3.11.20 \(\int \frac {(a+b x)^5}{(a c+b c x)^2} \, dx\) [1020]

Optimal. Leaf size=17 \[ \frac {(a+b x)^4}{4 b c^2} \]

[Out]

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

________________________________________________________________________________________

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)^4}{4 b c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5/(a*c + b*c*x)^2,x]

[Out]

(a + b*x)^4/(4*b*c^2)

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)^2} \, dx &=\frac {\int (a+b x)^3 \, dx}{c^2}\\ &=\frac {(a+b x)^4}{4 b c^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {(a+b x)^4}{4 b c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^5/(a*c + b*c*x)^2,x]

[Out]

(a + b*x)^4/(4*b*c^2)

________________________________________________________________________________________

Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(37\) vs. \(2(17)=34\).
time = 1.70, size = 35, normalized size = 2.06 \begin {gather*} \frac {x \left (4 a^3+6 a^2 b x+4 a b^2 x^2+b^3 x^3\right )}{4 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[(a + b*x)^5/(a*c + b*c*x)^2,x]')

[Out]

x (4 a ^ 3 + 6 a ^ 2 b x + 4 a b ^ 2 x ^ 2 + b ^ 3 x ^ 3) / (4 c ^ 2)

________________________________________________________________________________________

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

method	result	size
default	\(\frac {\left (b x +a \right )^{4}}{4 b \,c^{2}}\)	\(16\)
gosper	\(\frac {x \left (b^{3} x^{3}+4 a \,b^{2} x^{2}+6 a^{2} b x +4 a^{3}\right )}{4 c^{2}}\)	\(36\)
risch	\(\frac {b^{3} x^{4}}{4 c^{2}}+\frac {b^{2} a \,x^{3}}{c^{2}}+\frac {3 b \,a^{2} x^{2}}{2 c^{2}}+\frac {a^{3} x}{c^{2}}+\frac {a^{4}}{4 b \,c^{2}}\)	\(55\)
norman	\(\frac {\frac {a^{4} x}{c}+\frac {b^{4} x^{5}}{4 c}+\frac {5 a \,b^{3} x^{4}}{4 c}+\frac {5 a^{3} b \,x^{2}}{2 c}+\frac {5 b^{2} a^{2} x^{3}}{2 c}}{c \left (b x +a \right )}\)	\(70\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^5/(b*c*x+a*c)^2,x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (15) = 30\).
time = 0.26, size = 37, normalized size = 2.18 \begin {gather*} \frac {b^{3} x^{4} + 4 \, a b^{2} x^{3} + 6 \, a^{2} b x^{2} + 4 \, a^{3} x}{4 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/(b*c*x+a*c)^2,x, algorithm="maxima")

[Out]

1/4*(b^3*x^4 + 4*a*b^2*x^3 + 6*a^2*b*x^2 + 4*a^3*x)/c^2

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (15) = 30\).
time = 0.29, size = 37, normalized size = 2.18 \begin {gather*} \frac {b^{3} x^{4} + 4 \, a b^{2} x^{3} + 6 \, a^{2} b x^{2} + 4 \, a^{3} x}{4 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/(b*c*x+a*c)^2,x, algorithm="fricas")

[Out]

1/4*(b^3*x^4 + 4*a*b^2*x^3 + 6*a^2*b*x^2 + 4*a^3*x)/c^2

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (12) = 24\).
time = 0.05, size = 46, normalized size = 2.71 \begin {gather*} \frac {a^{3} x}{c^{2}} + \frac {3 a^{2} b x^{2}}{2 c^{2}} + \frac {a b^{2} x^{3}}{c^{2}} + \frac {b^{3} x^{4}}{4 c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**5/(b*c*x+a*c)**2,x)

[Out]

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

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (15) = 30\).
time = 0.00, size = 40, normalized size = 2.35 \begin {gather*} \frac {\frac {1}{4} x^{4} b^{3}+x^{3} b^{2} a+\frac {3}{2} x^{2} b a^{2}+x a^{3}}{c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/(b*c*x+a*c)^2,x)

[Out]

1/4*(b^3*x^4 + 4*a*b^2*x^3 + 6*a^2*b*x^2 + 4*a^3*x)/c^2

________________________________________________________________________________________

Mupad [B]
time = 0.05, size = 43, normalized size = 2.53 \begin {gather*} \frac {a^3\,x}{c^2}+\frac {b^3\,x^4}{4\,c^2}+\frac {3\,a^2\,b\,x^2}{2\,c^2}+\frac {a\,b^2\,x^3}{c^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^5/(a*c + b*c*x)^2,x)

[Out]

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

________________________________________________________________________________________