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

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

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 70, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.01, size = 70, normalized size = 1.00 \[ A b^3 x+\frac {1}{3} b^2 x^3 (3 A c+b B)+\frac {1}{7} c^2 x^7 (A c+3 b B)+\frac {3}{5} b c x^5 (A c+b B)+\frac {1}{9} B c^3 x^9 \]

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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fricas [A]  time = 0.83, size = 70, normalized size = 1.00 \[ \frac {1}{9} \, B c^{3} x^{9} + \frac {1}{7} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{7} + \frac {3}{5} \, {\left (B b^{2} c + A b c^{2}\right )} x^{5} + A b^{3} x + \frac {1}{3} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{3} \]

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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giac [A]  time = 0.17, size = 73, normalized size = 1.04 \[ \frac {1}{9} \, B c^{3} x^{9} + \frac {3}{7} \, B b c^{2} x^{7} + \frac {1}{7} \, A c^{3} x^{7} + \frac {3}{5} \, B b^{2} c x^{5} + \frac {3}{5} \, A b c^{2} x^{5} + \frac {1}{3} \, B b^{3} x^{3} + A b^{2} c x^{3} + A b^{3} x \]

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

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maple [A]  time = 0.04, size = 73, normalized size = 1.04 \[ \frac {B \,c^{3} x^{9}}{9}+\frac {\left (A \,c^{3}+3 B b \,c^{2}\right ) x^{7}}{7}+A \,b^{3} x +\frac {\left (3 A b \,c^{2}+3 B c \,b^{2}\right ) x^{5}}{5}+\frac {\left (3 A c \,b^{2}+B \,b^{3}\right ) x^{3}}{3} \]

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 = 1.28, size = 70, normalized size = 1.00 \[ \frac {1}{9} \, B c^{3} x^{9} + \frac {1}{7} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{7} + \frac {3}{5} \, {\left (B b^{2} c + A b c^{2}\right )} x^{5} + A b^{3} x + \frac {1}{3} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{3} \]

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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mupad [B]  time = 0.03, size = 65, normalized size = 0.93 \[ x^3\,\left (\frac {B\,b^3}{3}+A\,c\,b^2\right )+x^7\,\left (\frac {A\,c^3}{7}+\frac {3\,B\,b\,c^2}{7}\right )+\frac {B\,c^3\,x^9}{9}+A\,b^3\,x+\frac {3\,b\,c\,x^5\,\left (A\,c+B\,b\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.09, size = 76, normalized size = 1.09 \[ A b^{3} x + \frac {B c^{3} x^{9}}{9} + x^{7} \left (\frac {A c^{3}}{7} + \frac {3 B b c^{2}}{7}\right ) + x^{5} \left (\frac {3 A b c^{2}}{5} + \frac {3 B b^{2} c}{5}\right ) + x^{3} \left (A b^{2} c + \frac {B b^{3}}{3}\right ) \]

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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