3.319 \(\int x^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx\)

Optimal. Leaf size=96 \[ \frac{a^3 A x^{m+1}}{m+1}+\frac{a^2 x^{m+3} (a B+3 A b)}{m+3}+\frac{b^2 x^{m+7} (3 a B+A b)}{m+7}+\frac{3 a b x^{m+5} (a B+A b)}{m+5}+\frac{b^3 B x^{m+9}}{m+9} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.157444, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{a^3 A x^{m+1}}{m+1}+\frac{a^2 x^{m+3} (a B+3 A b)}{m+3}+\frac{b^2 x^{m+7} (3 a B+A b)}{m+7}+\frac{3 a b x^{m+5} (a B+A b)}{m+5}+\frac{b^3 B x^{m+9}}{m+9} \]

Antiderivative was successfully verified.

[In]  Int[x^m*(a + b*x^2)^3*(A + B*x^2),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 20.949, size = 87, normalized size = 0.91 \[ \frac{A a^{3} x^{m + 1}}{m + 1} + \frac{B b^{3} x^{m + 9}}{m + 9} + \frac{a^{2} x^{m + 3} \left (3 A b + B a\right )}{m + 3} + \frac{3 a b x^{m + 5} \left (A b + B a\right )}{m + 5} + \frac{b^{2} x^{m + 7} \left (A b + 3 B a\right )}{m + 7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m*(b*x**2+a)**3*(B*x**2+A),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.119591, size = 88, normalized size = 0.92 \[ x^m \left (\frac{a^3 A x}{m+1}+\frac{a^2 x^3 (a B+3 A b)}{m+3}+\frac{b^2 x^7 (3 a B+A b)}{m+7}+\frac{3 a b x^5 (a B+A b)}{m+5}+\frac{b^3 B x^9}{m+9}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^m*(a + b*x^2)^3*(A + B*x^2),x]

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.01, size = 474, normalized size = 4.9 \[{\frac{{x}^{1+m} \left ( B{b}^{3}{m}^{4}{x}^{8}+16\,B{b}^{3}{m}^{3}{x}^{8}+A{b}^{3}{m}^{4}{x}^{6}+3\,Ba{b}^{2}{m}^{4}{x}^{6}+86\,B{b}^{3}{m}^{2}{x}^{8}+18\,A{b}^{3}{m}^{3}{x}^{6}+54\,Ba{b}^{2}{m}^{3}{x}^{6}+176\,B{b}^{3}m{x}^{8}+3\,Aa{b}^{2}{m}^{4}{x}^{4}+104\,A{b}^{3}{m}^{2}{x}^{6}+3\,B{a}^{2}b{m}^{4}{x}^{4}+312\,Ba{b}^{2}{m}^{2}{x}^{6}+105\,B{b}^{3}{x}^{8}+60\,Aa{b}^{2}{m}^{3}{x}^{4}+222\,A{b}^{3}m{x}^{6}+60\,B{a}^{2}b{m}^{3}{x}^{4}+666\,Ba{b}^{2}m{x}^{6}+3\,A{a}^{2}b{m}^{4}{x}^{2}+390\,Aa{b}^{2}{m}^{2}{x}^{4}+135\,A{b}^{3}{x}^{6}+B{a}^{3}{m}^{4}{x}^{2}+390\,B{a}^{2}b{m}^{2}{x}^{4}+405\,Ba{b}^{2}{x}^{6}+66\,A{a}^{2}b{m}^{3}{x}^{2}+900\,Aa{b}^{2}m{x}^{4}+22\,B{a}^{3}{m}^{3}{x}^{2}+900\,B{a}^{2}bm{x}^{4}+A{a}^{3}{m}^{4}+492\,A{a}^{2}b{m}^{2}{x}^{2}+567\,Aa{b}^{2}{x}^{4}+164\,B{a}^{3}{m}^{2}{x}^{2}+567\,B{a}^{2}b{x}^{4}+24\,A{a}^{3}{m}^{3}+1374\,A{a}^{2}bm{x}^{2}+458\,B{a}^{3}m{x}^{2}+206\,A{a}^{3}{m}^{2}+945\,A{a}^{2}b{x}^{2}+315\,B{a}^{3}{x}^{2}+744\,A{a}^{3}m+945\,A{a}^{3} \right ) }{ \left ( 9+m \right ) \left ( 7+m \right ) \left ( 5+m \right ) \left ( 3+m \right ) \left ( 1+m \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(b*x^2+a)^3*(B*x^2+A),x)

[Out]

x^(1+m)*(B*b^3*m^4*x^8+16*B*b^3*m^3*x^8+A*b^3*m^4*x^6+3*B*a*b^2*m^4*x^6+86*B*b^3
*m^2*x^8+18*A*b^3*m^3*x^6+54*B*a*b^2*m^3*x^6+176*B*b^3*m*x^8+3*A*a*b^2*m^4*x^4+1
04*A*b^3*m^2*x^6+3*B*a^2*b*m^4*x^4+312*B*a*b^2*m^2*x^6+105*B*b^3*x^8+60*A*a*b^2*
m^3*x^4+222*A*b^3*m*x^6+60*B*a^2*b*m^3*x^4+666*B*a*b^2*m*x^6+3*A*a^2*b*m^4*x^2+3
90*A*a*b^2*m^2*x^4+135*A*b^3*x^6+B*a^3*m^4*x^2+390*B*a^2*b*m^2*x^4+405*B*a*b^2*x
^6+66*A*a^2*b*m^3*x^2+900*A*a*b^2*m*x^4+22*B*a^3*m^3*x^2+900*B*a^2*b*m*x^4+A*a^3
*m^4+492*A*a^2*b*m^2*x^2+567*A*a*b^2*x^4+164*B*a^3*m^2*x^2+567*B*a^2*b*x^4+24*A*
a^3*m^3+1374*A*a^2*b*m*x^2+458*B*a^3*m*x^2+206*A*a^3*m^2+945*A*a^2*b*x^2+315*B*a
^3*x^2+744*A*a^3*m+945*A*a^3)/(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)*(b*x^2 + a)^3*x^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.237212, size = 512, normalized size = 5.33 \[ \frac{{\left ({\left (B b^{3} m^{4} + 16 \, B b^{3} m^{3} + 86 \, B b^{3} m^{2} + 176 \, B b^{3} m + 105 \, B b^{3}\right )} x^{9} +{\left ({\left (3 \, B a b^{2} + A b^{3}\right )} m^{4} + 405 \, B a b^{2} + 135 \, A b^{3} + 18 \,{\left (3 \, B a b^{2} + A b^{3}\right )} m^{3} + 104 \,{\left (3 \, B a b^{2} + A b^{3}\right )} m^{2} + 222 \,{\left (3 \, B a b^{2} + A b^{3}\right )} m\right )} x^{7} + 3 \,{\left ({\left (B a^{2} b + A a b^{2}\right )} m^{4} + 189 \, B a^{2} b + 189 \, A a b^{2} + 20 \,{\left (B a^{2} b + A a b^{2}\right )} m^{3} + 130 \,{\left (B a^{2} b + A a b^{2}\right )} m^{2} + 300 \,{\left (B a^{2} b + A a b^{2}\right )} m\right )} x^{5} +{\left ({\left (B a^{3} + 3 \, A a^{2} b\right )} m^{4} + 315 \, B a^{3} + 945 \, A a^{2} b + 22 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} m^{3} + 164 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} m^{2} + 458 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} m\right )} x^{3} +{\left (A a^{3} m^{4} + 24 \, A a^{3} m^{3} + 206 \, A a^{3} m^{2} + 744 \, A a^{3} m + 945 \, A a^{3}\right )} x\right )} x^{m}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 9.57639, size = 2069, normalized size = 21.55 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**m*(b*x**2+a)**3*(B*x**2+A),x)

[Out]

Piecewise((-A*a**3/(8*x**8) - A*a**2*b/(2*x**6) - 3*A*a*b**2/(4*x**4) - A*b**3/(
2*x**2) - B*a**3/(6*x**6) - 3*B*a**2*b/(4*x**4) - 3*B*a*b**2/(2*x**2) + B*b**3*l
og(x), Eq(m, -9)), (-A*a**3/(6*x**6) - 3*A*a**2*b/(4*x**4) - 3*A*a*b**2/(2*x**2)
+ A*b**3*log(x) - B*a**3/(4*x**4) - 3*B*a**2*b/(2*x**2) + 3*B*a*b**2*log(x) + B
*b**3*x**2/2, Eq(m, -7)), (-A*a**3/(4*x**4) - 3*A*a**2*b/(2*x**2) + 3*A*a*b**2*l
og(x) + A*b**3*x**2/2 - B*a**3/(2*x**2) + 3*B*a**2*b*log(x) + 3*B*a*b**2*x**2/2
+ B*b**3*x**4/4, Eq(m, -5)), (-A*a**3/(2*x**2) + 3*A*a**2*b*log(x) + 3*A*a*b**2*
x**2/2 + A*b**3*x**4/4 + B*a**3*log(x) + 3*B*a**2*b*x**2/2 + 3*B*a*b**2*x**4/4 +
B*b**3*x**6/6, Eq(m, -3)), (A*a**3*log(x) + 3*A*a**2*b*x**2/2 + 3*A*a*b**2*x**4
/4 + A*b**3*x**6/6 + B*a**3*x**2/2 + 3*B*a**2*b*x**4/4 + B*a*b**2*x**6/2 + B*b**
3*x**8/8, Eq(m, -1)), (A*a**3*m**4*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2
+ 1689*m + 945) + 24*A*a**3*m**3*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 +
1689*m + 945) + 206*A*a**3*m**2*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1
689*m + 945) + 744*A*a**3*m*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*
m + 945) + 945*A*a**3*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 94
5) + 3*A*a**2*b*m**4*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m +
945) + 66*A*a**2*b*m**3*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m
+ 945) + 492*A*a**2*b*m**2*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 16
89*m + 945) + 1374*A*a**2*b*m*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 +
1689*m + 945) + 945*A*a**2*b*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1
689*m + 945) + 3*A*a*b**2*m**4*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 +
1689*m + 945) + 60*A*a*b**2*m**3*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**
2 + 1689*m + 945) + 390*A*a*b**2*m**2*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950
*m**2 + 1689*m + 945) + 900*A*a*b**2*m*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 95
0*m**2 + 1689*m + 945) + 567*A*a*b**2*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950
*m**2 + 1689*m + 945) + A*b**3*m**4*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m
**2 + 1689*m + 945) + 18*A*b**3*m**3*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*
m**2 + 1689*m + 945) + 104*A*b**3*m**2*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 95
0*m**2 + 1689*m + 945) + 222*A*b**3*m*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950
*m**2 + 1689*m + 945) + 135*A*b**3*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m*
*2 + 1689*m + 945) + B*a**3*m**4*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2
+ 1689*m + 945) + 22*B*a**3*m**3*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**
2 + 1689*m + 945) + 164*B*a**3*m**2*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m
**2 + 1689*m + 945) + 458*B*a**3*m*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m*
*2 + 1689*m + 945) + 315*B*a**3*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2
+ 1689*m + 945) + 3*B*a**2*b*m**4*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**
2 + 1689*m + 945) + 60*B*a**2*b*m**3*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*
m**2 + 1689*m + 945) + 390*B*a**2*b*m**2*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 +
950*m**2 + 1689*m + 945) + 900*B*a**2*b*m*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 +
950*m**2 + 1689*m + 945) + 567*B*a**2*b*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 +
950*m**2 + 1689*m + 945) + 3*B*a*b**2*m**4*x**7*x**m/(m**5 + 25*m**4 + 230*m**3
+ 950*m**2 + 1689*m + 945) + 54*B*a*b**2*m**3*x**7*x**m/(m**5 + 25*m**4 + 230*m*
*3 + 950*m**2 + 1689*m + 945) + 312*B*a*b**2*m**2*x**7*x**m/(m**5 + 25*m**4 + 23
0*m**3 + 950*m**2 + 1689*m + 945) + 666*B*a*b**2*m*x**7*x**m/(m**5 + 25*m**4 + 2
30*m**3 + 950*m**2 + 1689*m + 945) + 405*B*a*b**2*x**7*x**m/(m**5 + 25*m**4 + 23
0*m**3 + 950*m**2 + 1689*m + 945) + B*b**3*m**4*x**9*x**m/(m**5 + 25*m**4 + 230*
m**3 + 950*m**2 + 1689*m + 945) + 16*B*b**3*m**3*x**9*x**m/(m**5 + 25*m**4 + 230
*m**3 + 950*m**2 + 1689*m + 945) + 86*B*b**3*m**2*x**9*x**m/(m**5 + 25*m**4 + 23
0*m**3 + 950*m**2 + 1689*m + 945) + 176*B*b**3*m*x**9*x**m/(m**5 + 25*m**4 + 230
*m**3 + 950*m**2 + 1689*m + 945) + 105*B*b**3*x**9*x**m/(m**5 + 25*m**4 + 230*m*
*3 + 950*m**2 + 1689*m + 945), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.271332, size = 909, normalized size = 9.47 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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