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

Optimal. Leaf size=153 \[ -\frac{a^2 (a B+3 A b)}{x}-\frac{a^3 A}{2 x^2}+\frac{1}{2} x^2 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+c x^3 \left (a B c+A b c+b^2 B\right )+x \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+3 a \log (x) \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{4} c^2 x^4 (A c+3 b B)+\frac{1}{5} B c^3 x^5 \]

[Out]

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

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Rubi [A]  time = 0.124579, antiderivative size = 153, 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} \[ -\frac{a^2 (a B+3 A b)}{x}-\frac{a^3 A}{2 x^2}+\frac{1}{2} x^2 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+c x^3 \left (a B c+A b c+b^2 B\right )+x \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+3 a \log (x) \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{4} c^2 x^4 (A c+3 b B)+\frac{1}{5} B c^3 x^5 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0808553, size = 153, normalized size = 1. \[ -\frac{a^2 (a B+3 A b)}{x}-\frac{a^3 A}{2 x^2}+\frac{1}{2} x^2 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+c x^3 \left (a B c+A b c+b^2 B\right )+x \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+3 a \log (x) \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{4} c^2 x^4 (A c+3 b B)+\frac{1}{5} B c^3 x^5 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.007, size = 179, normalized size = 1.2 \begin{align*}{\frac{B{c}^{3}{x}^{5}}{5}}+{\frac{A{c}^{3}{x}^{4}}{4}}+{\frac{3\,B{x}^{4}b{c}^{2}}{4}}+A{x}^{3}b{c}^{2}+aB{c}^{2}{x}^{3}+B{x}^{3}{b}^{2}c+{\frac{3\,aA{c}^{2}{x}^{2}}{2}}+{\frac{3\,A{x}^{2}{b}^{2}c}{2}}+3\,B{x}^{2}abc+{\frac{{b}^{3}B{x}^{2}}{2}}+6\,Aabcx+A{b}^{3}x+3\,{a}^{2}Bcx+3\,Ba{b}^{2}x+3\,{a}^{2}Ac\ln \left ( x \right ) +3\,A\ln \left ( x \right ) a{b}^{2}+3\,B\ln \left ( x \right ){a}^{2}b-{\frac{A{a}^{3}}{2\,{x}^{2}}}-3\,{\frac{Ab{a}^{2}}{x}}-{\frac{B{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^3,x)

[Out]

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

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Maxima [A]  time = 1.06765, size = 217, normalized size = 1.42 \begin{align*} \frac{1}{5} \, B c^{3} x^{5} + \frac{1}{4} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{4} +{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{3} + \frac{1}{2} \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{2} +{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x + 3 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} \log \left (x\right ) - \frac{A a^{3} + 2 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{2 \, x^{2}} \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^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.28352, size = 377, normalized size = 2.46 \begin{align*} \frac{4 \, B c^{3} x^{7} + 5 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 20 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{5} + 10 \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} - 10 \, A a^{3} + 20 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + 60 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} \log \left (x\right ) - 20 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{20 \, x^{2}} \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^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.890518, size = 173, normalized size = 1.13 \begin{align*} \frac{B c^{3} x^{5}}{5} + 3 a \left (A a c + A b^{2} + B a b\right ) \log{\left (x \right )} + x^{4} \left (\frac{A c^{3}}{4} + \frac{3 B b c^{2}}{4}\right ) + x^{3} \left (A b c^{2} + B a c^{2} + B b^{2} c\right ) + x^{2} \left (\frac{3 A a c^{2}}{2} + \frac{3 A b^{2} c}{2} + 3 B a b c + \frac{B b^{3}}{2}\right ) + x \left (6 A a b c + A b^{3} + 3 B a^{2} c + 3 B a b^{2}\right ) - \frac{A a^{3} + x \left (6 A a^{2} b + 2 B a^{3}\right )}{2 x^{2}} \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**3,x)

[Out]

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

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Giac [A]  time = 1.32264, size = 235, normalized size = 1.54 \begin{align*} \frac{1}{5} \, B c^{3} x^{5} + \frac{3}{4} \, B b c^{2} x^{4} + \frac{1}{4} \, A c^{3} x^{4} + B b^{2} c x^{3} + B a c^{2} x^{3} + A b c^{2} x^{3} + \frac{1}{2} \, B b^{3} x^{2} + 3 \, B a b c x^{2} + \frac{3}{2} \, A b^{2} c x^{2} + \frac{3}{2} \, A a c^{2} x^{2} + 3 \, B a b^{2} x + A b^{3} x + 3 \, B a^{2} c x + 6 \, A a b c x + 3 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} \log \left ({\left | x \right |}\right ) - \frac{A a^{3} + 2 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{2 \, x^{2}} \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^3,x, algorithm="giac")

[Out]

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