Optimal. Leaf size=40 \[ \frac {2 (A b-a B) \sqrt {a+b x}}{b^2}+\frac {2 B (a+b x)^{3/2}}{3 b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45}
\begin {gather*} \frac {2 \sqrt {a+b x} (A b-a B)}{b^2}+\frac {2 B (a+b x)^{3/2}}{3 b^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rubi steps
\begin {align*} \int \frac {A+B x}{\sqrt {a+b x}} \, dx &=\int \left (\frac {A b-a B}{b \sqrt {a+b x}}+\frac {B \sqrt {a+b x}}{b}\right ) \, dx\\ &=\frac {2 (A b-a B) \sqrt {a+b x}}{b^2}+\frac {2 B (a+b x)^{3/2}}{3 b^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 29, normalized size = 0.72 \begin {gather*} \frac {2 \sqrt {a+b x} (3 A b-2 a B+b B x)}{3 b^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.06, size = 38, normalized size = 0.95
method | result | size |
gosper | \(\frac {2 \sqrt {b x +a}\, \left (b B x +3 A b -2 B a \right )}{3 b^{2}}\) | \(26\) |
trager | \(\frac {2 \sqrt {b x +a}\, \left (b B x +3 A b -2 B a \right )}{3 b^{2}}\) | \(26\) |
risch | \(\frac {2 \sqrt {b x +a}\, \left (b B x +3 A b -2 B a \right )}{3 b^{2}}\) | \(26\) |
derivativedivides | \(\frac {\frac {2 B \left (b x +a \right )^{\frac {3}{2}}}{3}+2 A b \sqrt {b x +a}-2 B a \sqrt {b x +a}}{b^{2}}\) | \(38\) |
default | \(\frac {\frac {2 B \left (b x +a \right )^{\frac {3}{2}}}{3}+2 A b \sqrt {b x +a}-2 B a \sqrt {b x +a}}{b^{2}}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 39, normalized size = 0.98 \begin {gather*} \frac {2 \, {\left (3 \, \sqrt {b x + a} A + \frac {{\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} B}{b}\right )}}{3 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.11, size = 25, normalized size = 0.62 \begin {gather*} \frac {2 \, {\left (B b x - 2 \, B a + 3 \, A b\right )} \sqrt {b x + a}}{3 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 121 vs.
\(2 (37) = 74\).
time = 2.21, size = 121, normalized size = 3.02 \begin {gather*} \begin {cases} \frac {- \frac {2 A a}{\sqrt {a + b x}} - 2 A \left (- \frac {a}{\sqrt {a + b x}} - \sqrt {a + b x}\right ) - \frac {2 B a \left (- \frac {a}{\sqrt {a + b x}} - \sqrt {a + b x}\right )}{b} - \frac {2 B \left (\frac {a^{2}}{\sqrt {a + b x}} + 2 a \sqrt {a + b x} - \frac {\left (a + b x\right )^{\frac {3}{2}}}{3}\right )}{b}}{b} & \text {for}\: b \neq 0 \\\frac {A x + \frac {B x^{2}}{2}}{\sqrt {a}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.14, size = 39, normalized size = 0.98 \begin {gather*} \frac {2 \, {\left (3 \, \sqrt {b x + a} A + \frac {{\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} B}{b}\right )}}{3 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.05, size = 28, normalized size = 0.70 \begin {gather*} \frac {2\,\sqrt {a+b\,x}\,\left (3\,A\,b-3\,B\,a+B\,\left (a+b\,x\right )\right )}{3\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________