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 = 0.02, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, 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 = 0.01, 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]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 29, normalized size = 0.78 \[ \frac {2 \, {\left (3 \, B b x^{4} + 7 \, {\left (B a + A b\right )} x^{2} - 21 \, A a\right )}}{21 \, \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.35, size = 29, normalized size = 0.78 \[ \frac {2}{7} \, B b x^{\frac {7}{2}} + \frac {2}{3} \, B a x^{\frac {3}{2}} + \frac {2}{3} \, A b x^{\frac {3}{2}} - \frac {2 \, A a}{\sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 32, normalized size = 0.86 \[ -\frac {2 \left (-3 B b \,x^{4}-7 A b \,x^{2}-7 B a \,x^{2}+21 A a \right )}{21 \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.10, size = 27, normalized size = 0.73 \[ \frac {2}{7} \, B b x^{\frac {7}{2}} + \frac {2}{3} \, {\left (B a + A b\right )} x^{\frac {3}{2}} - \frac {2 \, A a}{\sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.19, size = 31, normalized size = 0.84 \[ \frac {14\,A\,b\,x^2-42\,A\,a+14\,B\,a\,x^2+6\,B\,b\,x^4}{21\,\sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.99, size = 44, normalized size = 1.19 \[ - \frac {2 A a}{\sqrt {x}} + \frac {2 A b x^{\frac {3}{2}}}{3} + \frac {2 B a x^{\frac {3}{2}}}{3} + \frac {2 B b x^{\frac {7}{2}}}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________