Optimal. Leaf size=60 \[ \frac {1}{3} a^2 c x^2 \sqrt {c x^2}+\frac {1}{2} a b c x^3 \sqrt {c x^2}+\frac {1}{5} b^2 c x^4 \sqrt {c x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 43} \[ \frac {1}{3} a^2 c x^2 \sqrt {c x^2}+\frac {1}{2} a b c x^3 \sqrt {c x^2}+\frac {1}{5} b^2 c x^4 \sqrt {c x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 15
Rule 43
Rubi steps
\begin {align*} \int \frac {\left (c x^2\right )^{3/2} (a+b x)^2}{x} \, dx &=\frac {\left (c \sqrt {c x^2}\right ) \int x^2 (a+b x)^2 \, dx}{x}\\ &=\frac {\left (c \sqrt {c x^2}\right ) \int \left (a^2 x^2+2 a b x^3+b^2 x^4\right ) \, dx}{x}\\ &=\frac {1}{3} a^2 c x^2 \sqrt {c x^2}+\frac {1}{2} a b c x^3 \sqrt {c x^2}+\frac {1}{5} b^2 c x^4 \sqrt {c x^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 36, normalized size = 0.60 \[ \frac {1}{30} c x^2 \sqrt {c x^2} \left (10 a^2+15 a b x+6 b^2 x^2\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 36, normalized size = 0.60 \[ \frac {1}{30} \, {\left (6 \, b^{2} c x^{4} + 15 \, a b c x^{3} + 10 \, a^{2} c x^{2}\right )} \sqrt {c x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.95, size = 35, normalized size = 0.58 \[ \frac {1}{30} \, {\left (6 \, b^{2} x^{5} \mathrm {sgn}\relax (x) + 15 \, a b x^{4} \mathrm {sgn}\relax (x) + 10 \, a^{2} x^{3} \mathrm {sgn}\relax (x)\right )} c^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 29, normalized size = 0.48 \[ \frac {\left (6 b^{2} x^{2}+15 a b x +10 a^{2}\right ) \left (c \,x^{2}\right )^{\frac {3}{2}}}{30} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.30, size = 40, normalized size = 0.67 \[ \frac {1}{2} \, \left (c x^{2}\right )^{\frac {3}{2}} a b x + \frac {1}{3} \, \left (c x^{2}\right )^{\frac {3}{2}} a^{2} + \frac {\left (c x^{2}\right )^{\frac {5}{2}} b^{2}}{5 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (c\,x^2\right )}^{3/2}\,{\left (a+b\,x\right )}^2}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.79, size = 54, normalized size = 0.90 \[ \frac {a^{2} c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}}{3} + \frac {a b c^{\frac {3}{2}} x \left (x^{2}\right )^{\frac {3}{2}}}{2} + \frac {b^{2} c^{\frac {3}{2}} x^{2} \left (x^{2}\right )^{\frac {3}{2}}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________