3.29 \(\int \frac{(A+B x^2) (b x^2+c x^4)^3}{x^6} \, dx\)

Optimal. Leaf size=70 \[ \frac{1}{3} b^2 x^3 (3 A c+b B)+A b^3 x+\frac{1}{7} c^2 x^7 (A c+3 b B)+\frac{3}{5} b c x^5 (A c+b B)+\frac{1}{9} B c^3 x^9 \]

[Out]

A*b^3*x + (b^2*(b*B + 3*A*c)*x^3)/3 + (3*b*c*(b*B + A*c)*x^5)/5 + (c^2*(3*b*B + A*c)*x^7)/7 + (B*c^3*x^9)/9

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Rubi [A]  time = 0.0416425, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {1584, 373} \[ \frac{1}{3} b^2 x^3 (3 A c+b B)+A b^3 x+\frac{1}{7} c^2 x^7 (A c+3 b B)+\frac{3}{5} b c x^5 (A c+b B)+\frac{1}{9} B c^3 x^9 \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^6,x]

[Out]

A*b^3*x + (b^2*(b*B + 3*A*c)*x^3)/3 + (3*b*c*(b*B + A*c)*x^5)/5 + (c^2*(3*b*B + A*c)*x^7)/7 + (B*c^3*x^9)/9

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 373

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

Rubi steps

\begin{align*} \int \frac{\left (A+B x^2\right ) \left (b x^2+c x^4\right )^3}{x^6} \, dx &=\int \left (A+B x^2\right ) \left (b+c x^2\right )^3 \, dx\\ &=\int \left (A b^3+b^2 (b B+3 A c) x^2+3 b c (b B+A c) x^4+c^2 (3 b B+A c) x^6+B c^3 x^8\right ) \, dx\\ &=A b^3 x+\frac{1}{3} b^2 (b B+3 A c) x^3+\frac{3}{5} b c (b B+A c) x^5+\frac{1}{7} c^2 (3 b B+A c) x^7+\frac{1}{9} B c^3 x^9\\ \end{align*}

Mathematica [A]  time = 0.0107059, size = 70, normalized size = 1. \[ \frac{1}{3} b^2 x^3 (3 A c+b B)+A b^3 x+\frac{1}{7} c^2 x^7 (A c+3 b B)+\frac{3}{5} b c x^5 (A c+b B)+\frac{1}{9} B c^3 x^9 \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^6,x]

[Out]

A*b^3*x + (b^2*(b*B + 3*A*c)*x^3)/3 + (3*b*c*(b*B + A*c)*x^5)/5 + (c^2*(3*b*B + A*c)*x^7)/7 + (B*c^3*x^9)/9

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Maple [A]  time = 0.001, size = 73, normalized size = 1. \begin{align*}{\frac{B{c}^{3}{x}^{9}}{9}}+{\frac{ \left ( A{c}^{3}+3\,Bb{c}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,Ab{c}^{2}+3\,B{b}^{2}c \right ){x}^{5}}{5}}+{\frac{ \left ( 3\,A{b}^{2}c+B{b}^{3} \right ){x}^{3}}{3}}+A{b}^{3}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x^2+A)*(c*x^4+b*x^2)^3/x^6,x)

[Out]

1/9*B*c^3*x^9+1/7*(A*c^3+3*B*b*c^2)*x^7+1/5*(3*A*b*c^2+3*B*b^2*c)*x^5+1/3*(3*A*b^2*c+B*b^3)*x^3+A*b^3*x

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Maxima [A]  time = 1.08482, size = 95, normalized size = 1.36 \begin{align*} \frac{1}{9} \, B c^{3} x^{9} + \frac{1}{7} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{7} + \frac{3}{5} \,{\left (B b^{2} c + A b c^{2}\right )} x^{5} + A b^{3} x + \frac{1}{3} \,{\left (B b^{3} + 3 \, A b^{2} c\right )} 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)^3/x^6,x, algorithm="maxima")

[Out]

1/9*B*c^3*x^9 + 1/7*(3*B*b*c^2 + A*c^3)*x^7 + 3/5*(B*b^2*c + A*b*c^2)*x^5 + A*b^3*x + 1/3*(B*b^3 + 3*A*b^2*c)*
x^3

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Fricas [A]  time = 0.510629, size = 155, normalized size = 2.21 \begin{align*} \frac{1}{9} \, B c^{3} x^{9} + \frac{1}{7} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{7} + \frac{3}{5} \,{\left (B b^{2} c + A b c^{2}\right )} x^{5} + A b^{3} x + \frac{1}{3} \,{\left (B b^{3} + 3 \, A b^{2} c\right )} 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)^3/x^6,x, algorithm="fricas")

[Out]

1/9*B*c^3*x^9 + 1/7*(3*B*b*c^2 + A*c^3)*x^7 + 3/5*(B*b^2*c + A*b*c^2)*x^5 + A*b^3*x + 1/3*(B*b^3 + 3*A*b^2*c)*
x^3

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Sympy [A]  time = 0.074467, size = 76, normalized size = 1.09 \begin{align*} A b^{3} x + \frac{B c^{3} x^{9}}{9} + x^{7} \left (\frac{A c^{3}}{7} + \frac{3 B b c^{2}}{7}\right ) + x^{5} \left (\frac{3 A b c^{2}}{5} + \frac{3 B b^{2} c}{5}\right ) + x^{3} \left (A b^{2} c + \frac{B b^{3}}{3}\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)**3/x**6,x)

[Out]

A*b**3*x + B*c**3*x**9/9 + x**7*(A*c**3/7 + 3*B*b*c**2/7) + x**5*(3*A*b*c**2/5 + 3*B*b**2*c/5) + x**3*(A*b**2*
c + B*b**3/3)

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Giac [A]  time = 1.19985, size = 99, normalized size = 1.41 \begin{align*} \frac{1}{9} \, B c^{3} x^{9} + \frac{3}{7} \, B b c^{2} x^{7} + \frac{1}{7} \, A c^{3} x^{7} + \frac{3}{5} \, B b^{2} c x^{5} + \frac{3}{5} \, A b c^{2} x^{5} + \frac{1}{3} \, B b^{3} x^{3} + A b^{2} c x^{3} + A b^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)*(c*x^4+b*x^2)^3/x^6,x, algorithm="giac")

[Out]

1/9*B*c^3*x^9 + 3/7*B*b*c^2*x^7 + 1/7*A*c^3*x^7 + 3/5*B*b^2*c*x^5 + 3/5*A*b*c^2*x^5 + 1/3*B*b^3*x^3 + A*b^2*c*
x^3 + A*b^3*x