Optimal. Leaf size=41 \[ -\frac {a}{3 c x^2 \sqrt {c x^2}}-\frac {b}{2 c x \sqrt {c x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 45}
\begin {gather*} -\frac {a}{3 c x^2 \sqrt {c x^2}}-\frac {b}{2 c x \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 15
Rule 45
Rubi steps
\begin {align*} \int \frac {a+b x}{x \left (c x^2\right )^{3/2}} \, dx &=\frac {x \int \frac {a+b x}{x^4} \, dx}{c \sqrt {c x^2}}\\ &=\frac {x \int \left (\frac {a}{x^4}+\frac {b}{x^3}\right ) \, dx}{c \sqrt {c x^2}}\\ &=-\frac {a}{3 c x^2 \sqrt {c x^2}}-\frac {b}{2 c x \sqrt {c x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 25, normalized size = 0.61 \begin {gather*} \frac {c x^2 (-2 a-3 b x)}{6 \left (c x^2\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.02, size = 18, normalized size = 0.44
method | result | size |
gosper | \(-\frac {3 b x +2 a}{6 \left (c \,x^{2}\right )^{\frac {3}{2}}}\) | \(18\) |
default | \(-\frac {3 b x +2 a}{6 \left (c \,x^{2}\right )^{\frac {3}{2}}}\) | \(18\) |
risch | \(\frac {-\frac {b x}{2}-\frac {a}{3}}{c \,x^{2} \sqrt {c \,x^{2}}}\) | \(23\) |
trager | \(\frac {\left (-1+x \right ) \left (2 a \,x^{2}+3 x^{2} b +2 a x +3 b x +2 a \right ) \sqrt {c \,x^{2}}}{6 c^{2} x^{4}}\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.27, size = 19, normalized size = 0.46 \begin {gather*} -\frac {b}{2 \, c^{\frac {3}{2}} x^{2}} - \frac {a}{3 \, c^{\frac {3}{2}} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.00, size = 23, normalized size = 0.56 \begin {gather*} -\frac {\sqrt {c x^{2}} {\left (3 \, b x + 2 \, a\right )}}{6 \, c^{2} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.24, size = 26, normalized size = 0.63 \begin {gather*} - \frac {a}{3 \left (c x^{2}\right )^{\frac {3}{2}}} - \frac {b x}{2 \left (c x^{2}\right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 3.53, size = 20, normalized size = 0.49 \begin {gather*} -\frac {3 \, b x + 2 \, a}{6 \, c^{\frac {3}{2}} x^{3} \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 = 26, normalized size = 0.63 \begin {gather*} -\frac {2\,a\,\sqrt {x^2}+3\,b\,x\,\sqrt {x^2}}{6\,c^{3/2}\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________