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

Optimal. Leaf size=71 \[ \frac{a^2 A x^{m+1}}{m+1}+\frac{a x^{m+3} (a B+2 A b)}{m+3}+\frac{b x^{m+5} (2 a B+A b)}{m+5}+\frac{b^2 B x^{m+7}}{m+7} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.11568, antiderivative size = 71, 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 \[ \frac{a^2 A x^{m+1}}{m+1}+\frac{a x^{m+3} (a B+2 A b)}{m+3}+\frac{b x^{m+5} (2 a B+A b)}{m+5}+\frac{b^2 B x^{m+7}}{m+7} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 15.9866, size = 63, normalized size = 0.89 \[ \frac{A a^{2} x^{m + 1}}{m + 1} + \frac{B b^{2} x^{m + 7}}{m + 7} + \frac{a x^{m + 3} \left (2 A b + B a\right )}{m + 3} + \frac{b x^{m + 5} \left (A b + 2 B a\right )}{m + 5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0755131, size = 65, normalized size = 0.92 \[ x^m \left (\frac{a^2 A x}{m+1}+\frac{b x^5 (2 a B+A b)}{m+5}+\frac{a x^3 (a B+2 A b)}{m+3}+\frac{b^2 B x^7}{m+7}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.009, size = 262, normalized size = 3.7 \[{\frac{{x}^{1+m} \left ( B{b}^{2}{m}^{3}{x}^{6}+9\,B{b}^{2}{m}^{2}{x}^{6}+A{b}^{2}{m}^{3}{x}^{4}+2\,Bab{m}^{3}{x}^{4}+23\,B{b}^{2}m{x}^{6}+11\,A{b}^{2}{m}^{2}{x}^{4}+22\,Bab{m}^{2}{x}^{4}+15\,{b}^{2}B{x}^{6}+2\,Aab{m}^{3}{x}^{2}+31\,A{b}^{2}m{x}^{4}+B{a}^{2}{m}^{3}{x}^{2}+62\,Babm{x}^{4}+26\,Aab{m}^{2}{x}^{2}+21\,A{b}^{2}{x}^{4}+13\,B{a}^{2}{m}^{2}{x}^{2}+42\,Bab{x}^{4}+A{a}^{2}{m}^{3}+94\,Aabm{x}^{2}+47\,B{a}^{2}m{x}^{2}+15\,A{a}^{2}{m}^{2}+70\,aAb{x}^{2}+35\,B{a}^{2}{x}^{2}+71\,A{a}^{2}m+105\,{a}^{2}A \right ) }{ \left ( 7+m \right ) \left ( 5+m \right ) \left ( 3+m \right ) \left ( 1+m \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x^(1+m)*(B*b^2*m^3*x^6+9*B*b^2*m^2*x^6+A*b^2*m^3*x^4+2*B*a*b*m^3*x^4+23*B*b^2*m*
x^6+11*A*b^2*m^2*x^4+22*B*a*b*m^2*x^4+15*B*b^2*x^6+2*A*a*b*m^3*x^2+31*A*b^2*m*x^
4+B*a^2*m^3*x^2+62*B*a*b*m*x^4+26*A*a*b*m^2*x^2+21*A*b^2*x^4+13*B*a^2*m^2*x^2+42
*B*a*b*x^4+A*a^2*m^3+94*A*a*b*m*x^2+47*B*a^2*m*x^2+15*A*a^2*m^2+70*A*a*b*x^2+35*
B*a^2*x^2+71*A*a^2*m+105*A*a^2)/(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)^2*x^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.23495, size = 290, normalized size = 4.08 \[ \frac{{\left ({\left (B b^{2} m^{3} + 9 \, B b^{2} m^{2} + 23 \, B b^{2} m + 15 \, B b^{2}\right )} x^{7} +{\left ({\left (2 \, B a b + A b^{2}\right )} m^{3} + 42 \, B a b + 21 \, A b^{2} + 11 \,{\left (2 \, B a b + A b^{2}\right )} m^{2} + 31 \,{\left (2 \, B a b + A b^{2}\right )} m\right )} x^{5} +{\left ({\left (B a^{2} + 2 \, A a b\right )} m^{3} + 35 \, B a^{2} + 70 \, A a b + 13 \,{\left (B a^{2} + 2 \, A a b\right )} m^{2} + 47 \,{\left (B a^{2} + 2 \, A a b\right )} m\right )} x^{3} +{\left (A a^{2} m^{3} + 15 \, A a^{2} m^{2} + 71 \, A a^{2} m + 105 \, A a^{2}\right )} x\right )} x^{m}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((B*b^2*m^3 + 9*B*b^2*m^2 + 23*B*b^2*m + 15*B*b^2)*x^7 + ((2*B*a*b + A*b^2)*m^3
+ 42*B*a*b + 21*A*b^2 + 11*(2*B*a*b + A*b^2)*m^2 + 31*(2*B*a*b + A*b^2)*m)*x^5 +
((B*a^2 + 2*A*a*b)*m^3 + 35*B*a^2 + 70*A*a*b + 13*(B*a^2 + 2*A*a*b)*m^2 + 47*(B
*a^2 + 2*A*a*b)*m)*x^3 + (A*a^2*m^3 + 15*A*a^2*m^2 + 71*A*a^2*m + 105*A*a^2)*x)*
x^m/(m^4 + 16*m^3 + 86*m^2 + 176*m + 105)

_______________________________________________________________________________________

Sympy [A]  time = 5.53462, size = 1044, normalized size = 14.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-A*a**2/(6*x**6) - A*a*b/(2*x**4) - A*b**2/(2*x**2) - B*a**2/(4*x**4)
- B*a*b/x**2 + B*b**2*log(x), Eq(m, -7)), (-A*a**2/(4*x**4) - A*a*b/x**2 + A*b*
*2*log(x) - B*a**2/(2*x**2) + 2*B*a*b*log(x) + B*b**2*x**2/2, Eq(m, -5)), (-A*a*
*2/(2*x**2) + 2*A*a*b*log(x) + A*b**2*x**2/2 + B*a**2*log(x) + B*a*b*x**2 + B*b*
*2*x**4/4, Eq(m, -3)), (A*a**2*log(x) + A*a*b*x**2 + A*b**2*x**4/4 + B*a**2*x**2
/2 + B*a*b*x**4/2 + B*b**2*x**6/6, Eq(m, -1)), (A*a**2*m**3*x*x**m/(m**4 + 16*m*
*3 + 86*m**2 + 176*m + 105) + 15*A*a**2*m**2*x*x**m/(m**4 + 16*m**3 + 86*m**2 +
176*m + 105) + 71*A*a**2*m*x*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 105
*A*a**2*x*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 2*A*a*b*m**3*x**3*x**m
/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 26*A*a*b*m**2*x**3*x**m/(m**4 + 16*m
**3 + 86*m**2 + 176*m + 105) + 94*A*a*b*m*x**3*x**m/(m**4 + 16*m**3 + 86*m**2 +
176*m + 105) + 70*A*a*b*x**3*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + A*b
**2*m**3*x**5*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 11*A*b**2*m**2*x**
5*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 31*A*b**2*m*x**5*x**m/(m**4 +
16*m**3 + 86*m**2 + 176*m + 105) + 21*A*b**2*x**5*x**m/(m**4 + 16*m**3 + 86*m**2
+ 176*m + 105) + B*a**2*m**3*x**3*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105)
+ 13*B*a**2*m**2*x**3*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 47*B*a**2
*m*x**3*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 35*B*a**2*x**3*x**m/(m**
4 + 16*m**3 + 86*m**2 + 176*m + 105) + 2*B*a*b*m**3*x**5*x**m/(m**4 + 16*m**3 +
86*m**2 + 176*m + 105) + 22*B*a*b*m**2*x**5*x**m/(m**4 + 16*m**3 + 86*m**2 + 176
*m + 105) + 62*B*a*b*m*x**5*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 42*B
*a*b*x**5*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + B*b**2*m**3*x**7*x**m/
(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 9*B*b**2*m**2*x**7*x**m/(m**4 + 16*m*
*3 + 86*m**2 + 176*m + 105) + 23*B*b**2*m*x**7*x**m/(m**4 + 16*m**3 + 86*m**2 +
176*m + 105) + 15*B*b**2*x**7*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105), Tru
e))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.278326, size = 513, normalized size = 7.23 \[ \frac{B b^{2} m^{3} x^{7} e^{\left (m{\rm ln}\left (x\right )\right )} + 9 \, B b^{2} m^{2} x^{7} e^{\left (m{\rm ln}\left (x\right )\right )} + 2 \, B a b m^{3} x^{5} e^{\left (m{\rm ln}\left (x\right )\right )} + A b^{2} m^{3} x^{5} e^{\left (m{\rm ln}\left (x\right )\right )} + 23 \, B b^{2} m x^{7} e^{\left (m{\rm ln}\left (x\right )\right )} + 22 \, B a b m^{2} x^{5} e^{\left (m{\rm ln}\left (x\right )\right )} + 11 \, A b^{2} m^{2} x^{5} e^{\left (m{\rm ln}\left (x\right )\right )} + 15 \, B b^{2} x^{7} e^{\left (m{\rm ln}\left (x\right )\right )} + B a^{2} m^{3} x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 2 \, A a b m^{3} x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 62 \, B a b m x^{5} e^{\left (m{\rm ln}\left (x\right )\right )} + 31 \, A b^{2} m x^{5} e^{\left (m{\rm ln}\left (x\right )\right )} + 13 \, B a^{2} m^{2} x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 26 \, A a b m^{2} x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 42 \, B a b x^{5} e^{\left (m{\rm ln}\left (x\right )\right )} + 21 \, A b^{2} x^{5} e^{\left (m{\rm ln}\left (x\right )\right )} + A a^{2} m^{3} x e^{\left (m{\rm ln}\left (x\right )\right )} + 47 \, B a^{2} m x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 94 \, A a b m x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 15 \, A a^{2} m^{2} x e^{\left (m{\rm ln}\left (x\right )\right )} + 35 \, B a^{2} x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 70 \, A a b x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 71 \, A a^{2} m x e^{\left (m{\rm ln}\left (x\right )\right )} + 105 \, A a^{2} x e^{\left (m{\rm ln}\left (x\right )\right )}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(B*b^2*m^3*x^7*e^(m*ln(x)) + 9*B*b^2*m^2*x^7*e^(m*ln(x)) + 2*B*a*b*m^3*x^5*e^(m*
ln(x)) + A*b^2*m^3*x^5*e^(m*ln(x)) + 23*B*b^2*m*x^7*e^(m*ln(x)) + 22*B*a*b*m^2*x
^5*e^(m*ln(x)) + 11*A*b^2*m^2*x^5*e^(m*ln(x)) + 15*B*b^2*x^7*e^(m*ln(x)) + B*a^2
*m^3*x^3*e^(m*ln(x)) + 2*A*a*b*m^3*x^3*e^(m*ln(x)) + 62*B*a*b*m*x^5*e^(m*ln(x))
+ 31*A*b^2*m*x^5*e^(m*ln(x)) + 13*B*a^2*m^2*x^3*e^(m*ln(x)) + 26*A*a*b*m^2*x^3*e
^(m*ln(x)) + 42*B*a*b*x^5*e^(m*ln(x)) + 21*A*b^2*x^5*e^(m*ln(x)) + A*a^2*m^3*x*e
^(m*ln(x)) + 47*B*a^2*m*x^3*e^(m*ln(x)) + 94*A*a*b*m*x^3*e^(m*ln(x)) + 15*A*a^2*
m^2*x*e^(m*ln(x)) + 35*B*a^2*x^3*e^(m*ln(x)) + 70*A*a*b*x^3*e^(m*ln(x)) + 71*A*a
^2*m*x*e^(m*ln(x)) + 105*A*a^2*x*e^(m*ln(x)))/(m^4 + 16*m^3 + 86*m^2 + 176*m + 1
05)