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

Optimal. Leaf size=33 \[ \frac{1}{5} x^5 (A c+b B)+\frac{1}{3} A b x^3+\frac{1}{7} B c x^7 \]

[Out]

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

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Rubi [A]  time = 0.0242327, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {1593, 448} \[ \frac{1}{5} x^5 (A c+b B)+\frac{1}{3} A b x^3+\frac{1}{7} B c x^7 \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x^2)*(b*x^2 + c*x^4),x]

[Out]

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

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, 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 \left (A+B x^2\right ) \left (b x^2+c x^4\right ) \, dx &=\int x^2 \left (A+B x^2\right ) \left (b+c x^2\right ) \, dx\\ &=\int \left (A b x^2+(b B+A c) x^4+B c x^6\right ) \, dx\\ &=\frac{1}{3} A b x^3+\frac{1}{5} (b B+A c) x^5+\frac{1}{7} B c x^7\\ \end{align*}

Mathematica [A]  time = 0.0053023, size = 33, normalized size = 1. \[ \frac{1}{5} x^5 (A c+b B)+\frac{1}{3} A b x^3+\frac{1}{7} B c x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0., size = 28, normalized size = 0.9 \begin{align*}{\frac{Ab{x}^{3}}{3}}+{\frac{ \left ( Ac+Bb \right ){x}^{5}}{5}}+{\frac{Bc{x}^{7}}{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*A*b*x^3+1/5*(A*c+B*b)*x^5+1/7*B*c*x^7

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Maxima [A]  time = 1.13554, size = 36, normalized size = 1.09 \begin{align*} \frac{1}{7} \, B c x^{7} + \frac{1}{5} \,{\left (B b + A c\right )} x^{5} + \frac{1}{3} \, A b x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*B*c*x^7 + 1/5*(B*b + A*c)*x^5 + 1/3*A*b*x^3

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Fricas [A]  time = 0.395012, size = 74, normalized size = 2.24 \begin{align*} \frac{1}{7} x^{7} c B + \frac{1}{5} x^{5} b B + \frac{1}{5} x^{5} c A + \frac{1}{3} x^{3} b A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*x^7*c*B + 1/5*x^5*b*B + 1/5*x^5*c*A + 1/3*x^3*b*A

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Sympy [A]  time = 0.057483, size = 29, normalized size = 0.88 \begin{align*} \frac{A b x^{3}}{3} + \frac{B c x^{7}}{7} + x^{5} \left (\frac{A c}{5} + \frac{B b}{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15677, size = 39, normalized size = 1.18 \begin{align*} \frac{1}{7} \, B c x^{7} + \frac{1}{5} \, B b x^{5} + \frac{1}{5} \, A c x^{5} + \frac{1}{3} \, A b x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*B*c*x^7 + 1/5*B*b*x^5 + 1/5*A*c*x^5 + 1/3*A*b*x^3