Optimal. Leaf size=54 \[ \frac {\tan ^{-1}\left (\frac {x^4 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^8}}\right )}{4 \sqrt {a} \sqrt {b c-a d}} \]
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Rubi [A] time = 0.05, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {465, 377, 205} \[ \frac {\tan ^{-1}\left (\frac {x^4 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^8}}\right )}{4 \sqrt {a} \sqrt {b c-a d}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 377
Rule 465
Rubi steps
\begin {align*} \int \frac {x^3}{\left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx,x,x^4\right )\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x^4}{\sqrt {c+d x^8}}\right )\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {b c-a d} x^4}{\sqrt {a} \sqrt {c+d x^8}}\right )}{4 \sqrt {a} \sqrt {b c-a d}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 95, normalized size = 1.76 \[ \frac {x^4 \sqrt {\frac {d x^8}{c}+1} \tanh ^{-1}\left (\frac {\sqrt {\frac {d x^8}{c}-\frac {b x^8}{a}}}{\sqrt {\frac {d x^8}{c}+1}}\right )}{4 a \sqrt {c+d x^8} \sqrt {\frac {d x^8}{c}-\frac {b x^8}{a}}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.88, size = 245, normalized size = 4.54 \[ \left [-\frac {\sqrt {-a b c + a^{2} d} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{16} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{8} + a^{2} c^{2} - 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{12} - a c x^{4}\right )} \sqrt {d x^{8} + c} \sqrt {-a b c + a^{2} d}}{b^{2} x^{16} + 2 \, a b x^{8} + a^{2}}\right )}{16 \, {\left (a b c - a^{2} d\right )}}, \frac {\arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{8} - a c\right )} \sqrt {d x^{8} + c} \sqrt {a b c - a^{2} d}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{12} + {\left (a b c^{2} - a^{2} c d\right )} x^{4}\right )}}\right )}{8 \, \sqrt {a b c - a^{2} d}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 72, normalized size = 1.33 \[ -\frac {\sqrt {d} \arctan \left (\frac {{\left (\sqrt {d} x^{4} - \sqrt {d x^{8} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{4 \, \sqrt {a b c d - a^{2} d^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.62, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\left (b \,x^{8}+a \right ) \sqrt {d \,x^{8}+c}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{{\left (b x^{8} + a\right )} \sqrt {d x^{8} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^3}{\left (b\,x^8+a\right )\,\sqrt {d\,x^8+c}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\left (a + b x^{8}\right ) \sqrt {c + d x^{8}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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