Optimal. Leaf size=22 \[ -\frac {x}{b \sqrt {c x^2} (a+b x)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.00, antiderivative size = 22, 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 = {15, 32}
\begin {gather*} -\frac {x}{b \sqrt {c x^2} (a+b x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 15
Rule 32
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {c x^2} (a+b x)^2} \, dx &=\frac {x \int \frac {1}{(a+b x)^2} \, dx}{\sqrt {c x^2}}\\ &=-\frac {x}{b \sqrt {c x^2} (a+b x)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.00, size = 22, normalized size = 1.00 \begin {gather*} -\frac {x}{b \sqrt {c x^2} (a+b x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.12, size = 21, normalized size = 0.95
method | result | size |
gosper | \(-\frac {x}{b \left (b x +a \right ) \sqrt {c \,x^{2}}}\) | \(21\) |
default | \(-\frac {x}{b \left (b x +a \right ) \sqrt {c \,x^{2}}}\) | \(21\) |
risch | \(-\frac {x}{b \left (b x +a \right ) \sqrt {c \,x^{2}}}\) | \(21\) |
trager | \(\frac {\left (-1+x \right ) \sqrt {c \,x^{2}}}{c \left (b x +a \right ) \left (a +b \right ) x}\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 21, normalized size = 0.95 \begin {gather*} \frac {\sqrt {c x^{2}}}{a b c x + a^{2} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.40, size = 25, normalized size = 1.14 \begin {gather*} -\frac {\sqrt {c x^{2}}}{b^{2} c x^{2} + a b c x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 68 vs.
\(2 (17) = 34\).
time = 0.40, size = 68, normalized size = 3.09 \begin {gather*} \begin {cases} \frac {\tilde {\infty }}{\sqrt {c x^{2}}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {\tilde {\infty } x^{2}}{\sqrt {c x^{2}}} & \text {for}\: a = - b x \\\frac {x^{2}}{a^{2} \sqrt {c x^{2}}} & \text {for}\: b = 0 \\- \frac {x}{a b \sqrt {c x^{2}} + b^{2} x \sqrt {c x^{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.66, size = 32, normalized size = 1.45 \begin {gather*} \frac {\mathrm {sgn}\left (x\right )}{a b \sqrt {c}} - \frac {1}{{\left (b x + a\right )} b \sqrt {c} \mathrm {sgn}\left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.16, size = 25, normalized size = 1.14 \begin {gather*} -\frac {\sqrt {c\,x^2}}{b\,c\,x\,\left (a+b\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________