3.10 \(\int (e x)^m (a+b x^2) (A+B x^2) (c+d x^2)^2 \, dx\)

Optimal. Leaf size=144 \[ \frac{c (e x)^{m+3} (2 a A d+a B c+A b c)}{e^3 (m+3)}+\frac{(e x)^{m+5} (a d (A d+2 B c)+b c (2 A d+B c))}{e^5 (m+5)}+\frac{d (e x)^{m+7} (a B d+A b d+2 b B c)}{e^7 (m+7)}+\frac{a A c^2 (e x)^{m+1}}{e (m+1)}+\frac{b B d^2 (e x)^{m+9}}{e^9 (m+9)} \]

[Out]

(a*A*c^2*(e*x)^(1 + m))/(e*(1 + m)) + (c*(A*b*c + a*B*c + 2*a*A*d)*(e*x)^(3 + m))/(e^3*(3 + m)) + ((a*d*(2*B*c
+ A*d) + b*c*(B*c + 2*A*d))*(e*x)^(5 + m))/(e^5*(5 + m)) + (d*(2*b*B*c + A*b*d + a*B*d)*(e*x)^(7 + m))/(e^7*(
7 + m)) + (b*B*d^2*(e*x)^(9 + m))/(e^9*(9 + m))

________________________________________________________________________________________

Rubi [A]  time = 0.134343, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.034, Rules used = {570} \[ \frac{c (e x)^{m+3} (2 a A d+a B c+A b c)}{e^3 (m+3)}+\frac{(e x)^{m+5} (a d (A d+2 B c)+b c (2 A d+B c))}{e^5 (m+5)}+\frac{d (e x)^{m+7} (a B d+A b d+2 b B c)}{e^7 (m+7)}+\frac{a A c^2 (e x)^{m+1}}{e (m+1)}+\frac{b B d^2 (e x)^{m+9}}{e^9 (m+9)} \]

Antiderivative was successfully verified.

[In]

Int[(e*x)^m*(a + b*x^2)*(A + B*x^2)*(c + d*x^2)^2,x]

[Out]

(a*A*c^2*(e*x)^(1 + m))/(e*(1 + m)) + (c*(A*b*c + a*B*c + 2*a*A*d)*(e*x)^(3 + m))/(e^3*(3 + m)) + ((a*d*(2*B*c
+ A*d) + b*c*(B*c + 2*A*d))*(e*x)^(5 + m))/(e^5*(5 + m)) + (d*(2*b*B*c + A*b*d + a*B*d)*(e*x)^(7 + m))/(e^7*(
7 + m)) + (b*B*d^2*(e*x)^(9 + m))/(e^9*(9 + m))

Rule 570

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^
(r_.), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a,
b, c, d, e, f, g, m, n}, x] && IGtQ[p, -2] && IGtQ[q, 0] && IGtQ[r, 0]

Rubi steps

\begin{align*} \int (e x)^m \left (a+b x^2\right ) \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx &=\int \left (a A c^2 (e x)^m+\frac{c (A b c+a B c+2 a A d) (e x)^{2+m}}{e^2}+\frac{(a d (2 B c+A d)+b c (B c+2 A d)) (e x)^{4+m}}{e^4}+\frac{d (2 b B c+A b d+a B d) (e x)^{6+m}}{e^6}+\frac{b B d^2 (e x)^{8+m}}{e^8}\right ) \, dx\\ &=\frac{a A c^2 (e x)^{1+m}}{e (1+m)}+\frac{c (A b c+a B c+2 a A d) (e x)^{3+m}}{e^3 (3+m)}+\frac{(a d (2 B c+A d)+b c (B c+2 A d)) (e x)^{5+m}}{e^5 (5+m)}+\frac{d (2 b B c+A b d+a B d) (e x)^{7+m}}{e^7 (7+m)}+\frac{b B d^2 (e x)^{9+m}}{e^9 (9+m)}\\ \end{align*}

Mathematica [A]  time = 0.154997, size = 113, normalized size = 0.78 \[ x (e x)^m \left (\frac{d x^6 (a B d+A b d+2 b B c)}{m+7}+\frac{x^4 (a d (A d+2 B c)+b c (2 A d+B c))}{m+5}+\frac{c x^2 (2 a A d+a B c+A b c)}{m+3}+\frac{a A c^2}{m+1}+\frac{b B d^2 x^8}{m+9}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.006, size = 711, normalized size = 4.9 \begin{align*}{\frac{ \left ( Bb{d}^{2}{m}^{4}{x}^{8}+16\,Bb{d}^{2}{m}^{3}{x}^{8}+Ab{d}^{2}{m}^{4}{x}^{6}+Ba{d}^{2}{m}^{4}{x}^{6}+2\,Bbcd{m}^{4}{x}^{6}+86\,Bb{d}^{2}{m}^{2}{x}^{8}+18\,Ab{d}^{2}{m}^{3}{x}^{6}+18\,Ba{d}^{2}{m}^{3}{x}^{6}+36\,Bbcd{m}^{3}{x}^{6}+176\,Bb{d}^{2}m{x}^{8}+Aa{d}^{2}{m}^{4}{x}^{4}+2\,Abcd{m}^{4}{x}^{4}+104\,Ab{d}^{2}{m}^{2}{x}^{6}+2\,Bacd{m}^{4}{x}^{4}+104\,Ba{d}^{2}{m}^{2}{x}^{6}+Bb{c}^{2}{m}^{4}{x}^{4}+208\,Bbcd{m}^{2}{x}^{6}+105\,Bb{d}^{2}{x}^{8}+20\,Aa{d}^{2}{m}^{3}{x}^{4}+40\,Abcd{m}^{3}{x}^{4}+222\,Ab{d}^{2}m{x}^{6}+40\,Bacd{m}^{3}{x}^{4}+222\,Ba{d}^{2}m{x}^{6}+20\,Bb{c}^{2}{m}^{3}{x}^{4}+444\,Bbcdm{x}^{6}+2\,Aacd{m}^{4}{x}^{2}+130\,Aa{d}^{2}{m}^{2}{x}^{4}+Ab{c}^{2}{m}^{4}{x}^{2}+260\,Abcd{m}^{2}{x}^{4}+135\,Ab{d}^{2}{x}^{6}+Ba{c}^{2}{m}^{4}{x}^{2}+260\,Bacd{m}^{2}{x}^{4}+135\,Ba{d}^{2}{x}^{6}+130\,Bb{c}^{2}{m}^{2}{x}^{4}+270\,Bbcd{x}^{6}+44\,Aacd{m}^{3}{x}^{2}+300\,Aa{d}^{2}m{x}^{4}+22\,Ab{c}^{2}{m}^{3}{x}^{2}+600\,Abcdm{x}^{4}+22\,Ba{c}^{2}{m}^{3}{x}^{2}+600\,Bacdm{x}^{4}+300\,Bb{c}^{2}m{x}^{4}+Aa{c}^{2}{m}^{4}+328\,Aacd{m}^{2}{x}^{2}+189\,Aa{d}^{2}{x}^{4}+164\,Ab{c}^{2}{m}^{2}{x}^{2}+378\,Abcd{x}^{4}+164\,Ba{c}^{2}{m}^{2}{x}^{2}+378\,Bacd{x}^{4}+189\,Bb{c}^{2}{x}^{4}+24\,Aa{c}^{2}{m}^{3}+916\,Aacdm{x}^{2}+458\,Ab{c}^{2}m{x}^{2}+458\,Ba{c}^{2}m{x}^{2}+206\,Aa{c}^{2}{m}^{2}+630\,Aacd{x}^{2}+315\,Ab{c}^{2}{x}^{2}+315\,Ba{c}^{2}{x}^{2}+744\,Aa{c}^{2}m+945\,Aa{c}^{2} \right ) x \left ( ex \right ) ^{m}}{ \left ( 9+m \right ) \left ( 7+m \right ) \left ( 5+m \right ) \left ( 3+m \right ) \left ( 1+m \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(B*b*d^2*m^4*x^8+16*B*b*d^2*m^3*x^8+A*b*d^2*m^4*x^6+B*a*d^2*m^4*x^6+2*B*b*c*d*m^4*x^6+86*B*b*d^2*m^2*x^8+18*
A*b*d^2*m^3*x^6+18*B*a*d^2*m^3*x^6+36*B*b*c*d*m^3*x^6+176*B*b*d^2*m*x^8+A*a*d^2*m^4*x^4+2*A*b*c*d*m^4*x^4+104*
A*b*d^2*m^2*x^6+2*B*a*c*d*m^4*x^4+104*B*a*d^2*m^2*x^6+B*b*c^2*m^4*x^4+208*B*b*c*d*m^2*x^6+105*B*b*d^2*x^8+20*A
*a*d^2*m^3*x^4+40*A*b*c*d*m^3*x^4+222*A*b*d^2*m*x^6+40*B*a*c*d*m^3*x^4+222*B*a*d^2*m*x^6+20*B*b*c^2*m^3*x^4+44
4*B*b*c*d*m*x^6+2*A*a*c*d*m^4*x^2+130*A*a*d^2*m^2*x^4+A*b*c^2*m^4*x^2+260*A*b*c*d*m^2*x^4+135*A*b*d^2*x^6+B*a*
c^2*m^4*x^2+260*B*a*c*d*m^2*x^4+135*B*a*d^2*x^6+130*B*b*c^2*m^2*x^4+270*B*b*c*d*x^6+44*A*a*c*d*m^3*x^2+300*A*a
*d^2*m*x^4+22*A*b*c^2*m^3*x^2+600*A*b*c*d*m*x^4+22*B*a*c^2*m^3*x^2+600*B*a*c*d*m*x^4+300*B*b*c^2*m*x^4+A*a*c^2
*m^4+328*A*a*c*d*m^2*x^2+189*A*a*d^2*x^4+164*A*b*c^2*m^2*x^2+378*A*b*c*d*x^4+164*B*a*c^2*m^2*x^2+378*B*a*c*d*x
^4+189*B*b*c^2*x^4+24*A*a*c^2*m^3+916*A*a*c*d*m*x^2+458*A*b*c^2*m*x^2+458*B*a*c^2*m*x^2+206*A*a*c^2*m^2+630*A*
a*c*d*x^2+315*A*b*c^2*x^2+315*B*a*c^2*x^2+744*A*a*c^2*m+945*A*a*c^2)*(e*x)^m/(9+m)/(7+m)/(5+m)/(3+m)/(1+m)

________________________________________________________________________________________

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)^m*(b*x^2+a)*(B*x^2+A)*(d*x^2+c)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 1.65203, size = 1168, normalized size = 8.11 \begin{align*} \frac{{\left ({\left (B b d^{2} m^{4} + 16 \, B b d^{2} m^{3} + 86 \, B b d^{2} m^{2} + 176 \, B b d^{2} m + 105 \, B b d^{2}\right )} x^{9} +{\left ({\left (2 \, B b c d +{\left (B a + A b\right )} d^{2}\right )} m^{4} + 270 \, B b c d + 18 \,{\left (2 \, B b c d +{\left (B a + A b\right )} d^{2}\right )} m^{3} + 135 \,{\left (B a + A b\right )} d^{2} + 104 \,{\left (2 \, B b c d +{\left (B a + A b\right )} d^{2}\right )} m^{2} + 222 \,{\left (2 \, B b c d +{\left (B a + A b\right )} d^{2}\right )} m\right )} x^{7} +{\left ({\left (B b c^{2} + A a d^{2} + 2 \,{\left (B a + A b\right )} c d\right )} m^{4} + 189 \, B b c^{2} + 189 \, A a d^{2} + 20 \,{\left (B b c^{2} + A a d^{2} + 2 \,{\left (B a + A b\right )} c d\right )} m^{3} + 378 \,{\left (B a + A b\right )} c d + 130 \,{\left (B b c^{2} + A a d^{2} + 2 \,{\left (B a + A b\right )} c d\right )} m^{2} + 300 \,{\left (B b c^{2} + A a d^{2} + 2 \,{\left (B a + A b\right )} c d\right )} m\right )} x^{5} +{\left ({\left (2 \, A a c d +{\left (B a + A b\right )} c^{2}\right )} m^{4} + 630 \, A a c d + 22 \,{\left (2 \, A a c d +{\left (B a + A b\right )} c^{2}\right )} m^{3} + 315 \,{\left (B a + A b\right )} c^{2} + 164 \,{\left (2 \, A a c d +{\left (B a + A b\right )} c^{2}\right )} m^{2} + 458 \,{\left (2 \, A a c d +{\left (B a + A b\right )} c^{2}\right )} m\right )} x^{3} +{\left (A a c^{2} m^{4} + 24 \, A a c^{2} m^{3} + 206 \, A a c^{2} m^{2} + 744 \, A a c^{2} m + 945 \, A a c^{2}\right )} x\right )} \left (e x\right )^{m}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((B*b*d^2*m^4 + 16*B*b*d^2*m^3 + 86*B*b*d^2*m^2 + 176*B*b*d^2*m + 105*B*b*d^2)*x^9 + ((2*B*b*c*d + (B*a + A*b)
*d^2)*m^4 + 270*B*b*c*d + 18*(2*B*b*c*d + (B*a + A*b)*d^2)*m^3 + 135*(B*a + A*b)*d^2 + 104*(2*B*b*c*d + (B*a +
A*b)*d^2)*m^2 + 222*(2*B*b*c*d + (B*a + A*b)*d^2)*m)*x^7 + ((B*b*c^2 + A*a*d^2 + 2*(B*a + A*b)*c*d)*m^4 + 189
*B*b*c^2 + 189*A*a*d^2 + 20*(B*b*c^2 + A*a*d^2 + 2*(B*a + A*b)*c*d)*m^3 + 378*(B*a + A*b)*c*d + 130*(B*b*c^2 +
A*a*d^2 + 2*(B*a + A*b)*c*d)*m^2 + 300*(B*b*c^2 + A*a*d^2 + 2*(B*a + A*b)*c*d)*m)*x^5 + ((2*A*a*c*d + (B*a +
A*b)*c^2)*m^4 + 630*A*a*c*d + 22*(2*A*a*c*d + (B*a + A*b)*c^2)*m^3 + 315*(B*a + A*b)*c^2 + 164*(2*A*a*c*d + (B
*a + A*b)*c^2)*m^2 + 458*(2*A*a*c*d + (B*a + A*b)*c^2)*m)*x^3 + (A*a*c^2*m^4 + 24*A*a*c^2*m^3 + 206*A*a*c^2*m^
2 + 744*A*a*c^2*m + 945*A*a*c^2)*x)*(e*x)^m/(m^5 + 25*m^4 + 230*m^3 + 950*m^2 + 1689*m + 945)

________________________________________________________________________________________

Sympy [A]  time = 3.72834, size = 3373, normalized size = 23.42 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise(((-A*a*c**2/(8*x**8) - A*a*c*d/(3*x**6) - A*a*d**2/(4*x**4) - A*b*c**2/(6*x**6) - A*b*c*d/(2*x**4) -
A*b*d**2/(2*x**2) - B*a*c**2/(6*x**6) - B*a*c*d/(2*x**4) - B*a*d**2/(2*x**2) - B*b*c**2/(4*x**4) - B*b*c*d/x*
*2 + B*b*d**2*log(x))/e**9, Eq(m, -9)), ((-A*a*c**2/(6*x**6) - A*a*c*d/(2*x**4) - A*a*d**2/(2*x**2) - A*b*c**2
/(4*x**4) - A*b*c*d/x**2 + A*b*d**2*log(x) - B*a*c**2/(4*x**4) - B*a*c*d/x**2 + B*a*d**2*log(x) - B*b*c**2/(2*
x**2) + 2*B*b*c*d*log(x) + B*b*d**2*x**2/2)/e**7, Eq(m, -7)), ((-A*a*c**2/(4*x**4) - A*a*c*d/x**2 + A*a*d**2*l
og(x) - A*b*c**2/(2*x**2) + 2*A*b*c*d*log(x) + A*b*d**2*x**2/2 - B*a*c**2/(2*x**2) + 2*B*a*c*d*log(x) + B*a*d*
*2*x**2/2 + B*b*c**2*log(x) + B*b*c*d*x**2 + B*b*d**2*x**4/4)/e**5, Eq(m, -5)), ((-A*a*c**2/(2*x**2) + 2*A*a*c
*d*log(x) + A*a*d**2*x**2/2 + A*b*c**2*log(x) + A*b*c*d*x**2 + A*b*d**2*x**4/4 + B*a*c**2*log(x) + B*a*c*d*x**
2 + B*a*d**2*x**4/4 + B*b*c**2*x**2/2 + B*b*c*d*x**4/2 + B*b*d**2*x**6/6)/e**3, Eq(m, -3)), ((A*a*c**2*log(x)
+ A*a*c*d*x**2 + A*a*d**2*x**4/4 + A*b*c**2*x**2/2 + A*b*c*d*x**4/2 + A*b*d**2*x**6/6 + B*a*c**2*x**2/2 + B*a*
c*d*x**4/2 + B*a*d**2*x**6/6 + B*b*c**2*x**4/4 + B*b*c*d*x**6/3 + B*b*d**2*x**8/8)/e, Eq(m, -1)), (A*a*c**2*e*
*m*m**4*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 24*A*a*c**2*e**m*m**3*x*x**m/(m**5 + 25
*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 206*A*a*c**2*e**m*m**2*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m
**2 + 1689*m + 945) + 744*A*a*c**2*e**m*m*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 945*A
*a*c**2*e**m*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 2*A*a*c*d*e**m*m**4*x**3*x**m/(m**
5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 44*A*a*c*d*e**m*m**3*x**3*x**m/(m**5 + 25*m**4 + 230*m**3
+ 950*m**2 + 1689*m + 945) + 328*A*a*c*d*e**m*m**2*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m +
945) + 916*A*a*c*d*e**m*m*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 630*A*a*c*d*e**m*x
**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + A*a*d**2*e**m*m**4*x**5*x**m/(m**5 + 25*m**4
+ 230*m**3 + 950*m**2 + 1689*m + 945) + 20*A*a*d**2*e**m*m**3*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2
+ 1689*m + 945) + 130*A*a*d**2*e**m*m**2*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 300
*A*a*d**2*e**m*m*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 189*A*a*d**2*e**m*x**5*x**m
/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + A*b*c**2*e**m*m**4*x**3*x**m/(m**5 + 25*m**4 + 230*m*
*3 + 950*m**2 + 1689*m + 945) + 22*A*b*c**2*e**m*m**3*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m
+ 945) + 164*A*b*c**2*e**m*m**2*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 458*A*b*c**
2*e**m*m*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 315*A*b*c**2*e**m*x**3*x**m/(m**5 +
25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 2*A*b*c*d*e**m*m**4*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 95
0*m**2 + 1689*m + 945) + 40*A*b*c*d*e**m*m**3*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945)
+ 260*A*b*c*d*e**m*m**2*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 600*A*b*c*d*e**m*m*x
**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 378*A*b*c*d*e**m*x**5*x**m/(m**5 + 25*m**4 +
230*m**3 + 950*m**2 + 1689*m + 945) + A*b*d**2*e**m*m**4*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 168
9*m + 945) + 18*A*b*d**2*e**m*m**3*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 104*A*b*d
**2*e**m*m**2*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 222*A*b*d**2*e**m*m*x**7*x**m/
(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 135*A*b*d**2*e**m*x**7*x**m/(m**5 + 25*m**4 + 230*m**3
+ 950*m**2 + 1689*m + 945) + B*a*c**2*e**m*m**4*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 94
5) + 22*B*a*c**2*e**m*m**3*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 164*B*a*c**2*e**m
*m**2*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 458*B*a*c**2*e**m*m*x**3*x**m/(m**5 +
25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 315*B*a*c**2*e**m*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m
**2 + 1689*m + 945) + 2*B*a*c*d*e**m*m**4*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 40
*B*a*c*d*e**m*m**3*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 260*B*a*c*d*e**m*m**2*x**
5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 600*B*a*c*d*e**m*m*x**5*x**m/(m**5 + 25*m**4 +
230*m**3 + 950*m**2 + 1689*m + 945) + 378*B*a*c*d*e**m*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*
m + 945) + B*a*d**2*e**m*m**4*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 18*B*a*d**2*e*
*m*m**3*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 104*B*a*d**2*e**m*m**2*x**7*x**m/(m*
*5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 222*B*a*d**2*e**m*m*x**7*x**m/(m**5 + 25*m**4 + 230*m**3
+ 950*m**2 + 1689*m + 945) + 135*B*a*d**2*e**m*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945)
+ B*b*c**2*e**m*m**4*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 20*B*b*c**2*e**m*m**3*
x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 130*B*b*c**2*e**m*m**2*x**5*x**m/(m**5 + 25*
m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 300*B*b*c**2*e**m*m*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m*
*2 + 1689*m + 945) + 189*B*b*c**2*e**m*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 2*B*b
*c*d*e**m*m**4*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 36*B*b*c*d*e**m*m**3*x**7*x**
m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 208*B*b*c*d*e**m*m**2*x**7*x**m/(m**5 + 25*m**4 + 23
0*m**3 + 950*m**2 + 1689*m + 945) + 444*B*b*c*d*e**m*m*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*
m + 945) + 270*B*b*c*d*e**m*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + B*b*d**2*e**m*m*
*4*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 16*B*b*d**2*e**m*m**3*x**9*x**m/(m**5 + 2
5*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 86*B*b*d**2*e**m*m**2*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 + 95
0*m**2 + 1689*m + 945) + 176*B*b*d**2*e**m*m*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) +
105*B*b*d**2*e**m*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945), True))

________________________________________________________________________________________

Giac [B]  time = 1.27258, size = 1362, normalized size = 9.46 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(B*b*d^2*m^4*x^9*x^m*e^m + 16*B*b*d^2*m^3*x^9*x^m*e^m + 2*B*b*c*d*m^4*x^7*x^m*e^m + B*a*d^2*m^4*x^7*x^m*e^m +
A*b*d^2*m^4*x^7*x^m*e^m + 86*B*b*d^2*m^2*x^9*x^m*e^m + 36*B*b*c*d*m^3*x^7*x^m*e^m + 18*B*a*d^2*m^3*x^7*x^m*e^m
+ 18*A*b*d^2*m^3*x^7*x^m*e^m + 176*B*b*d^2*m*x^9*x^m*e^m + B*b*c^2*m^4*x^5*x^m*e^m + 2*B*a*c*d*m^4*x^5*x^m*e^
m + 2*A*b*c*d*m^4*x^5*x^m*e^m + A*a*d^2*m^4*x^5*x^m*e^m + 208*B*b*c*d*m^2*x^7*x^m*e^m + 104*B*a*d^2*m^2*x^7*x^
m*e^m + 104*A*b*d^2*m^2*x^7*x^m*e^m + 105*B*b*d^2*x^9*x^m*e^m + 20*B*b*c^2*m^3*x^5*x^m*e^m + 40*B*a*c*d*m^3*x^
5*x^m*e^m + 40*A*b*c*d*m^3*x^5*x^m*e^m + 20*A*a*d^2*m^3*x^5*x^m*e^m + 444*B*b*c*d*m*x^7*x^m*e^m + 222*B*a*d^2*
m*x^7*x^m*e^m + 222*A*b*d^2*m*x^7*x^m*e^m + B*a*c^2*m^4*x^3*x^m*e^m + A*b*c^2*m^4*x^3*x^m*e^m + 2*A*a*c*d*m^4*
x^3*x^m*e^m + 130*B*b*c^2*m^2*x^5*x^m*e^m + 260*B*a*c*d*m^2*x^5*x^m*e^m + 260*A*b*c*d*m^2*x^5*x^m*e^m + 130*A*
a*d^2*m^2*x^5*x^m*e^m + 270*B*b*c*d*x^7*x^m*e^m + 135*B*a*d^2*x^7*x^m*e^m + 135*A*b*d^2*x^7*x^m*e^m + 22*B*a*c
^2*m^3*x^3*x^m*e^m + 22*A*b*c^2*m^3*x^3*x^m*e^m + 44*A*a*c*d*m^3*x^3*x^m*e^m + 300*B*b*c^2*m*x^5*x^m*e^m + 600
*B*a*c*d*m*x^5*x^m*e^m + 600*A*b*c*d*m*x^5*x^m*e^m + 300*A*a*d^2*m*x^5*x^m*e^m + A*a*c^2*m^4*x*x^m*e^m + 164*B
*a*c^2*m^2*x^3*x^m*e^m + 164*A*b*c^2*m^2*x^3*x^m*e^m + 328*A*a*c*d*m^2*x^3*x^m*e^m + 189*B*b*c^2*x^5*x^m*e^m +
378*B*a*c*d*x^5*x^m*e^m + 378*A*b*c*d*x^5*x^m*e^m + 189*A*a*d^2*x^5*x^m*e^m + 24*A*a*c^2*m^3*x*x^m*e^m + 458*
B*a*c^2*m*x^3*x^m*e^m + 458*A*b*c^2*m*x^3*x^m*e^m + 916*A*a*c*d*m*x^3*x^m*e^m + 206*A*a*c^2*m^2*x*x^m*e^m + 31
5*B*a*c^2*x^3*x^m*e^m + 315*A*b*c^2*x^3*x^m*e^m + 630*A*a*c*d*x^3*x^m*e^m + 744*A*a*c^2*m*x*x^m*e^m + 945*A*a*
c^2*x*x^m*e^m)/(m^5 + 25*m^4 + 230*m^3 + 950*m^2 + 1689*m + 945)