Optimal. Leaf size=74 \[ \frac {a \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^8}}{\sqrt {b c-a d}}\right )}{4 b^{3/2} \sqrt {b c-a d}}+\frac {\sqrt {c+d x^8}}{4 b d} \]
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Rubi [A] time = 0.06, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {446, 80, 63, 208} \[ \frac {a \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^8}}{\sqrt {b c-a d}}\right )}{4 b^{3/2} \sqrt {b c-a d}}+\frac {\sqrt {c+d x^8}}{4 b d} \]
Antiderivative was successfully verified.
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Rule 63
Rule 80
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {x^{15}}{\left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx &=\frac {1}{8} \operatorname {Subst}\left (\int \frac {x}{(a+b x) \sqrt {c+d x}} \, dx,x,x^8\right )\\ &=\frac {\sqrt {c+d x^8}}{4 b d}-\frac {a \operatorname {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^8\right )}{8 b}\\ &=\frac {\sqrt {c+d x^8}}{4 b d}-\frac {a \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 b d}\\ &=\frac {\sqrt {c+d x^8}}{4 b d}+\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^8}}{\sqrt {b c-a d}}\right )}{4 b^{3/2} \sqrt {b c-a d}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 72, normalized size = 0.97 \[ \frac {1}{4} \left (\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^8}}{\sqrt {b c-a d}}\right )}{b^{3/2} \sqrt {b c-a d}}+\frac {\sqrt {c+d x^8}}{b d}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 205, normalized size = 2.77 \[ \left [\frac {\sqrt {b^{2} c - a b d} a d \log \left (\frac {b d x^{8} + 2 \, b c - a d + 2 \, \sqrt {d x^{8} + c} \sqrt {b^{2} c - a b d}}{b x^{8} + a}\right ) + 2 \, \sqrt {d x^{8} + c} {\left (b^{2} c - a b d\right )}}{8 \, {\left (b^{3} c d - a b^{2} d^{2}\right )}}, -\frac {\sqrt {-b^{2} c + a b d} a d \arctan \left (\frac {\sqrt {d x^{8} + c} \sqrt {-b^{2} c + a b d}}{b d x^{8} + b c}\right ) - \sqrt {d x^{8} + c} {\left (b^{2} c - a b d\right )}}{4 \, {\left (b^{3} c d - a b^{2} d^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 64, normalized size = 0.86 \[ -\frac {\frac {a d \arctan \left (\frac {\sqrt {d x^{8} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} b} - \frac {\sqrt {d x^{8} + c}}{b}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.64, size = 0, normalized size = 0.00 \[ \int \frac {x^{15}}{\left (b \,x^{8}+a \right ) \sqrt {d \,x^{8}+c}}\, dx \]
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 = 4.73, size = 58, normalized size = 0.78 \[ \frac {\sqrt {d\,x^8+c}}{4\,b\,d}-\frac {a\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d\,x^8+c}}{\sqrt {a\,d-b\,c}}\right )}{4\,b^{3/2}\,\sqrt {a\,d-b\,c}} \]
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 \[ \int \frac {x^{15}}{\left (a + b x^{8}\right ) \sqrt {c + d x^{8}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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