3.10.53 \(\int \frac {(a+b x)^5}{(a c+b c x)^4} \, dx\)

Optimal. Leaf size=18 \[ \frac {a x}{c^4}+\frac {b x^2}{2 c^4} \]

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Rubi [A]  time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {21} \begin {gather*} \frac {a x}{c^4}+\frac {b x^2}{2 c^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5/(a*c + b*c*x)^4,x]

[Out]

(a*x)/c^4 + (b*x^2)/(2*c^4)

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])

Rubi steps

\begin {align*} \int \frac {(a+b x)^5}{(a c+b c x)^4} \, dx &=\frac {\int (a+b x) \, dx}{c^4}\\ &=\frac {a x}{c^4}+\frac {b x^2}{2 c^4}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 16, normalized size = 0.89 \begin {gather*} \frac {a x+\frac {b x^2}{2}}{c^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^5/(a*c + b*c*x)^4,x]

[Out]

(a*x + (b*x^2)/2)/c^4

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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)^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)^5/(a*c + b*c*x)^4,x]

[Out]

IntegrateAlgebraic[(a + b*x)^5/(a*c + b*c*x)^4, x]

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fricas [A]  time = 1.36, size = 15, normalized size = 0.83 \begin {gather*} \frac {b x^{2} + 2 \, a x}{2 \, c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/(b*c*x+a*c)^4,x, algorithm="fricas")

[Out]

1/2*(b*x^2 + 2*a*x)/c^4

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giac [A]  time = 1.21, size = 15, normalized size = 0.83 \begin {gather*} \frac {b x^{2} + 2 \, a x}{2 \, c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/(b*c*x+a*c)^4,x, algorithm="giac")

[Out]

1/2*(b*x^2 + 2*a*x)/c^4

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maple [A]  time = 0.00, size = 15, normalized size = 0.83 \begin {gather*} \frac {\frac {1}{2} b \,x^{2}+a x}{c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^5/(b*c*x+a*c)^4,x)

[Out]

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

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maxima [A]  time = 1.38, size = 15, normalized size = 0.83 \begin {gather*} \frac {b x^{2} + 2 \, a x}{2 \, c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/(b*c*x+a*c)^4,x, algorithm="maxima")

[Out]

1/2*(b*x^2 + 2*a*x)/c^4

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mupad [B]  time = 0.02, size = 13, normalized size = 0.72 \begin {gather*} \frac {x\,\left (2\,a+b\,x\right )}{2\,c^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^5/(a*c + b*c*x)^4,x)

[Out]

(x*(2*a + b*x))/(2*c^4)

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sympy [A]  time = 0.11, size = 15, normalized size = 0.83 \begin {gather*} \frac {a x}{c^{4}} + \frac {b x^{2}}{2 c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**5/(b*c*x+a*c)**4,x)

[Out]

a*x/c**4 + b*x**2/(2*c**4)

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