### 3.16 $$\int \frac{(A+B x+C x^2) (a+b x^2+c x^4)^2}{x^3} \, dx$$

Optimal. Leaf size=149 $-\frac{a^2 A}{2 x^2}-\frac{a^2 B}{x}+\frac{1}{4} x^4 \left (C \left (2 a c+b^2\right )+2 A b c\right )+\frac{1}{2} x^2 \left (A \left (2 a c+b^2\right )+2 a b C\right )+a \log (x) (a C+2 A b)+\frac{1}{3} B x^3 \left (2 a c+b^2\right )+2 a b B x+\frac{1}{6} c x^6 (A c+2 b C)+\frac{2}{5} b B c x^5+\frac{1}{7} B c^2 x^7+\frac{1}{8} c^2 C x^8$

[Out]

-(a^2*A)/(2*x^2) - (a^2*B)/x + 2*a*b*B*x + ((A*(b^2 + 2*a*c) + 2*a*b*C)*x^2)/2 + (B*(b^2 + 2*a*c)*x^3)/3 + ((2
*A*b*c + (b^2 + 2*a*c)*C)*x^4)/4 + (2*b*B*c*x^5)/5 + (c*(A*c + 2*b*C)*x^6)/6 + (B*c^2*x^7)/7 + (c^2*C*x^8)/8 +
a*(2*A*b + a*C)*Log[x]

________________________________________________________________________________________

Rubi [A]  time = 0.122592, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 28, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.036, Rules used = {1628} $-\frac{a^2 A}{2 x^2}-\frac{a^2 B}{x}+\frac{1}{4} x^4 \left (C \left (2 a c+b^2\right )+2 A b c\right )+\frac{1}{2} x^2 \left (A \left (2 a c+b^2\right )+2 a b C\right )+a \log (x) (a C+2 A b)+\frac{1}{3} B x^3 \left (2 a c+b^2\right )+2 a b B x+\frac{1}{6} c x^6 (A c+2 b C)+\frac{2}{5} b B c x^5+\frac{1}{7} B c^2 x^7+\frac{1}{8} c^2 C x^8$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(a^2*A)/(2*x^2) - (a^2*B)/x + 2*a*b*B*x + ((A*(b^2 + 2*a*c) + 2*a*b*C)*x^2)/2 + (B*(b^2 + 2*a*c)*x^3)/3 + ((2
*A*b*c + (b^2 + 2*a*c)*C)*x^4)/4 + (2*b*B*c*x^5)/5 + (c*(A*c + 2*b*C)*x^6)/6 + (B*c^2*x^7)/7 + (c^2*C*x^8)/8 +
a*(2*A*b + a*C)*Log[x]

Rule 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

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

Mathematica [A]  time = 0.108635, size = 139, normalized size = 0.93 $-\frac{a^2 (A+2 B x)}{2 x^2}+\frac{1}{6} a x \left (c x \left (6 A+4 B x+3 C x^2\right )+6 b (2 B+C x)\right )+a \log (x) (a C+2 A b)+\frac{1}{840} x^2 \left (140 A \left (3 b^2+3 b c x^2+c^2 x^4\right )+70 b^2 x (4 B+3 C x)+56 b c x^3 (6 B+5 C x)+15 c^2 x^5 (8 B+7 C x)\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(a^2*(A + 2*B*x))/(2*x^2) + (a*x*(6*b*(2*B + C*x) + c*x*(6*A + 4*B*x + 3*C*x^2)))/6 + (x^2*(70*b^2*x*(4*B + 3
*C*x) + 56*b*c*x^3*(6*B + 5*C*x) + 15*c^2*x^5*(8*B + 7*C*x) + 140*A*(3*b^2 + 3*b*c*x^2 + c^2*x^4)))/840 + a*(2
*A*b + a*C)*Log[x]

________________________________________________________________________________________

Maple [A]  time = 0.008, size = 148, normalized size = 1. \begin{align*}{\frac{{c}^{2}C{x}^{8}}{8}}+{\frac{B{c}^{2}{x}^{7}}{7}}+{\frac{A{x}^{6}{c}^{2}}{6}}+{\frac{C{x}^{6}bc}{3}}+{\frac{2\,bBc{x}^{5}}{5}}+{\frac{A{x}^{4}bc}{2}}+{\frac{C{x}^{4}ac}{2}}+{\frac{C{x}^{4}{b}^{2}}{4}}+{\frac{2\,B{x}^{3}ac}{3}}+{\frac{B{x}^{3}{b}^{2}}{3}}+A{x}^{2}ac+{\frac{A{x}^{2}{b}^{2}}{2}}+C{x}^{2}ab+2\,abBx-{\frac{B{a}^{2}}{x}}-{\frac{A{a}^{2}}{2\,{x}^{2}}}+2\,A\ln \left ( x \right ) ab+C\ln \left ( x \right ){a}^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*c^2*C*x^8+1/7*B*c^2*x^7+1/6*A*x^6*c^2+1/3*C*x^6*b*c+2/5*b*B*c*x^5+1/2*A*x^4*b*c+1/2*C*x^4*a*c+1/4*C*x^4*b^
2+2/3*B*x^3*a*c+1/3*B*x^3*b^2+A*x^2*a*c+1/2*A*x^2*b^2+C*x^2*a*b+2*a*b*B*x-a^2*B/x-1/2*a^2*A/x^2+2*A*ln(x)*a*b+
C*ln(x)*a^2

________________________________________________________________________________________

Maxima [A]  time = 0.943822, size = 188, normalized size = 1.26 \begin{align*} \frac{1}{8} \, C c^{2} x^{8} + \frac{1}{7} \, B c^{2} x^{7} + \frac{2}{5} \, B b c x^{5} + \frac{1}{6} \,{\left (2 \, C b c + A c^{2}\right )} x^{6} + \frac{1}{4} \,{\left (C b^{2} + 2 \,{\left (C a + A b\right )} c\right )} x^{4} + 2 \, B a b x + \frac{1}{3} \,{\left (B b^{2} + 2 \, B a c\right )} x^{3} + \frac{1}{2} \,{\left (2 \, C a b + A b^{2} + 2 \, A a c\right )} x^{2} +{\left (C a^{2} + 2 \, A a b\right )} \log \left (x\right ) - \frac{2 \, B a^{2} x + A a^{2}}{2 \, x^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)*(c*x^4+b*x^2+a)^2/x^3,x, algorithm="maxima")

[Out]

1/8*C*c^2*x^8 + 1/7*B*c^2*x^7 + 2/5*B*b*c*x^5 + 1/6*(2*C*b*c + A*c^2)*x^6 + 1/4*(C*b^2 + 2*(C*a + A*b)*c)*x^4
+ 2*B*a*b*x + 1/3*(B*b^2 + 2*B*a*c)*x^3 + 1/2*(2*C*a*b + A*b^2 + 2*A*a*c)*x^2 + (C*a^2 + 2*A*a*b)*log(x) - 1/2
*(2*B*a^2*x + A*a^2)/x^2

________________________________________________________________________________________

Fricas [A]  time = 1.30031, size = 362, normalized size = 2.43 \begin{align*} \frac{105 \, C c^{2} x^{10} + 120 \, B c^{2} x^{9} + 336 \, B b c x^{7} + 140 \,{\left (2 \, C b c + A c^{2}\right )} x^{8} + 210 \,{\left (C b^{2} + 2 \,{\left (C a + A b\right )} c\right )} x^{6} + 1680 \, B a b x^{3} + 280 \,{\left (B b^{2} + 2 \, B a c\right )} x^{5} + 420 \,{\left (2 \, C a b + A b^{2} + 2 \, A a c\right )} x^{4} - 840 \, B a^{2} x + 840 \,{\left (C a^{2} + 2 \, A a b\right )} x^{2} \log \left (x\right ) - 420 \, A a^{2}}{840 \, x^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)*(c*x^4+b*x^2+a)^2/x^3,x, algorithm="fricas")

[Out]

1/840*(105*C*c^2*x^10 + 120*B*c^2*x^9 + 336*B*b*c*x^7 + 140*(2*C*b*c + A*c^2)*x^8 + 210*(C*b^2 + 2*(C*a + A*b)
*c)*x^6 + 1680*B*a*b*x^3 + 280*(B*b^2 + 2*B*a*c)*x^5 + 420*(2*C*a*b + A*b^2 + 2*A*a*c)*x^4 - 840*B*a^2*x + 840
*(C*a^2 + 2*A*a*b)*x^2*log(x) - 420*A*a^2)/x^2

________________________________________________________________________________________

Sympy [A]  time = 0.641083, size = 151, normalized size = 1.01 \begin{align*} 2 B a b x + \frac{2 B b c x^{5}}{5} + \frac{B c^{2} x^{7}}{7} + \frac{C c^{2} x^{8}}{8} + a \left (2 A b + C a\right ) \log{\left (x \right )} + x^{6} \left (\frac{A c^{2}}{6} + \frac{C b c}{3}\right ) + x^{4} \left (\frac{A b c}{2} + \frac{C a c}{2} + \frac{C b^{2}}{4}\right ) + x^{3} \left (\frac{2 B a c}{3} + \frac{B b^{2}}{3}\right ) + x^{2} \left (A a c + \frac{A b^{2}}{2} + C a b\right ) - \frac{A a^{2} + 2 B a^{2} x}{2 x^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x**2+B*x+A)*(c*x**4+b*x**2+a)**2/x**3,x)

[Out]

2*B*a*b*x + 2*B*b*c*x**5/5 + B*c**2*x**7/7 + C*c**2*x**8/8 + a*(2*A*b + C*a)*log(x) + x**6*(A*c**2/6 + C*b*c/3
) + x**4*(A*b*c/2 + C*a*c/2 + C*b**2/4) + x**3*(2*B*a*c/3 + B*b**2/3) + x**2*(A*a*c + A*b**2/2 + C*a*b) - (A*a
**2 + 2*B*a**2*x)/(2*x**2)

________________________________________________________________________________________

Giac [A]  time = 1.13592, size = 200, normalized size = 1.34 \begin{align*} \frac{1}{8} \, C c^{2} x^{8} + \frac{1}{7} \, B c^{2} x^{7} + \frac{1}{3} \, C b c x^{6} + \frac{1}{6} \, A c^{2} x^{6} + \frac{2}{5} \, B b c x^{5} + \frac{1}{4} \, C b^{2} x^{4} + \frac{1}{2} \, C a c x^{4} + \frac{1}{2} \, A b c x^{4} + \frac{1}{3} \, B b^{2} x^{3} + \frac{2}{3} \, B a c x^{3} + C a b x^{2} + \frac{1}{2} \, A b^{2} x^{2} + A a c x^{2} + 2 \, B a b x +{\left (C a^{2} + 2 \, A a b\right )} \log \left ({\left | x \right |}\right ) - \frac{2 \, B a^{2} x + A a^{2}}{2 \, x^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)*(c*x^4+b*x^2+a)^2/x^3,x, algorithm="giac")

[Out]

1/8*C*c^2*x^8 + 1/7*B*c^2*x^7 + 1/3*C*b*c*x^6 + 1/6*A*c^2*x^6 + 2/5*B*b*c*x^5 + 1/4*C*b^2*x^4 + 1/2*C*a*c*x^4
+ 1/2*A*b*c*x^4 + 1/3*B*b^2*x^3 + 2/3*B*a*c*x^3 + C*a*b*x^2 + 1/2*A*b^2*x^2 + A*a*c*x^2 + 2*B*a*b*x + (C*a^2 +
2*A*a*b)*log(abs(x)) - 1/2*(2*B*a^2*x + A*a^2)/x^2