3.874 \(\int \frac{(A+B x) (a+b x+c x^2)^3}{x^5} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.123429, 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} \[ -\frac{a^2 (a B+3 A b)}{3 x^3}-\frac{a^3 A}{4 x^4}+\log (x) \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )-\frac{3 a \left (A \left (a c+b^2\right )+a b B\right )}{2 x^2}+3 c x \left (a B c+A b c+b^2 B\right )-\frac{A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )}{x}+\frac{1}{2} c^2 x^2 (A c+3 b B)+\frac{1}{3} B c^3 x^3 \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + b*x + c*x^2)^3)/x^5,x]

[Out]

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

Mathematica [A]  time = 0.0633141, size = 154, normalized size = 0.99 \[ \frac{-4 a^2 x (a B+3 A b)-3 a^3 A+12 x^4 \log (x) \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+36 c x^5 \left (a B c+A b c+b^2 B\right )-12 x^3 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )-18 a x^2 \left (A \left (a c+b^2\right )+a b B\right )+6 c^2 x^6 (A c+3 b B)+4 B c^3 x^7}{12 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + b*x + c*x^2)^3)/x^5,x]

[Out]

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

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Maple [A]  time = 0.007, size = 183, normalized size = 1.2 \begin{align*}{\frac{B{c}^{3}{x}^{3}}{3}}+{\frac{A{x}^{2}{c}^{3}}{2}}+{\frac{3\,B{x}^{2}b{c}^{2}}{2}}+3\,Ab{c}^{2}x+3\,Ba{c}^{2}x+3\,B{b}^{2}cx+3\,A\ln \left ( x \right ) a{c}^{2}+3\,A\ln \left ( x \right ){b}^{2}c+6\,B\ln \left ( x \right ) abc+{b}^{3}B\ln \left ( x \right ) -{\frac{Ab{a}^{2}}{{x}^{3}}}-{\frac{B{a}^{3}}{3\,{x}^{3}}}-{\frac{3\,A{a}^{2}c}{2\,{x}^{2}}}-{\frac{3\,Aa{b}^{2}}{2\,{x}^{2}}}-{\frac{3\,B{a}^{2}b}{2\,{x}^{2}}}-6\,{\frac{Aabc}{x}}-{\frac{A{b}^{3}}{x}}-3\,{\frac{B{a}^{2}c}{x}}-3\,{\frac{Ba{b}^{2}}{x}}-{\frac{A{a}^{3}}{4\,{x}^{4}}} \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^5,x)

[Out]

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

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Maxima [A]  time = 1.1645, size = 220, normalized size = 1.41 \begin{align*} \frac{1}{3} \, B c^{3} x^{3} + \frac{1}{2} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{2} + 3 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x +{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} \log \left (x\right ) - \frac{3 \, A a^{3} + 12 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + 18 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 4 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{12 \, x^{4}} \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^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.18062, size = 374, normalized size = 2.4 \begin{align*} \frac{4 \, B c^{3} x^{7} + 6 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 36 \,{\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} \log \left (x\right ) - 3 \, A a^{3} - 12 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} - 18 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} - 4 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{12 \, x^{4}} \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^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 5.87902, size = 184, normalized size = 1.18 \begin{align*} \frac{B c^{3} x^{3}}{3} + x^{2} \left (\frac{A c^{3}}{2} + \frac{3 B b c^{2}}{2}\right ) + x \left (3 A b c^{2} + 3 B a c^{2} + 3 B b^{2} c\right ) + \left (3 A a c^{2} + 3 A b^{2} c + 6 B a b c + B b^{3}\right ) \log{\left (x \right )} - \frac{3 A a^{3} + x^{3} \left (72 A a b c + 12 A b^{3} + 36 B a^{2} c + 36 B a b^{2}\right ) + x^{2} \left (18 A a^{2} c + 18 A a b^{2} + 18 B a^{2} b\right ) + x \left (12 A a^{2} b + 4 B a^{3}\right )}{12 x^{4}} \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**5,x)

[Out]

B*c**3*x**3/3 + x**2*(A*c**3/2 + 3*B*b*c**2/2) + x*(3*A*b*c**2 + 3*B*a*c**2 + 3*B*b**2*c) + (3*A*a*c**2 + 3*A*
b**2*c + 6*B*a*b*c + B*b**3)*log(x) - (3*A*a**3 + x**3*(72*A*a*b*c + 12*A*b**3 + 36*B*a**2*c + 36*B*a*b**2) +
x**2*(18*A*a**2*c + 18*A*a*b**2 + 18*B*a**2*b) + x*(12*A*a**2*b + 4*B*a**3))/(12*x**4)

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Giac [A]  time = 1.14299, size = 224, normalized size = 1.44 \begin{align*} \frac{1}{3} \, B c^{3} x^{3} + \frac{3}{2} \, B b c^{2} x^{2} + \frac{1}{2} \, A c^{3} x^{2} + 3 \, B b^{2} c x + 3 \, B a c^{2} x + 3 \, A b c^{2} x +{\left (B b^{3} + 6 \, B a b c + 3 \, A b^{2} c + 3 \, A a c^{2}\right )} \log \left ({\left | x \right |}\right ) - \frac{3 \, A a^{3} + 12 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \, B a^{2} c + 6 \, A a b c\right )} x^{3} + 18 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 4 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{12 \, x^{4}} \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^5,x, algorithm="giac")

[Out]

1/3*B*c^3*x^3 + 3/2*B*b*c^2*x^2 + 1/2*A*c^3*x^2 + 3*B*b^2*c*x + 3*B*a*c^2*x + 3*A*b*c^2*x + (B*b^3 + 6*B*a*b*c
+ 3*A*b^2*c + 3*A*a*c^2)*log(abs(x)) - 1/12*(3*A*a^3 + 12*(3*B*a*b^2 + A*b^3 + 3*B*a^2*c + 6*A*a*b*c)*x^3 + 1
8*(B*a^2*b + A*a*b^2 + A*a^2*c)*x^2 + 4*(B*a^3 + 3*A*a^2*b)*x)/x^4