∖int(ex)m∖left(a + bx2∖right)2∖left(A + Bx2∖right)∖left(c + dx2∖right)2∖,dx

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

Optimal. Leaf size=216 \[ \frac{(e x)^{m+5} \left (A \left (a^2 d^2+4 a b c d+b^2 c^2\right )+2 a B c (a d+b c)\right )}{e^5 (m+5)}+\frac{(e x)^{m+7} \left (a^2 B d^2+2 a b d (A d+2 B c)+b^2 c (2 A d+B c)\right )}{e^7 (m+7)}+\frac{a^2 A c^2 (e x)^{m+1}}{e (m+1)}+\frac{b d (e x)^{m+9} (2 a B d+A b d+2 b B c)}{e^9 (m+9)}+\frac{a c (e x)^{m+3} (2 A (a d+b c)+a B c)}{e^3 (m+3)}+\frac{b^2 B d^2 (e x)^{m+11}}{e^{11} (m+11)} \]

[Out]

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

+ m))/(e^3*(3 + m)) + ((2*a*B*c*(b*c + a*d) + A*(b^2*c^2 + 4*a*b*c*d + a^2*d^2))

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

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

^(9 + m))/(e^9*(9 + m)) + (b^2*B*d^2*(e*x)^(11 + m))/(e^11*(11 + m))

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Rubi [A] time = 0.559959, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.032 \[ \frac{(e x)^{m+5} \left (A \left (a^2 d^2+4 a b c d+b^2 c^2\right )+2 a B c (a d+b c)\right )}{e^5 (m+5)}+\frac{(e x)^{m+7} \left (a^2 B d^2+2 a b d (A d+2 B c)+b^2 c (2 A d+B c)\right )}{e^7 (m+7)}+\frac{a^2 A c^2 (e x)^{m+1}}{e (m+1)}+\frac{b d (e x)^{m+9} (2 a B d+A b d+2 b B c)}{e^9 (m+9)}+\frac{a c (e x)^{m+3} (2 A (a d+b c)+a B c)}{e^3 (m+3)}+\frac{b^2 B d^2 (e x)^{m+11}}{e^{11} (m+11)} \]

Antiderivative was successfully veriﬁed.

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

[Out]

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

+ m))/(e^3*(3 + m)) + ((2*a*B*c*(b*c + a*d) + A*(b^2*c^2 + 4*a*b*c*d + a^2*d^2))

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

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

^(9 + m))/(e^9*(9 + m)) + (b^2*B*d^2*(e*x)^(11 + m))/(e^11*(11 + m))

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Rubi in Sympy [A] time = 104.884, size = 235, normalized size = 1.09 \[ \frac{A a^{2} c^{2} \left (e x\right )^{m + 1}}{e \left (m + 1\right )} + \frac{B b^{2} d^{2} \left (e x\right )^{m + 11}}{e^{11} \left (m + 11\right )} + \frac{a c \left (e x\right )^{m + 3} \left (2 A a d + 2 A b c + B a c\right )}{e^{3} \left (m + 3\right )} + \frac{b d \left (e x\right )^{m + 9} \left (A b d + 2 B a d + 2 B b c\right )}{e^{9} \left (m + 9\right )} + \frac{\left (e x\right )^{m + 5} \left (A a^{2} d^{2} + 4 A a b c d + A b^{2} c^{2} + 2 B a^{2} c d + 2 B a b c^{2}\right )}{e^{5} \left (m + 5\right )} + \frac{\left (e x\right )^{m + 7} \left (2 A a b d^{2} + 2 A b^{2} c d + B a^{2} d^{2} + 4 B a b c d + B b^{2} c^{2}\right )}{e^{7} \left (m + 7\right )} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a**2*c**2*(e*x)**(m + 1)/(e*(m + 1)) + B*b**2*d**2*(e*x)**(m + 11)/(e**11*(m +

11)) + a*c*(e*x)**(m + 3)*(2*A*a*d + 2*A*b*c + B*a*c)/(e**3*(m + 3)) + b*d*(e*x

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

**2 + 4*A*a*b*c*d + A*b**2*c**2 + 2*B*a**2*c*d + 2*B*a*b*c**2)/(e**5*(m + 5)) +

(e*x)**(m + 7)*(2*A*a*b*d**2 + 2*A*b**2*c*d + B*a**2*d**2 + 4*B*a*b*c*d + B*b**2

*c**2)/(e**7*(m + 7))

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Mathematica [A] time = 0.779022, size = 178, normalized size = 0.82 \[ (e x)^m \left (\frac{x^5 \left (A \left (a^2 d^2+4 a b c d+b^2 c^2\right )+2 a B c (a d+b c)\right )}{m+5}+\frac{x^7 \left (a^2 B d^2+2 a b d (A d+2 B c)+b^2 c (2 A d+B c)\right )}{m+7}+\frac{a^2 A c^2 x}{m+1}+\frac{b d x^9 (2 a B d+A b d+2 b B c)}{m+9}+\frac{a c x^3 (2 A (a d+b c)+a B c)}{m+3}+\frac{b^2 B d^2 x^{11}}{m+11}\right ) \]

Antiderivative was successfully veriﬁed.

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

[Out]

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

(2*a*B*c*(b*c + a*d) + A*(b^2*c^2 + 4*a*b*c*d + a^2*d^2))*x^5)/(5 + m) + ((a^2*B

*d^2 + 2*a*b*d*(2*B*c + A*d) + b^2*c*(B*c + 2*A*d))*x^7)/(7 + m) + (b*d*(2*b*B*c

+ A*b*d + 2*a*B*d)*x^9)/(9 + m) + (b^2*B*d^2*x^11)/(11 + m))

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Maple [B] time = 0.012, size = 1471, normalized size = 6.8 \[ \text{result too large to display} \]

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)^2,x)

[Out]

x*(B*b^2*d^2*m^5*x^10+25*B*b^2*d^2*m^4*x^10+A*b^2*d^2*m^5*x^8+2*B*a*b*d^2*m^5*x^

8+2*B*b^2*c*d*m^5*x^8+230*B*b^2*d^2*m^3*x^10+27*A*b^2*d^2*m^4*x^8+54*B*a*b*d^2*m

^4*x^8+54*B*b^2*c*d*m^4*x^8+950*B*b^2*d^2*m^2*x^10+2*A*a*b*d^2*m^5*x^6+2*A*b^2*c

*d*m^5*x^6+262*A*b^2*d^2*m^3*x^8+B*a^2*d^2*m^5*x^6+4*B*a*b*c*d*m^5*x^6+524*B*a*b

*d^2*m^3*x^8+B*b^2*c^2*m^5*x^6+524*B*b^2*c*d*m^3*x^8+1689*B*b^2*d^2*m*x^10+58*A*

a*b*d^2*m^4*x^6+58*A*b^2*c*d*m^4*x^6+1122*A*b^2*d^2*m^2*x^8+29*B*a^2*d^2*m^4*x^6

+116*B*a*b*c*d*m^4*x^6+2244*B*a*b*d^2*m^2*x^8+29*B*b^2*c^2*m^4*x^6+2244*B*b^2*c*

d*m^2*x^8+945*B*b^2*d^2*x^10+A*a^2*d^2*m^5*x^4+4*A*a*b*c*d*m^5*x^4+604*A*a*b*d^2

*m^3*x^6+A*b^2*c^2*m^5*x^4+604*A*b^2*c*d*m^3*x^6+2041*A*b^2*d^2*m*x^8+2*B*a^2*c*

d*m^5*x^4+302*B*a^2*d^2*m^3*x^6+2*B*a*b*c^2*m^5*x^4+1208*B*a*b*c*d*m^3*x^6+4082*

B*a*b*d^2*m*x^8+302*B*b^2*c^2*m^3*x^6+4082*B*b^2*c*d*m*x^8+31*A*a^2*d^2*m^4*x^4+

124*A*a*b*c*d*m^4*x^4+2732*A*a*b*d^2*m^2*x^6+31*A*b^2*c^2*m^4*x^4+2732*A*b^2*c*d

*m^2*x^6+1155*A*b^2*d^2*x^8+62*B*a^2*c*d*m^4*x^4+1366*B*a^2*d^2*m^2*x^6+62*B*a*b

*c^2*m^4*x^4+5464*B*a*b*c*d*m^2*x^6+2310*B*a*b*d^2*x^8+1366*B*b^2*c^2*m^2*x^6+23

10*B*b^2*c*d*x^8+2*A*a^2*c*d*m^5*x^2+350*A*a^2*d^2*m^3*x^4+2*A*a*b*c^2*m^5*x^2+1

400*A*a*b*c*d*m^3*x^4+5154*A*a*b*d^2*m*x^6+350*A*b^2*c^2*m^3*x^4+5154*A*b^2*c*d*

m*x^6+B*a^2*c^2*m^5*x^2+700*B*a^2*c*d*m^3*x^4+2577*B*a^2*d^2*m*x^6+700*B*a*b*c^2

*m^3*x^4+10308*B*a*b*c*d*m*x^6+2577*B*b^2*c^2*m*x^6+66*A*a^2*c*d*m^4*x^2+1730*A*

a^2*d^2*m^2*x^4+66*A*a*b*c^2*m^4*x^2+6920*A*a*b*c*d*m^2*x^4+2970*A*a*b*d^2*x^6+1

730*A*b^2*c^2*m^2*x^4+2970*A*b^2*c*d*x^6+33*B*a^2*c^2*m^4*x^2+3460*B*a^2*c*d*m^2

*x^4+1485*B*a^2*d^2*x^6+3460*B*a*b*c^2*m^2*x^4+5940*B*a*b*c*d*x^6+1485*B*b^2*c^2

*x^6+A*a^2*c^2*m^5+812*A*a^2*c*d*m^3*x^2+3489*A*a^2*d^2*m*x^4+812*A*a*b*c^2*m^3*

x^2+13956*A*a*b*c*d*m*x^4+3489*A*b^2*c^2*m*x^4+406*B*a^2*c^2*m^3*x^2+6978*B*a^2*

c*d*m*x^4+6978*B*a*b*c^2*m*x^4+35*A*a^2*c^2*m^4+4524*A*a^2*c*d*m^2*x^2+2079*A*a^

2*d^2*x^4+4524*A*a*b*c^2*m^2*x^2+8316*A*a*b*c*d*x^4+2079*A*b^2*c^2*x^4+2262*B*a^

2*c^2*m^2*x^2+4158*B*a^2*c*d*x^4+4158*B*a*b*c^2*x^4+470*A*a^2*c^2*m^3+10706*A*a^

2*c*d*m*x^2+10706*A*a*b*c^2*m*x^2+5353*B*a^2*c^2*m*x^2+3010*A*a^2*c^2*m^2+6930*A

*a^2*c*d*x^2+6930*A*a*b*c^2*x^2+3465*B*a^2*c^2*x^2+9129*A*a^2*c^2*m+10395*A*a^2*

c^2)*(e*x)^m/(11+m)/(9+m)/(7+m)/(5+m)/(3+m)/(1+m)

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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A] time = 0.264621, size = 1408, normalized size = 6.52 \[ \text{result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

((B*b^2*d^2*m^5 + 25*B*b^2*d^2*m^4 + 230*B*b^2*d^2*m^3 + 950*B*b^2*d^2*m^2 + 168

9*B*b^2*d^2*m + 945*B*b^2*d^2)*x^11 + ((2*B*b^2*c*d + (2*B*a*b + A*b^2)*d^2)*m^5

+ 2310*B*b^2*c*d + 27*(2*B*b^2*c*d + (2*B*a*b + A*b^2)*d^2)*m^4 + 262*(2*B*b^2*

c*d + (2*B*a*b + A*b^2)*d^2)*m^3 + 1155*(2*B*a*b + A*b^2)*d^2 + 1122*(2*B*b^2*c*

d + (2*B*a*b + A*b^2)*d^2)*m^2 + 2041*(2*B*b^2*c*d + (2*B*a*b + A*b^2)*d^2)*m)*x

^9 + ((B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2)*m^5 + 1485*B

*b^2*c^2 + 29*(B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2)*m^4

+ 302*(B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2)*m^3 + 2970*(

2*B*a*b + A*b^2)*c*d + 1485*(B*a^2 + 2*A*a*b)*d^2 + 1366*(B*b^2*c^2 + 2*(2*B*a*b

+ A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2)*m^2 + 2577*(B*b^2*c^2 + 2*(2*B*a*b + A*b^

2)*c*d + (B*a^2 + 2*A*a*b)*d^2)*m)*x^7 + ((A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2

*(B*a^2 + 2*A*a*b)*c*d)*m^5 + 2079*A*a^2*d^2 + 31*(A*a^2*d^2 + (2*B*a*b + A*b^2)

*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d)*m^4 + 350*(A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2

*(B*a^2 + 2*A*a*b)*c*d)*m^3 + 2079*(2*B*a*b + A*b^2)*c^2 + 4158*(B*a^2 + 2*A*a*b

)*c*d + 1730*(A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d)*m^2 +

3489*(A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d)*m)*x^5 + ((2

*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*m^5 + 6930*A*a^2*c*d + 33*(2*A*a^2*c*d + (B*

a^2 + 2*A*a*b)*c^2)*m^4 + 406*(2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*m^3 + 3465*(

B*a^2 + 2*A*a*b)*c^2 + 2262*(2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*m^2 + 5353*(2*

A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*m)*x^3 + (A*a^2*c^2*m^5 + 35*A*a^2*c^2*m^4 +

470*A*a^2*c^2*m^3 + 3010*A*a^2*c^2*m^2 + 9129*A*a^2*c^2*m + 10395*A*a^2*c^2)*x)*

(e*x)^m/(m^6 + 36*m^5 + 505*m^4 + 3480*m^3 + 12139*m^2 + 19524*m + 10395)

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Sympy [A] time = 11.0196, size = 7019, normalized size = 32.5 \[ \text{result too large to display} \]

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)**2,x)

[Out]

Piecewise(((-A*a**2*c**2/(10*x**10) - A*a**2*c*d/(4*x**8) - A*a**2*d**2/(6*x**6)

- A*a*b*c**2/(4*x**8) - 2*A*a*b*c*d/(3*x**6) - A*a*b*d**2/(2*x**4) - A*b**2*c**

2/(6*x**6) - A*b**2*c*d/(2*x**4) - A*b**2*d**2/(2*x**2) - B*a**2*c**2/(8*x**8) -

B*a**2*c*d/(3*x**6) - B*a**2*d**2/(4*x**4) - B*a*b*c**2/(3*x**6) - B*a*b*c*d/x*

*4 - B*a*b*d**2/x**2 - B*b**2*c**2/(4*x**4) - B*b**2*c*d/x**2 + B*b**2*d**2*log(

x))/e**11, Eq(m, -11)), ((-A*a**2*c**2/(8*x**8) - A*a**2*c*d/(3*x**6) - A*a**2*d

**2/(4*x**4) - A*a*b*c**2/(3*x**6) - A*a*b*c*d/x**4 - A*a*b*d**2/x**2 - A*b**2*c

**2/(4*x**4) - A*b**2*c*d/x**2 + A*b**2*d**2*log(x) - B*a**2*c**2/(6*x**6) - B*a

**2*c*d/(2*x**4) - B*a**2*d**2/(2*x**2) - B*a*b*c**2/(2*x**4) - 2*B*a*b*c*d/x**2

+ 2*B*a*b*d**2*log(x) - B*b**2*c**2/(2*x**2) + 2*B*b**2*c*d*log(x) + B*b**2*d**

2*x**2/2)/e**9, Eq(m, -9)), ((-A*a**2*c**2/(6*x**6) - A*a**2*c*d/(2*x**4) - A*a*

*2*d**2/(2*x**2) - A*a*b*c**2/(2*x**4) - 2*A*a*b*c*d/x**2 + 2*A*a*b*d**2*log(x)

- A*b**2*c**2/(2*x**2) + 2*A*b**2*c*d*log(x) + A*b**2*d**2*x**2/2 - B*a**2*c**2/

(4*x**4) - B*a**2*c*d/x**2 + B*a**2*d**2*log(x) - B*a*b*c**2/x**2 + 4*B*a*b*c*d*

log(x) + B*a*b*d**2*x**2 + B*b**2*c**2*log(x) + B*b**2*c*d*x**2 + B*b**2*d**2*x*

*4/4)/e**7, Eq(m, -7)), ((-A*a**2*c**2/(4*x**4) - A*a**2*c*d/x**2 + A*a**2*d**2*

log(x) - A*a*b*c**2/x**2 + 4*A*a*b*c*d*log(x) + A*a*b*d**2*x**2 + A*b**2*c**2*lo

g(x) + A*b**2*c*d*x**2 + A*b**2*d**2*x**4/4 - B*a**2*c**2/(2*x**2) + 2*B*a**2*c*

d*log(x) + B*a**2*d**2*x**2/2 + 2*B*a*b*c**2*log(x) + 2*B*a*b*c*d*x**2 + B*a*b*d

**2*x**4/2 + B*b**2*c**2*x**2/2 + B*b**2*c*d*x**4/2 + B*b**2*d**2*x**6/6)/e**5,

Eq(m, -5)), ((-A*a**2*c**2/(2*x**2) + 2*A*a**2*c*d*log(x) + A*a**2*d**2*x**2/2 +

2*A*a*b*c**2*log(x) + 2*A*a*b*c*d*x**2 + A*a*b*d**2*x**4/2 + A*b**2*c**2*x**2/2

+ A*b**2*c*d*x**4/2 + A*b**2*d**2*x**6/6 + B*a**2*c**2*log(x) + B*a**2*c*d*x**2

+ B*a**2*d**2*x**4/4 + B*a*b*c**2*x**2 + B*a*b*c*d*x**4 + B*a*b*d**2*x**6/3 + B

*b**2*c**2*x**4/4 + B*b**2*c*d*x**6/3 + B*b**2*d**2*x**8/8)/e**3, Eq(m, -3)), ((

A*a**2*c**2*log(x) + A*a**2*c*d*x**2 + A*a**2*d**2*x**4/4 + A*a*b*c**2*x**2 + A*

a*b*c*d*x**4 + A*a*b*d**2*x**6/3 + A*b**2*c**2*x**4/4 + A*b**2*c*d*x**6/3 + A*b*

*2*d**2*x**8/8 + B*a**2*c**2*x**2/2 + B*a**2*c*d*x**4/2 + B*a**2*d**2*x**6/6 + B

*a*b*c**2*x**4/2 + 2*B*a*b*c*d*x**6/3 + B*a*b*d**2*x**8/4 + B*b**2*c**2*x**6/6 +

B*b**2*c*d*x**8/4 + B*b**2*d**2*x**10/10)/e, Eq(m, -1)), (A*a**2*c**2*e**m*m**5

*x*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) +

35*A*a**2*c**2*e**m*m**4*x*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*

m**2 + 19524*m + 10395) + 470*A*a**2*c**2*e**m*m**3*x*x**m/(m**6 + 36*m**5 + 505

*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 3010*A*a**2*c**2*e**m*m**2*x

*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 9

129*A*a**2*c**2*e**m*m*x*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**

2 + 19524*m + 10395) + 10395*A*a**2*c**2*e**m*x*x**m/(m**6 + 36*m**5 + 505*m**4

+ 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2*A*a**2*c*d*e**m*m**5*x**3*x**m/(

m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 66*A*a**

2*c*d*e**m*m**4*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 +

19524*m + 10395) + 812*A*a**2*c*d*e**m*m**3*x**3*x**m/(m**6 + 36*m**5 + 505*m**4

+ 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 4524*A*a**2*c*d*e**m*m**2*x**3*x*

*m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 1070

6*A*a**2*c*d*e**m*m*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**

2 + 19524*m + 10395) + 6930*A*a**2*c*d*e**m*x**3*x**m/(m**6 + 36*m**5 + 505*m**4

+ 3480*m**3 + 12139*m**2 + 19524*m + 10395) + A*a**2*d**2*e**m*m**5*x**5*x**m/(

m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 31*A*a**

2*d**2*e**m*m**4*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 +

19524*m + 10395) + 350*A*a**2*d**2*e**m*m**3*x**5*x**m/(m**6 + 36*m**5 + 505*m*

*4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 1730*A*a**2*d**2*e**m*m**2*x**5

*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 3

489*A*a**2*d**2*e**m*m*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*

m**2 + 19524*m + 10395) + 2079*A*a**2*d**2*e**m*x**5*x**m/(m**6 + 36*m**5 + 505*

m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2*A*a*b*c**2*e**m*m**5*x**3*x

**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 66*

A*a*b*c**2*e**m*m**4*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m*

*2 + 19524*m + 10395) + 812*A*a*b*c**2*e**m*m**3*x**3*x**m/(m**6 + 36*m**5 + 505

*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 4524*A*a*b*c**2*e**m*m**2*x*

*3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) +

10706*A*a*b*c**2*e**m*m*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 1213

9*m**2 + 19524*m + 10395) + 6930*A*a*b*c**2*e**m*x**3*x**m/(m**6 + 36*m**5 + 505

*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 4*A*a*b*c*d*e**m*m**5*x**5*x

**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 124

*A*a*b*c*d*e**m*m**4*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m*

*2 + 19524*m + 10395) + 1400*A*a*b*c*d*e**m*m**3*x**5*x**m/(m**6 + 36*m**5 + 505

*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 6920*A*a*b*c*d*e**m*m**2*x**

5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) +

13956*A*a*b*c*d*e**m*m*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*

m**2 + 19524*m + 10395) + 8316*A*a*b*c*d*e**m*x**5*x**m/(m**6 + 36*m**5 + 505*m*

*4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2*A*a*b*d**2*e**m*m**5*x**7*x**

m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 58*A*

a*b*d**2*e**m*m**4*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2

+ 19524*m + 10395) + 604*A*a*b*d**2*e**m*m**3*x**7*x**m/(m**6 + 36*m**5 + 505*m

**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2732*A*a*b*d**2*e**m*m**2*x**7

*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 5

154*A*a*b*d**2*e**m*m*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m

**2 + 19524*m + 10395) + 2970*A*a*b*d**2*e**m*x**7*x**m/(m**6 + 36*m**5 + 505*m*

*4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + A*b**2*c**2*e**m*m**5*x**5*x**m

/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 31*A*b

**2*c**2*e**m*m**4*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2

+ 19524*m + 10395) + 350*A*b**2*c**2*e**m*m**3*x**5*x**m/(m**6 + 36*m**5 + 505*

m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 1730*A*b**2*c**2*e**m*m**2*x*

*5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) +

3489*A*b**2*c**2*e**m*m*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 1213

9*m**2 + 19524*m + 10395) + 2079*A*b**2*c**2*e**m*x**5*x**m/(m**6 + 36*m**5 + 50

5*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2*A*b**2*c*d*e**m*m**5*x**7

*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 5

8*A*b**2*c*d*e**m*m**4*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*

m**2 + 19524*m + 10395) + 604*A*b**2*c*d*e**m*m**3*x**7*x**m/(m**6 + 36*m**5 + 5

05*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2732*A*b**2*c*d*e**m*m**2*

x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395)

+ 5154*A*b**2*c*d*e**m*m*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 121

39*m**2 + 19524*m + 10395) + 2970*A*b**2*c*d*e**m*x**7*x**m/(m**6 + 36*m**5 + 50

5*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + A*b**2*d**2*e**m*m**5*x**9*

x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 27

*A*b**2*d**2*e**m*m**4*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*

m**2 + 19524*m + 10395) + 262*A*b**2*d**2*e**m*m**3*x**9*x**m/(m**6 + 36*m**5 +

505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 1122*A*b**2*d**2*e**m*m**

2*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 1039

5) + 2041*A*b**2*d**2*e**m*m*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 +

12139*m**2 + 19524*m + 10395) + 1155*A*b**2*d**2*e**m*x**9*x**m/(m**6 + 36*m**5

+ 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + B*a**2*c**2*e**m*m**5*x

**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395)

+ 33*B*a**2*c**2*e**m*m**4*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12

139*m**2 + 19524*m + 10395) + 406*B*a**2*c**2*e**m*m**3*x**3*x**m/(m**6 + 36*m**

5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2262*B*a**2*c**2*e**m

*m**2*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m +

10395) + 5353*B*a**2*c**2*e**m*m*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**

3 + 12139*m**2 + 19524*m + 10395) + 3465*B*a**2*c**2*e**m*x**3*x**m/(m**6 + 36*m

**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2*B*a**2*c*d*e**m*m

**5*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10

395) + 62*B*a**2*c*d*e**m*m**4*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3

+ 12139*m**2 + 19524*m + 10395) + 700*B*a**2*c*d*e**m*m**3*x**5*x**m/(m**6 + 36*

m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 3460*B*a**2*c*d*e*

*m*m**2*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m

+ 10395) + 6978*B*a**2*c*d*e**m*m*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m*

*3 + 12139*m**2 + 19524*m + 10395) + 4158*B*a**2*c*d*e**m*x**5*x**m/(m**6 + 36*m

**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + B*a**2*d**2*e**m*m*

*5*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 103

95) + 29*B*a**2*d**2*e**m*m**4*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3

+ 12139*m**2 + 19524*m + 10395) + 302*B*a**2*d**2*e**m*m**3*x**7*x**m/(m**6 + 36

*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 1366*B*a**2*d**2*

e**m*m**2*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*

m + 10395) + 2577*B*a**2*d**2*e**m*m*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480

*m**3 + 12139*m**2 + 19524*m + 10395) + 1485*B*a**2*d**2*e**m*x**7*x**m/(m**6 +

36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2*B*a*b*c**2*e*

*m*m**5*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m

+ 10395) + 62*B*a*b*c**2*e**m*m**4*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m

**3 + 12139*m**2 + 19524*m + 10395) + 700*B*a*b*c**2*e**m*m**3*x**5*x**m/(m**6 +

36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 3460*B*a*b*c**

2*e**m*m**2*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 1952

4*m + 10395) + 6978*B*a*b*c**2*e**m*m*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 348

0*m**3 + 12139*m**2 + 19524*m + 10395) + 4158*B*a*b*c**2*e**m*x**5*x**m/(m**6 +

36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 4*B*a*b*c*d*e**

m*m**5*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m +

10395) + 116*B*a*b*c*d*e**m*m**4*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m*

*3 + 12139*m**2 + 19524*m + 10395) + 1208*B*a*b*c*d*e**m*m**3*x**7*x**m/(m**6 +

36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 5464*B*a*b*c*d*

e**m*m**2*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*

m + 10395) + 10308*B*a*b*c*d*e**m*m*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*

m**3 + 12139*m**2 + 19524*m + 10395) + 5940*B*a*b*c*d*e**m*x**7*x**m/(m**6 + 36*

m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2*B*a*b*d**2*e**m*

m**5*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 1

0395) + 54*B*a*b*d**2*e**m*m**4*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3

+ 12139*m**2 + 19524*m + 10395) + 524*B*a*b*d**2*e**m*m**3*x**9*x**m/(m**6 + 36

*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2244*B*a*b*d**2*e

**m*m**2*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m

+ 10395) + 4082*B*a*b*d**2*e**m*m*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m

**3 + 12139*m**2 + 19524*m + 10395) + 2310*B*a*b*d**2*e**m*x**9*x**m/(m**6 + 36*

m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + B*b**2*c**2*e**m*m

**5*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10

395) + 29*B*b**2*c**2*e**m*m**4*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3

+ 12139*m**2 + 19524*m + 10395) + 302*B*b**2*c**2*e**m*m**3*x**7*x**m/(m**6 + 3

6*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 1366*B*b**2*c**2

*e**m*m**2*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524

*m + 10395) + 2577*B*b**2*c**2*e**m*m*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 348

0*m**3 + 12139*m**2 + 19524*m + 10395) + 1485*B*b**2*c**2*e**m*x**7*x**m/(m**6 +

36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2*B*b**2*c*d*e

**m*m**5*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m

+ 10395) + 54*B*b**2*c*d*e**m*m**4*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*

m**3 + 12139*m**2 + 19524*m + 10395) + 524*B*b**2*c*d*e**m*m**3*x**9*x**m/(m**6

+ 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2244*B*b**2*c

*d*e**m*m**2*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 195

24*m + 10395) + 4082*B*b**2*c*d*e**m*m*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 34

80*m**3 + 12139*m**2 + 19524*m + 10395) + 2310*B*b**2*c*d*e**m*x**9*x**m/(m**6 +

36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + B*b**2*d**2*e*

*m*m**5*x**11*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m

+ 10395) + 25*B*b**2*d**2*e**m*m**4*x**11*x**m/(m**6 + 36*m**5 + 505*m**4 + 348

0*m**3 + 12139*m**2 + 19524*m + 10395) + 230*B*b**2*d**2*e**m*m**3*x**11*x**m/(m

**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 950*B*b**

2*d**2*e**m*m**2*x**11*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2

+ 19524*m + 10395) + 1689*B*b**2*d**2*e**m*m*x**11*x**m/(m**6 + 36*m**5 + 505*m*

*4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 945*B*b**2*d**2*e**m*x**11*x**m

/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395), True))

_______________________________________________________________________________________

GIAC/XCAS [A] time = 0.228532, size = 1, normalized size = 0. \[ \mathit{Done} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

Done

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