\(\int \frac {(a+b x)^5 (A+B x)}{x} \, dx\) [125]

Optimal result

Integrand size = 16, antiderivative size = 80 \[ \int \frac {(a+b x)^5 (A+B x)}{x} \, dx=5 a^4 A b x+5 a^3 A b^2 x^2+\frac {10}{3} a^2 A b^3 x^3+\frac {5}{4} a A b^4 x^4+\frac {1}{5} A b^5 x^5+\frac {B (a+b x)^6}{6 b}+a^5 A \log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {81, 45} \[ \int \frac {(a+b x)^5 (A+B x)}{x} \, dx=a^5 A \log (x)+5 a^4 A b x+5 a^3 A b^2 x^2+\frac {10}{3} a^2 A b^3 x^3+\frac {5}{4} a A b^4 x^4+\frac {B (a+b x)^6}{6 b}+\frac {1}{5} A b^5 x^5 \]

[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 45

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])

Rule 81

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]

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.02 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.35 \[ \int \frac {(a+b x)^5 (A+B x)}{x} \, dx=a^4 (5 A b+a B) x+\frac {5}{2} a^3 b (2 A b+a B) x^2+\frac {10}{3} a^2 b^2 (A b+a B) x^3+\frac {5}{4} a b^3 (A b+2 a B) x^4+\frac {1}{5} b^4 (A b+5 a B) x^5+\frac {1}{6} b^5 B x^6+a^5 A \log (x) \]

[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]

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.44

method	result	size
norman	\(\left (\frac {1}{5} b^{5} A +a \,b^{4} B \right ) x^{5}+\left (\frac {5}{4} a \,b^{4} A +\frac {5}{2} a^{2} b^{3} B \right ) x^{4}+\left (\frac {10}{3} a^{2} b^{3} A +\frac {10}{3} a^{3} b^{2} B \right ) x^{3}+\left (5 a^{3} b^{2} A +\frac {5}{2} a^{4} b B \right ) x^{2}+\left (5 a^{4} b A +a^{5} B \right ) x +\frac {b^{5} B \,x^{6}}{6}+a^{5} A \ln \left (x \right )\)	\(115\)
default	\(\frac {b^{5} B \,x^{6}}{6}+\frac {A \,b^{5} x^{5}}{5}+B a \,b^{4} x^{5}+\frac {5 a A \,b^{4} x^{4}}{4}+\frac {5 B \,a^{2} b^{3} x^{4}}{2}+\frac {10 a^{2} A \,b^{3} x^{3}}{3}+\frac {10 B \,a^{3} b^{2} x^{3}}{3}+5 a^{3} A \,b^{2} x^{2}+\frac {5 B \,a^{4} b \,x^{2}}{2}+5 a^{4} A b x +a^{5} B x +a^{5} A \ln \left (x \right )\)	\(118\)
risch	\(\frac {b^{5} B \,x^{6}}{6}+\frac {A \,b^{5} x^{5}}{5}+B a \,b^{4} x^{5}+\frac {5 a A \,b^{4} x^{4}}{4}+\frac {5 B \,a^{2} b^{3} x^{4}}{2}+\frac {10 a^{2} A \,b^{3} x^{3}}{3}+\frac {10 B \,a^{3} b^{2} x^{3}}{3}+5 a^{3} A \,b^{2} x^{2}+\frac {5 B \,a^{4} b \,x^{2}}{2}+5 a^{4} A b x +a^{5} B x +a^{5} A \ln \left (x \right )\)	\(118\)
parallelrisch	\(\frac {b^{5} B \,x^{6}}{6}+\frac {A \,b^{5} x^{5}}{5}+B a \,b^{4} x^{5}+\frac {5 a A \,b^{4} x^{4}}{4}+\frac {5 B \,a^{2} b^{3} x^{4}}{2}+\frac {10 a^{2} A \,b^{3} x^{3}}{3}+\frac {10 B \,a^{3} b^{2} x^{3}}{3}+5 a^{3} A \,b^{2} x^{2}+\frac {5 B \,a^{4} b \,x^{2}}{2}+5 a^{4} A b x +a^{5} B x +a^{5} A \ln \left (x \right )\)	\(118\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.42 \[ \int \frac {(a+b x)^5 (A+B x)}{x} \, dx=\frac {1}{6} \, B b^{5} x^{6} + A a^{5} \log \left (x\right ) + \frac {1}{5} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + \frac {5}{4} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + \frac {10}{3} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + \frac {5}{2} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + {\left (B a^{5} + 5 \, A a^{4} b\right )} x \]

[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

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.58 \[ \int \frac {(a+b x)^5 (A+B x)}{x} \, dx=A a^{5} \log {\left (x \right )} + \frac {B b^{5} x^{6}}{6} + x^{5} \left (\frac {A b^{5}}{5} + B a b^{4}\right ) + x^{4} \cdot \left (\frac {5 A a b^{4}}{4} + \frac {5 B a^{2} b^{3}}{2}\right ) + x^{3} \cdot \left (\frac {10 A a^{2} b^{3}}{3} + \frac {10 B a^{3} b^{2}}{3}\right ) + x^{2} \cdot \left (5 A a^{3} b^{2} + \frac {5 B a^{4} b}{2}\right ) + x \left (5 A a^{4} b + B a^{5}\right ) \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.42 \[ \int \frac {(a+b x)^5 (A+B x)}{x} \, dx=\frac {1}{6} \, B b^{5} x^{6} + A a^{5} \log \left (x\right ) + \frac {1}{5} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + \frac {5}{4} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + \frac {10}{3} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + \frac {5}{2} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + {\left (B a^{5} + 5 \, A a^{4} b\right )} x \]

[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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.48 \[ \int \frac {(a+b x)^5 (A+B x)}{x} \, dx=\frac {1}{6} \, B b^{5} x^{6} + B a b^{4} x^{5} + \frac {1}{5} \, A b^{5} x^{5} + \frac {5}{2} \, B a^{2} b^{3} x^{4} + \frac {5}{4} \, A a b^{4} x^{4} + \frac {10}{3} \, B a^{3} b^{2} x^{3} + \frac {10}{3} \, A a^{2} b^{3} x^{3} + \frac {5}{2} \, B a^{4} b x^{2} + 5 \, A a^{3} b^{2} x^{2} + B a^{5} x + 5 \, A a^{4} b x + A a^{5} \log \left ({\left | x \right |}\right ) \]

[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))

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.26 \[ \int \frac {(a+b x)^5 (A+B x)}{x} \, dx=x\,\left (B\,a^5+5\,A\,b\,a^4\right )+x^5\,\left (\frac {A\,b^5}{5}+B\,a\,b^4\right )+\frac {B\,b^5\,x^6}{6}+A\,a^5\,\ln \left (x\right )+\frac {10\,a^2\,b^2\,x^3\,\left (A\,b+B\,a\right )}{3}+\frac {5\,a^3\,b\,x^2\,\left (2\,A\,b+B\,a\right )}{2}+\frac {5\,a\,b^3\,x^4\,\left (A\,b+2\,B\,a\right )}{4} \]

[In]

int(((A + B*x)*(a + b*x)^5)/x,x)

[Out]

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