3.11.34 \(\int \frac {a+b x}{(a c-b c x)^3} \, dx\) [1034]

Optimal. Leaf size=13 \[ \frac {x}{c^3 (a-b x)^2} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {34} \begin {gather*} \frac {x}{c^3 (a-b x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/(a*c - b*c*x)^3,x]

[Out]

x/(c^3*(a - b*x)^2)

Rule 34

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_)), x_Symbol] :> Simp[d*x*((a + b*x)^(m + 1)/(b*(m + 2))), x] /
; FreeQ[{a, b, c, d, m}, x] && EqQ[a*d - b*c*(m + 2), 0]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 13, normalized size = 1.00 \begin {gather*} \frac {x}{c^3 (a-b x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x/(c^3*(a - b*x)^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(31\) vs. \(2(13)=26\).
time = 0.14, size = 32, normalized size = 2.46

method	result	size
gosper	\(\frac {x}{c^{3} \left (-b x +a \right )^{2}}\)	\(14\)
norman	\(\frac {x}{c^{3} \left (-b x +a \right )^{2}}\)	\(14\)
risch	\(\frac {x}{c^{3} \left (-b x +a \right )^{2}}\)	\(14\)
default	\(\frac {\frac {a}{b \left (-b x +a \right )^{2}}-\frac {1}{b \left (-b x +a \right )}}{c^{3}}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)/(-b*c*x+a*c)^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 30 vs. \(2 (14) = 28\).
time = 0.29, size = 30, normalized size = 2.31 \begin {gather*} \frac {x}{b^{2} c^{3} x^{2} - 2 \, a b c^{3} x + a^{2} c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 30 vs. \(2 (14) = 28\).
time = 0.60, size = 30, normalized size = 2.31 \begin {gather*} \frac {x}{b^{2} c^{3} x^{2} - 2 \, a b c^{3} x + a^{2} c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (10) = 20\).
time = 0.09, size = 27, normalized size = 2.08 \begin {gather*} \frac {x}{a^{2} c^{3} - 2 a b c^{3} x + b^{2} c^{3} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.62, size = 14, normalized size = 1.08 \begin {gather*} \frac {x}{{\left (b x - a\right )}^{2} c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/((b*x - a)^2*c^3)

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Mupad [B]
time = 0.15, size = 13, normalized size = 1.00 \begin {gather*} \frac {x}{c^3\,{\left (a-b\,x\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)/(a*c - b*c*x)^3,x)

[Out]

x/(c^3*(a - b*x)^2)

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