Optimal. Leaf size=28 \[ -\frac {(c+d x)^2}{2 (b c-a d) (a+b x)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.00, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {37}
\begin {gather*} -\frac {(c+d x)^2}{2 (a+b x)^2 (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 37
Rubi steps
\begin {align*} \int \frac {c+d x}{(a+b x)^3} \, dx &=-\frac {(c+d x)^2}{2 (b c-a d) (a+b x)^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 26, normalized size = 0.93 \begin {gather*} -\frac {a d+b (c+2 d x)}{2 b^2 (a+b x)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Mathics [A]
time = 1.86, size = 37, normalized size = 1.32 \begin {gather*} \frac {-a d-b c-2 b d x}{2 b^2 \left (a^2+2 a b x+b^2 x^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.12, size = 35, normalized size = 1.25
method | result | size |
gosper | \(-\frac {2 b d x +a d +b c}{2 \left (b x +a \right )^{2} b^{2}}\) | \(25\) |
risch | \(\frac {-\frac {d x}{b}-\frac {a d +b c}{2 b^{2}}}{\left (b x +a \right )^{2}}\) | \(29\) |
norman | \(\frac {-\frac {d x}{b}+\frac {-a d -b c}{2 b^{2}}}{\left (b x +a \right )^{2}}\) | \(31\) |
default | \(-\frac {d}{b^{2} \left (b x +a \right )}-\frac {-a d +b c}{2 b^{2} \left (b x +a \right )^{2}}\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.27, size = 38, normalized size = 1.36 \begin {gather*} -\frac {2 \, b d x + b c + a d}{2 \, {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.30, size = 38, normalized size = 1.36 \begin {gather*} -\frac {2 \, b d x + b c + a d}{2 \, {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.15, size = 39, normalized size = 1.39 \begin {gather*} \frac {- a d - b c - 2 b d x}{2 a^{2} b^{2} + 4 a b^{3} x + 2 b^{4} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.00, size = 29, normalized size = 1.04 \begin {gather*} \frac {-2 x d b-d a-c b}{2 b^{2} \left (x b+a\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.16, size = 39, normalized size = 1.39 \begin {gather*} -\frac {\frac {a\,d+b\,c}{2\,b^2}+\frac {d\,x}{b}}{a^2+2\,a\,b\,x+b^2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________