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

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

[Out]

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

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Rubi [A]  time = 0.121412, 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} \[ \frac{a^2 A c (e x)^{m+1}}{e (m+1)}+\frac{a (e x)^{m+3} (a A d+a B c+2 A b c)}{e^3 (m+3)}+\frac{(e x)^{m+5} (A b (2 a d+b c)+a B (a d+2 b c))}{e^5 (m+5)}+\frac{b (e x)^{m+7} (2 a B d+A b d+b B c)}{e^7 (m+7)}+\frac{b^2 B d (e x)^{m+9}}{e^9 (m+9)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.126008, size = 113, normalized size = 0.78 \[ x (e x)^m \left (\frac{a^2 A c}{m+1}+\frac{b x^6 (2 a B d+A b d+b B c)}{m+7}+\frac{x^4 (A b (2 a d+b c)+a B (a d+2 b c))}{m+5}+\frac{a x^2 (a A d+a B c+2 A b c)}{m+3}+\frac{b^2 B d x^8}{m+9}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.006, size = 711, normalized size = 4.9 \begin{align*}{\frac{ \left ( B{b}^{2}d{m}^{4}{x}^{8}+16\,B{b}^{2}d{m}^{3}{x}^{8}+A{b}^{2}d{m}^{4}{x}^{6}+2\,Babd{m}^{4}{x}^{6}+B{b}^{2}c{m}^{4}{x}^{6}+86\,B{b}^{2}d{m}^{2}{x}^{8}+18\,A{b}^{2}d{m}^{3}{x}^{6}+36\,Babd{m}^{3}{x}^{6}+18\,B{b}^{2}c{m}^{3}{x}^{6}+176\,B{b}^{2}dm{x}^{8}+2\,Aabd{m}^{4}{x}^{4}+A{b}^{2}c{m}^{4}{x}^{4}+104\,A{b}^{2}d{m}^{2}{x}^{6}+B{a}^{2}d{m}^{4}{x}^{4}+2\,Babc{m}^{4}{x}^{4}+208\,Babd{m}^{2}{x}^{6}+104\,B{b}^{2}c{m}^{2}{x}^{6}+105\,Bd{b}^{2}{x}^{8}+40\,Aabd{m}^{3}{x}^{4}+20\,A{b}^{2}c{m}^{3}{x}^{4}+222\,A{b}^{2}dm{x}^{6}+20\,B{a}^{2}d{m}^{3}{x}^{4}+40\,Babc{m}^{3}{x}^{4}+444\,Babdm{x}^{6}+222\,B{b}^{2}cm{x}^{6}+A{a}^{2}d{m}^{4}{x}^{2}+2\,Aabc{m}^{4}{x}^{2}+260\,Aabd{m}^{2}{x}^{4}+130\,A{b}^{2}c{m}^{2}{x}^{4}+135\,A{b}^{2}d{x}^{6}+B{a}^{2}c{m}^{4}{x}^{2}+130\,B{a}^{2}d{m}^{2}{x}^{4}+260\,Babc{m}^{2}{x}^{4}+270\,Babd{x}^{6}+135\,B{b}^{2}c{x}^{6}+22\,A{a}^{2}d{m}^{3}{x}^{2}+44\,Aabc{m}^{3}{x}^{2}+600\,Aabdm{x}^{4}+300\,A{b}^{2}cm{x}^{4}+22\,B{a}^{2}c{m}^{3}{x}^{2}+300\,B{a}^{2}dm{x}^{4}+600\,Babcm{x}^{4}+A{a}^{2}c{m}^{4}+164\,A{a}^{2}d{m}^{2}{x}^{2}+328\,Aabc{m}^{2}{x}^{2}+378\,Aabd{x}^{4}+189\,A{b}^{2}c{x}^{4}+164\,B{a}^{2}c{m}^{2}{x}^{2}+189\,B{a}^{2}d{x}^{4}+378\,Babc{x}^{4}+24\,A{a}^{2}c{m}^{3}+458\,A{a}^{2}dm{x}^{2}+916\,Aabcm{x}^{2}+458\,B{a}^{2}cm{x}^{2}+206\,A{a}^{2}c{m}^{2}+315\,A{a}^{2}d{x}^{2}+630\,Aabc{x}^{2}+315\,B{a}^{2}c{x}^{2}+744\,A{a}^{2}cm+945\,Ac{a}^{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)^2*(B*x^2+A)*(d*x^2+c),x)

[Out]

x*(B*b^2*d*m^4*x^8+16*B*b^2*d*m^3*x^8+A*b^2*d*m^4*x^6+2*B*a*b*d*m^4*x^6+B*b^2*c*m^4*x^6+86*B*b^2*d*m^2*x^8+18*
A*b^2*d*m^3*x^6+36*B*a*b*d*m^3*x^6+18*B*b^2*c*m^3*x^6+176*B*b^2*d*m*x^8+2*A*a*b*d*m^4*x^4+A*b^2*c*m^4*x^4+104*
A*b^2*d*m^2*x^6+B*a^2*d*m^4*x^4+2*B*a*b*c*m^4*x^4+208*B*a*b*d*m^2*x^6+104*B*b^2*c*m^2*x^6+105*B*b^2*d*x^8+40*A
*a*b*d*m^3*x^4+20*A*b^2*c*m^3*x^4+222*A*b^2*d*m*x^6+20*B*a^2*d*m^3*x^4+40*B*a*b*c*m^3*x^4+444*B*a*b*d*m*x^6+22
2*B*b^2*c*m*x^6+A*a^2*d*m^4*x^2+2*A*a*b*c*m^4*x^2+260*A*a*b*d*m^2*x^4+130*A*b^2*c*m^2*x^4+135*A*b^2*d*x^6+B*a^
2*c*m^4*x^2+130*B*a^2*d*m^2*x^4+260*B*a*b*c*m^2*x^4+270*B*a*b*d*x^6+135*B*b^2*c*x^6+22*A*a^2*d*m^3*x^2+44*A*a*
b*c*m^3*x^2+600*A*a*b*d*m*x^4+300*A*b^2*c*m*x^4+22*B*a^2*c*m^3*x^2+300*B*a^2*d*m*x^4+600*B*a*b*c*m*x^4+A*a^2*c
*m^4+164*A*a^2*d*m^2*x^2+328*A*a*b*c*m^2*x^2+378*A*a*b*d*x^4+189*A*b^2*c*x^4+164*B*a^2*c*m^2*x^2+189*B*a^2*d*x
^4+378*B*a*b*c*x^4+24*A*a^2*c*m^3+458*A*a^2*d*m*x^2+916*A*a*b*c*m*x^2+458*B*a^2*c*m*x^2+206*A*a^2*c*m^2+315*A*
a^2*d*x^2+630*A*a*b*c*x^2+315*B*a^2*c*x^2+744*A*a^2*c*m+945*A*a^2*c)*(e*x)^m/(9+m)/(7+m)/(5+m)/(3+m)/(1+m)

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.62932, size = 1224, normalized size = 8.5 \begin{align*} \frac{{\left ({\left (B b^{2} d m^{4} + 16 \, B b^{2} d m^{3} + 86 \, B b^{2} d m^{2} + 176 \, B b^{2} d m + 105 \, B b^{2} d\right )} x^{9} +{\left ({\left (B b^{2} c +{\left (2 \, B a b + A b^{2}\right )} d\right )} m^{4} + 135 \, B b^{2} c + 18 \,{\left (B b^{2} c +{\left (2 \, B a b + A b^{2}\right )} d\right )} m^{3} + 104 \,{\left (B b^{2} c +{\left (2 \, B a b + A b^{2}\right )} d\right )} m^{2} + 135 \,{\left (2 \, B a b + A b^{2}\right )} d + 222 \,{\left (B b^{2} c +{\left (2 \, B a b + A b^{2}\right )} d\right )} m\right )} x^{7} +{\left ({\left ({\left (2 \, B a b + A b^{2}\right )} c +{\left (B a^{2} + 2 \, A a b\right )} d\right )} m^{4} + 20 \,{\left ({\left (2 \, B a b + A b^{2}\right )} c +{\left (B a^{2} + 2 \, A a b\right )} d\right )} m^{3} + 130 \,{\left ({\left (2 \, B a b + A b^{2}\right )} c +{\left (B a^{2} + 2 \, A a b\right )} d\right )} m^{2} + 189 \,{\left (2 \, B a b + A b^{2}\right )} c + 189 \,{\left (B a^{2} + 2 \, A a b\right )} d + 300 \,{\left ({\left (2 \, B a b + A b^{2}\right )} c +{\left (B a^{2} + 2 \, A a b\right )} d\right )} m\right )} x^{5} +{\left ({\left (A a^{2} d +{\left (B a^{2} + 2 \, A a b\right )} c\right )} m^{4} + 315 \, A a^{2} d + 22 \,{\left (A a^{2} d +{\left (B a^{2} + 2 \, A a b\right )} c\right )} m^{3} + 164 \,{\left (A a^{2} d +{\left (B a^{2} + 2 \, A a b\right )} c\right )} m^{2} + 315 \,{\left (B a^{2} + 2 \, A a b\right )} c + 458 \,{\left (A a^{2} d +{\left (B a^{2} + 2 \, A a b\right )} c\right )} m\right )} x^{3} +{\left (A a^{2} c m^{4} + 24 \, A a^{2} c m^{3} + 206 \, A a^{2} c m^{2} + 744 \, A a^{2} c m + 945 \, A a^{2} c\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)^2*(B*x^2+A)*(d*x^2+c),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 3.74501, 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)**2*(B*x**2+A)*(d*x**2+c),x)

[Out]

Piecewise(((-A*a**2*c/(8*x**8) - A*a**2*d/(6*x**6) - A*a*b*c/(3*x**6) - A*a*b*d/(2*x**4) - A*b**2*c/(4*x**4) -
A*b**2*d/(2*x**2) - B*a**2*c/(6*x**6) - B*a**2*d/(4*x**4) - B*a*b*c/(2*x**4) - B*a*b*d/x**2 - B*b**2*c/(2*x**
2) + B*b**2*d*log(x))/e**9, Eq(m, -9)), ((-A*a**2*c/(6*x**6) - A*a**2*d/(4*x**4) - A*a*b*c/(2*x**4) - A*a*b*d/
x**2 - A*b**2*c/(2*x**2) + A*b**2*d*log(x) - B*a**2*c/(4*x**4) - B*a**2*d/(2*x**2) - B*a*b*c/x**2 + 2*B*a*b*d*
log(x) + B*b**2*c*log(x) + B*b**2*d*x**2/2)/e**7, Eq(m, -7)), ((-A*a**2*c/(4*x**4) - A*a**2*d/(2*x**2) - A*a*b
*c/x**2 + 2*A*a*b*d*log(x) + A*b**2*c*log(x) + A*b**2*d*x**2/2 - B*a**2*c/(2*x**2) + B*a**2*d*log(x) + 2*B*a*b
*c*log(x) + B*a*b*d*x**2 + B*b**2*c*x**2/2 + B*b**2*d*x**4/4)/e**5, Eq(m, -5)), ((-A*a**2*c/(2*x**2) + A*a**2*
d*log(x) + 2*A*a*b*c*log(x) + A*a*b*d*x**2 + A*b**2*c*x**2/2 + A*b**2*d*x**4/4 + B*a**2*c*log(x) + B*a**2*d*x*
*2/2 + B*a*b*c*x**2 + B*a*b*d*x**4/2 + B*b**2*c*x**4/4 + B*b**2*d*x**6/6)/e**3, Eq(m, -3)), ((A*a**2*c*log(x)
+ A*a**2*d*x**2/2 + A*a*b*c*x**2 + A*a*b*d*x**4/2 + A*b**2*c*x**4/4 + A*b**2*d*x**6/6 + B*a**2*c*x**2/2 + B*a*
*2*d*x**4/4 + B*a*b*c*x**4/2 + B*a*b*d*x**6/3 + B*b**2*c*x**6/6 + B*b**2*d*x**8/8)/e, Eq(m, -1)), (A*a**2*c*e*
*m*m**4*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 24*A*a**2*c*e**m*m**3*x*x**m/(m**5 + 25
*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 206*A*a**2*c*e**m*m**2*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m
**2 + 1689*m + 945) + 744*A*a**2*c*e**m*m*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 945*A
*a**2*c*e**m*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + A*a**2*d*e**m*m**4*x**3*x**m/(m**5
+ 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 22*A*a**2*d*e**m*m**3*x**3*x**m/(m**5 + 25*m**4 + 230*m**3
+ 950*m**2 + 1689*m + 945) + 164*A*a**2*d*e**m*m**2*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m +
945) + 458*A*a**2*d*e**m*m*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 315*A*a**2*d*e**
m*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 2*A*a*b*c*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*b*c*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*b*c*e**m*m**2*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 9
16*A*a*b*c*e**m*m*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 630*A*a*b*c*e**m*x**3*x**m
/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 2*A*a*b*d*e**m*m**4*x**5*x**m/(m**5 + 25*m**4 + 230*m
**3 + 950*m**2 + 1689*m + 945) + 40*A*a*b*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*a*b*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*a*b*d*
e**m*m*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 378*A*a*b*d*e**m*x**5*x**m/(m**5 + 25
*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + A*b**2*c*e**m*m**4*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m*
*2 + 1689*m + 945) + 20*A*b**2*c*e**m*m**3*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 1
30*A*b**2*c*e**m*m**2*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 300*A*b**2*c*e**m*m*x*
*5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 189*A*b**2*c*e**m*x**5*x**m/(m**5 + 25*m**4 +
230*m**3 + 950*m**2 + 1689*m + 945) + A*b**2*d*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**2*d*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**
2*d*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**2*d*e**m*m*x**7*x**m/
(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 135*A*b**2*d*e**m*x**7*x**m/(m**5 + 25*m**4 + 230*m**3
+ 950*m**2 + 1689*m + 945) + B*a**2*c*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**2*c*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**2*c*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**2*c*e**m*m*x**3*x**m/(m**5 +
25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 315*B*a**2*c*e**m*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m
**2 + 1689*m + 945) + B*a**2*d*e**m*m**4*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 20*
B*a**2*d*e**m*m**3*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 130*B*a**2*d*e**m*m**2*x*
*5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 300*B*a**2*d*e**m*m*x**5*x**m/(m**5 + 25*m**4
+ 230*m**3 + 950*m**2 + 1689*m + 945) + 189*B*a**2*d*e**m*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 16
89*m + 945) + 2*B*a*b*c*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*b*c
*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*b*c*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*b*c*e**m*m*x**5*x**m/(m**5 + 25*m**4 + 230*m**3
+ 950*m**2 + 1689*m + 945) + 378*B*a*b*c*e**m*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945)
+ 2*B*a*b*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*a*b*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*a*b*d*e**m*m**2*x**7*x**m/(m**5 + 25*m
**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 444*B*a*b*d*e**m*m*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2
+ 1689*m + 945) + 270*B*a*b*d*e**m*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + B*b**2*c
*e**m*m**4*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 18*B*b**2*c*e**m*m**3*x**7*x**m/(
m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 104*B*b**2*c*e**m*m**2*x**7*x**m/(m**5 + 25*m**4 + 230*
m**3 + 950*m**2 + 1689*m + 945) + 222*B*b**2*c*e**m*m*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m
+ 945) + 135*B*b**2*c*e**m*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + B*b**2*d*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**2*d*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**2*d*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**2*d*e**m*m*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) +
105*B*b**2*d*e**m*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945), True))

________________________________________________________________________________________

Giac [B]  time = 1.29024, 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)^2*(B*x^2+A)*(d*x^2+c),x, algorithm="giac")

[Out]

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