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

Optimal. Leaf size=50 \[ A b^2 x+\frac {1}{5} c x^5 (A c+2 b B)+\frac {1}{3} b x^3 (2 A c+b B)+\frac {1}{7} B c^2 x^7 \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 50, 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} \[ A b^2 x+\frac {1}{5} c x^5 (A c+2 b B)+\frac {1}{3} b x^3 (2 A c+b B)+\frac {1}{7} B c^2 x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.01, size = 50, normalized size = 1.00 \[ A b^2 x+\frac {1}{5} c x^5 (A c+2 b B)+\frac {1}{3} b x^3 (2 A c+b B)+\frac {1}{7} B c^2 x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.92, size = 48, normalized size = 0.96 \[ \frac {1}{7} \, B c^{2} x^{7} + \frac {1}{5} \, {\left (2 \, B b c + A c^{2}\right )} x^{5} + A b^{2} x + \frac {1}{3} \, {\left (B b^{2} + 2 \, A b c\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.17, size = 50, normalized size = 1.00 \[ \frac {1}{7} \, B c^{2} x^{7} + \frac {2}{5} \, B b c x^{5} + \frac {1}{5} \, A c^{2} x^{5} + \frac {1}{3} \, B b^{2} x^{3} + \frac {2}{3} \, A b c x^{3} + A b^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 49, normalized size = 0.98 \[ \frac {B \,c^{2} x^{7}}{7}+\frac {\left (A \,c^{2}+2 b B c \right ) x^{5}}{5}+A \,b^{2} x +\frac {\left (2 A b c +B \,b^{2}\right ) x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.24, size = 48, normalized size = 0.96 \[ \frac {1}{7} \, B c^{2} x^{7} + \frac {1}{5} \, {\left (2 \, B b c + A c^{2}\right )} x^{5} + A b^{2} x + \frac {1}{3} \, {\left (B b^{2} + 2 \, A b c\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.05, size = 48, normalized size = 0.96 \[ x^3\,\left (\frac {B\,b^2}{3}+\frac {2\,A\,c\,b}{3}\right )+x^5\,\left (\frac {A\,c^2}{5}+\frac {2\,B\,b\,c}{5}\right )+\frac {B\,c^2\,x^7}{7}+A\,b^2\,x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.08, size = 53, normalized size = 1.06 \[ A b^{2} x + \frac {B c^{2} x^{7}}{7} + x^{5} \left (\frac {A c^{2}}{5} + \frac {2 B b c}{5}\right ) + x^{3} \left (\frac {2 A b c}{3} + \frac {B b^{2}}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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