Optimal. Leaf size=85 \[ \frac {\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^8}}{\sqrt {b c-a d}}\right )}{4 a \sqrt {b c-a d}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x^8}}{\sqrt {c}}\right )}{4 a \sqrt {c}} \]
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Rubi [A] time = 0.08, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {446, 86, 63, 208} \[ \frac {\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^8}}{\sqrt {b c-a d}}\right )}{4 a \sqrt {b c-a d}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x^8}}{\sqrt {c}}\right )}{4 a \sqrt {c}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 86
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {1}{x \left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx &=\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{x (a+b x) \sqrt {c+d x}} \, dx,x,x^8\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^8\right )}{8 a}-\frac {b \operatorname {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^8\right )}{8 a}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^8}\right )}{4 a d}-\frac {b \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 a d}\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x^8}}{\sqrt {c}}\right )}{4 a \sqrt {c}}+\frac {\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^8}}{\sqrt {b c-a d}}\right )}{4 a \sqrt {b c-a d}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 81, normalized size = 0.95 \[ \frac {\frac {\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^8}}{\sqrt {b c-a d}}\right )}{\sqrt {b c-a d}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x^8}}{\sqrt {c}}\right )}{\sqrt {c}}}{4 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 431, normalized size = 5.07 \[ \left [\frac {c \sqrt {\frac {b}{b c - a d}} \log \left (\frac {b d x^{8} + 2 \, b c - a d + 2 \, \sqrt {d x^{8} + c} {\left (b c - a d\right )} \sqrt {\frac {b}{b c - a d}}}{b x^{8} + a}\right ) + \sqrt {c} \log \left (\frac {d x^{8} - 2 \, \sqrt {d x^{8} + c} \sqrt {c} + 2 \, c}{x^{8}}\right )}{8 \, a c}, \frac {2 \, c \sqrt {-\frac {b}{b c - a d}} \arctan \left (-\frac {\sqrt {d x^{8} + c} {\left (b c - a d\right )} \sqrt {-\frac {b}{b c - a d}}}{b d x^{8} + b c}\right ) + \sqrt {c} \log \left (\frac {d x^{8} - 2 \, \sqrt {d x^{8} + c} \sqrt {c} + 2 \, c}{x^{8}}\right )}{8 \, a c}, \frac {c \sqrt {\frac {b}{b c - a d}} \log \left (\frac {b d x^{8} + 2 \, b c - a d + 2 \, \sqrt {d x^{8} + c} {\left (b c - a d\right )} \sqrt {\frac {b}{b c - a d}}}{b x^{8} + a}\right ) + 2 \, \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{8} + c} \sqrt {-c}}{c}\right )}{8 \, a c}, \frac {c \sqrt {-\frac {b}{b c - a d}} \arctan \left (-\frac {\sqrt {d x^{8} + c} {\left (b c - a d\right )} \sqrt {-\frac {b}{b c - a d}}}{b d x^{8} + b c}\right ) + \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{8} + c} \sqrt {-c}}{c}\right )}{4 \, a c}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 71, normalized size = 0.84 \[ -\frac {b \arctan \left (\frac {\sqrt {d x^{8} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{4 \, \sqrt {-b^{2} c + a b d} a} + \frac {\arctan \left (\frac {\sqrt {d x^{8} + c}}{\sqrt {-c}}\right )}{4 \, a \sqrt {-c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.68, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{8}+a \right ) \sqrt {d \,x^{8}+c}\, x}\, dx \]
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 \[ \int \frac {1}{{\left (b x^{8} + a\right )} \sqrt {d x^{8} + c} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.81, size = 652, normalized size = 7.67 \[ -\frac {\mathrm {atanh}\left (\frac {\sqrt {d\,x^8+c}}{\sqrt {c}}\right )}{4\,a\,\sqrt {c}}-\frac {\mathrm {atan}\left (\frac {\frac {\sqrt {b^2\,c-a\,b\,d}\,\left (\frac {b^3\,d^2\,\sqrt {d\,x^8+c}}{4}-\frac {\sqrt {b^2\,c-a\,b\,d}\,\left (a^2\,b^2\,d^3-\frac {\left (8\,a^3\,b^2\,d^3-16\,a^2\,b^3\,c\,d^2\right )\,\sqrt {d\,x^8+c}\,\sqrt {b^2\,c-a\,b\,d}}{8\,\left (a^2\,d-a\,b\,c\right )}\right )}{8\,\left (a^2\,d-a\,b\,c\right )}\right )\,1{}\mathrm {i}}{8\,\left (a^2\,d-a\,b\,c\right )}+\frac {\sqrt {b^2\,c-a\,b\,d}\,\left (\frac {b^3\,d^2\,\sqrt {d\,x^8+c}}{4}+\frac {\sqrt {b^2\,c-a\,b\,d}\,\left (a^2\,b^2\,d^3+\frac {\left (8\,a^3\,b^2\,d^3-16\,a^2\,b^3\,c\,d^2\right )\,\sqrt {d\,x^8+c}\,\sqrt {b^2\,c-a\,b\,d}}{8\,\left (a^2\,d-a\,b\,c\right )}\right )}{8\,\left (a^2\,d-a\,b\,c\right )}\right )\,1{}\mathrm {i}}{8\,\left (a^2\,d-a\,b\,c\right )}}{\frac {\sqrt {b^2\,c-a\,b\,d}\,\left (\frac {b^3\,d^2\,\sqrt {d\,x^8+c}}{4}-\frac {\sqrt {b^2\,c-a\,b\,d}\,\left (a^2\,b^2\,d^3-\frac {\left (8\,a^3\,b^2\,d^3-16\,a^2\,b^3\,c\,d^2\right )\,\sqrt {d\,x^8+c}\,\sqrt {b^2\,c-a\,b\,d}}{8\,\left (a^2\,d-a\,b\,c\right )}\right )}{8\,\left (a^2\,d-a\,b\,c\right )}\right )}{8\,\left (a^2\,d-a\,b\,c\right )}-\frac {\sqrt {b^2\,c-a\,b\,d}\,\left (\frac {b^3\,d^2\,\sqrt {d\,x^8+c}}{4}+\frac {\sqrt {b^2\,c-a\,b\,d}\,\left (a^2\,b^2\,d^3+\frac {\left (8\,a^3\,b^2\,d^3-16\,a^2\,b^3\,c\,d^2\right )\,\sqrt {d\,x^8+c}\,\sqrt {b^2\,c-a\,b\,d}}{8\,\left (a^2\,d-a\,b\,c\right )}\right )}{8\,\left (a^2\,d-a\,b\,c\right )}\right )}{8\,\left (a^2\,d-a\,b\,c\right )}}\right )\,\sqrt {b^2\,c-a\,b\,d}\,1{}\mathrm {i}}{4\,\left (a^2\,d-a\,b\,c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 37.86, size = 66, normalized size = 0.78 \[ - \frac {\operatorname {atan}{\left (\frac {\sqrt {c + d x^{8}}}{\sqrt {\frac {a d - b c}{b}}} \right )}}{4 a \sqrt {\frac {a d - b c}{b}}} + \frac {\operatorname {atan}{\left (\frac {\sqrt {c + d x^{8}}}{\sqrt {- c}} \right )}}{4 a \sqrt {- c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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