3.10.52 \(\int \frac {(a+b x)^5}{(a c+b c x)^3} \, dx\)

Optimal. Leaf size=17 \[ \frac {(a+b x)^3}{3 b c^3} \]

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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)^3}{3 b c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5/(a*c + b*c*x)^3,x]

[Out]

(a + b*x)^3/(3*b*c^3)

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

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {(a+b x)^3}{3 b c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^5/(a*c + b*c*x)^3,x]

[Out]

(a + b*x)^3/(3*b*c^3)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^5}{(a c+b c x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)^5/(a*c + b*c*x)^3,x]

[Out]

IntegrateAlgebraic[(a + b*x)^5/(a*c + b*c*x)^3, x]

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fricas [A]  time = 0.71, size = 26, normalized size = 1.53 \begin {gather*} \frac {b^{2} x^{3} + 3 \, a b x^{2} + 3 \, a^{2} x}{3 \, c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/(b*c*x+a*c)^3,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 1.00, size = 26, normalized size = 1.53 \begin {gather*} \frac {b^{2} x^{3} + 3 \, a b x^{2} + 3 \, a^{2} x}{3 \, c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/(b*c*x+a*c)^3,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.00, size = 16, normalized size = 0.94 \begin {gather*} \frac {\left (b x +a \right )^{3}}{3 b \,c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^5/(b*c*x+a*c)^3,x)

[Out]

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

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maxima [A]  time = 1.31, size = 26, normalized size = 1.53 \begin {gather*} \frac {b^{2} x^{3} + 3 \, a b x^{2} + 3 \, a^{2} x}{3 \, c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/(b*c*x+a*c)^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.04, size = 24, normalized size = 1.41 \begin {gather*} \frac {x\,\left (3\,a^2+3\,a\,b\,x+b^2\,x^2\right )}{3\,c^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^5/(a*c + b*c*x)^3,x)

[Out]

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

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sympy [B]  time = 0.11, size = 29, normalized size = 1.71 \begin {gather*} \frac {a^{2} x}{c^{3}} + \frac {a b x^{2}}{c^{3}} + \frac {b^{2} x^{3}}{3 c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**5/(b*c*x+a*c)**3,x)

[Out]

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

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