3.18 \(\int (e x)^m \left (A+B x^2\right ) \left (c+d x^2\right )^3 \, dx\)

Optimal. Leaf size=121 \[ \frac{c^2 (e x)^{m+3} (3 A d+B c)}{e^3 (m+3)}+\frac{d^2 (e x)^{m+7} (A d+3 B c)}{e^7 (m+7)}+\frac{3 c d (e x)^{m+5} (A d+B c)}{e^5 (m+5)}+\frac{A c^3 (e x)^{m+1}}{e (m+1)}+\frac{B d^3 (e x)^{m+9}}{e^9 (m+9)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.230133, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{c^2 (e x)^{m+3} (3 A d+B c)}{e^3 (m+3)}+\frac{d^2 (e x)^{m+7} (A d+3 B c)}{e^7 (m+7)}+\frac{3 c d (e x)^{m+5} (A d+B c)}{e^5 (m+5)}+\frac{A c^3 (e x)^{m+1}}{e (m+1)}+\frac{B d^3 (e x)^{m+9}}{e^9 (m+9)} \]

Antiderivative was successfully verified.

[In]  Int[(e*x)^m*(A + B*x^2)*(c + d*x^2)^3,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 31.1922, size = 110, normalized size = 0.91 \[ \frac{A c^{3} \left (e x\right )^{m + 1}}{e \left (m + 1\right )} + \frac{B d^{3} \left (e x\right )^{m + 9}}{e^{9} \left (m + 9\right )} + \frac{c^{2} \left (e x\right )^{m + 3} \left (3 A d + B c\right )}{e^{3} \left (m + 3\right )} + \frac{3 c d \left (e x\right )^{m + 5} \left (A d + B c\right )}{e^{5} \left (m + 5\right )} + \frac{d^{2} \left (e x\right )^{m + 7} \left (A d + 3 B c\right )}{e^{7} \left (m + 7\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**m*(B*x**2+A)*(d*x**2+c)**3,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.10516, size = 90, normalized size = 0.74 \[ (e x)^m \left (\frac{c^2 x^3 (3 A d+B c)}{m+3}+\frac{d^2 x^7 (A d+3 B c)}{m+7}+\frac{3 c d x^5 (A d+B c)}{m+5}+\frac{A c^3 x}{m+1}+\frac{B d^3 x^9}{m+9}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(e*x)^m*(A + B*x^2)*(c + d*x^2)^3,x]

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.01, size = 475, normalized size = 3.9 \[{\frac{ \left ( B{d}^{3}{m}^{4}{x}^{8}+16\,B{d}^{3}{m}^{3}{x}^{8}+A{d}^{3}{m}^{4}{x}^{6}+3\,Bc{d}^{2}{m}^{4}{x}^{6}+86\,B{d}^{3}{m}^{2}{x}^{8}+18\,A{d}^{3}{m}^{3}{x}^{6}+54\,Bc{d}^{2}{m}^{3}{x}^{6}+176\,B{d}^{3}m{x}^{8}+3\,Ac{d}^{2}{m}^{4}{x}^{4}+104\,A{d}^{3}{m}^{2}{x}^{6}+3\,B{c}^{2}d{m}^{4}{x}^{4}+312\,Bc{d}^{2}{m}^{2}{x}^{6}+105\,B{d}^{3}{x}^{8}+60\,Ac{d}^{2}{m}^{3}{x}^{4}+222\,A{d}^{3}m{x}^{6}+60\,B{c}^{2}d{m}^{3}{x}^{4}+666\,Bc{d}^{2}m{x}^{6}+3\,A{c}^{2}d{m}^{4}{x}^{2}+390\,Ac{d}^{2}{m}^{2}{x}^{4}+135\,A{d}^{3}{x}^{6}+B{c}^{3}{m}^{4}{x}^{2}+390\,B{c}^{2}d{m}^{2}{x}^{4}+405\,Bc{d}^{2}{x}^{6}+66\,A{c}^{2}d{m}^{3}{x}^{2}+900\,Ac{d}^{2}m{x}^{4}+22\,B{c}^{3}{m}^{3}{x}^{2}+900\,B{c}^{2}dm{x}^{4}+A{c}^{3}{m}^{4}+492\,A{c}^{2}d{m}^{2}{x}^{2}+567\,Ac{d}^{2}{x}^{4}+164\,B{c}^{3}{m}^{2}{x}^{2}+567\,B{c}^{2}d{x}^{4}+24\,A{c}^{3}{m}^{3}+1374\,A{c}^{2}dm{x}^{2}+458\,B{c}^{3}m{x}^{2}+206\,A{c}^{3}{m}^{2}+945\,A{c}^{2}d{x}^{2}+315\,B{c}^{3}{x}^{2}+744\,A{c}^{3}m+945\,A{c}^{3} \right ) x \left ( ex \right ) ^{m}}{ \left ( 9+m \right ) \left ( 7+m \right ) \left ( 5+m \right ) \left ( 3+m \right ) \left ( 1+m \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^m*(B*x^2+A)*(d*x^2+c)^3,x)

[Out]

x*(B*d^3*m^4*x^8+16*B*d^3*m^3*x^8+A*d^3*m^4*x^6+3*B*c*d^2*m^4*x^6+86*B*d^3*m^2*x
^8+18*A*d^3*m^3*x^6+54*B*c*d^2*m^3*x^6+176*B*d^3*m*x^8+3*A*c*d^2*m^4*x^4+104*A*d
^3*m^2*x^6+3*B*c^2*d*m^4*x^4+312*B*c*d^2*m^2*x^6+105*B*d^3*x^8+60*A*c*d^2*m^3*x^
4+222*A*d^3*m*x^6+60*B*c^2*d*m^3*x^4+666*B*c*d^2*m*x^6+3*A*c^2*d*m^4*x^2+390*A*c
*d^2*m^2*x^4+135*A*d^3*x^6+B*c^3*m^4*x^2+390*B*c^2*d*m^2*x^4+405*B*c*d^2*x^6+66*
A*c^2*d*m^3*x^2+900*A*c*d^2*m*x^4+22*B*c^3*m^3*x^2+900*B*c^2*d*m*x^4+A*c^3*m^4+4
92*A*c^2*d*m^2*x^2+567*A*c*d^2*x^4+164*B*c^3*m^2*x^2+567*B*c^2*d*x^4+24*A*c^3*m^
3+1374*A*c^2*d*m*x^2+458*B*c^3*m*x^2+206*A*c^3*m^2+945*A*c^2*d*x^2+315*B*c^3*x^2
+744*A*c^3*m+945*A*c^3)*(e*x)^m/(9+m)/(7+m)/(5+m)/(3+m)/(1+m)

_______________________________________________________________________________________

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)*(d*x^2 + c)^3*(e*x)^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.260655, size = 514, normalized size = 4.25 \[ \frac{{\left ({\left (B d^{3} m^{4} + 16 \, B d^{3} m^{3} + 86 \, B d^{3} m^{2} + 176 \, B d^{3} m + 105 \, B d^{3}\right )} x^{9} +{\left ({\left (3 \, B c d^{2} + A d^{3}\right )} m^{4} + 405 \, B c d^{2} + 135 \, A d^{3} + 18 \,{\left (3 \, B c d^{2} + A d^{3}\right )} m^{3} + 104 \,{\left (3 \, B c d^{2} + A d^{3}\right )} m^{2} + 222 \,{\left (3 \, B c d^{2} + A d^{3}\right )} m\right )} x^{7} + 3 \,{\left ({\left (B c^{2} d + A c d^{2}\right )} m^{4} + 189 \, B c^{2} d + 189 \, A c d^{2} + 20 \,{\left (B c^{2} d + A c d^{2}\right )} m^{3} + 130 \,{\left (B c^{2} d + A c d^{2}\right )} m^{2} + 300 \,{\left (B c^{2} d + A c d^{2}\right )} m\right )} x^{5} +{\left ({\left (B c^{3} + 3 \, A c^{2} d\right )} m^{4} + 315 \, B c^{3} + 945 \, A c^{2} d + 22 \,{\left (B c^{3} + 3 \, A c^{2} d\right )} m^{3} + 164 \,{\left (B c^{3} + 3 \, A c^{2} d\right )} m^{2} + 458 \,{\left (B c^{3} + 3 \, A c^{2} d\right )} m\right )} x^{3} +{\left (A c^{3} m^{4} + 24 \, A c^{3} m^{3} + 206 \, A c^{3} m^{2} + 744 \, A c^{3} m + 945 \, A c^{3}\right )} x\right )} \left (e x\right )^{m}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(d*x^2 + c)^3*(e*x)^m,x, algorithm="fricas")

[Out]

((B*d^3*m^4 + 16*B*d^3*m^3 + 86*B*d^3*m^2 + 176*B*d^3*m + 105*B*d^3)*x^9 + ((3*B
*c*d^2 + A*d^3)*m^4 + 405*B*c*d^2 + 135*A*d^3 + 18*(3*B*c*d^2 + A*d^3)*m^3 + 104
*(3*B*c*d^2 + A*d^3)*m^2 + 222*(3*B*c*d^2 + A*d^3)*m)*x^7 + 3*((B*c^2*d + A*c*d^
2)*m^4 + 189*B*c^2*d + 189*A*c*d^2 + 20*(B*c^2*d + A*c*d^2)*m^3 + 130*(B*c^2*d +
A*c*d^2)*m^2 + 300*(B*c^2*d + A*c*d^2)*m)*x^5 + ((B*c^3 + 3*A*c^2*d)*m^4 + 315*
B*c^3 + 945*A*c^2*d + 22*(B*c^3 + 3*A*c^2*d)*m^3 + 164*(B*c^3 + 3*A*c^2*d)*m^2 +
458*(B*c^3 + 3*A*c^2*d)*m)*x^3 + (A*c^3*m^4 + 24*A*c^3*m^3 + 206*A*c^3*m^2 + 74
4*A*c^3*m + 945*A*c^3)*x)*(e*x)^m/(m^5 + 25*m^4 + 230*m^3 + 950*m^2 + 1689*m + 9
45)

_______________________________________________________________________________________

Sympy [A]  time = 4.83388, size = 2220, normalized size = 18.35 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)**m*(B*x**2+A)*(d*x**2+c)**3,x)

[Out]

Piecewise(((-A*c**3/(8*x**8) - A*c**2*d/(2*x**6) - 3*A*c*d**2/(4*x**4) - A*d**3/
(2*x**2) - B*c**3/(6*x**6) - 3*B*c**2*d/(4*x**4) - 3*B*c*d**2/(2*x**2) + B*d**3*
log(x))/e**9, Eq(m, -9)), ((-A*c**3/(6*x**6) - 3*A*c**2*d/(4*x**4) - 3*A*c*d**2/
(2*x**2) + A*d**3*log(x) - B*c**3/(4*x**4) - 3*B*c**2*d/(2*x**2) + 3*B*c*d**2*lo
g(x) + B*d**3*x**2/2)/e**7, Eq(m, -7)), ((-A*c**3/(4*x**4) - 3*A*c**2*d/(2*x**2)
+ 3*A*c*d**2*log(x) + A*d**3*x**2/2 - B*c**3/(2*x**2) + 3*B*c**2*d*log(x) + 3*B
*c*d**2*x**2/2 + B*d**3*x**4/4)/e**5, Eq(m, -5)), ((-A*c**3/(2*x**2) + 3*A*c**2*
d*log(x) + 3*A*c*d**2*x**2/2 + A*d**3*x**4/4 + B*c**3*log(x) + 3*B*c**2*d*x**2/2
+ 3*B*c*d**2*x**4/4 + B*d**3*x**6/6)/e**3, Eq(m, -3)), ((A*c**3*log(x) + 3*A*c*
*2*d*x**2/2 + 3*A*c*d**2*x**4/4 + A*d**3*x**6/6 + B*c**3*x**2/2 + 3*B*c**2*d*x**
4/4 + B*c*d**2*x**6/2 + B*d**3*x**8/8)/e, Eq(m, -1)), (A*c**3*e**m*m**4*x*x**m/(
m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 24*A*c**3*e**m*m**3*x*x**
m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 206*A*c**3*e**m*m**2*x
*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 744*A*c**3*e**m*m*
x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 945*A*c**3*e**m*x
*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 3*A*c**2*d*e**m*m*
*4*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 66*A*c**2*d
*e**m*m**3*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 492
*A*c**2*d*e**m*m**2*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 9
45) + 1374*A*c**2*d*e**m*m*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 168
9*m + 945) + 945*A*c**2*d*e**m*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 +
1689*m + 945) + 3*A*c*d**2*e**m*m**4*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950
*m**2 + 1689*m + 945) + 60*A*c*d**2*e**m*m**3*x**5*x**m/(m**5 + 25*m**4 + 230*m*
*3 + 950*m**2 + 1689*m + 945) + 390*A*c*d**2*e**m*m**2*x**5*x**m/(m**5 + 25*m**4
+ 230*m**3 + 950*m**2 + 1689*m + 945) + 900*A*c*d**2*e**m*m*x**5*x**m/(m**5 + 2
5*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 567*A*c*d**2*e**m*x**5*x**m/(m**5
+ 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + A*d**3*e**m*m**4*x**7*x**m/(m
**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 18*A*d**3*e**m*m**3*x**7*x
**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 104*A*d**3*e**m*m**2
*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 222*A*d**3*e*
*m*m*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 135*A*d**
3*e**m*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + B*c**3*
e**m*m**4*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 22*B
*c**3*e**m*m**3*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945)
+ 164*B*c**3*e**m*m**2*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m
+ 945) + 458*B*c**3*e**m*m*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 168
9*m + 945) + 315*B*c**3*e**m*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1
689*m + 945) + 3*B*c**2*d*e**m*m**4*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m
**2 + 1689*m + 945) + 60*B*c**2*d*e**m*m**3*x**5*x**m/(m**5 + 25*m**4 + 230*m**3
+ 950*m**2 + 1689*m + 945) + 390*B*c**2*d*e**m*m**2*x**5*x**m/(m**5 + 25*m**4 +
230*m**3 + 950*m**2 + 1689*m + 945) + 900*B*c**2*d*e**m*m*x**5*x**m/(m**5 + 25*
m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 567*B*c**2*d*e**m*x**5*x**m/(m**5 +
25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 3*B*c*d**2*e**m*m**4*x**7*x**m/
(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 54*B*c*d**2*e**m*m**3*x*
*7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 312*B*c*d**2*e**
m*m**2*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 666*B*c
*d**2*e**m*m*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 4
05*B*c*d**2*e**m*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945)
+ B*d**3*e**m*m**4*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 9
45) + 16*B*d**3*e**m*m**3*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689
*m + 945) + 86*B*d**3*e**m*m**2*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2
+ 1689*m + 945) + 176*B*d**3*e**m*m*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m
**2 + 1689*m + 945) + 105*B*d**3*e**m*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 + 950
*m**2 + 1689*m + 945), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.24371, size = 1017, normalized size = 8.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(d*x^2 + c)^3*(e*x)^m,x, algorithm="giac")

[Out]

(B*d^3*m^4*x^9*e^(m*ln(x) + m) + 16*B*d^3*m^3*x^9*e^(m*ln(x) + m) + 3*B*c*d^2*m^
4*x^7*e^(m*ln(x) + m) + A*d^3*m^4*x^7*e^(m*ln(x) + m) + 86*B*d^3*m^2*x^9*e^(m*ln
(x) + m) + 54*B*c*d^2*m^3*x^7*e^(m*ln(x) + m) + 18*A*d^3*m^3*x^7*e^(m*ln(x) + m)
+ 176*B*d^3*m*x^9*e^(m*ln(x) + m) + 3*B*c^2*d*m^4*x^5*e^(m*ln(x) + m) + 3*A*c*d
^2*m^4*x^5*e^(m*ln(x) + m) + 312*B*c*d^2*m^2*x^7*e^(m*ln(x) + m) + 104*A*d^3*m^2
*x^7*e^(m*ln(x) + m) + 105*B*d^3*x^9*e^(m*ln(x) + m) + 60*B*c^2*d*m^3*x^5*e^(m*l
n(x) + m) + 60*A*c*d^2*m^3*x^5*e^(m*ln(x) + m) + 666*B*c*d^2*m*x^7*e^(m*ln(x) +
m) + 222*A*d^3*m*x^7*e^(m*ln(x) + m) + B*c^3*m^4*x^3*e^(m*ln(x) + m) + 3*A*c^2*d
*m^4*x^3*e^(m*ln(x) + m) + 390*B*c^2*d*m^2*x^5*e^(m*ln(x) + m) + 390*A*c*d^2*m^2
*x^5*e^(m*ln(x) + m) + 405*B*c*d^2*x^7*e^(m*ln(x) + m) + 135*A*d^3*x^7*e^(m*ln(x
) + m) + 22*B*c^3*m^3*x^3*e^(m*ln(x) + m) + 66*A*c^2*d*m^3*x^3*e^(m*ln(x) + m) +
900*B*c^2*d*m*x^5*e^(m*ln(x) + m) + 900*A*c*d^2*m*x^5*e^(m*ln(x) + m) + A*c^3*m
^4*x*e^(m*ln(x) + m) + 164*B*c^3*m^2*x^3*e^(m*ln(x) + m) + 492*A*c^2*d*m^2*x^3*e
^(m*ln(x) + m) + 567*B*c^2*d*x^5*e^(m*ln(x) + m) + 567*A*c*d^2*x^5*e^(m*ln(x) +
m) + 24*A*c^3*m^3*x*e^(m*ln(x) + m) + 458*B*c^3*m*x^3*e^(m*ln(x) + m) + 1374*A*c
^2*d*m*x^3*e^(m*ln(x) + m) + 206*A*c^3*m^2*x*e^(m*ln(x) + m) + 315*B*c^3*x^3*e^(
m*ln(x) + m) + 945*A*c^2*d*x^3*e^(m*ln(x) + m) + 744*A*c^3*m*x*e^(m*ln(x) + m) +
945*A*c^3*x*e^(m*ln(x) + m))/(m^5 + 25*m^4 + 230*m^3 + 950*m^2 + 1689*m + 945)