\(\int \frac {(a+b x)^5}{a c+b c x} \, dx\) [1019]

Optimal result

Integrand size = 18, antiderivative size = 17 \[ \int \frac {(a+b x)^5}{a c+b c x} \, dx=\frac {(a+b x)^5}{5 b c} \]

[Out]

1/5*(b*x+a)^5/b/c

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {21, 32} \[ \int \frac {(a+b x)^5}{a c+b c x} \, dx=\frac {(a+b x)^5}{5 b c} \]

[In]

Int[(a + b*x)^5/(a*c + b*c*x),x]

[Out]

(a + b*x)^5/(5*b*c)

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {\int (a+b x)^4 \, dx}{c} \\ & = \frac {(a+b x)^5}{5 b c} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^5}{a c+b c x} \, dx=\frac {(a+b x)^5}{5 b c} \]

[In]

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

[Out]

(a + b*x)^5/(5*b*c)

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94

method	result	size
default	\(\frac {\left (b x +a \right )^{5}}{5 b c}\)	\(16\)
gosper	\(\frac {x \left (b^{4} x^{4}+5 a \,b^{3} x^{3}+10 a^{2} b^{2} x^{2}+10 a^{3} b x +5 a^{4}\right )}{5 c}\)	\(47\)
parallelrisch	\(\frac {b^{4} x^{5}+5 a \,b^{3} x^{4}+10 a^{2} b^{2} x^{3}+10 a^{3} b \,x^{2}+5 a^{4} x}{5 c}\)	\(49\)
norman	\(\frac {a^{4} x}{c}+\frac {a \,b^{3} x^{4}}{c}+\frac {b^{4} x^{5}}{5 c}+\frac {2 a^{2} b^{2} x^{3}}{c}+\frac {2 a^{3} b \,x^{2}}{c}\)	\(58\)
risch	\(\frac {b^{4} x^{5}}{5 c}+\frac {a \,b^{3} x^{4}}{c}+\frac {2 a^{2} b^{2} x^{3}}{c}+\frac {2 a^{3} b \,x^{2}}{c}+\frac {a^{4} x}{c}+\frac {a^{5}}{5 b c}\)	\(69\)

[In]

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

[Out]

1/5*(b*x+a)^5/b/c

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (15) = 30\).

Time = 0.22 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.82 \[ \int \frac {(a+b x)^5}{a c+b c x} \, dx=\frac {b^{4} x^{5} + 5 \, a b^{3} x^{4} + 10 \, a^{2} b^{2} x^{3} + 10 \, a^{3} b x^{2} + 5 \, a^{4} x}{5 \, c} \]

[In]

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

[Out]

1/5*(b^4*x^5 + 5*a*b^3*x^4 + 10*a^2*b^2*x^3 + 10*a^3*b*x^2 + 5*a^4*x)/c

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (10) = 20\).

Time = 0.09 (sec) , antiderivative size = 51, normalized size of antiderivative = 3.00 \[ \int \frac {(a+b x)^5}{a c+b c x} \, dx=\frac {a^{4} x}{c} + \frac {2 a^{3} b x^{2}}{c} + \frac {2 a^{2} b^{2} x^{3}}{c} + \frac {a b^{3} x^{4}}{c} + \frac {b^{4} x^{5}}{5 c} \]

[In]

integrate((b*x+a)**5/(b*c*x+a*c),x)

[Out]

a**4*x/c + 2*a**3*b*x**2/c + 2*a**2*b**2*x**3/c + a*b**3*x**4/c + b**4*x**5/(5*c)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (15) = 30\).

Time = 0.21 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.82 \[ \int \frac {(a+b x)^5}{a c+b c x} \, dx=\frac {b^{4} x^{5} + 5 \, a b^{3} x^{4} + 10 \, a^{2} b^{2} x^{3} + 10 \, a^{3} b x^{2} + 5 \, a^{4} x}{5 \, c} \]

[In]

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

[Out]

1/5*(b^4*x^5 + 5*a*b^3*x^4 + 10*a^2*b^2*x^3 + 10*a^3*b*x^2 + 5*a^4*x)/c

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (15) = 30\).

Time = 0.30 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.82 \[ \int \frac {(a+b x)^5}{a c+b c x} \, dx=\frac {b^{4} x^{5} + 5 \, a b^{3} x^{4} + 10 \, a^{2} b^{2} x^{3} + 10 \, a^{3} b x^{2} + 5 \, a^{4} x}{5 \, c} \]

[In]

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

[Out]

1/5*(b^4*x^5 + 5*a*b^3*x^4 + 10*a^2*b^2*x^3 + 10*a^3*b*x^2 + 5*a^4*x)/c

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 3.35 \[ \int \frac {(a+b x)^5}{a c+b c x} \, dx=\frac {a^4\,x}{c}+\frac {b^4\,x^5}{5\,c}+\frac {2\,a^3\,b\,x^2}{c}+\frac {a\,b^3\,x^4}{c}+\frac {2\,a^2\,b^2\,x^3}{c} \]

[In]

int((a + b*x)^5/(a*c + b*c*x),x)

[Out]

(a^4*x)/c + (b^4*x^5)/(5*c) + (2*a^3*b*x^2)/c + (a*b^3*x^4)/c + (2*a^2*b^2*x^3)/c