Optimal. Leaf size=38 \[ \frac {3 a}{2 b^2 \sqrt [3]{a+b x^2}}+\frac {3 \left (a+b x^2\right )^{2/3}}{4 b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac {3 a}{2 b^2 \sqrt [3]{a+b x^2}}+\frac {3 \left (a+b x^2\right )^{2/3}}{4 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^3}{\left (a+b x^2\right )^{4/3}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{(a+b x)^{4/3}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {a}{b (a+b x)^{4/3}}+\frac {1}{b \sqrt [3]{a+b x}}\right ) \, dx,x,x^2\right )\\ &=\frac {3 a}{2 b^2 \sqrt [3]{a+b x^2}}+\frac {3 \left (a+b x^2\right )^{2/3}}{4 b^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 27, normalized size = 0.71 \[ \frac {3 \left (3 a+b x^2\right )}{4 b^2 \sqrt [3]{a+b x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.86, size = 35, normalized size = 0.92 \[ \frac {3 \, {\left (b x^{2} + 3 \, a\right )} {\left (b x^{2} + a\right )}^{\frac {2}{3}}}{4 \, {\left (b^{3} x^{2} + a b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.57, size = 34, normalized size = 0.89 \[ \frac {3 \, {\left (\frac {{\left (b x^{2} + a\right )}^{\frac {2}{3}}}{b} + \frac {2 \, a}{{\left (b x^{2} + a\right )}^{\frac {1}{3}} b}\right )}}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 24, normalized size = 0.63 \[ \frac {\frac {3 b \,x^{2}}{4}+\frac {9 a}{4}}{\left (b \,x^{2}+a \right )^{\frac {1}{3}} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.35, size = 30, normalized size = 0.79 \[ \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {2}{3}}}{4 \, b^{2}} + \frac {3 \, a}{2 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 5.59, size = 24, normalized size = 0.63 \[ \frac {3\,b\,x^2+9\,a}{4\,b^2\,{\left (b\,x^2+a\right )}^{1/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.70, size = 46, normalized size = 1.21 \[ \begin {cases} \frac {9 a}{4 b^{2} \sqrt [3]{a + b x^{2}}} + \frac {3 x^{2}}{4 b \sqrt [3]{a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{4}}{4 a^{\frac {4}{3}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________