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

Optimal. Leaf size=157 \[ a^2 x (a B+3 A b)+a^3 A \log (x)+\frac{1}{4} x^4 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{3}{5} c x^5 \left (a B c+A b c+b^2 B\right )+\frac{1}{3} x^3 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{3}{2} a x^2 \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{6} c^2 x^6 (A c+3 b B)+\frac{1}{7} B c^3 x^7 \]

[Out]

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

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Rubi [A]  time = 0.110567, antiderivative size = 157, 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} \[ a^2 x (a B+3 A b)+a^3 A \log (x)+\frac{1}{4} x^4 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{3}{5} c x^5 \left (a B c+A b c+b^2 B\right )+\frac{1}{3} x^3 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{3}{2} a x^2 \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{6} c^2 x^6 (A c+3 b B)+\frac{1}{7} B c^3 x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

a^2*(3*A*b + a*B)*x + (3*a*(a*b*B + A*(b^2 + a*c))*x^2)/2 + ((3*a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))*x^3)/3 +
((b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^4)/4 + (3*c*(b^2*B + A*b*c + a*B*c)*x^5)/5 + (c^2*(3*b*B + A*c)
*x^6)/6 + (B*c^3*x^7)/7 + a^3*A*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} \, dx &=\int \left (a^2 (3 A b+a B)+\frac{a^3 A}{x}+3 a \left (a b B+A \left (b^2+a c\right )\right ) x+\left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^2+\left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^3+3 c \left (b^2 B+A b c+a B c\right ) x^4+c^2 (3 b B+A c) x^5+B c^3 x^6\right ) \, dx\\ &=a^2 (3 A b+a B) x+\frac{3}{2} a \left (a b B+A \left (b^2+a c\right )\right ) x^2+\frac{1}{3} \left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^3+\frac{1}{4} \left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^4+\frac{3}{5} c \left (b^2 B+A b c+a B c\right ) x^5+\frac{1}{6} c^2 (3 b B+A c) x^6+\frac{1}{7} B c^3 x^7+a^3 A \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0650266, size = 157, normalized size = 1. \[ a^2 x (a B+3 A b)+a^3 A \log (x)+\frac{1}{4} x^4 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{3}{5} c x^5 \left (a B c+A b c+b^2 B\right )+\frac{1}{3} x^3 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{3}{2} a x^2 \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{6} c^2 x^6 (A c+3 b B)+\frac{1}{7} B c^3 x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.004, size = 185, normalized size = 1.2 \begin{align*}{\frac{B{c}^{3}{x}^{7}}{7}}+{\frac{A{c}^{3}{x}^{6}}{6}}+{\frac{B{x}^{6}b{c}^{2}}{2}}+{\frac{3\,A{x}^{5}b{c}^{2}}{5}}+{\frac{3\,aB{c}^{2}{x}^{5}}{5}}+{\frac{3\,B{x}^{5}{b}^{2}c}{5}}+{\frac{3\,aA{c}^{2}{x}^{4}}{4}}+{\frac{3\,A{x}^{4}{b}^{2}c}{4}}+{\frac{3\,B{x}^{4}abc}{2}}+{\frac{B{x}^{4}{b}^{3}}{4}}+2\,A{x}^{3}abc+{\frac{A{b}^{3}{x}^{3}}{3}}+{a}^{2}Bc{x}^{3}+B{x}^{3}a{b}^{2}+{\frac{3\,{a}^{2}Ac{x}^{2}}{2}}+{\frac{3\,A{x}^{2}a{b}^{2}}{2}}+{\frac{3\,B{x}^{2}{a}^{2}b}{2}}+3\,A{a}^{2}bx+{a}^{3}Bx+{a}^{3}A\ln \left ( x \right ) \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,x)

[Out]

1/7*B*c^3*x^7+1/6*A*c^3*x^6+1/2*B*x^6*b*c^2+3/5*A*x^5*b*c^2+3/5*a*B*c^2*x^5+3/5*B*x^5*b^2*c+3/4*a*A*c^2*x^4+3/
4*A*x^4*b^2*c+3/2*B*x^4*a*b*c+1/4*B*x^4*b^3+2*A*x^3*a*b*c+1/3*A*b^3*x^3+a^2*B*c*x^3+B*x^3*a*b^2+3/2*a^2*A*c*x^
2+3/2*A*x^2*a*b^2+3/2*B*x^2*a^2*b+3*A*a^2*b*x+a^3*B*x+a^3*A*ln(x)

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Maxima [A]  time = 1.04642, size = 217, normalized size = 1.38 \begin{align*} \frac{1}{7} \, B c^{3} x^{7} + \frac{1}{6} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + \frac{3}{5} \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{5} + \frac{1}{4} \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} + A a^{3} \log \left (x\right ) + \frac{1}{3} \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + \frac{3}{2} \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} +{\left (B a^{3} + 3 \, A a^{2} b\right )} 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,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.26851, size = 365, normalized size = 2.32 \begin{align*} \frac{1}{7} \, B c^{3} x^{7} + \frac{1}{6} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + \frac{3}{5} \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{5} + \frac{1}{4} \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} + A a^{3} \log \left (x\right ) + \frac{1}{3} \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + \frac{3}{2} \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} +{\left (B a^{3} + 3 \, A a^{2} b\right )} 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,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.491705, size = 192, normalized size = 1.22 \begin{align*} A a^{3} \log{\left (x \right )} + \frac{B c^{3} x^{7}}{7} + x^{6} \left (\frac{A c^{3}}{6} + \frac{B b c^{2}}{2}\right ) + x^{5} \left (\frac{3 A b c^{2}}{5} + \frac{3 B a c^{2}}{5} + \frac{3 B b^{2} c}{5}\right ) + x^{4} \left (\frac{3 A a c^{2}}{4} + \frac{3 A b^{2} c}{4} + \frac{3 B a b c}{2} + \frac{B b^{3}}{4}\right ) + x^{3} \left (2 A a b c + \frac{A b^{3}}{3} + B a^{2} c + B a b^{2}\right ) + x^{2} \left (\frac{3 A a^{2} c}{2} + \frac{3 A a b^{2}}{2} + \frac{3 B a^{2} b}{2}\right ) + x \left (3 A a^{2} b + B a^{3}\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,x)

[Out]

A*a**3*log(x) + B*c**3*x**7/7 + x**6*(A*c**3/6 + B*b*c**2/2) + x**5*(3*A*b*c**2/5 + 3*B*a*c**2/5 + 3*B*b**2*c/
5) + x**4*(3*A*a*c**2/4 + 3*A*b**2*c/4 + 3*B*a*b*c/2 + B*b**3/4) + x**3*(2*A*a*b*c + A*b**3/3 + B*a**2*c + B*a
*b**2) + x**2*(3*A*a**2*c/2 + 3*A*a*b**2/2 + 3*B*a**2*b/2) + x*(3*A*a**2*b + B*a**3)

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Giac [A]  time = 1.17793, size = 250, normalized size = 1.59 \begin{align*} \frac{1}{7} \, B c^{3} x^{7} + \frac{1}{2} \, B b c^{2} x^{6} + \frac{1}{6} \, A c^{3} x^{6} + \frac{3}{5} \, B b^{2} c x^{5} + \frac{3}{5} \, B a c^{2} x^{5} + \frac{3}{5} \, A b c^{2} x^{5} + \frac{1}{4} \, B b^{3} x^{4} + \frac{3}{2} \, B a b c x^{4} + \frac{3}{4} \, A b^{2} c x^{4} + \frac{3}{4} \, A a c^{2} x^{4} + B a b^{2} x^{3} + \frac{1}{3} \, A b^{3} x^{3} + B a^{2} c x^{3} + 2 \, A a b c x^{3} + \frac{3}{2} \, B a^{2} b x^{2} + \frac{3}{2} \, A a b^{2} x^{2} + \frac{3}{2} \, A a^{2} c x^{2} + B a^{3} x + 3 \, A a^{2} b x + A a^{3} \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,x, algorithm="giac")

[Out]

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