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

Optimal. Leaf size=156 \[ a^2 x (a B+3 A b)-\frac{a^3 A}{x}+\frac{1}{7} x^7 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{1}{3} c x^9 \left (a B c+A b c+b^2 B\right )+\frac{1}{5} x^5 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+a x^3 \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{11} c^2 x^{11} (A c+3 b B)+\frac{1}{13} B c^3 x^{13} \]

[Out]

-((a^3*A)/x) + a^2*(3*A*b + a*B)*x + a*(a*b*B + A*(b^2 + a*c))*x^3 + ((3*a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))*
x^5)/5 + ((b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^7)/7 + (c*(b^2*B + A*b*c + a*B*c)*x^9)/3 + (c^2*(3*b*B
+ A*c)*x^11)/11 + (B*c^3*x^13)/13

________________________________________________________________________________________

Rubi [A]  time = 0.108046, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {1261} \[ a^2 x (a B+3 A b)-\frac{a^3 A}{x}+\frac{1}{7} x^7 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{1}{3} c x^9 \left (a B c+A b c+b^2 B\right )+\frac{1}{5} x^5 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+a x^3 \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{11} c^2 x^{11} (A c+3 b B)+\frac{1}{13} B c^3 x^{13} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a^3*A)/x) + a^2*(3*A*b + a*B)*x + a*(a*b*B + A*(b^2 + a*c))*x^3 + ((3*a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))*
x^5)/5 + ((b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^7)/7 + (c*(b^2*B + A*b*c + a*B*c)*x^9)/3 + (c^2*(3*b*B
+ A*c)*x^11)/11 + (B*c^3*x^13)/13

Rule 1261

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] &&
NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rubi steps

\begin{align*} \int \frac{\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^3}{x^2} \, dx &=\int \left (a^2 (3 A b+a B)+\frac{a^3 A}{x^2}+3 a \left (a b B+A \left (b^2+a c\right )\right ) x^2+\left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^4+\left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^6+3 c \left (b^2 B+A b c+a B c\right ) x^8+c^2 (3 b B+A c) x^{10}+B c^3 x^{12}\right ) \, dx\\ &=-\frac{a^3 A}{x}+a^2 (3 A b+a B) x+a \left (a b B+A \left (b^2+a c\right )\right ) x^3+\frac{1}{5} \left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^5+\frac{1}{7} \left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^7+\frac{1}{3} c \left (b^2 B+A b c+a B c\right ) x^9+\frac{1}{11} c^2 (3 b B+A c) x^{11}+\frac{1}{13} B c^3 x^{13}\\ \end{align*}

Mathematica [A]  time = 0.0944426, size = 156, normalized size = 1. \[ a^2 x (a B+3 A b)-\frac{a^3 A}{x}+\frac{1}{7} x^7 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{1}{3} c x^9 \left (a B c+A b c+b^2 B\right )+\frac{1}{5} x^5 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+a x^3 \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{11} c^2 x^{11} (A c+3 b B)+\frac{1}{13} B c^3 x^{13} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a^3*A)/x) + a^2*(3*A*b + a*B)*x + a*(a*b*B + A*(b^2 + a*c))*x^3 + ((3*a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))*
x^5)/5 + ((b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^7)/7 + (c*(b^2*B + A*b*c + a*B*c)*x^9)/3 + (c^2*(3*b*B
+ A*c)*x^11)/11 + (B*c^3*x^13)/13

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 186, normalized size = 1.2 \begin{align*}{\frac{B{c}^{3}{x}^{13}}{13}}+{\frac{A{x}^{11}{c}^{3}}{11}}+{\frac{3\,B{x}^{11}b{c}^{2}}{11}}+{\frac{A{x}^{9}b{c}^{2}}{3}}+{\frac{B{x}^{9}a{c}^{2}}{3}}+{\frac{B{x}^{9}{b}^{2}c}{3}}+{\frac{3\,A{x}^{7}a{c}^{2}}{7}}+{\frac{3\,A{x}^{7}{b}^{2}c}{7}}+{\frac{6\,B{x}^{7}abc}{7}}+{\frac{B{x}^{7}{b}^{3}}{7}}+{\frac{6\,A{x}^{5}abc}{5}}+{\frac{A{x}^{5}{b}^{3}}{5}}+{\frac{3\,B{x}^{5}{a}^{2}c}{5}}+{\frac{3\,B{x}^{5}a{b}^{2}}{5}}+A{x}^{3}{a}^{2}c+A{x}^{3}a{b}^{2}+B{x}^{3}{a}^{2}b+3\,A{a}^{2}bx+B{a}^{3}x-{\frac{A{a}^{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+a)^3/x^2,x)

[Out]

1/13*B*c^3*x^13+1/11*A*x^11*c^3+3/11*B*x^11*b*c^2+1/3*A*x^9*b*c^2+1/3*B*x^9*a*c^2+1/3*B*x^9*b^2*c+3/7*A*x^7*a*
c^2+3/7*A*x^7*b^2*c+6/7*B*x^7*a*b*c+1/7*B*x^7*b^3+6/5*A*x^5*a*b*c+1/5*A*x^5*b^3+3/5*B*x^5*a^2*c+3/5*B*x^5*a*b^
2+A*x^3*a^2*c+A*x^3*a*b^2+B*x^3*a^2*b+3*A*a^2*b*x+B*a^3*x-a^3*A/x

________________________________________________________________________________________

Maxima [A]  time = 0.987112, size = 219, normalized size = 1.4 \begin{align*} \frac{1}{13} \, B c^{3} x^{13} + \frac{1}{11} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{11} + \frac{1}{3} \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{9} + \frac{1}{7} \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{7} + \frac{1}{5} \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{5} +{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{3} - \frac{A a^{3}}{x} +{\left (B a^{3} + 3 \, A a^{2} b\right )} 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+a)^3/x^2,x, algorithm="maxima")

[Out]

1/13*B*c^3*x^13 + 1/11*(3*B*b*c^2 + A*c^3)*x^11 + 1/3*(B*b^2*c + (B*a + A*b)*c^2)*x^9 + 1/7*(B*b^3 + 3*A*a*c^2
+ 3*(2*B*a*b + A*b^2)*c)*x^7 + 1/5*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^5 + (B*a^2*b + A*a*b^2 + A*a
^2*c)*x^3 - A*a^3/x + (B*a^3 + 3*A*a^2*b)*x

________________________________________________________________________________________

Fricas [A]  time = 1.48085, size = 404, normalized size = 2.59 \begin{align*} \frac{1155 \, B c^{3} x^{14} + 1365 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{12} + 5005 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{10} + 2145 \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{8} + 3003 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{6} + 15015 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{4} - 15015 \, A a^{3} + 15015 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2}}{15015 \, 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+a)^3/x^2,x, algorithm="fricas")

[Out]

1/15015*(1155*B*c^3*x^14 + 1365*(3*B*b*c^2 + A*c^3)*x^12 + 5005*(B*b^2*c + (B*a + A*b)*c^2)*x^10 + 2145*(B*b^3
+ 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^8 + 3003*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^6 + 15015*(B*a^
2*b + A*a*b^2 + A*a^2*c)*x^4 - 15015*A*a^3 + 15015*(B*a^3 + 3*A*a^2*b)*x^2)/x

________________________________________________________________________________________

Sympy [A]  time = 0.478917, size = 185, normalized size = 1.19 \begin{align*} - \frac{A a^{3}}{x} + \frac{B c^{3} x^{13}}{13} + x^{11} \left (\frac{A c^{3}}{11} + \frac{3 B b c^{2}}{11}\right ) + x^{9} \left (\frac{A b c^{2}}{3} + \frac{B a c^{2}}{3} + \frac{B b^{2} c}{3}\right ) + x^{7} \left (\frac{3 A a c^{2}}{7} + \frac{3 A b^{2} c}{7} + \frac{6 B a b c}{7} + \frac{B b^{3}}{7}\right ) + x^{5} \left (\frac{6 A a b c}{5} + \frac{A b^{3}}{5} + \frac{3 B a^{2} c}{5} + \frac{3 B a b^{2}}{5}\right ) + x^{3} \left (A a^{2} c + A a b^{2} + B a^{2} b\right ) + x \left (3 A a^{2} b + B a^{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+a)**3/x**2,x)

[Out]

-A*a**3/x + B*c**3*x**13/13 + x**11*(A*c**3/11 + 3*B*b*c**2/11) + x**9*(A*b*c**2/3 + B*a*c**2/3 + B*b**2*c/3)
+ x**7*(3*A*a*c**2/7 + 3*A*b**2*c/7 + 6*B*a*b*c/7 + B*b**3/7) + x**5*(6*A*a*b*c/5 + A*b**3/5 + 3*B*a**2*c/5 +
3*B*a*b**2/5) + x**3*(A*a**2*c + A*a*b**2 + B*a**2*b) + x*(3*A*a**2*b + B*a**3)

________________________________________________________________________________________

Giac [A]  time = 1.13561, size = 250, normalized size = 1.6 \begin{align*} \frac{1}{13} \, B c^{3} x^{13} + \frac{3}{11} \, B b c^{2} x^{11} + \frac{1}{11} \, A c^{3} x^{11} + \frac{1}{3} \, B b^{2} c x^{9} + \frac{1}{3} \, B a c^{2} x^{9} + \frac{1}{3} \, A b c^{2} x^{9} + \frac{1}{7} \, B b^{3} x^{7} + \frac{6}{7} \, B a b c x^{7} + \frac{3}{7} \, A b^{2} c x^{7} + \frac{3}{7} \, A a c^{2} x^{7} + \frac{3}{5} \, B a b^{2} x^{5} + \frac{1}{5} \, A b^{3} x^{5} + \frac{3}{5} \, B a^{2} c x^{5} + \frac{6}{5} \, A a b c x^{5} + B a^{2} b x^{3} + A a b^{2} x^{3} + A a^{2} c x^{3} + B a^{3} x + 3 \, A a^{2} b x - \frac{A a^{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+a)^3/x^2,x, algorithm="giac")

[Out]

1/13*B*c^3*x^13 + 3/11*B*b*c^2*x^11 + 1/11*A*c^3*x^11 + 1/3*B*b^2*c*x^9 + 1/3*B*a*c^2*x^9 + 1/3*A*b*c^2*x^9 +
1/7*B*b^3*x^7 + 6/7*B*a*b*c*x^7 + 3/7*A*b^2*c*x^7 + 3/7*A*a*c^2*x^7 + 3/5*B*a*b^2*x^5 + 1/5*A*b^3*x^5 + 3/5*B*
a^2*c*x^5 + 6/5*A*a*b*c*x^5 + B*a^2*b*x^3 + A*a*b^2*x^3 + A*a^2*c*x^3 + B*a^3*x + 3*A*a^2*b*x - A*a^3/x