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

Optimal. Leaf size=156 \[ a^2 \log (x) (a B+3 A b)-\frac{a^3 A}{x}+\frac{1}{3} x^3 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{3}{4} c x^4 \left (a B c+A b c+b^2 B\right )+\frac{1}{2} x^2 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+3 a x \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{5} c^2 x^5 (A c+3 b B)+\frac{1}{6} B c^3 x^6 \]

[Out]

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

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Rubi [A]  time = 0.107991, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {765} \[ a^2 \log (x) (a B+3 A b)-\frac{a^3 A}{x}+\frac{1}{3} x^3 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{3}{4} c x^4 \left (a B c+A b c+b^2 B\right )+\frac{1}{2} x^2 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+3 a x \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{5} c^2 x^5 (A c+3 b B)+\frac{1}{6} B c^3 x^6 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 765

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand
Integrand[(e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, e, f, g, m}, x] && IntegerQ[p] && (
GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

Mathematica [A]  time = 0.0848783, size = 156, normalized size = 1. \[ a^2 \log (x) (a B+3 A b)-\frac{a^3 A}{x}+\frac{1}{3} x^3 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{3}{4} c x^4 \left (a B c+A b c+b^2 B\right )+\frac{1}{2} x^2 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+3 a x \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{5} c^2 x^5 (A c+3 b B)+\frac{1}{6} B c^3 x^6 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.006, size = 183, normalized size = 1.2 \begin{align*}{\frac{B{c}^{3}{x}^{6}}{6}}+{\frac{A{c}^{3}{x}^{5}}{5}}+{\frac{3\,B{x}^{5}b{c}^{2}}{5}}+{\frac{3\,A{x}^{4}b{c}^{2}}{4}}+{\frac{3\,aB{c}^{2}{x}^{4}}{4}}+{\frac{3\,B{x}^{4}{b}^{2}c}{4}}+aA{c}^{2}{x}^{3}+A{x}^{3}{b}^{2}c+2\,B{x}^{3}abc+{\frac{{b}^{3}B{x}^{3}}{3}}+3\,A{x}^{2}abc+{\frac{A{b}^{3}{x}^{2}}{2}}+{\frac{3\,{a}^{2}Bc{x}^{2}}{2}}+{\frac{3\,B{x}^{2}a{b}^{2}}{2}}+3\,{a}^{2}Acx+3\,Aa{b}^{2}x+3\,B{a}^{2}bx+3\,A\ln \left ( x \right ){a}^{2}b+{a}^{3}B\ln \left ( x \right ) -{\frac{A{a}^{3}}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02829, size = 219, normalized size = 1.4 \begin{align*} \frac{1}{6} \, B c^{3} x^{6} + \frac{1}{5} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{5} + \frac{3}{4} \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{4} + \frac{1}{3} \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{3} - \frac{A a^{3}}{x} + \frac{1}{2} \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{2} + 3 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x +{\left (B a^{3} + 3 \, A a^{2} b\right )} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.26357, size = 378, normalized size = 2.42 \begin{align*} \frac{10 \, B c^{3} x^{7} + 12 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 45 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{5} + 20 \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} - 60 \, A a^{3} + 30 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + 180 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 60 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x \log \left (x\right )}{60 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(10*B*c^3*x^7 + 12*(3*B*b*c^2 + A*c^3)*x^6 + 45*(B*b^2*c + (B*a + A*b)*c^2)*x^5 + 20*(B*b^3 + 3*A*a*c^2 +
3*(2*B*a*b + A*b^2)*c)*x^4 - 60*A*a^3 + 30*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^3 + 180*(B*a^2*b + A
*a*b^2 + A*a^2*c)*x^2 + 60*(B*a^3 + 3*A*a^2*b)*x*log(x))/x

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Sympy [A]  time = 0.61875, size = 184, normalized size = 1.18 \begin{align*} - \frac{A a^{3}}{x} + \frac{B c^{3} x^{6}}{6} + a^{2} \left (3 A b + B a\right ) \log{\left (x \right )} + x^{5} \left (\frac{A c^{3}}{5} + \frac{3 B b c^{2}}{5}\right ) + x^{4} \left (\frac{3 A b c^{2}}{4} + \frac{3 B a c^{2}}{4} + \frac{3 B b^{2} c}{4}\right ) + x^{3} \left (A a c^{2} + A b^{2} c + 2 B a b c + \frac{B b^{3}}{3}\right ) + x^{2} \left (3 A a b c + \frac{A b^{3}}{2} + \frac{3 B a^{2} c}{2} + \frac{3 B a b^{2}}{2}\right ) + x \left (3 A a^{2} c + 3 A a b^{2} + 3 B a^{2} b\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.21102, size = 247, normalized size = 1.58 \begin{align*} \frac{1}{6} \, B c^{3} x^{6} + \frac{3}{5} \, B b c^{2} x^{5} + \frac{1}{5} \, A c^{3} x^{5} + \frac{3}{4} \, B b^{2} c x^{4} + \frac{3}{4} \, B a c^{2} x^{4} + \frac{3}{4} \, A b c^{2} x^{4} + \frac{1}{3} \, B b^{3} x^{3} + 2 \, B a b c x^{3} + A b^{2} c x^{3} + A a c^{2} x^{3} + \frac{3}{2} \, B a b^{2} x^{2} + \frac{1}{2} \, A b^{3} x^{2} + \frac{3}{2} \, B a^{2} c x^{2} + 3 \, A a b c x^{2} + 3 \, B a^{2} b x + 3 \, A a b^{2} x + 3 \, A a^{2} c x - \frac{A a^{3}}{x} +{\left (B a^{3} + 3 \, A a^{2} b\right )} \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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