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

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)

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Rubi [A]  time = 0.0421679, 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, Rules used = {448} \[ \frac{a^2 A x^{m+1}}{m+1}+\frac{a x^{m+3} (a B+2 A b)}{m+3}+\frac{b x^{m+5} (2 a B+A b)}{m+5}+\frac{b^2 B x^{m+7}}{m+7} \]

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)

Rule 448

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

Rubi steps

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

Mathematica [A]  time = 0.0500152, size = 66, normalized size = 0.93 \[ x^{m+1} \left (\frac{a^2 A}{m+1}+\frac{b x^4 (2 a B+A b)}{m+5}+\frac{a x^2 (a B+2 A b)}{m+3}+\frac{b^2 B x^6}{m+7}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

x^(1 + m)*((a^2*A)/(1 + m) + (a*(2*A*b + a*B)*x^2)/(3 + m) + (b*(A*b + 2*a*B)*x^4)/(5 + m) + (b^2*B*x^6)/(7 +
m))

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Maple [B]  time = 0.004, size = 262, normalized size = 3.7 \begin{align*}{\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{b}^{2}{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\,B{x}^{4}ab+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{x}^{2}{a}^{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 ) }} \end{align*}

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)

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.07757, size = 490, normalized size = 6.9 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(b*x^2+a)^2*(B*x^2+A),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)

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Sympy [A]  time = 1.71208, size = 1044, normalized size = 14.7 \begin{align*} \text{result too large to display} \end{align*}

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 + 10
5) + 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), True))

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Giac [B]  time = 1.23412, size = 448, normalized size = 6.31 \begin{align*} \frac{B b^{2} m^{3} x^{7} x^{m} + 9 \, B b^{2} m^{2} x^{7} x^{m} + 2 \, B a b m^{3} x^{5} x^{m} + A b^{2} m^{3} x^{5} x^{m} + 23 \, B b^{2} m x^{7} x^{m} + 22 \, B a b m^{2} x^{5} x^{m} + 11 \, A b^{2} m^{2} x^{5} x^{m} + 15 \, B b^{2} x^{7} x^{m} + B a^{2} m^{3} x^{3} x^{m} + 2 \, A a b m^{3} x^{3} x^{m} + 62 \, B a b m x^{5} x^{m} + 31 \, A b^{2} m x^{5} x^{m} + 13 \, B a^{2} m^{2} x^{3} x^{m} + 26 \, A a b m^{2} x^{3} x^{m} + 42 \, B a b x^{5} x^{m} + 21 \, A b^{2} x^{5} x^{m} + A a^{2} m^{3} x x^{m} + 47 \, B a^{2} m x^{3} x^{m} + 94 \, A a b m x^{3} x^{m} + 15 \, A a^{2} m^{2} x x^{m} + 35 \, B a^{2} x^{3} x^{m} + 70 \, A a b x^{3} x^{m} + 71 \, A a^{2} m x x^{m} + 105 \, A a^{2} x x^{m}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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