Optimal. Leaf size=38 \[ \frac {a}{2 b^2 \sqrt {a+b x^4}}+\frac {\sqrt {a+b x^4}}{2 b^2} \]
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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} \[ \frac {a}{2 b^2 \sqrt {a+b x^4}}+\frac {\sqrt {a+b x^4}}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^7}{\left (a+b x^4\right )^{3/2}} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {x}{(a+b x)^{3/2}} \, dx,x,x^4\right )\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \left (-\frac {a}{b (a+b x)^{3/2}}+\frac {1}{b \sqrt {a+b x}}\right ) \, dx,x,x^4\right )\\ &=\frac {a}{2 b^2 \sqrt {a+b x^4}}+\frac {\sqrt {a+b x^4}}{2 b^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 27, normalized size = 0.71 \[ \frac {2 a+b x^4}{2 b^2 \sqrt {a+b x^4}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 35, normalized size = 0.92 \[ \frac {{\left (b x^{4} + 2 \, a\right )} \sqrt {b x^{4} + a}}{2 \, {\left (b^{3} x^{4} + a b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 33, normalized size = 0.87 \[ \frac {\frac {\sqrt {b x^{4} + a}}{b} + \frac {a}{\sqrt {b x^{4} + a} b}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 24, normalized size = 0.63 \[ \frac {b \,x^{4}+2 a}{2 \sqrt {b \,x^{4}+a}\, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, size = 30, normalized size = 0.79 \[ \frac {\sqrt {b x^{4} + a}}{2 \, b^{2}} + \frac {a}{2 \, \sqrt {b x^{4} + a} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.21, size = 21, normalized size = 0.55 \[ \frac {\frac {b\,x^4}{2}+a}{b^2\,\sqrt {b\,x^4+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.48, size = 41, normalized size = 1.08 \[ \begin {cases} \frac {a}{b^{2} \sqrt {a + b x^{4}}} + \frac {x^{4}}{2 b \sqrt {a + b x^{4}}} & \text {for}\: b \neq 0 \\\frac {x^{8}}{8 a^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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