Optimal. Leaf size=59 \[ \frac {a^2 \sqrt {a+b x^4}}{2 b^3}-\frac {a \left (a+b x^4\right )^{3/2}}{3 b^3}+\frac {\left (a+b x^4\right )^{5/2}}{10 b^3} \]
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Rubi [A]
time = 0.03, antiderivative size = 59, 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 {a^2 \sqrt {a+b x^4}}{2 b^3}+\frac {\left (a+b x^4\right )^{5/2}}{10 b^3}-\frac {a \left (a+b x^4\right )^{3/2}}{3 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {x^{11}}{\sqrt {a+b x^4}} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {x^2}{\sqrt {a+b x}} \, dx,x,x^4\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \left (\frac {a^2}{b^2 \sqrt {a+b x}}-\frac {2 a \sqrt {a+b x}}{b^2}+\frac {(a+b x)^{3/2}}{b^2}\right ) \, dx,x,x^4\right )\\ &=\frac {a^2 \sqrt {a+b x^4}}{2 b^3}-\frac {a \left (a+b x^4\right )^{3/2}}{3 b^3}+\frac {\left (a+b x^4\right )^{5/2}}{10 b^3}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 39, normalized size = 0.66 \begin {gather*} \frac {\sqrt {a+b x^4} \left (8 a^2-4 a b x^4+3 b^2 x^8\right )}{30 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 36, normalized size = 0.61
method | result | size |
gosper | \(\frac {\sqrt {b \,x^{4}+a}\, \left (3 b^{2} x^{8}-4 a b \,x^{4}+8 a^{2}\right )}{30 b^{3}}\) | \(36\) |
default | \(\frac {\sqrt {b \,x^{4}+a}\, \left (3 b^{2} x^{8}-4 a b \,x^{4}+8 a^{2}\right )}{30 b^{3}}\) | \(36\) |
trager | \(\frac {\sqrt {b \,x^{4}+a}\, \left (3 b^{2} x^{8}-4 a b \,x^{4}+8 a^{2}\right )}{30 b^{3}}\) | \(36\) |
risch | \(\frac {\sqrt {b \,x^{4}+a}\, \left (3 b^{2} x^{8}-4 a b \,x^{4}+8 a^{2}\right )}{30 b^{3}}\) | \(36\) |
elliptic | \(\frac {\sqrt {b \,x^{4}+a}\, \left (3 b^{2} x^{8}-4 a b \,x^{4}+8 a^{2}\right )}{30 b^{3}}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 47, normalized size = 0.80 \begin {gather*} \frac {{\left (b x^{4} + a\right )}^{\frac {5}{2}}}{10 \, b^{3}} - \frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}} a}{3 \, b^{3}} + \frac {\sqrt {b x^{4} + a} a^{2}}{2 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 35, normalized size = 0.59 \begin {gather*} \frac {{\left (3 \, b^{2} x^{8} - 4 \, a b x^{4} + 8 \, a^{2}\right )} \sqrt {b x^{4} + a}}{30 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.43, size = 68, normalized size = 1.15 \begin {gather*} \begin {cases} \frac {4 a^{2} \sqrt {a + b x^{4}}}{15 b^{3}} - \frac {2 a x^{4} \sqrt {a + b x^{4}}}{15 b^{2}} + \frac {x^{8} \sqrt {a + b x^{4}}}{10 b} & \text {for}\: b \neq 0 \\\frac {x^{12}}{12 \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 = 1.42, size = 47, normalized size = 0.80 \begin {gather*} \frac {\sqrt {b x^{4} + a} a^{2}}{2 \, b^{3}} + \frac {3 \, {\left (b x^{4} + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x^{4} + a\right )}^{\frac {3}{2}} a}{30 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.20, size = 36, normalized size = 0.61 \begin {gather*} \sqrt {b\,x^4+a}\,\left (\frac {4\,a^2}{15\,b^3}+\frac {x^8}{10\,b}-\frac {2\,a\,x^4}{15\,b^2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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