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

Optimal. Leaf size=17 \[ \frac {(a+b x)^5}{5 b c} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {21, 32} \begin {gather*} \frac {(a+b x)^5}{5 b c} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {(a+b x)^5}{a c+b c x} \, dx &=\frac {\int (a+b x)^4 \, dx}{c}\\ &=\frac {(a+b x)^5}{5 b c}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {(a+b x)^5}{5 b c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a + b*x)^5/(5*b*c)

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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(48\) vs. \(2(17)=34\).
time = 1.75, size = 46, normalized size = 2.71 \begin {gather*} \frac {x \left (5 a^4+10 a^3 b x+10 a^2 b^2 x^2+5 a b^3 x^3+b^4 x^4\right )}{5 c} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[(a + b*x)^5/(a*c + b*c*x)^1,x]')

[Out]

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

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Maple [A]
time = 0.15, size = 16, normalized size = 0.94

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (15) = 30\).
time = 0.28, size = 48, normalized size = 2.82 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (15) = 30\).
time = 0.29, size = 48, normalized size = 2.82 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (10) = 20\).
time = 0.05, size = 51, normalized size = 3.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (15) = 30\).
time = 0.00, size = 47, normalized size = 2.76 \begin {gather*} \frac {\frac {1}{5} x^{5} b^{4}+x^{4} b^{3} a+2 x^{3} b^{2} a^{2}+2 x^{2} b a^{3}+x a^{4}}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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

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Mupad [B]
time = 0.03, size = 57, normalized size = 3.35 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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