3.722 \(\int (d+e x)^m (a+c x^2)^2 \, dx\)

Optimal. Leaf size=140 \[ \frac{\left (a e^2+c d^2\right )^2 (d+e x)^{m+1}}{e^5 (m+1)}-\frac{4 c d \left (a e^2+c d^2\right ) (d+e x)^{m+2}}{e^5 (m+2)}+\frac{2 c \left (a e^2+3 c d^2\right ) (d+e x)^{m+3}}{e^5 (m+3)}-\frac{4 c^2 d (d+e x)^{m+4}}{e^5 (m+4)}+\frac{c^2 (d+e x)^{m+5}}{e^5 (m+5)} \]

[Out]

((c*d^2 + a*e^2)^2*(d + e*x)^(1 + m))/(e^5*(1 + m)) - (4*c*d*(c*d^2 + a*e^2)*(d + e*x)^(2 + m))/(e^5*(2 + m))
+ (2*c*(3*c*d^2 + a*e^2)*(d + e*x)^(3 + m))/(e^5*(3 + m)) - (4*c^2*d*(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 = 0.0687169, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {697} \[ \frac{\left (a e^2+c d^2\right )^2 (d+e x)^{m+1}}{e^5 (m+1)}-\frac{4 c d \left (a e^2+c d^2\right ) (d+e x)^{m+2}}{e^5 (m+2)}+\frac{2 c \left (a e^2+3 c d^2\right ) (d+e x)^{m+3}}{e^5 (m+3)}-\frac{4 c^2 d (d+e x)^{m+4}}{e^5 (m+4)}+\frac{c^2 (d+e x)^{m+5}}{e^5 (m+5)} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^m*(a + c*x^2)^2,x]

[Out]

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

Rule 697

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int (d+e x)^m \left (a+c x^2\right )^2 \, dx &=\int \left (\frac{\left (c d^2+a e^2\right )^2 (d+e x)^m}{e^4}-\frac{4 c d \left (c d^2+a e^2\right ) (d+e x)^{1+m}}{e^4}+\frac{2 c \left (3 c d^2+a e^2\right ) (d+e x)^{2+m}}{e^4}-\frac{4 c^2 d (d+e x)^{3+m}}{e^4}+\frac{c^2 (d+e x)^{4+m}}{e^4}\right ) \, dx\\ &=\frac{\left (c d^2+a e^2\right )^2 (d+e x)^{1+m}}{e^5 (1+m)}-\frac{4 c d \left (c d^2+a e^2\right ) (d+e x)^{2+m}}{e^5 (2+m)}+\frac{2 c \left (3 c d^2+a e^2\right ) (d+e x)^{3+m}}{e^5 (3+m)}-\frac{4 c^2 d (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 = 0.188468, size = 176, normalized size = 1.26 \[ \frac{(d+e x)^{m+1} \left (\frac{4 \left (a e^2+c d^2\right ) \left (a e^2 \left (m^2+5 m+6\right )+c \left (2 d^2-2 d e (m+1) x+e^2 \left (m^2+3 m+2\right ) x^2\right )\right )}{e^4 (m+1) (m+2) (m+3)}-\frac{4 c d (d+e x) \left (a e^2 \left (m^2+7 m+12\right )+c \left (2 d^2-2 d e (m+2) x+e^2 \left (m^2+5 m+6\right ) x^2\right )\right )}{e^4 (m+2) (m+3) (m+4)}+\left (a+c x^2\right )^2\right )}{e (m+5)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((d + e*x)^(1 + m)*((a + c*x^2)^2 + (4*(c*d^2 + a*e^2)*(a*e^2*(6 + 5*m + m^2) + c*(2*d^2 - 2*d*e*(1 + m)*x + e
^2*(2 + 3*m + m^2)*x^2)))/(e^4*(1 + m)*(2 + m)*(3 + m)) - (4*c*d*(d + e*x)*(a*e^2*(12 + 7*m + m^2) + c*(2*d^2
- 2*d*e*(2 + m)*x + e^2*(6 + 5*m + m^2)*x^2)))/(e^4*(2 + m)*(3 + m)*(4 + m))))/(e*(5 + m))

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Maple [B]  time = 0.047, size = 420, normalized size = 3. \begin{align*}{\frac{ \left ( ex+d \right ) ^{1+m} \left ({c}^{2}{e}^{4}{m}^{4}{x}^{4}+10\,{c}^{2}{e}^{4}{m}^{3}{x}^{4}+2\,ac{e}^{4}{m}^{4}{x}^{2}-4\,{c}^{2}d{e}^{3}{m}^{3}{x}^{3}+35\,{c}^{2}{e}^{4}{m}^{2}{x}^{4}+24\,ac{e}^{4}{m}^{3}{x}^{2}-24\,{c}^{2}d{e}^{3}{m}^{2}{x}^{3}+50\,{c}^{2}{e}^{4}m{x}^{4}+{a}^{2}{e}^{4}{m}^{4}-4\,acd{e}^{3}{m}^{3}x+98\,ac{e}^{4}{m}^{2}{x}^{2}+12\,{c}^{2}{d}^{2}{e}^{2}{m}^{2}{x}^{2}-44\,{c}^{2}d{e}^{3}m{x}^{3}+24\,{x}^{4}{c}^{2}{e}^{4}+14\,{a}^{2}{e}^{4}{m}^{3}-40\,acd{e}^{3}{m}^{2}x+156\,ac{e}^{4}m{x}^{2}+36\,{c}^{2}{d}^{2}{e}^{2}m{x}^{2}-24\,{x}^{3}{c}^{2}d{e}^{3}+71\,{a}^{2}{e}^{4}{m}^{2}+4\,ac{d}^{2}{e}^{2}{m}^{2}-116\,acd{e}^{3}mx+80\,{x}^{2}ac{e}^{4}-24\,{c}^{2}{d}^{3}emx+24\,{x}^{2}{c}^{2}{d}^{2}{e}^{2}+154\,{a}^{2}{e}^{4}m+36\,ac{d}^{2}{e}^{2}m-80\,xacd{e}^{3}-24\,x{c}^{2}{d}^{3}e+120\,{a}^{2}{e}^{4}+80\,ac{d}^{2}{e}^{2}+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+a)^2,x)

[Out]

(e*x+d)^(1+m)*(c^2*e^4*m^4*x^4+10*c^2*e^4*m^3*x^4+2*a*c*e^4*m^4*x^2-4*c^2*d*e^3*m^3*x^3+35*c^2*e^4*m^2*x^4+24*
a*c*e^4*m^3*x^2-24*c^2*d*e^3*m^2*x^3+50*c^2*e^4*m*x^4+a^2*e^4*m^4-4*a*c*d*e^3*m^3*x+98*a*c*e^4*m^2*x^2+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+14*a^2*e^4*m^3-40*a*c*d*e^3*m^2*x+156*a*c*e^4*m*x^2+36*c^2*d
^2*e^2*m*x^2-24*c^2*d*e^3*x^3+71*a^2*e^4*m^2+4*a*c*d^2*e^2*m^2-116*a*c*d*e^3*m*x+80*a*c*e^4*x^2-24*c^2*d^3*e*m
*x+24*c^2*d^2*e^2*x^2+154*a^2*e^4*m+36*a*c*d^2*e^2*m-80*a*c*d*e^3*x-24*c^2*d^3*e*x+120*a^2*e^4+80*a*c*d^2*e^2+
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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.22426, size = 1092, normalized size = 7.8 \begin{align*} \frac{{\left (a^{2} d e^{4} m^{4} + 14 \, a^{2} d e^{4} m^{3} + 24 \, c^{2} d^{5} + 80 \, a c d^{3} e^{2} + 120 \, a^{2} d e^{4} +{\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 (c^{2} d e^{4} m^{4} + 6 \, c^{2} d e^{4} m^{3} + 11 \, c^{2} d e^{4} m^{2} + 6 \, c^{2} d e^{4} m\right )} x^{4} + 2 \,{\left (a c e^{5} m^{4} + 40 \, a c e^{5} - 2 \,{\left (c^{2} d^{2} e^{3} - 6 \, a c e^{5}\right )} m^{3} -{\left (6 \, c^{2} d^{2} e^{3} - 49 \, a c e^{5}\right )} m^{2} - 2 \,{\left (2 \, c^{2} d^{2} e^{3} - 39 \, a c e^{5}\right )} m\right )} x^{3} +{\left (4 \, a c d^{3} e^{2} + 71 \, a^{2} d e^{4}\right )} m^{2} + 2 \,{\left (a c d e^{4} m^{4} + 10 \, a c d e^{4} m^{3} +{\left (6 \, c^{2} d^{3} e^{2} + 29 \, a c d e^{4}\right )} m^{2} + 2 \,{\left (3 \, c^{2} d^{3} e^{2} + 10 \, a c d e^{4}\right )} m\right )} x^{2} + 2 \,{\left (18 \, a c d^{3} e^{2} + 77 \, a^{2} d e^{4}\right )} m +{\left (a^{2} e^{5} m^{4} + 120 \, a^{2} e^{5} - 2 \,{\left (2 \, a c d^{2} e^{3} - 7 \, a^{2} e^{5}\right )} m^{3} -{\left (36 \, a c d^{2} e^{3} - 71 \, a^{2} e^{5}\right )} m^{2} - 2 \,{\left (12 \, c^{2} d^{4} e + 40 \, a c d^{2} e^{3} - 77 \, a^{2} e^{5}\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+a)^2,x, algorithm="fricas")

[Out]

(a^2*d*e^4*m^4 + 14*a^2*d*e^4*m^3 + 24*c^2*d^5 + 80*a*c*d^3*e^2 + 120*a^2*d*e^4 + (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 + (c^2*d*e^4*m^4 + 6*c^2*d*e^4*m^3 + 11*c^2*d*e^4*m^2 + 6*
c^2*d*e^4*m)*x^4 + 2*(a*c*e^5*m^4 + 40*a*c*e^5 - 2*(c^2*d^2*e^3 - 6*a*c*e^5)*m^3 - (6*c^2*d^2*e^3 - 49*a*c*e^5
)*m^2 - 2*(2*c^2*d^2*e^3 - 39*a*c*e^5)*m)*x^3 + (4*a*c*d^3*e^2 + 71*a^2*d*e^4)*m^2 + 2*(a*c*d*e^4*m^4 + 10*a*c
*d*e^4*m^3 + (6*c^2*d^3*e^2 + 29*a*c*d*e^4)*m^2 + 2*(3*c^2*d^3*e^2 + 10*a*c*d*e^4)*m)*x^2 + 2*(18*a*c*d^3*e^2
+ 77*a^2*d*e^4)*m + (a^2*e^5*m^4 + 120*a^2*e^5 - 2*(2*a*c*d^2*e^3 - 7*a^2*e^5)*m^3 - (36*a*c*d^2*e^3 - 71*a^2*
e^5)*m^2 - 2*(12*c^2*d^4*e + 40*a*c*d^2*e^3 - 77*a^2*e^5)*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 = 5.7532, size = 5049, normalized size = 36.06 \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+a)**2,x)

[Out]

Piecewise((d**m*(a**2*x + 2*a*c*x**3/3 + c**2*x**5/5), Eq(e, 0)), (-3*a**2*d**2*e**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) + 8*a*c*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) + 2*a*c*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) + 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 + 4
8*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/(1
2*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)), (-a**2*d*e**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) + 2*a*c*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) - 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)), (-a**2*e**4/
(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 4*a*c*d**2*e**2*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**
2) + 6*a*c*d**2*e**2/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 8*a*c*d*e**3*x*log(d/e + x)/(2*d**2*e**5 + 4*d
*e**6*x + 2*e**7*x**2) + 8*a*c*d*e**3*x/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 4*a*c*e**4*x**2*log(d/e + x
)/(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)), (-3*a**2*e**4/(3*d*e**5 + 3*e**6*x) -
12*a*c*d**2*e**2*log(d/e + x)/(3*d*e**5 + 3*e**6*x) - 12*a*c*d**2*e**2/(3*d*e**5 + 3*e**6*x) - 12*a*c*d*e**3*
x*log(d/e + x)/(3*d*e**5 + 3*e**6*x) + 6*a*c*e**4*x**2/(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)), (a**2*log(d/e + x)/e + 2*a*c*d**2*log(d/e + x)/e**3 - 2*a*c*d*x/e**2 + a*c*x**2/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)), (a**2*d*e**4*m**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) + 14*a**2*d*e**4*m**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) + 71*a**2*d*e**4*m**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 22
5*e**5*m**2 + 274*e**5*m + 120*e**5) + 154*a**2*d*e**4*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) + 120*a**2*d*e**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) + a**2*e**5*m**4*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) + 14*a**2*e**5*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) + 71*a**2*e**5*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) + 154*a**2*e**5*m*x*(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) + 120*a**2*e**5*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) + 4*a*c*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) + 36*a*c*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) + 80*a*c*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) - 4*a*c*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) - 36*a*c*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)
- 80*a*c*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 + 12
0*e**5) + 2*a*c*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) + 20*a*c*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) + 58*a*c*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) + 40*a*c*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) + 2*a*c*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) + 24*a*c*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) + 98*a*c*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) + 156*a*c*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) + 80*a*c*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) + 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 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 12
0*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 + 2
74*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 + 22
5*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 + 1
5*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 + 225*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 = 1.15116, size = 1145, normalized size = 8.18 \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+a)^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 + 10*(x*e + d)^m*c^2*m^3*x^5*e^5 + 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 + 2*(x*e + d)^m*a*c*m^4*x^3*e^5 + 35*(x*e + d)^m*c^2*m^2
*x^5*e^5 + 2*(x*e + d)^m*a*c*d*m^4*x^2*e^4 + 11*(x*e + d)^m*c^2*d*m^2*x^4*e^4 - 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 + 24*(x*e + d)^m*a*c*m^3*x^3*e^5 + 50*(x*e + d)^m*c^2*m*x^5*e^5 + 20
*(x*e + d)^m*a*c*d*m^3*x^2*e^4 + 6*(x*e + d)^m*c^2*d*m*x^4*e^4 - 4*(x*e + d)^m*a*c*d^2*m^3*x*e^3 - 8*(x*e + d)
^m*c^2*d^2*m*x^3*e^3 + 12*(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 + (x*e + d)^m*a^2*m^4*x
*e^5 + 98*(x*e + d)^m*a*c*m^2*x^3*e^5 + 24*(x*e + d)^m*c^2*x^5*e^5 + (x*e + d)^m*a^2*d*m^4*e^4 + 58*(x*e + d)^
m*a*c*d*m^2*x^2*e^4 - 36*(x*e + d)^m*a*c*d^2*m^2*x*e^3 + 4*(x*e + d)^m*a*c*d^3*m^2*e^2 + 24*(x*e + d)^m*c^2*d^
5 + 14*(x*e + d)^m*a^2*m^3*x*e^5 + 156*(x*e + d)^m*a*c*m*x^3*e^5 + 14*(x*e + d)^m*a^2*d*m^3*e^4 + 40*(x*e + d)
^m*a*c*d*m*x^2*e^4 - 80*(x*e + d)^m*a*c*d^2*m*x*e^3 + 36*(x*e + d)^m*a*c*d^3*m*e^2 + 71*(x*e + d)^m*a^2*m^2*x*
e^5 + 80*(x*e + d)^m*a*c*x^3*e^5 + 71*(x*e + d)^m*a^2*d*m^2*e^4 + 80*(x*e + d)^m*a*c*d^3*e^2 + 154*(x*e + d)^m
*a^2*m*x*e^5 + 154*(x*e + d)^m*a^2*d*m*e^4 + 120*(x*e + d)^m*a^2*x*e^5 + 120*(x*e + d)^m*a^2*d*e^4)/(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)