Optimal. Leaf size=38 \[ -\frac {a \sqrt {a+b x^4}}{2 b^2}+\frac {\left (a+b x^4\right )^{3/2}}{6 b^2} \]
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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 = {272, 45}
\begin {gather*} \frac {\left (a+b x^4\right )^{3/2}}{6 b^2}-\frac {a \sqrt {a+b x^4}}{2 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {x^7}{\sqrt {a+b x^4}} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {x}{\sqrt {a+b x}} \, dx,x,x^4\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \left (-\frac {a}{b \sqrt {a+b x}}+\frac {\sqrt {a+b x}}{b}\right ) \, dx,x,x^4\right )\\ &=-\frac {a \sqrt {a+b x^4}}{2 b^2}+\frac {\left (a+b x^4\right )^{3/2}}{6 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 27, normalized size = 0.71 \begin {gather*} \frac {\left (-2 a+b x^4\right ) \sqrt {a+b x^4}}{6 b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 25, normalized size = 0.66
method | result | size |
gosper | \(-\frac {\sqrt {b \,x^{4}+a}\, \left (-b \,x^{4}+2 a \right )}{6 b^{2}}\) | \(25\) |
default | \(-\frac {\sqrt {b \,x^{4}+a}\, \left (-b \,x^{4}+2 a \right )}{6 b^{2}}\) | \(25\) |
trager | \(-\frac {\sqrt {b \,x^{4}+a}\, \left (-b \,x^{4}+2 a \right )}{6 b^{2}}\) | \(25\) |
risch | \(-\frac {\sqrt {b \,x^{4}+a}\, \left (-b \,x^{4}+2 a \right )}{6 b^{2}}\) | \(25\) |
elliptic | \(-\frac {\sqrt {b \,x^{4}+a}\, \left (-b \,x^{4}+2 a \right )}{6 b^{2}}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 30, normalized size = 0.79 \begin {gather*} \frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}}}{6 \, b^{2}} - \frac {\sqrt {b x^{4} + a} a}{2 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 23, normalized size = 0.61 \begin {gather*} \frac {\sqrt {b x^{4} + a} {\left (b x^{4} - 2 \, a\right )}}{6 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.29, size = 42, normalized size = 1.11 \begin {gather*} \begin {cases} - \frac {a \sqrt {a + b x^{4}}}{3 b^{2}} + \frac {x^{4} \sqrt {a + b x^{4}}}{6 b} & \text {for}\: b \neq 0 \\\frac {x^{8}}{8 \sqrt {a}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.14, size = 30, normalized size = 0.79 \begin {gather*} \frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}}}{6 \, b^{2}} - \frac {\sqrt {b x^{4} + a} a}{2 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.15, size = 24, normalized size = 0.63 \begin {gather*} -\frac {\sqrt {b\,x^4+a}\,\left (2\,a-b\,x^4\right )}{6\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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