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

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

[Out]

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

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Rubi [A]  time = 0.040332, 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} \[ -\frac{a^2 (a B+3 A b)}{x}-\frac{a^3 A}{2 x^2}+b^2 x (3 a B+A b)+3 a b \log (x) (a B+A b)+\frac{1}{2} b^3 B x^2 \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^3*(A + B*x))/x^3,x]

[Out]

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

Mathematica [A]  time = 0.0256161, size = 62, normalized size = 0.95 \[ \frac{1}{2} \left (-\frac{6 a^2 A b}{x}-\frac{a^3 (A+2 B x)}{x^2}+6 a b \log (x) (a B+A b)+6 a b^2 B x+b^3 x (2 A+B x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.987261, size = 93, normalized size = 1.43 \begin{align*} \frac{1}{2} \, B b^{3} x^{2} +{\left (3 \, B a b^{2} + A b^{3}\right )} x + 3 \,{\left (B a^{2} b + A a b^{2}\right )} \log \left (x\right ) - \frac{A a^{3} + 2 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.55146, size = 159, normalized size = 2.45 \begin{align*} \frac{B b^{3} x^{4} - A a^{3} + 2 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 6 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} \log \left (x\right ) - 2 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.593348, size = 66, normalized size = 1.02 \begin{align*} \frac{B b^{3} x^{2}}{2} + 3 a b \left (A b + B a\right ) \log{\left (x \right )} + x \left (A b^{3} + 3 B a b^{2}\right ) - \frac{A a^{3} + x \left (6 A a^{2} b + 2 B a^{3}\right )}{2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*b**3*x**2/2 + 3*a*b*(A*b + B*a)*log(x) + x*(A*b**3 + 3*B*a*b**2) - (A*a**3 + x*(6*A*a**2*b + 2*B*a**3))/(2*x
**2)

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Giac [A]  time = 1.22381, size = 93, normalized size = 1.43 \begin{align*} \frac{1}{2} \, B b^{3} x^{2} + 3 \, B a b^{2} x + A b^{3} x + 3 \,{\left (B a^{2} b + A a b^{2}\right )} \log \left ({\left | x \right |}\right ) - \frac{A a^{3} + 2 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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