Optimal. Leaf size=38 \[ \frac {\left (a+b x^2\right )^{5/2}}{5 b^2}-\frac {a \left (a+b x^2\right )^{3/2}}{3 b^2} \]
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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} \[ \frac {\left (a+b x^2\right )^{5/2}}{5 b^2}-\frac {a \left (a+b x^2\right )^{3/2}}{3 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^3 \sqrt {a+b x^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x \sqrt {a+b x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {a \sqrt {a+b x}}{b}+\frac {(a+b x)^{3/2}}{b}\right ) \, dx,x,x^2\right )\\ &=-\frac {a \left (a+b x^2\right )^{3/2}}{3 b^2}+\frac {\left (a+b x^2\right )^{5/2}}{5 b^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 0.74 \[ \frac {\left (a+b x^2\right )^{3/2} \left (3 b x^2-2 a\right )}{15 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 34, normalized size = 0.89 \[ \frac {{\left (3 \, b^{2} x^{4} + a b x^{2} - 2 \, a^{2}\right )} \sqrt {b x^{2} + a}}{15 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.60, size = 29, normalized size = 0.76 \[ \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} - 5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a}{15 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 25, normalized size = 0.66 \[ -\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (-3 b \,x^{2}+2 a \right )}{15 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 33, normalized size = 0.87 \[ \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} x^{2}}{5 \, b} - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a}{15 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.65, size = 33, normalized size = 0.87 \[ \sqrt {b\,x^2+a}\,\left (\frac {x^4}{5}-\frac {2\,a^2}{15\,b^2}+\frac {a\,x^2}{15\,b}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.32, size = 63, normalized size = 1.66 \[ \begin {cases} - \frac {2 a^{2} \sqrt {a + b x^{2}}}{15 b^{2}} + \frac {a x^{2} \sqrt {a + b x^{2}}}{15 b} + \frac {x^{4} \sqrt {a + b x^{2}}}{5} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{4}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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