3.875 \(\int \frac{(A+B x) (a+b x+c x^2)^3}{x^6} \, dx\)

Optimal. Leaf size=154 \[ -\frac{a^2 (a B+3 A b)}{4 x^4}-\frac{a^3 A}{5 x^5}-\frac{3 a A c^2+6 a b B c+3 A b^2 c+b^3 B}{x}-\frac{a \left (A \left (a c+b^2\right )+a b B\right )}{x^3}-\frac{A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )}{2 x^2}+3 c \log (x) \left (a B c+A b c+b^2 B\right )+c^2 x (A c+3 b B)+\frac{1}{2} B c^3 x^2 \]

[Out]

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

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Rubi [A]  time = 0.115829, antiderivative size = 154, 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} \[ -\frac{a^2 (a B+3 A b)}{4 x^4}-\frac{a^3 A}{5 x^5}-\frac{3 a A c^2+6 a b B c+3 A b^2 c+b^3 B}{x}-\frac{a \left (A \left (a c+b^2\right )+a b B\right )}{x^3}-\frac{A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )}{2 x^2}+3 c \log (x) \left (a B c+A b c+b^2 B\right )+c^2 x (A c+3 b B)+\frac{1}{2} B c^3 x^2 \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + b*x + c*x^2)^3)/x^6,x]

[Out]

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

Mathematica [A]  time = 0.0890341, size = 161, normalized size = 1.05 \[ -\frac{5 a^2 x \left (3 A b+4 A c x+4 b B x+6 B c x^2\right )+a^3 (4 A+5 B x)+10 a x^2 \left (2 A \left (b^2+3 b c x+3 c^2 x^2\right )+3 b B x (b+4 c x)\right )-60 c x^5 \log (x) \left (a B c+A b c+b^2 B\right )+10 x^3 \left (A \left (6 b^2 c x+b^3-2 c^3 x^3\right )-B x \left (-2 b^3+6 b c^2 x^2+c^3 x^3\right )\right )}{20 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + b*x + c*x^2)^3)/x^6,x]

[Out]

-(a^3*(4*A + 5*B*x) + 5*a^2*x*(3*A*b + 4*b*B*x + 4*A*c*x + 6*B*c*x^2) + 10*a*x^2*(3*b*B*x*(b + 4*c*x) + 2*A*(b
^2 + 3*b*c*x + 3*c^2*x^2)) + 10*x^3*(A*(b^3 + 6*b^2*c*x - 2*c^3*x^3) - B*x*(-2*b^3 + 6*b*c^2*x^2 + c^3*x^3)) -
60*c*(b^2*B + A*b*c + a*B*c)*x^5*Log[x])/(20*x^5)

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Maple [A]  time = 0.009, size = 186, normalized size = 1.2 \begin{align*}{\frac{B{c}^{3}{x}^{2}}{2}}+A{c}^{3}x+3\,Bb{c}^{2}x+3\,A\ln \left ( x \right ) b{c}^{2}+3\,B\ln \left ( x \right ) a{c}^{2}+3\,B\ln \left ( x \right ){b}^{2}c-{\frac{A{a}^{2}c}{{x}^{3}}}-{\frac{Aa{b}^{2}}{{x}^{3}}}-{\frac{B{a}^{2}b}{{x}^{3}}}-3\,{\frac{Aabc}{{x}^{2}}}-{\frac{A{b}^{3}}{2\,{x}^{2}}}-{\frac{3\,B{a}^{2}c}{2\,{x}^{2}}}-{\frac{3\,Ba{b}^{2}}{2\,{x}^{2}}}-3\,{\frac{aA{c}^{2}}{x}}-3\,{\frac{A{b}^{2}c}{x}}-6\,{\frac{abBc}{x}}-{\frac{{b}^{3}B}{x}}-{\frac{A{a}^{3}}{5\,{x}^{5}}}-{\frac{3\,Ab{a}^{2}}{4\,{x}^{4}}}-{\frac{B{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^6,x)

[Out]

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

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Maxima [A]  time = 1.09298, size = 220, normalized size = 1.43 \begin{align*} \frac{1}{2} \, B c^{3} x^{2} +{\left (3 \, B b c^{2} + A c^{3}\right )} x + 3 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} \log \left (x\right ) - \frac{20 \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} + 4 \, A a^{3} + 10 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + 20 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 5 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{20 \, x^{5}} \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^6,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.15193, size = 377, normalized size = 2.45 \begin{align*} \frac{10 \, B c^{3} x^{7} + 20 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 60 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{5} \log \left (x\right ) - 20 \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} - 4 \, A a^{3} - 10 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} - 20 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} - 5 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{20 \, x^{5}} \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^6,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 13.9309, size = 175, normalized size = 1.14 \begin{align*} \frac{B c^{3} x^{2}}{2} + 3 c \left (A b c + B a c + B b^{2}\right ) \log{\left (x \right )} + x \left (A c^{3} + 3 B b c^{2}\right ) - \frac{4 A a^{3} + x^{4} \left (60 A a c^{2} + 60 A b^{2} c + 120 B a b c + 20 B b^{3}\right ) + x^{3} \left (60 A a b c + 10 A b^{3} + 30 B a^{2} c + 30 B a b^{2}\right ) + x^{2} \left (20 A a^{2} c + 20 A a b^{2} + 20 B a^{2} b\right ) + x \left (15 A a^{2} b + 5 B a^{3}\right )}{20 x^{5}} \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**6,x)

[Out]

B*c**3*x**2/2 + 3*c*(A*b*c + B*a*c + B*b**2)*log(x) + x*(A*c**3 + 3*B*b*c**2) - (4*A*a**3 + x**4*(60*A*a*c**2
+ 60*A*b**2*c + 120*B*a*b*c + 20*B*b**3) + x**3*(60*A*a*b*c + 10*A*b**3 + 30*B*a**2*c + 30*B*a*b**2) + x**2*(2
0*A*a**2*c + 20*A*a*b**2 + 20*B*a**2*b) + x*(15*A*a**2*b + 5*B*a**3))/(20*x**5)

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Giac [A]  time = 1.28801, size = 219, normalized size = 1.42 \begin{align*} \frac{1}{2} \, B c^{3} x^{2} + 3 \, B b c^{2} x + A c^{3} x + 3 \,{\left (B b^{2} c + B a c^{2} + A b c^{2}\right )} \log \left ({\left | x \right |}\right ) - \frac{20 \,{\left (B b^{3} + 6 \, B a b c + 3 \, A b^{2} c + 3 \, A a c^{2}\right )} x^{4} + 4 \, A a^{3} + 10 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \, B a^{2} c + 6 \, A a b c\right )} x^{3} + 20 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 5 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{20 \, x^{5}} \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^6,x, algorithm="giac")

[Out]

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