Optimal. Leaf size=41 \[ -\frac{2 a A}{\sqrt{x}}+2 a B \sqrt{x}+\frac{2}{3} A c x^{3/2}+\frac{2}{5} B c x^{5/2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0118859, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {766} \[ -\frac{2 a A}{\sqrt{x}}+2 a B \sqrt{x}+\frac{2}{3} A c x^{3/2}+\frac{2}{5} B c x^{5/2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 766
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a+c x^2\right )}{x^{3/2}} \, dx &=\int \left (\frac{a A}{x^{3/2}}+\frac{a B}{\sqrt{x}}+A c \sqrt{x}+B c x^{3/2}\right ) \, dx\\ &=-\frac{2 a A}{\sqrt{x}}+2 a B \sqrt{x}+\frac{2}{3} A c x^{3/2}+\frac{2}{5} B c x^{5/2}\\ \end{align*}
Mathematica [A] time = 0.0141049, size = 32, normalized size = 0.78 \[ \frac{2 \left (c x^2 (5 A+3 B x)-15 a (A-B x)\right )}{15 \sqrt{x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.004, size = 30, normalized size = 0.7 \begin{align*} -{\frac{-6\,Bc{x}^{3}-10\,Ac{x}^{2}-30\,aBx+30\,aA}{15}{\frac{1}{\sqrt{x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.00983, size = 39, normalized size = 0.95 \begin{align*} \frac{2}{5} \, B c x^{\frac{5}{2}} + \frac{2}{3} \, A c x^{\frac{3}{2}} + 2 \, B a \sqrt{x} - \frac{2 \, A a}{\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.35885, size = 78, normalized size = 1.9 \begin{align*} \frac{2 \,{\left (3 \, B c x^{3} + 5 \, A c x^{2} + 15 \, B a x - 15 \, A a\right )}}{15 \, \sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.924955, size = 42, normalized size = 1.02 \begin{align*} - \frac{2 A a}{\sqrt{x}} + \frac{2 A c x^{\frac{3}{2}}}{3} + 2 B a \sqrt{x} + \frac{2 B c x^{\frac{5}{2}}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.26966, size = 39, normalized size = 0.95 \begin{align*} \frac{2}{5} \, B c x^{\frac{5}{2}} + \frac{2}{3} \, A c x^{\frac{3}{2}} + 2 \, B a \sqrt{x} - \frac{2 \, A a}{\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]