Optimal. Leaf size=55 \[ -\frac {a^2}{b^3 \sqrt {a+b x^2}}-\frac {2 a \sqrt {a+b x^2}}{b^3}+\frac {\left (a+b x^2\right )^{3/2}}{3 b^3} \]
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Rubi [A] time = 0.03, antiderivative size = 55, 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} \[ -\frac {a^2}{b^3 \sqrt {a+b x^2}}-\frac {2 a \sqrt {a+b x^2}}{b^3}+\frac {\left (a+b x^2\right )^{3/2}}{3 b^3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^5}{\left (a+b x^2\right )^{3/2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2}{(a+b x)^{3/2}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {a^2}{b^2 (a+b x)^{3/2}}-\frac {2 a}{b^2 \sqrt {a+b x}}+\frac {\sqrt {a+b x}}{b^2}\right ) \, dx,x,x^2\right )\\ &=-\frac {a^2}{b^3 \sqrt {a+b x^2}}-\frac {2 a \sqrt {a+b x^2}}{b^3}+\frac {\left (a+b x^2\right )^{3/2}}{3 b^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 38, normalized size = 0.69 \[ \frac {-8 a^2-4 a b x^2+b^2 x^4}{3 b^3 \sqrt {a+b x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 46, normalized size = 0.84 \[ \frac {{\left (b^{2} x^{4} - 4 \, a b x^{2} - 8 \, a^{2}\right )} \sqrt {b x^{2} + a}}{3 \, {\left (b^{4} x^{2} + a b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.98, size = 52, normalized size = 0.95 \[ -\frac {a^{2}}{\sqrt {b x^{2} + a} b^{3}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{6} - 6 \, \sqrt {b x^{2} + a} a b^{6}}{3 \, b^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 36, normalized size = 0.65 \[ -\frac {-b^{2} x^{4}+4 a b \,x^{2}+8 a^{2}}{3 \sqrt {b \,x^{2}+a}\, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 53, normalized size = 0.96 \[ \frac {x^{4}}{3 \, \sqrt {b x^{2} + a} b} - \frac {4 \, a x^{2}}{3 \, \sqrt {b x^{2} + a} b^{2}} - \frac {8 \, a^{2}}{3 \, \sqrt {b x^{2} + a} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.72, size = 41, normalized size = 0.75 \[ -\frac {6\,a\,\left (b\,x^2+a\right )-{\left (b\,x^2+a\right )}^2+3\,a^2}{3\,b^3\,\sqrt {b\,x^2+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.94, size = 68, normalized size = 1.24 \[ \begin {cases} - \frac {8 a^{2}}{3 b^{3} \sqrt {a + b x^{2}}} - \frac {4 a x^{2}}{3 b^{2} \sqrt {a + b x^{2}}} + \frac {x^{4}}{3 b \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{6}}{6 a^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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