + 9*d**2*e**7*x**2 + 3*d*e**8*x**3) - 36*c**2*d**3*e**2*x**2*log(d/e + x)/(3*d**4*e**5 + 9*d**3*e**
6*x + 9*d**2*e**7*x**2 + 3*d*e**8*x**3) - 12*c**2*d**2*e**3*x**3*log(d/e + x)/(3*d**4*e**5 + 9*d**3*e**6*x + 9
*d**2*e**7*x**2 + 3*d*e**8*x**3) + 12*c**2*d**2*e**3*x**3/(3*d**4*e**5 + 9*d**3*e**6*x + 9*d**2*e**7*x**2 + 3*
d*e**8*x**3) + 3*c**2*d*e**4*x**4/(3*d**4*e**5 + 9*d**3*e**6*x + 9*d**2*e**7*x**2 + 3*d*e**8*x**3), Eq(m, -4))
, (2*b**2*d**2*e**2*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 3*b**2*d**2*e**2/(2*d**2*e**5 + 4*
d*e**6*x + 2*e**7*x**2) + 4*b**2*d*e**3*x*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 4*b**2*d*e**
3*x/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 2*b**2*e**4*x**2*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**
7*x**2) - 12*b*c*d**3*e*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 18*b*c*d**3*e/(2*d**2*e**5 + 4
*d*e**6*x + 2*e**7*x**2) - 24*b*c*d**2*e**2*x*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 24*b*c*d
**2*e**2*x/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 12*b*c*d*e**3*x**2*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*
x + 2*e**7*x**2) + 4*b*c*e**4*x**3/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 12*c**2*d**4*log(d/e + x)/(2*d**
2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 18*c**2*d**4/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 24*c**2*d**3*e*x*
log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 24*c**2*d**3*e*x/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x*
*2) + 12*c**2*d**2*e**2*x**2*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 4*c**2*d*e**3*x**3/(2*d**
2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + c**2*e**4*x**4/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2), Eq(m, -3)), (-6*
b**2*d**2*e**2*log(d/e + x)/(3*d*e**5 + 3*e**6*x) - 6*b**2*d**2*e**2/(3*d*e**5 + 3*e**6*x) - 6*b**2*d*e**3*x*l
og(d/e + x)/(3*d*e**5 + 3*e**6*x) + 3*b**2*e**4*x**2/(3*d*e**5 + 3*e**6*x) + 18*b*c*d**3*e*log(d/e + x)/(3*d*e
**5 + 3*e**6*x) + 18*b*c*d**3*e/(3*d*e**5 + 3*e**6*x) + 18*b*c*d**2*e**2*x*log(d/e + x)/(3*d*e**5 + 3*e**6*x)
- 9*b*c*d*e**3*x**2/(3*d*e**5 + 3*e**6*x) + 3*b*c*e**4*x**3/(3*d*e**5 + 3*e**6*x) - 12*c**2*d**4*log(d/e + x)/
(3*d*e**5 + 3*e**6*x) - 12*c**2*d**4/(3*d*e**5 + 3*e**6*x) - 12*c**2*d**3*e*x*log(d/e + x)/(3*d*e**5 + 3*e**6*
x) + 6*c**2*d**2*e**2*x**2/(3*d*e**5 + 3*e**6*x) - 2*c**2*d*e**3*x**3/(3*d*e**5 + 3*e**6*x) + c**2*e**4*x**4/(
3*d*e**5 + 3*e**6*x), Eq(m, -2)), (b**2*d**2*log(d/e + x)/e**3 - b**2*d*x/e**2 + b**2*x**2/(2*e) - 2*b*c*d**3*
log(d/e + x)/e**4 + 2*b*c*d**2*x/e**3 - b*c*d*x**2/e**2 + 2*b*c*x**3/(3*e) + c**2*d**4*log(d/e + x)/e**5 - c**
2*d**3*x/e**4 + c**2*d**2*x**2/(2*e**3) - c**2*d*x**3/(3*e**2) + c**2*x**4/(4*e), Eq(m, -1)), (2*b**2*d**3*e**
2*m**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 18*b**
2*d**3*e**2*m*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) +
40*b**2*d**3*e**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e*
*5) - 2*b**2*d**2*e**3*m**3*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5
*m + 120*e**5) - 18*b**2*d**2*e**3*m**2*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**
2 + 274*e**5*m + 120*e**5) - 40*b**2*d**2*e**3*m*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225
*e**5*m**2 + 274*e**5*m + 120*e**5) + b**2*d*e**4*m**4*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m
**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 10*b**2*d*e**4*m**3*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4
+ 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 29*b**2*d*e**4*m**2*x**2*(d + e*x)**m/(e**5*m**5 +
15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 20*b**2*d*e**4*m*x**2*(d + e*x)**m/(e**
5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + b**2*e**5*m**4*x**3*(d + e*x)*
*m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 12*b**2*e**5*m**3*x**3*
(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 49*b**2*e**5*
m**2*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 78*
b**2*e**5*m*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5
) + 40*b**2*e**5*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120
*e**5) - 12*b*c*d**4*e*m*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m +
120*e**5) - 60*b*c*d**4*e*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m +
120*e**5) + 12*b*c*d**3*e**2*m**2*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 2
74*e**5*m + 120*e**5) + 60*b*c*d**3*e**2*m*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*
m**2 + 274*e**5*m + 120*e**5) - 6*b*c*d**2*e**3*m**3*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**
3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 36*b*c*d**2*e**3*m**2*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4
+ 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 30*b*c*d**2*e**3*m*x**2*(d + e*x)**m/(e**5*m**5 + 1
5*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 2*b*c*d*e**4*m**4*x**3*(d + e*x)**m/(e**
5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 16*b*c*d*e**4*m**3*x**3*(d + e
*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 34*b*c*d*e**4*m**2*
x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 20*b*c*d
*e**4*m*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) +
2*b*c*e**5*m**4*x**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*
e**5) + 22*b*c*e**5*m**3*x**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5
*m + 120*e**5) + 82*b*c*e**5*m**2*x**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 +
274*e**5*m + 120*e**5) + 122*b*c*e**5*m*x**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5
*m**2 + 274*e**5*m + 120*e**5) + 60*b*c*e**5*x**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*
e**5*m**2 + 274*e**5*m + 120*e**5) + 24*c**2*d**5*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*
e**5*m**2 + 274*e**5*m + 120*e**5) - 24*c**2*d**4*e*m*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3
+ 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 12*c**2*d**3*e**2*m**2*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4
+ 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 12*c**2*d**3*e**2*m*x**2*(d + e*x)**m/(e**5*m**5 + 1
5*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 4*c**2*d**2*e**3*m**3*x**3*(d + e*x)**m/
(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 12*c**2*d**2*e**3*m**2*x**
3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 8*c**2*d**2
*e**3*m*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) +
c**2*d*e**4*m**4*x**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120
*e**5) + 6*c**2*d*e**4*m**3*x**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e
**5*m + 120*e**5) + 11*c**2*d*e**4*m**2*x**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*
m**2 + 274*e**5*m + 120*e**5) + 6*c**2*d*e**4*m*x**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 2
25*e**5*m**2 + 274*e**5*m + 120*e**5) + c**2*e**5*m**4*x**5*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m
**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 10*c**2*e**5*m**3*x**5*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 +
85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 35*c**2*e**5*m**2*x**5*(d + e*x)**m/(e**5*m**5 + 15*e
**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 50*c**2*e**5*m*x**5*(d + e*x)**m/(e**5*m**5
+ 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 24*c**2*e**5*x**5*(d + e*x)**m/(e**5
*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5), True))

________________________________________________________________________________________

Giac [B]  time = 1.34414, size = 1353, normalized size = 8.51 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(c*x^2+b*x)^2,x, algorithm="giac")

[Out]

((x*e + d)^m*c^2*m^4*x^5*e^5 + (x*e + d)^m*c^2*d*m^4*x^4*e^4 + 2*(x*e + d)^m*b*c*m^4*x^4*e^5 + 10*(x*e + d)^m*
c^2*m^3*x^5*e^5 + 2*(x*e + d)^m*b*c*d*m^4*x^3*e^4 + 6*(x*e + d)^m*c^2*d*m^3*x^4*e^4 - 4*(x*e + d)^m*c^2*d^2*m^
3*x^3*e^3 + (x*e + d)^m*b^2*m^4*x^3*e^5 + 22*(x*e + d)^m*b*c*m^3*x^4*e^5 + 35*(x*e + d)^m*c^2*m^2*x^5*e^5 + (x
*e + d)^m*b^2*d*m^4*x^2*e^4 + 16*(x*e + d)^m*b*c*d*m^3*x^3*e^4 + 11*(x*e + d)^m*c^2*d*m^2*x^4*e^4 - 6*(x*e + d
)^m*b*c*d^2*m^3*x^2*e^3 - 12*(x*e + d)^m*c^2*d^2*m^2*x^3*e^3 + 12*(x*e + d)^m*c^2*d^3*m^2*x^2*e^2 + 12*(x*e +
d)^m*b^2*m^3*x^3*e^5 + 82*(x*e + d)^m*b*c*m^2*x^4*e^5 + 50*(x*e + d)^m*c^2*m*x^5*e^5 + 10*(x*e + d)^m*b^2*d*m^
3*x^2*e^4 + 34*(x*e + d)^m*b*c*d*m^2*x^3*e^4 + 6*(x*e + d)^m*c^2*d*m*x^4*e^4 - 2*(x*e + d)^m*b^2*d^2*m^3*x*e^3
- 36*(x*e + d)^m*b*c*d^2*m^2*x^2*e^3 - 8*(x*e + d)^m*c^2*d^2*m*x^3*e^3 + 12*(x*e + d)^m*b*c*d^3*m^2*x*e^2 + 1
2*(x*e + d)^m*c^2*d^3*m*x^2*e^2 - 24*(x*e + d)^m*c^2*d^4*m*x*e + 49*(x*e + d)^m*b^2*m^2*x^3*e^5 + 122*(x*e + d
)^m*b*c*m*x^4*e^5 + 24*(x*e + d)^m*c^2*x^5*e^5 + 29*(x*e + d)^m*b^2*d*m^2*x^2*e^4 + 20*(x*e + d)^m*b*c*d*m*x^3
*e^4 - 18*(x*e + d)^m*b^2*d^2*m^2*x*e^3 - 30*(x*e + d)^m*b*c*d^2*m*x^2*e^3 + 2*(x*e + d)^m*b^2*d^3*m^2*e^2 + 6
0*(x*e + d)^m*b*c*d^3*m*x*e^2 - 12*(x*e + d)^m*b*c*d^4*m*e + 24*(x*e + d)^m*c^2*d^5 + 78*(x*e + d)^m*b^2*m*x^3
*e^5 + 60*(x*e + d)^m*b*c*x^4*e^5 + 20*(x*e + d)^m*b^2*d*m*x^2*e^4 - 40*(x*e + d)^m*b^2*d^2*m*x*e^3 + 18*(x*e
+ d)^m*b^2*d^3*m*e^2 - 60*(x*e + d)^m*b*c*d^4*e + 40*(x*e + d)^m*b^2*x^3*e^5 + 40*(x*e + d)^m*b^2*d^3*e^2)/(m^
5*e^5 + 15*m^4*e^5 + 85*m^3*e^5 + 225*m^2*e^5 + 274*m*e^5 + 120*e^5)