Optimal. Leaf size=24 \[ \frac {x (a+b x)^3}{3 b \sqrt {c x^2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.00, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 32} \begin {gather*} \frac {x (a+b x)^3}{3 b \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 15
Rule 32
Rubi steps
\begin {align*} \int \frac {x (a+b x)^2}{\sqrt {c x^2}} \, dx &=\frac {x \int (a+b x)^2 \, dx}{\sqrt {c x^2}}\\ &=\frac {x (a+b x)^3}{3 b \sqrt {c x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 24, normalized size = 1.00 \begin {gather*} \frac {x (a+b x)^3}{3 b \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.03, size = 34, normalized size = 1.42 \begin {gather*} \frac {\sqrt {c x^2} \left (3 a^2+3 a b x+b^2 x^2\right )}{3 c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.16, size = 30, normalized size = 1.25 \begin {gather*} \frac {{\left (b^{2} x^{2} + 3 \, a b x + 3 \, a^{2}\right )} \sqrt {c x^{2}}}{3 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.15, size = 36, normalized size = 1.50 \begin {gather*} \frac {1}{3} \, \sqrt {c x^{2}} {\left ({\left (\frac {b^{2} x}{c} + \frac {3 \, a b}{c}\right )} x + \frac {3 \, a^{2}}{c}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 31, normalized size = 1.29 \begin {gather*} \frac {\left (b^{2} x^{2}+3 a b x +3 a^{2}\right ) x^{2}}{3 \sqrt {c \,x^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 1.37, size = 42, normalized size = 1.75 \begin {gather*} \frac {\sqrt {c x^{2}} b^{2} x^{2}}{3 \, c} + \frac {a b x^{2}}{\sqrt {c}} + \frac {\sqrt {c x^{2}} a^{2}}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {x\,{\left (a+b\,x\right )}^2}{\sqrt {c\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.54, size = 56, normalized size = 2.33 \begin {gather*} \frac {a^{2} x^{2}}{\sqrt {c} \sqrt {x^{2}}} + \frac {a b x^{3}}{\sqrt {c} \sqrt {x^{2}}} + \frac {b^{2} x^{4}}{3 \sqrt {c} \sqrt {x^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________