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

Optimal. Leaf size=54 \[ 3 a^2 A b x+a^3 A \log (x)+\frac{3}{2} a A b^2 x^2+\frac{B (a+b x)^4}{4 b}+\frac{1}{3} A b^3 x^3 \]

[Out]

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

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Rubi [A]  time = 0.0161688, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {80, 43} \[ 3 a^2 A b x+a^3 A \log (x)+\frac{3}{2} a A b^2 x^2+\frac{B (a+b x)^4}{4 b}+\frac{1}{3} A b^3 x^3 \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^3*(A + B*x))/x,x]

[Out]

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

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0235164, size = 63, normalized size = 1.17 \[ \frac{1}{12} x \left (18 a^2 b (2 A+B x)+12 a^3 B+6 a b^2 x (3 A+2 B x)+b^3 x^2 (4 A+3 B x)\right )+a^3 A \log (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.002, size = 70, normalized size = 1.3 \begin{align*}{\frac{B{b}^{3}{x}^{4}}{4}}+{\frac{A{b}^{3}{x}^{3}}{3}}+B{x}^{3}a{b}^{2}+{\frac{3\,aA{b}^{2}{x}^{2}}{2}}+{\frac{3\,B{x}^{2}{a}^{2}b}{2}}+3\,{a}^{2}Abx+{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)^3*(B*x+A)/x,x)

[Out]

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

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Maxima [A]  time = 1.04244, size = 92, normalized size = 1.7 \begin{align*} \frac{1}{4} \, B b^{3} x^{4} + A a^{3} \log \left (x\right ) + \frac{1}{3} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + \frac{3}{2} \,{\left (B a^{2} b + A a b^{2}\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)^3*(B*x+A)/x,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.64857, size = 154, normalized size = 2.85 \begin{align*} \frac{1}{4} \, B b^{3} x^{4} + A a^{3} \log \left (x\right ) + \frac{1}{3} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + \frac{3}{2} \,{\left (B a^{2} b + A a b^{2}\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)^3*(B*x+A)/x,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.36449, size = 73, normalized size = 1.35 \begin{align*} A a^{3} \log{\left (x \right )} + \frac{B b^{3} x^{4}}{4} + x^{3} \left (\frac{A b^{3}}{3} + B a b^{2}\right ) + x^{2} \left (\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)**3*(B*x+A)/x,x)

[Out]

A*a**3*log(x) + B*b**3*x**4/4 + x**3*(A*b**3/3 + B*a*b**2) + x**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 = 1.28789, size = 95, normalized size = 1.76 \begin{align*} \frac{1}{4} \, B b^{3} x^{4} + B a b^{2} x^{3} + \frac{1}{3} \, A b^{3} x^{3} + \frac{3}{2} \, B a^{2} b x^{2} + \frac{3}{2} \, A a b^{2} 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)^3*(B*x+A)/x,x, algorithm="giac")

[Out]

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