Optimal. Leaf size=69 \[ \frac {3 b}{4 a^2 \sqrt {a+\frac {b}{x^4}}}+\frac {x^4}{4 a \sqrt {a+\frac {b}{x^4}}}-\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )}{4 a^{5/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {272, 44, 53, 65,
214} \begin {gather*} -\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )}{4 a^{5/2}}+\frac {3 b}{4 a^2 \sqrt {a+\frac {b}{x^4}}}+\frac {x^4}{4 a \sqrt {a+\frac {b}{x^4}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 53
Rule 65
Rule 214
Rule 272
Rubi steps
\begin {align*} \int \frac {x^3}{\left (a+\frac {b}{x^4}\right )^{3/2}} \, dx &=-\left (\frac {1}{4} \text {Subst}\left (\int \frac {1}{x^2 (a+b x)^{3/2}} \, dx,x,\frac {1}{x^4}\right )\right )\\ &=-\frac {x^4}{2 a \sqrt {a+\frac {b}{x^4}}}-\frac {3 \text {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,\frac {1}{x^4}\right )}{4 a}\\ &=-\frac {x^4}{2 a \sqrt {a+\frac {b}{x^4}}}+\frac {3 \sqrt {a+\frac {b}{x^4}} x^4}{4 a^2}+\frac {(3 b) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x^4}\right )}{8 a^2}\\ &=-\frac {x^4}{2 a \sqrt {a+\frac {b}{x^4}}}+\frac {3 \sqrt {a+\frac {b}{x^4}} x^4}{4 a^2}+\frac {3 \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x^4}}\right )}{4 a^2}\\ &=-\frac {x^4}{2 a \sqrt {a+\frac {b}{x^4}}}+\frac {3 \sqrt {a+\frac {b}{x^4}} x^4}{4 a^2}-\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )}{4 a^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 77, normalized size = 1.12 \begin {gather*} \frac {\sqrt {a} x^2 \left (3 b+a x^4\right )-3 b \sqrt {b+a x^4} \tanh ^{-1}\left (\frac {\sqrt {b+a x^4}}{\sqrt {a} x^2}\right )}{4 a^{5/2} \sqrt {a+\frac {b}{x^4}} x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 80, normalized size = 1.16
method | result | size |
default | \(-\frac {\left (a \,x^{4}+b \right ) \left (-x^{6} a^{\frac {7}{2}}-3 b \,x^{2} a^{\frac {5}{2}}+3 b \ln \left (x^{2} \sqrt {a}+\sqrt {a \,x^{4}+b}\right ) a^{2} \sqrt {a \,x^{4}+b}\right )}{4 \left (\frac {a \,x^{4}+b}{x^{4}}\right )^{\frac {3}{2}} x^{6} a^{\frac {9}{2}}}\) | \(80\) |
risch | \(\frac {a \,x^{4}+b}{4 a^{2} \sqrt {\frac {a \,x^{4}+b}{x^{4}}}}+\frac {\left (\frac {b \,x^{2}}{2 a^{2} \sqrt {a \,x^{4}+b}}-\frac {3 b \ln \left (x^{2} \sqrt {a}+\sqrt {a \,x^{4}+b}\right )}{4 a^{\frac {5}{2}}}\right ) \sqrt {a \,x^{4}+b}}{\sqrt {\frac {a \,x^{4}+b}{x^{4}}}\, x^{2}}\) | \(96\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 86, normalized size = 1.25 \begin {gather*} \frac {3 \, {\left (a + \frac {b}{x^{4}}\right )} b - 2 \, a b}{4 \, {\left ({\left (a + \frac {b}{x^{4}}\right )}^{\frac {3}{2}} a^{2} - \sqrt {a + \frac {b}{x^{4}}} a^{3}\right )}} + \frac {3 \, b \log \left (\frac {\sqrt {a + \frac {b}{x^{4}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{4}}} + \sqrt {a}}\right )}{8 \, a^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 192, normalized size = 2.78 \begin {gather*} \left [\frac {3 \, {\left (a b x^{4} + b^{2}\right )} \sqrt {a} \log \left (-2 \, a x^{4} + 2 \, \sqrt {a} x^{4} \sqrt {\frac {a x^{4} + b}{x^{4}}} - b\right ) + 2 \, {\left (a^{2} x^{8} + 3 \, a b x^{4}\right )} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{8 \, {\left (a^{4} x^{4} + a^{3} b\right )}}, \frac {3 \, {\left (a b x^{4} + b^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} x^{4} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{a x^{4} + b}\right ) + {\left (a^{2} x^{8} + 3 \, a b x^{4}\right )} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{4 \, {\left (a^{4} x^{4} + a^{3} b\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.78, size = 75, normalized size = 1.09 \begin {gather*} \frac {x^{6}}{4 a \sqrt {b} \sqrt {\frac {a x^{4}}{b} + 1}} + \frac {3 \sqrt {b} x^{2}}{4 a^{2} \sqrt {\frac {a x^{4}}{b} + 1}} - \frac {3 b \operatorname {asinh}{\left (\frac {\sqrt {a} x^{2}}{\sqrt {b}} \right )}}{4 a^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.51, size = 55, normalized size = 0.80 \begin {gather*} \frac {{\left (\frac {x^{4}}{a} + \frac {3 \, b}{a^{2}}\right )} x^{2}}{4 \, \sqrt {a x^{4} + b}} + \frac {3 \, b \log \left ({\left | -\sqrt {a} x^{2} + \sqrt {a x^{4} + b} \right |}\right )}{4 \, a^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.76, size = 53, normalized size = 0.77 \begin {gather*} \frac {3\,b}{4\,a^2\,\sqrt {a+\frac {b}{x^4}}}+\frac {x^4}{4\,a\,\sqrt {a+\frac {b}{x^4}}}-\frac {3\,b\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )}{4\,a^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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