Optimal. Leaf size=74 \[ -\frac {x^6}{2 b \sqrt {a+b x^4}}+\frac {3 x^2 \sqrt {a+b x^4}}{4 b^2}-\frac {3 a \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{4 b^{5/2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 74, 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 = {281, 294, 327,
223, 212} \begin {gather*} -\frac {3 a \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{4 b^{5/2}}+\frac {3 x^2 \sqrt {a+b x^4}}{4 b^2}-\frac {x^6}{2 b \sqrt {a+b x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
Rule 281
Rule 294
Rule 327
Rubi steps
\begin {align*} \int \frac {x^9}{\left (a+b x^4\right )^{3/2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^4}{\left (a+b x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=-\frac {x^6}{2 b \sqrt {a+b x^4}}+\frac {3 \text {Subst}\left (\int \frac {x^2}{\sqrt {a+b x^2}} \, dx,x,x^2\right )}{2 b}\\ &=-\frac {x^6}{2 b \sqrt {a+b x^4}}+\frac {3 x^2 \sqrt {a+b x^4}}{4 b^2}-\frac {(3 a) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,x^2\right )}{4 b^2}\\ &=-\frac {x^6}{2 b \sqrt {a+b x^4}}+\frac {3 x^2 \sqrt {a+b x^4}}{4 b^2}-\frac {(3 a) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x^2}{\sqrt {a+b x^4}}\right )}{4 b^2}\\ &=-\frac {x^6}{2 b \sqrt {a+b x^4}}+\frac {3 x^2 \sqrt {a+b x^4}}{4 b^2}-\frac {3 a \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{4 b^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 62, normalized size = 0.84 \begin {gather*} \frac {3 a x^2+b x^6}{4 b^2 \sqrt {a+b x^4}}-\frac {3 a \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {b} x^2}\right )}{4 b^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 61, normalized size = 0.82
method | result | size |
default | \(\frac {x^{6}}{4 b \sqrt {b \,x^{4}+a}}+\frac {3 a \,x^{2}}{4 b^{2} \sqrt {b \,x^{4}+a}}-\frac {3 a \ln \left (x^{2} \sqrt {b}+\sqrt {b \,x^{4}+a}\right )}{4 b^{\frac {5}{2}}}\) | \(61\) |
risch | \(\frac {x^{2} \sqrt {b \,x^{4}+a}}{4 b^{2}}+\frac {a \,x^{2}}{2 b^{2} \sqrt {b \,x^{4}+a}}-\frac {3 a \ln \left (x^{2} \sqrt {b}+\sqrt {b \,x^{4}+a}\right )}{4 b^{\frac {5}{2}}}\) | \(61\) |
elliptic | \(\frac {x^{6}}{4 b \sqrt {b \,x^{4}+a}}+\frac {3 a \,x^{2}}{4 b^{2} \sqrt {b \,x^{4}+a}}-\frac {3 a \ln \left (x^{2} \sqrt {b}+\sqrt {b \,x^{4}+a}\right )}{4 b^{\frac {5}{2}}}\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 103, normalized size = 1.39 \begin {gather*} \frac {2 \, a b - \frac {3 \, {\left (b x^{4} + a\right )} a}{x^{4}}}{4 \, {\left (\frac {\sqrt {b x^{4} + a} b^{3}}{x^{2}} - \frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}} b^{2}}{x^{6}}\right )}} + \frac {3 \, a \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x^{4} + a}}{x^{2}}}{\sqrt {b} + \frac {\sqrt {b x^{4} + a}}{x^{2}}}\right )}{8 \, b^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 167, normalized size = 2.26 \begin {gather*} \left [\frac {3 \, {\left (a b x^{4} + a^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{4} + 2 \, \sqrt {b x^{4} + a} \sqrt {b} x^{2} - a\right ) + 2 \, {\left (b^{2} x^{6} + 3 \, a b x^{2}\right )} \sqrt {b x^{4} + a}}{8 \, {\left (b^{4} x^{4} + a b^{3}\right )}}, \frac {3 \, {\left (a b x^{4} + a^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x^{2}}{\sqrt {b x^{4} + a}}\right ) + {\left (b^{2} x^{6} + 3 \, a b x^{2}\right )} \sqrt {b x^{4} + a}}{4 \, {\left (b^{4} x^{4} + a b^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.79, size = 75, normalized size = 1.01 \begin {gather*} \frac {3 \sqrt {a} x^{2}}{4 b^{2} \sqrt {1 + \frac {b x^{4}}{a}}} - \frac {3 a \operatorname {asinh}{\left (\frac {\sqrt {b} x^{2}}{\sqrt {a}} \right )}}{4 b^{\frac {5}{2}}} + \frac {x^{6}}{4 \sqrt {a} b \sqrt {1 + \frac {b x^{4}}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.73, size = 55, normalized size = 0.74 \begin {gather*} \frac {{\left (\frac {x^{4}}{b} + \frac {3 \, a}{b^{2}}\right )} x^{2}}{4 \, \sqrt {b x^{4} + a}} + \frac {3 \, a \log \left ({\left | -\sqrt {b} x^{2} + \sqrt {b x^{4} + a} \right |}\right )}{4 \, b^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^9}{{\left (b\,x^4+a\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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