Optimal. Leaf size=37 \[ \frac{2}{3} x^{3/2} (a B+A b)-\frac{2 a A}{\sqrt{x}}+\frac{2}{7} b B x^{7/2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.015213, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {448} \[ \frac{2}{3} x^{3/2} (a B+A b)-\frac{2 a A}{\sqrt{x}}+\frac{2}{7} b B x^{7/2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 448
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right ) \left (A+B x^2\right )}{x^{3/2}} \, dx &=\int \left (\frac{a A}{x^{3/2}}+(A b+a B) \sqrt{x}+b B x^{5/2}\right ) \, dx\\ &=-\frac{2 a A}{\sqrt{x}}+\frac{2}{3} (A b+a B) x^{3/2}+\frac{2}{7} b B x^{7/2}\\ \end{align*}
Mathematica [A] time = 0.0104049, size = 35, normalized size = 0.95 \[ \frac{2 \left (-21 a A+7 a B x^2+7 A b x^2+3 b B x^4\right )}{21 \sqrt{x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.002, size = 32, normalized size = 0.9 \begin{align*} -{\frac{-6\,bB{x}^{4}-14\,A{x}^{2}b-14\,B{x}^{2}a+42\,Aa}{21}{\frac{1}{\sqrt{x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.00286, size = 36, normalized size = 0.97 \begin{align*} \frac{2}{7} \, B b x^{\frac{7}{2}} + \frac{2}{3} \,{\left (B a + A b\right )} x^{\frac{3}{2}} - \frac{2 \, A a}{\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.852547, size = 74, normalized size = 2. \begin{align*} \frac{2 \,{\left (3 \, B b x^{4} + 7 \,{\left (B a + A b\right )} x^{2} - 21 \, A a\right )}}{21 \, \sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.983058, size = 44, normalized size = 1.19 \begin{align*} - \frac{2 A a}{\sqrt{x}} + \frac{2 A b x^{\frac{3}{2}}}{3} + \frac{2 B a x^{\frac{3}{2}}}{3} + \frac{2 B b x^{\frac{7}{2}}}{7} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.1786, size = 39, normalized size = 1.05 \begin{align*} \frac{2}{7} \, B b x^{\frac{7}{2}} + \frac{2}{3} \, B a x^{\frac{3}{2}} + \frac{2}{3} \, A b x^{\frac{3}{2}} - \frac{2 \, A a}{\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]