Optimal. Leaf size=51 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^8}}{\sqrt {b c-a d}}\right )}{4 \sqrt {b} \sqrt {b c-a d}} \]
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Rubi [A] time = 0.05, antiderivative size = 51, 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 = {444, 63, 208} \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^8}}{\sqrt {b c-a d}}\right )}{4 \sqrt {b} \sqrt {b c-a d}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 444
Rubi steps
\begin {align*} \int \frac {x^7}{\left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx &=\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^8\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x^8}\right )}{4 d}\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^8}}{\sqrt {b c-a d}}\right )}{4 \sqrt {b} \sqrt {b c-a d}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 51, normalized size = 1.00 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^8}}{\sqrt {b c-a d}}\right )}{4 \sqrt {b} \sqrt {b c-a d}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 130, normalized size = 2.55 \[ \left [\frac {\log \left (\frac {b d x^{8} + 2 \, b c - a d - 2 \, \sqrt {d x^{8} + c} \sqrt {b^{2} c - a b d}}{b x^{8} + a}\right )}{8 \, \sqrt {b^{2} c - a b d}}, \frac {\sqrt {-b^{2} c + a b d} \arctan \left (\frac {\sqrt {d x^{8} + c} \sqrt {-b^{2} c + a b d}}{b d x^{8} + b c}\right )}{4 \, {\left (b^{2} c - a b d\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 40, normalized size = 0.78 \[ \frac {\arctan \left (\frac {\sqrt {d x^{8} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{4 \, \sqrt {-b^{2} c + a b d}} \]
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 \[ \int \frac {x^{7}}{\left (b \,x^{8}+a \right ) \sqrt {d \,x^{8}+c}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.59, size = 40, normalized size = 0.78 \[ \frac {\mathrm {atan}\left (\frac {b\,\sqrt {d\,x^8+c}}{\sqrt {a\,b\,d-b^2\,c}}\right )}{4\,\sqrt {a\,b\,d-b^2\,c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 44.57, size = 37, normalized size = 0.73 \[ \frac {\operatorname {atan}{\left (\frac {\sqrt {c + d x^{8}}}{\sqrt {\frac {a d - b c}{b}}} \right )}}{4 b \sqrt {\frac {a d - b c}{b}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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