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

Optimal. Leaf size=65 \[ a^2 \log (x) (a B+3 A b)-\frac{a^3 A}{x}+\frac{1}{2} b^2 x^2 (3 a B+A b)+3 a b x (a B+A b)+\frac{1}{3} b^3 B x^3 \]

[Out]

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

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Rubi [A]  time = 0.0385159, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {76} \[ a^2 \log (x) (a B+3 A b)-\frac{a^3 A}{x}+\frac{1}{2} b^2 x^2 (3 a B+A b)+3 a b x (a B+A b)+\frac{1}{3} b^3 B x^3 \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^3*(A + B*x))/x^2,x]

[Out]

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

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

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

Mathematica [A]  time = 0.0256477, size = 67, normalized size = 1.03 \[ \log (x) \left (3 a^2 A b+a^3 B\right )-\frac{a^3 A}{x}+\frac{1}{2} b^2 x^2 (3 a B+A b)+3 a b x (a B+A b)+\frac{1}{3} b^3 B x^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.005, size = 71, normalized size = 1.1 \begin{align*}{\frac{{b}^{3}B{x}^{3}}{3}}+{\frac{A{x}^{2}{b}^{3}}{2}}+{\frac{3\,B{x}^{2}a{b}^{2}}{2}}+3\,a{b}^{2}Ax+3\,{a}^{2}bBx+3\,A\ln \left ( x \right ){a}^{2}b+B\ln \left ( x \right ){a}^{3}-{\frac{A{a}^{3}}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3*(B*x+A)/x^2,x)

[Out]

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

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Maxima [A]  time = 1.02371, size = 93, normalized size = 1.43 \begin{align*} \frac{1}{3} \, B b^{3} x^{3} - \frac{A a^{3}}{x} + \frac{1}{2} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{2} + 3 \,{\left (B a^{2} b + A a b^{2}\right )} x +{\left (B a^{3} + 3 \, A a^{2} b\right )} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/x^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.66924, size = 163, normalized size = 2.51 \begin{align*} \frac{2 \, B b^{3} x^{4} - 6 \, A a^{3} + 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 18 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} + 6 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x \log \left (x\right )}{6 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/x^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.406116, size = 70, normalized size = 1.08 \begin{align*} - \frac{A a^{3}}{x} + \frac{B b^{3} x^{3}}{3} + a^{2} \left (3 A b + B a\right ) \log{\left (x \right )} + x^{2} \left (\frac{A b^{3}}{2} + \frac{3 B a b^{2}}{2}\right ) + x \left (3 A a b^{2} + 3 B a^{2} b\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3*(B*x+A)/x**2,x)

[Out]

-A*a**3/x + B*b**3*x**3/3 + a**2*(3*A*b + B*a)*log(x) + x**2*(A*b**3/2 + 3*B*a*b**2/2) + x*(3*A*a*b**2 + 3*B*a
**2*b)

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Giac [A]  time = 1.20815, size = 96, normalized size = 1.48 \begin{align*} \frac{1}{3} \, B b^{3} x^{3} + \frac{3}{2} \, B a b^{2} x^{2} + \frac{1}{2} \, A b^{3} x^{2} + 3 \, B a^{2} b x + 3 \, A a b^{2} x - \frac{A a^{3}}{x} +{\left (B a^{3} + 3 \, A a^{2} b\right )} \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/x^2,x, algorithm="giac")

[Out]

1/3*B*b^3*x^3 + 3/2*B*a*b^2*x^2 + 1/2*A*b^3*x^2 + 3*B*a^2*b*x + 3*A*a*b^2*x - A*a^3/x + (B*a^3 + 3*A*a^2*b)*lo
g(abs(x))