Optimal. Leaf size=61 \[ \frac {2 B \sqrt {x} \sqrt {b x+c x^2}}{3 c}-\frac {2 \sqrt {b x+c x^2} (2 b B-3 A c)}{3 c^2 \sqrt {x}} \]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {794, 648} \begin {gather*} \frac {2 B \sqrt {x} \sqrt {b x+c x^2}}{3 c}-\frac {2 \sqrt {b x+c x^2} (2 b B-3 A c)}{3 c^2 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 648
Rule 794
Rubi steps
\begin {align*} \int \frac {\sqrt {x} (A+B x)}{\sqrt {b x+c x^2}} \, dx &=\frac {2 B \sqrt {x} \sqrt {b x+c x^2}}{3 c}+\frac {\left (2 \left (\frac {1}{2} (-b B+A c)+\frac {1}{2} (-b B+2 A c)\right )\right ) \int \frac {\sqrt {x}}{\sqrt {b x+c x^2}} \, dx}{3 c}\\ &=-\frac {2 (2 b B-3 A c) \sqrt {b x+c x^2}}{3 c^2 \sqrt {x}}+\frac {2 B \sqrt {x} \sqrt {b x+c x^2}}{3 c}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 36, normalized size = 0.59 \begin {gather*} \frac {2 \sqrt {x (b+c x)} (3 A c-2 b B+B c x)}{3 c^2 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.08, size = 38, normalized size = 0.62 \begin {gather*} \frac {2 \sqrt {b x+c x^2} (3 A c-2 b B+B c x)}{3 c^2 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.40, size = 32, normalized size = 0.52 \begin {gather*} \frac {2 \, {\left (B c x - 2 \, B b + 3 \, A c\right )} \sqrt {c x^{2} + b x}}{3 \, c^{2} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 53, normalized size = 0.87 \begin {gather*} \frac {2 \, {\left (c x + b\right )}^{\frac {3}{2}} B}{3 \, c^{2}} - \frac {2 \, {\left (B b - A c\right )} \sqrt {c x + b}}{c^{2}} + \frac {2 \, {\left (2 \, B b^{\frac {3}{2}} - 3 \, A \sqrt {b} c\right )}}{3 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 38, normalized size = 0.62 \begin {gather*} \frac {2 \left (c x +b \right ) \left (B c x +3 A c -2 b B \right ) \sqrt {x}}{3 \sqrt {c \,x^{2}+b x}\, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.62, size = 45, normalized size = 0.74 \begin {gather*} \frac {2 \, \sqrt {c x + b} A}{c} + \frac {2 \, {\left (c^{2} x^{2} - b c x - 2 \, b^{2}\right )} B}{3 \, \sqrt {c x + b} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {x}\,\left (A+B\,x\right )}{\sqrt {c\,x^2+b\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x} \left (A + B x\right )}{\sqrt {x \left (b + c x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________