3.24 \(\int \frac{(A+B x^2) (b x^2+c x^4)^2}{x^{12}} \, dx\)

Optimal. Leaf size=53 \[ -\frac{A b^2}{7 x^7}-\frac{b (2 A c+b B)}{5 x^5}-\frac{c (A c+2 b B)}{3 x^3}-\frac{B c^2}{x} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0358008, antiderivative size = 53, 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, 448} \[ -\frac{A b^2}{7 x^7}-\frac{b (2 A c+b B)}{5 x^5}-\frac{c (A c+2 b B)}{3 x^3}-\frac{B c^2}{x} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x^2)*(b*x^2 + c*x^4)^2)/x^12,x]

[Out]

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

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 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 \frac{\left (A+B x^2\right ) \left (b x^2+c x^4\right )^2}{x^{12}} \, dx &=\int \frac{\left (A+B x^2\right ) \left (b+c x^2\right )^2}{x^8} \, dx\\ &=\int \left (\frac{A b^2}{x^8}+\frac{b (b B+2 A c)}{x^6}+\frac{c (2 b B+A c)}{x^4}+\frac{B c^2}{x^2}\right ) \, dx\\ &=-\frac{A b^2}{7 x^7}-\frac{b (b B+2 A c)}{5 x^5}-\frac{c (2 b B+A c)}{3 x^3}-\frac{B c^2}{x}\\ \end{align*}

Mathematica [A]  time = 0.0168103, size = 59, normalized size = 1.11 \[ \frac{b^2 (-B)-2 A b c}{5 x^5}-\frac{A b^2}{7 x^7}+\frac{-A c^2-2 b B c}{3 x^3}-\frac{B c^2}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^2)/x^12,x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 48, normalized size = 0.9 \begin{align*} -{\frac{A{b}^{2}}{7\,{x}^{7}}}-{\frac{b \left ( 2\,Ac+Bb \right ) }{5\,{x}^{5}}}-{\frac{c \left ( Ac+2\,Bb \right ) }{3\,{x}^{3}}}-{\frac{B{c}^{2}}{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)^2/x^12,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.16759, size = 72, normalized size = 1.36 \begin{align*} -\frac{105 \, B c^{2} x^{6} + 35 \,{\left (2 \, B b c + A c^{2}\right )} x^{4} + 15 \, A b^{2} + 21 \,{\left (B b^{2} + 2 \, A b c\right )} x^{2}}{105 \, 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^12,x, algorithm="maxima")

[Out]

-1/105*(105*B*c^2*x^6 + 35*(2*B*b*c + A*c^2)*x^4 + 15*A*b^2 + 21*(B*b^2 + 2*A*b*c)*x^2)/x^7

________________________________________________________________________________________

Fricas [A]  time = 0.479817, size = 126, normalized size = 2.38 \begin{align*} -\frac{105 \, B c^{2} x^{6} + 35 \,{\left (2 \, B b c + A c^{2}\right )} x^{4} + 15 \, A b^{2} + 21 \,{\left (B b^{2} + 2 \, A b c\right )} x^{2}}{105 \, 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^12,x, algorithm="fricas")

[Out]

-1/105*(105*B*c^2*x^6 + 35*(2*B*b*c + A*c^2)*x^4 + 15*A*b^2 + 21*(B*b^2 + 2*A*b*c)*x^2)/x^7

________________________________________________________________________________________

Sympy [A]  time = 1.16941, size = 56, normalized size = 1.06 \begin{align*} - \frac{15 A b^{2} + 105 B c^{2} x^{6} + x^{4} \left (35 A c^{2} + 70 B b c\right ) + x^{2} \left (42 A b c + 21 B b^{2}\right )}{105 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**12,x)

[Out]

-(15*A*b**2 + 105*B*c**2*x**6 + x**4*(35*A*c**2 + 70*B*b*c) + x**2*(42*A*b*c + 21*B*b**2))/(105*x**7)

________________________________________________________________________________________

Giac [A]  time = 1.24865, size = 74, normalized size = 1.4 \begin{align*} -\frac{105 \, B c^{2} x^{6} + 70 \, B b c x^{4} + 35 \, A c^{2} x^{4} + 21 \, B b^{2} x^{2} + 42 \, A b c x^{2} + 15 \, A b^{2}}{105 \, 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^12,x, algorithm="giac")

[Out]

-1/105*(105*B*c^2*x^6 + 70*B*b*c*x^4 + 35*A*c^2*x^4 + 21*B*b^2*x^2 + 42*A*b*c*x^2 + 15*A*b^2)/x^7