Optimal. Leaf size=59 \[ -\frac {1}{2} a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+c x^4}}{\sqrt {a}}\right )+\frac {1}{2} a \sqrt {a+c x^4}+\frac {1}{6} \left (a+c x^4\right )^{3/2} \]
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Rubi [A] time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {266, 50, 63, 208} \[ -\frac {1}{2} a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+c x^4}}{\sqrt {a}}\right )+\frac {1}{2} a \sqrt {a+c x^4}+\frac {1}{6} \left (a+c x^4\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+c x^4\right )^{3/2}}{x} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {(a+c x)^{3/2}}{x} \, dx,x,x^4\right )\\ &=\frac {1}{6} \left (a+c x^4\right )^{3/2}+\frac {1}{4} a \operatorname {Subst}\left (\int \frac {\sqrt {a+c x}}{x} \, dx,x,x^4\right )\\ &=\frac {1}{2} a \sqrt {a+c x^4}+\frac {1}{6} \left (a+c x^4\right )^{3/2}+\frac {1}{4} a^2 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^4\right )\\ &=\frac {1}{2} a \sqrt {a+c x^4}+\frac {1}{6} \left (a+c x^4\right )^{3/2}+\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^4}\right )}{2 c}\\ &=\frac {1}{2} a \sqrt {a+c x^4}+\frac {1}{6} \left (a+c x^4\right )^{3/2}-\frac {1}{2} a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+c x^4}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 51, normalized size = 0.86 \[ \frac {1}{6} \left (\sqrt {a+c x^4} \left (4 a+c x^4\right )-3 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+c x^4}}{\sqrt {a}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 103, normalized size = 1.75 \[ \left [\frac {1}{4} \, a^{\frac {3}{2}} \log \left (\frac {c x^{4} - 2 \, \sqrt {c x^{4} + a} \sqrt {a} + 2 \, a}{x^{4}}\right ) + \frac {1}{6} \, {\left (c x^{4} + 4 \, a\right )} \sqrt {c x^{4} + a}, \frac {1}{2} \, \sqrt {-a} a \arctan \left (\frac {\sqrt {c x^{4} + a} \sqrt {-a}}{a}\right ) + \frac {1}{6} \, {\left (c x^{4} + 4 \, a\right )} \sqrt {c x^{4} + a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 50, normalized size = 0.85 \[ \frac {a^{2} \arctan \left (\frac {\sqrt {c x^{4} + a}}{\sqrt {-a}}\right )}{2 \, \sqrt {-a}} + \frac {1}{6} \, {\left (c x^{4} + a\right )}^{\frac {3}{2}} + \frac {1}{2} \, \sqrt {c x^{4} + a} a \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 57, normalized size = 0.97 \[ \frac {\sqrt {c \,x^{4}+a}\, c \,x^{4}}{6}-\frac {a^{\frac {3}{2}} \ln \left (\frac {2 a +2 \sqrt {c \,x^{4}+a}\, \sqrt {a}}{x^{2}}\right )}{2}+\frac {2 \sqrt {c \,x^{4}+a}\, a}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.00, size = 61, normalized size = 1.03 \[ \frac {1}{4} \, a^{\frac {3}{2}} \log \left (\frac {\sqrt {c x^{4} + a} - \sqrt {a}}{\sqrt {c x^{4} + a} + \sqrt {a}}\right ) + \frac {1}{6} \, {\left (c x^{4} + a\right )}^{\frac {3}{2}} + \frac {1}{2} \, \sqrt {c x^{4} + a} a \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.17, size = 43, normalized size = 0.73 \[ \frac {a\,\sqrt {c\,x^4+a}}{2}-\frac {a^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {c\,x^4+a}}{\sqrt {a}}\right )}{2}+\frac {{\left (c\,x^4+a\right )}^{3/2}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.64, size = 80, normalized size = 1.36 \[ \frac {2 a^{\frac {3}{2}} \sqrt {1 + \frac {c x^{4}}{a}}}{3} + \frac {a^{\frac {3}{2}} \log {\left (\frac {c x^{4}}{a} \right )}}{4} - \frac {a^{\frac {3}{2}} \log {\left (\sqrt {1 + \frac {c x^{4}}{a}} + 1 \right )}}{2} + \frac {\sqrt {a} c x^{4} \sqrt {1 + \frac {c x^{4}}{a}}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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