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 = 0.0077756, antiderivative size = 41, normalized size of antiderivative = 1., 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, 43} \[ -\frac{a}{3 c x^2 \sqrt{c x^2}}-\frac{b}{2 c x \sqrt{c x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 15
Rule 43
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 = 0.0087747, size = 25, normalized size = 0.61 \[ \frac{c x^2 (-2 a-3 b x)}{6 \left (c x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.003, size = 18, normalized size = 0.4 \begin{align*} -{\frac{3\,bx+2\,a}{6} \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.12331, size = 26, normalized size = 0.63 \begin{align*} -\frac{b}{2 \, c^{\frac{3}{2}} x^{2}} - \frac{a}{3 \, c^{\frac{3}{2}} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.45478, size = 57, normalized size = 1.39 \begin{align*} -\frac{\sqrt{c x^{2}}{\left (3 \, b x + 2 \, a\right )}}{6 \, c^{2} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.637379, size = 32, normalized size = 0.78 \begin{align*} - \frac{a}{3 c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} - \frac{b x}{2 c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b x + a}{\left (c x^{2}\right )^{\frac{3}{2}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]