Optimal. Leaf size=57 \[ -\frac {a^2}{2 b^3 \sqrt {a+b x^4}}-\frac {a \sqrt {a+b x^4}}{b^3}+\frac {\left (a+b x^4\right )^{3/2}}{6 b^3} \]
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Rubi [A]
time = 0.03, antiderivative size = 57, 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}{2 b^3 \sqrt {a+b x^4}}-\frac {a \sqrt {a+b x^4}}{b^3}+\frac {\left (a+b x^4\right )^{3/2}}{6 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {x^{11}}{\left (a+b x^4\right )^{3/2}} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {x^2}{(a+b x)^{3/2}} \, dx,x,x^4\right )\\ &=\frac {1}{4} \text {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^4\right )\\ &=-\frac {a^2}{2 b^3 \sqrt {a+b x^4}}-\frac {a \sqrt {a+b x^4}}{b^3}+\frac {\left (a+b x^4\right )^{3/2}}{6 b^3}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 38, normalized size = 0.67 \begin {gather*} \frac {-8 a^2-4 a b x^4+b^2 x^8}{6 b^3 \sqrt {a+b x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 36, normalized size = 0.63
method | result | size |
gosper | \(-\frac {-b^{2} x^{8}+4 a b \,x^{4}+8 a^{2}}{6 \sqrt {b \,x^{4}+a}\, b^{3}}\) | \(36\) |
default | \(-\frac {-b^{2} x^{8}+4 a b \,x^{4}+8 a^{2}}{6 \sqrt {b \,x^{4}+a}\, b^{3}}\) | \(36\) |
trager | \(-\frac {-b^{2} x^{8}+4 a b \,x^{4}+8 a^{2}}{6 \sqrt {b \,x^{4}+a}\, b^{3}}\) | \(36\) |
elliptic | \(-\frac {-b^{2} x^{8}+4 a b \,x^{4}+8 a^{2}}{6 \sqrt {b \,x^{4}+a}\, b^{3}}\) | \(36\) |
risch | \(-\frac {\left (-b \,x^{4}+5 a \right ) \sqrt {b \,x^{4}+a}}{6 b^{3}}-\frac {a^{2}}{2 b^{3} \sqrt {b \,x^{4}+a}}\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 47, normalized size = 0.82 \begin {gather*} \frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}}}{6 \, b^{3}} - \frac {\sqrt {b x^{4} + a} a}{b^{3}} - \frac {a^{2}}{2 \, \sqrt {b x^{4} + a} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 46, normalized size = 0.81 \begin {gather*} \frac {{\left (b^{2} x^{8} - 4 \, a b x^{4} - 8 \, a^{2}\right )} \sqrt {b x^{4} + a}}{6 \, {\left (b^{4} x^{4} + a b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.53, size = 68, normalized size = 1.19 \begin {gather*} \begin {cases} - \frac {4 a^{2}}{3 b^{3} \sqrt {a + b x^{4}}} - \frac {2 a x^{4}}{3 b^{2} \sqrt {a + b x^{4}}} + \frac {x^{8}}{6 b \sqrt {a + b x^{4}}} & \text {for}\: b \neq 0 \\\frac {x^{12}}{12 a^{\frac {3}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.79, size = 52, normalized size = 0.91 \begin {gather*} -\frac {a^{2}}{2 \, \sqrt {b x^{4} + a} b^{3}} + \frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}} b^{6} - 6 \, \sqrt {b x^{4} + a} a b^{6}}{6 \, b^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.30, size = 41, normalized size = 0.72 \begin {gather*} -\frac {6\,a\,\left (b\,x^4+a\right )-{\left (b\,x^4+a\right )}^2+3\,a^2}{6\,b^3\,\sqrt {b\,x^4+a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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