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

Optimal. Leaf size=55 \[ \frac{1}{5} A b^2 x^5+\frac{1}{9} c x^9 (A c+2 b B)+\frac{1}{7} b x^7 (2 A c+b B)+\frac{1}{11} B c^2 x^{11} \]

[Out]

(A*b^2*x^5)/5 + (b*(b*B + 2*A*c)*x^7)/7 + (c*(2*b*B + A*c)*x^9)/9 + (B*c^2*x^11)/11

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Rubi [A]  time = 0.0475207, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {1593, 448} \[ \frac{1}{5} A b^2 x^5+\frac{1}{9} c x^9 (A c+2 b B)+\frac{1}{7} b x^7 (2 A c+b B)+\frac{1}{11} B c^2 x^{11} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x^2)*(b*x^2 + c*x^4)^2,x]

[Out]

(A*b^2*x^5)/5 + (b*(b*B + 2*A*c)*x^7)/7 + (c*(2*b*B + A*c)*x^9)/9 + (B*c^2*x^11)/11

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 )^2 \, dx &=\int x^4 \left (A+B x^2\right ) \left (b+c x^2\right )^2 \, dx\\ &=\int \left (A b^2 x^4+b (b B+2 A c) x^6+c (2 b B+A c) x^8+B c^2 x^{10}\right ) \, dx\\ &=\frac{1}{5} A b^2 x^5+\frac{1}{7} b (b B+2 A c) x^7+\frac{1}{9} c (2 b B+A c) x^9+\frac{1}{11} B c^2 x^{11}\\ \end{align*}

Mathematica [A]  time = 0.0096894, size = 55, normalized size = 1. \[ \frac{1}{5} A b^2 x^5+\frac{1}{9} c x^9 (A c+2 b B)+\frac{1}{7} b x^7 (2 A c+b B)+\frac{1}{11} B c^2 x^{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(A*b^2*x^5)/5 + (b*(b*B + 2*A*c)*x^7)/7 + (c*(2*b*B + A*c)*x^9)/9 + (B*c^2*x^11)/11

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Maple [A]  time = 0., size = 52, normalized size = 1. \begin{align*}{\frac{B{c}^{2}{x}^{11}}{11}}+{\frac{ \left ( A{c}^{2}+2\,Bbc \right ){x}^{9}}{9}}+{\frac{ \left ( 2\,Abc+B{b}^{2} \right ){x}^{7}}{7}}+{\frac{A{b}^{2}{x}^{5}}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*B*c^2*x^11+1/9*(A*c^2+2*B*b*c)*x^9+1/7*(2*A*b*c+B*b^2)*x^7+1/5*A*b^2*x^5

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Maxima [A]  time = 1.1433, size = 69, normalized size = 1.25 \begin{align*} \frac{1}{11} \, B c^{2} x^{11} + \frac{1}{9} \,{\left (2 \, B b c + A c^{2}\right )} x^{9} + \frac{1}{5} \, A b^{2} x^{5} + \frac{1}{7} \,{\left (B b^{2} + 2 \, A b c\right )} x^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*B*c^2*x^11 + 1/9*(2*B*b*c + A*c^2)*x^9 + 1/5*A*b^2*x^5 + 1/7*(B*b^2 + 2*A*b*c)*x^7

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Fricas [A]  time = 0.41931, size = 131, normalized size = 2.38 \begin{align*} \frac{1}{11} x^{11} c^{2} B + \frac{2}{9} x^{9} c b B + \frac{1}{9} x^{9} c^{2} A + \frac{1}{7} x^{7} b^{2} B + \frac{2}{7} x^{7} c b A + \frac{1}{5} x^{5} b^{2} 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)^2,x, algorithm="fricas")

[Out]

1/11*x^11*c^2*B + 2/9*x^9*c*b*B + 1/9*x^9*c^2*A + 1/7*x^7*b^2*B + 2/7*x^7*c*b*A + 1/5*x^5*b^2*A

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Sympy [A]  time = 0.070626, size = 56, normalized size = 1.02 \begin{align*} \frac{A b^{2} x^{5}}{5} + \frac{B c^{2} x^{11}}{11} + x^{9} \left (\frac{A c^{2}}{9} + \frac{2 B b c}{9}\right ) + x^{7} \left (\frac{2 A b c}{7} + \frac{B b^{2}}{7}\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)**2,x)

[Out]

A*b**2*x**5/5 + B*c**2*x**11/11 + x**9*(A*c**2/9 + 2*B*b*c/9) + x**7*(2*A*b*c/7 + B*b**2/7)

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Giac [A]  time = 1.20551, size = 72, normalized size = 1.31 \begin{align*} \frac{1}{11} \, B c^{2} x^{11} + \frac{2}{9} \, B b c x^{9} + \frac{1}{9} \, A c^{2} x^{9} + \frac{1}{7} \, B b^{2} x^{7} + \frac{2}{7} \, A b c x^{7} + \frac{1}{5} \, A b^{2} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*B*c^2*x^11 + 2/9*B*b*c*x^9 + 1/9*A*c^2*x^9 + 1/7*B*b^2*x^7 + 2/7*A*b*c*x^7 + 1/5*A*b^2*x^5