Optimal. Leaf size=32 \[ \frac {2 a}{b c^2 (a-b x)}+\frac {\log (a-b x)}{b c^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {45}
\begin {gather*} \frac {2 a}{b c^2 (a-b x)}+\frac {\log (a-b x)}{b c^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rubi steps
\begin {align*} \int \frac {a+b x}{(a c-b c x)^2} \, dx &=\int \left (\frac {2 a}{c^2 (a-b x)^2}-\frac {1}{c^2 (a-b x)}\right ) \, dx\\ &=\frac {2 a}{b c^2 (a-b x)}+\frac {\log (a-b x)}{b c^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 28, normalized size = 0.88 \begin {gather*} \frac {\frac {2 a}{a-b x}+\log (c (a-b x))}{b c^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.12, size = 31, normalized size = 0.97
method | result | size |
default | \(\frac {\frac {2 a}{b \left (-b x +a \right )}+\frac {\ln \left (-b x +a \right )}{b}}{c^{2}}\) | \(31\) |
norman | \(\frac {2 a}{b \,c^{2} \left (-b x +a \right )}+\frac {\ln \left (-b x +a \right )}{b \,c^{2}}\) | \(33\) |
risch | \(\frac {2 a}{b \,c^{2} \left (-b x +a \right )}+\frac {\ln \left (-b x +a \right )}{b \,c^{2}}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 37, normalized size = 1.16 \begin {gather*} -\frac {2 \, a}{b^{2} c^{2} x - a b c^{2}} + \frac {\log \left (b x - a\right )}{b c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.63, size = 39, normalized size = 1.22 \begin {gather*} \frac {{\left (b x - a\right )} \log \left (b x - a\right ) - 2 \, a}{b^{2} c^{2} x - a b c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.07, size = 29, normalized size = 0.91 \begin {gather*} - \frac {2 a}{- a b c^{2} + b^{2} c^{2} x} + \frac {\log {\left (- a + b x \right )}}{b c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 81 vs.
\(2 (33) = 66\).
time = 2.07, size = 81, normalized size = 2.53 \begin {gather*} -\frac {\frac {a}{{\left (b c x - a c\right )} b} + \frac {\log \left (\frac {{\left | b c x - a c \right |}}{{\left (b c x - a c\right )}^{2} {\left | b \right |} {\left | c \right |}}\right )}{b c}}{c} - \frac {a}{{\left (b c x - a c\right )} b c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.05, size = 37, normalized size = 1.16 \begin {gather*} \frac {\ln \left (b\,x-a\right )}{b\,c^2}+\frac {2\,a}{b\,\left (a\,c^2-b\,c^2\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________