3.125 \(\int \frac{(a+b x)^5 (A+B x)}{x} \, dx\)

Optimal. Leaf size=80 \[ \frac{10}{3} a^2 A b^3 x^3+5 a^3 A b^2 x^2+5 a^4 A b x+a^5 A \log (x)+\frac{5}{4} a A b^4 x^4+\frac{B (a+b x)^6}{6 b}+\frac{1}{5} A b^5 x^5 \]

[Out]

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

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Rubi [A]  time = 0.0294455, antiderivative size = 80, 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} \[ \frac{10}{3} a^2 A b^3 x^3+5 a^3 A b^2 x^2+5 a^4 A b x+a^5 A \log (x)+\frac{5}{4} a A b^4 x^4+\frac{B (a+b x)^6}{6 b}+\frac{1}{5} A b^5 x^5 \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^5*(A + B*x))/x,x]

[Out]

5*a^4*A*b*x + 5*a^3*A*b^2*x^2 + (10*a^2*A*b^3*x^3)/3 + (5*a*A*b^4*x^4)/4 + (A*b^5*x^5)/5 + (B*(a + b*x)^6)/(6*
b) + a^5*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)^5 (A+B x)}{x} \, dx &=\frac{B (a+b x)^6}{6 b}+A \int \frac{(a+b x)^5}{x} \, dx\\ &=\frac{B (a+b x)^6}{6 b}+A \int \left (5 a^4 b+\frac{a^5}{x}+10 a^3 b^2 x+10 a^2 b^3 x^2+5 a b^4 x^3+b^5 x^4\right ) \, dx\\ &=5 a^4 A b x+5 a^3 A b^2 x^2+\frac{10}{3} a^2 A b^3 x^3+\frac{5}{4} a A b^4 x^4+\frac{1}{5} A b^5 x^5+\frac{B (a+b x)^6}{6 b}+a^5 A \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0344768, size = 108, normalized size = 1.35 \[ \frac{10}{3} a^2 b^2 x^3 (a B+A b)+\frac{5}{2} a^3 b x^2 (a B+2 A b)+a^4 x (a B+5 A b)+a^5 A \log (x)+\frac{1}{5} b^4 x^5 (5 a B+A b)+\frac{5}{4} a b^3 x^4 (2 a B+A b)+\frac{1}{6} b^5 B x^6 \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^5*(A + B*x))/x,x]

[Out]

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

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Maple [A]  time = 0.001, size = 118, normalized size = 1.5 \begin{align*}{\frac{B{b}^{5}{x}^{6}}{6}}+{\frac{A{b}^{5}{x}^{5}}{5}}+B{x}^{5}a{b}^{4}+{\frac{5\,aA{b}^{4}{x}^{4}}{4}}+{\frac{5\,B{x}^{4}{a}^{2}{b}^{3}}{2}}+{\frac{10\,{a}^{2}A{b}^{3}{x}^{3}}{3}}+{\frac{10\,B{x}^{3}{a}^{3}{b}^{2}}{3}}+5\,{a}^{3}A{b}^{2}{x}^{2}+{\frac{5\,B{x}^{2}{a}^{4}b}{2}}+5\,{a}^{4}Abx+{a}^{5}Bx+{a}^{5}A\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^5*(B*x+A)/x,x)

[Out]

1/6*B*b^5*x^6+1/5*A*b^5*x^5+B*x^5*a*b^4+5/4*a*A*b^4*x^4+5/2*B*x^4*a^2*b^3+10/3*a^2*A*b^3*x^3+10/3*B*x^3*a^3*b^
2+5*a^3*A*b^2*x^2+5/2*B*x^2*a^4*b+5*a^4*A*b*x+a^5*B*x+a^5*A*ln(x)

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Maxima [A]  time = 1.03581, size = 154, normalized size = 1.92 \begin{align*} \frac{1}{6} \, B b^{5} x^{6} + A a^{5} \log \left (x\right ) + \frac{1}{5} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + \frac{5}{4} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + \frac{10}{3} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + \frac{5}{2} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} +{\left (B a^{5} + 5 \, A a^{4} b\right )} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(B*x+A)/x,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.76294, size = 252, normalized size = 3.15 \begin{align*} \frac{1}{6} \, B b^{5} x^{6} + A a^{5} \log \left (x\right ) + \frac{1}{5} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + \frac{5}{4} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + \frac{10}{3} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + \frac{5}{2} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} +{\left (B a^{5} + 5 \, A a^{4} b\right )} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(B*x+A)/x,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.437466, size = 126, normalized size = 1.58 \begin{align*} A a^{5} \log{\left (x \right )} + \frac{B b^{5} x^{6}}{6} + x^{5} \left (\frac{A b^{5}}{5} + B a b^{4}\right ) + x^{4} \left (\frac{5 A a b^{4}}{4} + \frac{5 B a^{2} b^{3}}{2}\right ) + x^{3} \left (\frac{10 A a^{2} b^{3}}{3} + \frac{10 B a^{3} b^{2}}{3}\right ) + x^{2} \left (5 A a^{3} b^{2} + \frac{5 B a^{4} b}{2}\right ) + x \left (5 A a^{4} b + B a^{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**5*(B*x+A)/x,x)

[Out]

A*a**5*log(x) + B*b**5*x**6/6 + x**5*(A*b**5/5 + B*a*b**4) + x**4*(5*A*a*b**4/4 + 5*B*a**2*b**3/2) + x**3*(10*
A*a**2*b**3/3 + 10*B*a**3*b**2/3) + x**2*(5*A*a**3*b**2 + 5*B*a**4*b/2) + x*(5*A*a**4*b + B*a**5)

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Giac [A]  time = 1.24238, size = 159, normalized size = 1.99 \begin{align*} \frac{1}{6} \, B b^{5} x^{6} + B a b^{4} x^{5} + \frac{1}{5} \, A b^{5} x^{5} + \frac{5}{2} \, B a^{2} b^{3} x^{4} + \frac{5}{4} \, A a b^{4} x^{4} + \frac{10}{3} \, B a^{3} b^{2} x^{3} + \frac{10}{3} \, A a^{2} b^{3} x^{3} + \frac{5}{2} \, B a^{4} b x^{2} + 5 \, A a^{3} b^{2} x^{2} + B a^{5} x + 5 \, A a^{4} b x + A a^{5} \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(B*x+A)/x,x, algorithm="giac")

[Out]

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