Optimal. Leaf size=35 \[ 2 \sqrt {x} (a B+A b)-\frac {2 a A}{\sqrt {x}}+\frac {2}{3} b B x^{3/2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {76} \[ 2 \sqrt {x} (a B+A b)-\frac {2 a A}{\sqrt {x}}+\frac {2}{3} b B x^{3/2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 76
Rubi steps
\begin {align*} \int \frac {(a+b x) (A+B x)}{x^{3/2}} \, dx &=\int \left (\frac {a A}{x^{3/2}}+\frac {A b+a B}{\sqrt {x}}+b B \sqrt {x}\right ) \, dx\\ &=-\frac {2 a A}{\sqrt {x}}+2 (A b+a B) \sqrt {x}+\frac {2}{3} b B x^{3/2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 29, normalized size = 0.83 \[ \frac {2 (b x (3 A+B x)-3 a (A-B x))}{3 \sqrt {x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.91, size = 26, normalized size = 0.74 \[ \frac {2 \, {\left (B b x^{2} - 3 \, A a + 3 \, {\left (B a + A b\right )} x\right )}}{3 \, \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.21, size = 29, normalized size = 0.83 \[ \frac {2}{3} \, B b x^{\frac {3}{2}} + 2 \, B a \sqrt {x} + 2 \, A b \sqrt {x} - \frac {2 \, A a}{\sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 28, normalized size = 0.80 \[ -\frac {2 \left (-B b \,x^{2}-3 A b x -3 B a x +3 A a \right )}{3 \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.81, size = 27, normalized size = 0.77 \[ \frac {2}{3} \, B b x^{\frac {3}{2}} - \frac {2 \, A a}{\sqrt {x}} + 2 \, {\left (B a + A b\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.34, size = 27, normalized size = 0.77 \[ \frac {6\,A\,b\,x-6\,A\,a+6\,B\,a\,x+2\,B\,b\,x^2}{3\,\sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.43, size = 41, normalized size = 1.17 \[ - \frac {2 A a}{\sqrt {x}} + 2 A b \sqrt {x} + 2 B a \sqrt {x} + \frac {2 B b x^{\frac {3}{2}}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________