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

Optimal. Leaf size=75 \[ \frac{1}{5} b^2 x^5 (3 A c+b B)+\frac{1}{3} A b^3 x^3+\frac{1}{9} c^2 x^9 (A c+3 b B)+\frac{3}{7} b c x^7 (A c+b B)+\frac{1}{11} B c^3 x^{11} \]

[Out]

(A*b^3*x^3)/3 + (b^2*(b*B + 3*A*c)*x^5)/5 + (3*b*c*(b*B + A*c)*x^7)/7 + (c^2*(3*b*B + A*c)*x^9)/9 + (B*c^3*x^1
1)/11

________________________________________________________________________________________

Rubi [A]  time = 0.0552321, antiderivative size = 75, 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} \[ \frac{1}{5} b^2 x^5 (3 A c+b B)+\frac{1}{3} A b^3 x^3+\frac{1}{9} c^2 x^9 (A c+3 b B)+\frac{3}{7} b c x^7 (A c+b B)+\frac{1}{11} B c^3 x^{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(A*b^3*x^3)/3 + (b^2*(b*B + 3*A*c)*x^5)/5 + (3*b*c*(b*B + A*c)*x^7)/7 + (c^2*(3*b*B + A*c)*x^9)/9 + (B*c^3*x^1
1)/11

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 )^3}{x^4} \, dx &=\int x^2 \left (A+B x^2\right ) \left (b+c x^2\right )^3 \, dx\\ &=\int \left (A b^3 x^2+b^2 (b B+3 A c) x^4+3 b c (b B+A c) x^6+c^2 (3 b B+A c) x^8+B c^3 x^{10}\right ) \, dx\\ &=\frac{1}{3} A b^3 x^3+\frac{1}{5} b^2 (b B+3 A c) x^5+\frac{3}{7} b c (b B+A c) x^7+\frac{1}{9} c^2 (3 b B+A c) x^9+\frac{1}{11} B c^3 x^{11}\\ \end{align*}

Mathematica [A]  time = 0.0118488, size = 75, normalized size = 1. \[ \frac{1}{5} b^2 x^5 (3 A c+b B)+\frac{1}{3} A b^3 x^3+\frac{1}{9} c^2 x^9 (A c+3 b B)+\frac{3}{7} b c x^7 (A c+b B)+\frac{1}{11} B c^3 x^{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(A*b^3*x^3)/3 + (b^2*(b*B + 3*A*c)*x^5)/5 + (3*b*c*(b*B + A*c)*x^7)/7 + (c^2*(3*b*B + A*c)*x^9)/9 + (B*c^3*x^1
1)/11

________________________________________________________________________________________

Maple [A]  time = 0.001, size = 76, normalized size = 1. \begin{align*}{\frac{B{c}^{3}{x}^{11}}{11}}+{\frac{ \left ( A{c}^{3}+3\,Bb{c}^{2} \right ){x}^{9}}{9}}+{\frac{ \left ( 3\,Ab{c}^{2}+3\,B{b}^{2}c \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,A{b}^{2}c+B{b}^{3} \right ){x}^{5}}{5}}+{\frac{A{b}^{3}{x}^{3}}{3}} \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^4,x)

[Out]

1/11*B*c^3*x^11+1/9*(A*c^3+3*B*b*c^2)*x^9+1/7*(3*A*b*c^2+3*B*b^2*c)*x^7+1/5*(3*A*b^2*c+B*b^3)*x^5+1/3*A*b^3*x^
3

________________________________________________________________________________________

Maxima [A]  time = 1.12863, size = 99, normalized size = 1.32 \begin{align*} \frac{1}{11} \, B c^{3} x^{11} + \frac{1}{9} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{9} + \frac{3}{7} \,{\left (B b^{2} c + A b c^{2}\right )} x^{7} + \frac{1}{3} \, A b^{3} x^{3} + \frac{1}{5} \,{\left (B b^{3} + 3 \, A b^{2} c\right )} 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)^3/x^4,x, algorithm="maxima")

[Out]

1/11*B*c^3*x^11 + 1/9*(3*B*b*c^2 + A*c^3)*x^9 + 3/7*(B*b^2*c + A*b*c^2)*x^7 + 1/3*A*b^3*x^3 + 1/5*(B*b^3 + 3*A
*b^2*c)*x^5

________________________________________________________________________________________

Fricas [A]  time = 0.442372, size = 166, normalized size = 2.21 \begin{align*} \frac{1}{11} \, B c^{3} x^{11} + \frac{1}{9} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{9} + \frac{3}{7} \,{\left (B b^{2} c + A b c^{2}\right )} x^{7} + \frac{1}{3} \, A b^{3} x^{3} + \frac{1}{5} \,{\left (B b^{3} + 3 \, A b^{2} c\right )} 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)^3/x^4,x, algorithm="fricas")

[Out]

1/11*B*c^3*x^11 + 1/9*(3*B*b*c^2 + A*c^3)*x^9 + 3/7*(B*b^2*c + A*b*c^2)*x^7 + 1/3*A*b^3*x^3 + 1/5*(B*b^3 + 3*A
*b^2*c)*x^5

________________________________________________________________________________________

Sympy [A]  time = 0.074925, size = 82, normalized size = 1.09 \begin{align*} \frac{A b^{3} x^{3}}{3} + \frac{B c^{3} x^{11}}{11} + x^{9} \left (\frac{A c^{3}}{9} + \frac{B b c^{2}}{3}\right ) + x^{7} \left (\frac{3 A b c^{2}}{7} + \frac{3 B b^{2} c}{7}\right ) + x^{5} \left (\frac{3 A b^{2} c}{5} + \frac{B b^{3}}{5}\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**4,x)

[Out]

A*b**3*x**3/3 + B*c**3*x**11/11 + x**9*(A*c**3/9 + B*b*c**2/3) + x**7*(3*A*b*c**2/7 + 3*B*b**2*c/7) + x**5*(3*
A*b**2*c/5 + B*b**3/5)

________________________________________________________________________________________

Giac [A]  time = 1.21324, size = 104, normalized size = 1.39 \begin{align*} \frac{1}{11} \, B c^{3} x^{11} + \frac{1}{3} \, B b c^{2} x^{9} + \frac{1}{9} \, A c^{3} x^{9} + \frac{3}{7} \, B b^{2} c x^{7} + \frac{3}{7} \, A b c^{2} x^{7} + \frac{1}{5} \, B b^{3} x^{5} + \frac{3}{5} \, A b^{2} c x^{5} + \frac{1}{3} \, A b^{3} 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^4,x, algorithm="giac")

[Out]

1/11*B*c^3*x^11 + 1/3*B*b*c^2*x^9 + 1/9*A*c^3*x^9 + 3/7*B*b^2*c*x^7 + 3/7*A*b*c^2*x^7 + 1/5*B*b^3*x^5 + 3/5*A*
b^2*c*x^5 + 1/3*A*b^3*x^3