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

Optimal. Leaf size=155 \[ -\frac{a^2 (a B+3 A b)}{2 x^2}-\frac{a^3 A}{3 x^3}+x \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{3}{2} c x^2 \left (a B c+A b c+b^2 B\right )-\frac{3 a \left (A \left (a c+b^2\right )+a b B\right )}{x}+\log (x) \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{1}{3} c^2 x^3 (A c+3 b B)+\frac{1}{4} B c^3 x^4 \]

[Out]

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

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Rubi [A]  time = 0.136451, antiderivative size = 155, 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} \[ -\frac{a^2 (a B+3 A b)}{2 x^2}-\frac{a^3 A}{3 x^3}+x \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{3}{2} c x^2 \left (a B c+A b c+b^2 B\right )-\frac{3 a \left (A \left (a c+b^2\right )+a b B\right )}{x}+\log (x) \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{1}{3} c^2 x^3 (A c+3 b B)+\frac{1}{4} B c^3 x^4 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0621291, size = 160, normalized size = 1.03 \[ \frac{-18 a^2 x (A (b+2 c x)+2 b B x)-2 a^3 (2 A+3 B x)+18 a x^2 \left (B c x^2 (4 b+c x)-2 A \left (b^2-c^2 x^2\right )\right )+12 x^3 \log (x) \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+x^4 \left (18 b^2 c (2 A+B x)+6 b c^2 x (3 A+2 B x)+c^3 x^2 (4 A+3 B x)+12 b^3 B\right )}{12 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^3*(2*A + 3*B*x) + x^4*(12*b^3*B + 18*b^2*c*(2*A + B*x) + 6*b*c^2*x*(3*A + 2*B*x) + c^3*x^2*(4*A + 3*B*x)
) - 18*a^2*x*(2*b*B*x + A*(b + 2*c*x)) + 18*a*x^2*(B*c*x^2*(4*b + c*x) - 2*A*(b^2 - c^2*x^2)) + 12*(3*a*B*(b^2
+ a*c) + A*(b^3 + 6*a*b*c))*x^3*Log[x])/(12*x^3)

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Maple [A]  time = 0.007, size = 179, normalized size = 1.2 \begin{align*}{\frac{B{c}^{3}{x}^{4}}{4}}+{\frac{A{x}^{3}{c}^{3}}{3}}+B{x}^{3}b{c}^{2}+{\frac{3\,A{x}^{2}b{c}^{2}}{2}}+{\frac{3\,B{x}^{2}a{c}^{2}}{2}}+{\frac{3\,B{x}^{2}{b}^{2}c}{2}}+3\,aA{c}^{2}x+3\,A{b}^{2}cx+6\,abBcx+{b}^{3}Bx+6\,A\ln \left ( x \right ) abc+A{b}^{3}\ln \left ( x \right ) +3\,B\ln \left ( x \right ){a}^{2}c+3\,B\ln \left ( x \right ) a{b}^{2}-{\frac{A{a}^{3}}{3\,{x}^{3}}}-{\frac{3\,Ab{a}^{2}}{2\,{x}^{2}}}-{\frac{B{a}^{3}}{2\,{x}^{2}}}-3\,{\frac{A{a}^{2}c}{x}}-3\,{\frac{Aa{b}^{2}}{x}}-3\,{\frac{B{a}^{2}b}{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^4,x)

[Out]

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

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Maxima [A]  time = 1.08541, size = 219, normalized size = 1.41 \begin{align*} \frac{1}{4} \, B c^{3} x^{4} + \frac{1}{3} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{3} + \frac{3}{2} \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{2} +{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x +{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} \log \left (x\right ) - \frac{2 \, A a^{3} + 18 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 3 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{6 \, x^{3}} \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^4,x, algorithm="maxima")

[Out]

1/4*B*c^3*x^4 + 1/3*(3*B*b*c^2 + A*c^3)*x^3 + 3/2*(B*b^2*c + (B*a + A*b)*c^2)*x^2 + (B*b^3 + 3*A*a*c^2 + 3*(2*
B*a*b + A*b^2)*c)*x + (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*log(x) - 1/6*(2*A*a^3 + 18*(B*a^2*b + A*a*b^
2 + A*a^2*c)*x^2 + 3*(B*a^3 + 3*A*a^2*b)*x)/x^3

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Fricas [A]  time = 1.27572, size = 374, normalized size = 2.41 \begin{align*} \frac{3 \, B c^{3} x^{7} + 4 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 18 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{5} + 12 \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} + 12 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} \log \left (x\right ) - 4 \, A a^{3} - 36 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} - 6 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{12 \, x^{3}} \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^4,x, algorithm="fricas")

[Out]

1/12*(3*B*c^3*x^7 + 4*(3*B*b*c^2 + A*c^3)*x^6 + 18*(B*b^2*c + (B*a + A*b)*c^2)*x^5 + 12*(B*b^3 + 3*A*a*c^2 + 3
*(2*B*a*b + A*b^2)*c)*x^4 + 12*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^3*log(x) - 4*A*a^3 - 36*(B*a^2*b
+ A*a*b^2 + A*a^2*c)*x^2 - 6*(B*a^3 + 3*A*a^2*b)*x)/x^3

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Sympy [A]  time = 1.86521, size = 184, normalized size = 1.19 \begin{align*} \frac{B c^{3} x^{4}}{4} + x^{3} \left (\frac{A c^{3}}{3} + B b c^{2}\right ) + x^{2} \left (\frac{3 A b c^{2}}{2} + \frac{3 B a c^{2}}{2} + \frac{3 B b^{2} c}{2}\right ) + x \left (3 A a c^{2} + 3 A b^{2} c + 6 B a b c + B b^{3}\right ) + \left (6 A a b c + A b^{3} + 3 B a^{2} c + 3 B a b^{2}\right ) \log{\left (x \right )} - \frac{2 A a^{3} + x^{2} \left (18 A a^{2} c + 18 A a b^{2} + 18 B a^{2} b\right ) + x \left (9 A a^{2} b + 3 B a^{3}\right )}{6 x^{3}} \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**4,x)

[Out]

B*c**3*x**4/4 + x**3*(A*c**3/3 + B*b*c**2) + x**2*(3*A*b*c**2/2 + 3*B*a*c**2/2 + 3*B*b**2*c/2) + x*(3*A*a*c**2
+ 3*A*b**2*c + 6*B*a*b*c + B*b**3) + (6*A*a*b*c + A*b**3 + 3*B*a**2*c + 3*B*a*b**2)*log(x) - (2*A*a**3 + x**2
*(18*A*a**2*c + 18*A*a*b**2 + 18*B*a**2*b) + x*(9*A*a**2*b + 3*B*a**3))/(6*x**3)

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Giac [A]  time = 1.31961, size = 228, normalized size = 1.47 \begin{align*} \frac{1}{4} \, B c^{3} x^{4} + B b c^{2} x^{3} + \frac{1}{3} \, A c^{3} x^{3} + \frac{3}{2} \, B b^{2} c x^{2} + \frac{3}{2} \, B a c^{2} x^{2} + \frac{3}{2} \, A b c^{2} x^{2} + B b^{3} x + 6 \, B a b c x + 3 \, A b^{2} c x + 3 \, A a c^{2} x +{\left (3 \, B a b^{2} + A b^{3} + 3 \, B a^{2} c + 6 \, A a b c\right )} \log \left ({\left | x \right |}\right ) - \frac{2 \, A a^{3} + 18 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 3 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{6 \, x^{3}} \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^4,x, algorithm="giac")

[Out]

1/4*B*c^3*x^4 + B*b*c^2*x^3 + 1/3*A*c^3*x^3 + 3/2*B*b^2*c*x^2 + 3/2*B*a*c^2*x^2 + 3/2*A*b*c^2*x^2 + B*b^3*x +
6*B*a*b*c*x + 3*A*b^2*c*x + 3*A*a*c^2*x + (3*B*a*b^2 + A*b^3 + 3*B*a^2*c + 6*A*a*b*c)*log(abs(x)) - 1/6*(2*A*a
^3 + 18*(B*a^2*b + A*a*b^2 + A*a^2*c)*x^2 + 3*(B*a^3 + 3*A*a^2*b)*x)/x^3