Optimal. Leaf size=52 \[ \frac {\left (b x^2+c x^4\right )^{5/2}}{7 c x^3}-\frac {2 b \left (b x^2+c x^4\right )^{5/2}}{35 c^2 x^5} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2002, 2014} \[ \frac {\left (b x^2+c x^4\right )^{5/2}}{7 c x^3}-\frac {2 b \left (b x^2+c x^4\right )^{5/2}}{35 c^2 x^5} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2002
Rule 2014
Rubi steps
\begin {align*} \int \left (b x^2+c x^4\right )^{3/2} \, dx &=\frac {\left (b x^2+c x^4\right )^{5/2}}{7 c x^3}-\frac {(2 b) \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^2} \, dx}{7 c}\\ &=-\frac {2 b \left (b x^2+c x^4\right )^{5/2}}{35 c^2 x^5}+\frac {\left (b x^2+c x^4\right )^{5/2}}{7 c x^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 42, normalized size = 0.81 \[ \frac {x \left (b+c x^2\right )^3 \left (5 c x^2-2 b\right )}{35 c^2 \sqrt {x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.76, size = 52, normalized size = 1.00 \[ \frac {{\left (5 \, c^{3} x^{6} + 8 \, b c^{2} x^{4} + b^{2} c x^{2} - 2 \, b^{3}\right )} \sqrt {c x^{4} + b x^{2}}}{35 \, c^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 44, normalized size = 0.85 \[ \frac {2 \, b^{\frac {7}{2}} \mathrm {sgn}\relax (x)}{35 \, c^{2}} + \frac {5 \, {\left (c x^{2} + b\right )}^{\frac {7}{2}} \mathrm {sgn}\relax (x) - 7 \, {\left (c x^{2} + b\right )}^{\frac {5}{2}} b \mathrm {sgn}\relax (x)}{35 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 39, normalized size = 0.75 \[ -\frac {\left (c \,x^{2}+b \right ) \left (-5 c \,x^{2}+2 b \right ) \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}}}{35 c^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.50, size = 45, normalized size = 0.87 \[ \frac {{\left (5 \, c^{3} x^{6} + 8 \, b c^{2} x^{4} + b^{2} c x^{2} - 2 \, b^{3}\right )} \sqrt {c x^{2} + b}}{35 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.16, size = 40, normalized size = 0.77 \[ -\frac {{\left (c\,x^2+b\right )}^2\,\sqrt {c\,x^4+b\,x^2}\,\left (2\,b-5\,c\,x^2\right )}{35\,c^2\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b x^{2} + c x^{4}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________