3.271 \(\int x^m (A+B x^2) (b x^2+c x^4) \, dx\)

Optimal. Leaf size=45 \[ \frac{x^{m+5} (A c+b B)}{m+5}+\frac{A b x^{m+3}}{m+3}+\frac{B c x^{m+7}}{m+7} \]

[Out]

(A*b*x^(3 + m))/(3 + m) + ((b*B + A*c)*x^(5 + m))/(5 + m) + (B*c*x^(7 + m))/(7 + m)

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Rubi [A]  time = 0.0298884, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {1584, 448} \[ \frac{x^{m+5} (A c+b B)}{m+5}+\frac{A b x^{m+3}}{m+3}+\frac{B c x^{m+7}}{m+7} \]

Antiderivative was successfully verified.

[In]

Int[x^m*(A + B*x^2)*(b*x^2 + c*x^4),x]

[Out]

(A*b*x^(3 + m))/(3 + m) + ((b*B + A*c)*x^(5 + m))/(5 + m) + (B*c*x^(7 + m))/(7 + m)

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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

Mathematica [A]  time = 0.0399325, size = 42, normalized size = 0.93 \[ x^{m+3} \left (\frac{x^2 (A c+b B)}{m+5}+\frac{A b}{m+3}+\frac{B c x^4}{m+7}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^m*(A + B*x^2)*(b*x^2 + c*x^4),x]

[Out]

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

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Maple [B]  time = 0.003, size = 110, normalized size = 2.4 \begin{align*}{\frac{{x}^{3+m} \left ( Bc{m}^{2}{x}^{4}+8\,Bcm{x}^{4}+Ac{m}^{2}{x}^{2}+Bb{m}^{2}{x}^{2}+15\,Bc{x}^{4}+10\,Acm{x}^{2}+10\,Bbm{x}^{2}+Ab{m}^{2}+21\,A{x}^{2}c+21\,B{x}^{2}b+12\,Abm+35\,Ab \right ) }{ \left ( 7+m \right ) \left ( 5+m \right ) \left ( 3+m \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(B*x^2+A)*(c*x^4+b*x^2),x)

[Out]

x^(3+m)*(B*c*m^2*x^4+8*B*c*m*x^4+A*c*m^2*x^2+B*b*m^2*x^2+15*B*c*x^4+10*A*c*m*x^2+10*B*b*m*x^2+A*b*m^2+21*A*c*x
^2+21*B*b*x^2+12*A*b*m+35*A*b)/(7+m)/(5+m)/(3+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)*(c*x^4+b*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.31435, size = 223, normalized size = 4.96 \begin{align*} \frac{{\left ({\left (B c m^{2} + 8 \, B c m + 15 \, B c\right )} x^{7} +{\left ({\left (B b + A c\right )} m^{2} + 21 \, B b + 21 \, A c + 10 \,{\left (B b + A c\right )} m\right )} x^{5} +{\left (A b m^{2} + 12 \, A b m + 35 \, A b\right )} x^{3}\right )} x^{m}}{m^{3} + 15 \, m^{2} + 71 \, m + 105} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(B*x^2+A)*(c*x^4+b*x^2),x, algorithm="fricas")

[Out]

((B*c*m^2 + 8*B*c*m + 15*B*c)*x^7 + ((B*b + A*c)*m^2 + 21*B*b + 21*A*c + 10*(B*b + A*c)*m)*x^5 + (A*b*m^2 + 12
*A*b*m + 35*A*b)*x^3)*x^m/(m^3 + 15*m^2 + 71*m + 105)

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Sympy [A]  time = 1.8105, size = 415, normalized size = 9.22 \begin{align*} \begin{cases} - \frac{A b}{4 x^{4}} - \frac{A c}{2 x^{2}} - \frac{B b}{2 x^{2}} + B c \log{\left (x \right )} & \text{for}\: m = -7 \\- \frac{A b}{2 x^{2}} + A c \log{\left (x \right )} + B b \log{\left (x \right )} + \frac{B c x^{2}}{2} & \text{for}\: m = -5 \\A b \log{\left (x \right )} + \frac{A c x^{2}}{2} + \frac{B b x^{2}}{2} + \frac{B c x^{4}}{4} & \text{for}\: m = -3 \\\frac{A b m^{2} x^{3} x^{m}}{m^{3} + 15 m^{2} + 71 m + 105} + \frac{12 A b m x^{3} x^{m}}{m^{3} + 15 m^{2} + 71 m + 105} + \frac{35 A b x^{3} x^{m}}{m^{3} + 15 m^{2} + 71 m + 105} + \frac{A c m^{2} x^{5} x^{m}}{m^{3} + 15 m^{2} + 71 m + 105} + \frac{10 A c m x^{5} x^{m}}{m^{3} + 15 m^{2} + 71 m + 105} + \frac{21 A c x^{5} x^{m}}{m^{3} + 15 m^{2} + 71 m + 105} + \frac{B b m^{2} x^{5} x^{m}}{m^{3} + 15 m^{2} + 71 m + 105} + \frac{10 B b m x^{5} x^{m}}{m^{3} + 15 m^{2} + 71 m + 105} + \frac{21 B b x^{5} x^{m}}{m^{3} + 15 m^{2} + 71 m + 105} + \frac{B c m^{2} x^{7} x^{m}}{m^{3} + 15 m^{2} + 71 m + 105} + \frac{8 B c m x^{7} x^{m}}{m^{3} + 15 m^{2} + 71 m + 105} + \frac{15 B c x^{7} x^{m}}{m^{3} + 15 m^{2} + 71 m + 105} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*(B*x**2+A)*(c*x**4+b*x**2),x)

[Out]

Piecewise((-A*b/(4*x**4) - A*c/(2*x**2) - B*b/(2*x**2) + B*c*log(x), Eq(m, -7)), (-A*b/(2*x**2) + A*c*log(x) +
B*b*log(x) + B*c*x**2/2, Eq(m, -5)), (A*b*log(x) + A*c*x**2/2 + B*b*x**2/2 + B*c*x**4/4, Eq(m, -3)), (A*b*m**
2*x**3*x**m/(m**3 + 15*m**2 + 71*m + 105) + 12*A*b*m*x**3*x**m/(m**3 + 15*m**2 + 71*m + 105) + 35*A*b*x**3*x**
m/(m**3 + 15*m**2 + 71*m + 105) + A*c*m**2*x**5*x**m/(m**3 + 15*m**2 + 71*m + 105) + 10*A*c*m*x**5*x**m/(m**3
+ 15*m**2 + 71*m + 105) + 21*A*c*x**5*x**m/(m**3 + 15*m**2 + 71*m + 105) + B*b*m**2*x**5*x**m/(m**3 + 15*m**2
+ 71*m + 105) + 10*B*b*m*x**5*x**m/(m**3 + 15*m**2 + 71*m + 105) + 21*B*b*x**5*x**m/(m**3 + 15*m**2 + 71*m + 1
05) + B*c*m**2*x**7*x**m/(m**3 + 15*m**2 + 71*m + 105) + 8*B*c*m*x**7*x**m/(m**3 + 15*m**2 + 71*m + 105) + 15*
B*c*x**7*x**m/(m**3 + 15*m**2 + 71*m + 105), True))

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Giac [B]  time = 1.40694, size = 201, normalized size = 4.47 \begin{align*} \frac{B c m^{2} x^{7} x^{m} + 8 \, B c m x^{7} x^{m} + B b m^{2} x^{5} x^{m} + A c m^{2} x^{5} x^{m} + 15 \, B c x^{7} x^{m} + 10 \, B b m x^{5} x^{m} + 10 \, A c m x^{5} x^{m} + A b m^{2} x^{3} x^{m} + 21 \, B b x^{5} x^{m} + 21 \, A c x^{5} x^{m} + 12 \, A b m x^{3} x^{m} + 35 \, A b x^{3} x^{m}}{m^{3} + 15 \, m^{2} + 71 \, m + 105} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(B*x^2+A)*(c*x^4+b*x^2),x, algorithm="giac")

[Out]

(B*c*m^2*x^7*x^m + 8*B*c*m*x^7*x^m + B*b*m^2*x^5*x^m + A*c*m^2*x^5*x^m + 15*B*c*x^7*x^m + 10*B*b*m*x^5*x^m + 1
0*A*c*m*x^5*x^m + A*b*m^2*x^3*x^m + 21*B*b*x^5*x^m + 21*A*c*x^5*x^m + 12*A*b*m*x^3*x^m + 35*A*b*x^3*x^m)/(m^3
+ 15*m^2 + 71*m + 105)