Optimal. Leaf size=80 \[ -\frac {b \tan ^{-1}\left (\frac {x^4 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^8}}\right )}{4 a^{3/2} \sqrt {b c-a d}}-\frac {\sqrt {c+d x^8}}{4 a c x^4} \]
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Rubi [A] time = 0.09, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {465, 480, 12, 377, 205} \begin {gather*} -\frac {b \tan ^{-1}\left (\frac {x^4 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^8}}\right )}{4 a^{3/2} \sqrt {b c-a d}}-\frac {\sqrt {c+d x^8}}{4 a c x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 205
Rule 377
Rule 465
Rule 480
Rubi steps
\begin {align*} \int \frac {1}{x^5 \left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{x^2 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx,x,x^4\right )\\ &=-\frac {\sqrt {c+d x^8}}{4 a c x^4}-\frac {\operatorname {Subst}\left (\int \frac {b c}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx,x,x^4\right )}{4 a c}\\ &=-\frac {\sqrt {c+d x^8}}{4 a c x^4}-\frac {b \operatorname {Subst}\left (\int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx,x,x^4\right )}{4 a}\\ &=-\frac {\sqrt {c+d x^8}}{4 a c x^4}-\frac {b \operatorname {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x^4}{\sqrt {c+d x^8}}\right )}{4 a}\\ &=-\frac {\sqrt {c+d x^8}}{4 a c x^4}-\frac {b \tan ^{-1}\left (\frac {\sqrt {b c-a d} x^4}{\sqrt {a} \sqrt {c+d x^8}}\right )}{4 a^{3/2} \sqrt {b c-a d}}\\ \end {align*}
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Mathematica [C] time = 0.83, size = 179, normalized size = 2.24 \begin {gather*} -\frac {\left (\frac {d x^8}{c}+1\right ) \left (\frac {4 x^8 \left (c+d x^8\right ) (b c-a d) \, _2F_1\left (2,2;\frac {5}{2};\frac {(b c-a d) x^8}{c \left (b x^8+a\right )}\right )}{3 c^2 \left (a+b x^8\right )}+\frac {\left (c+2 d x^8\right ) \sin ^{-1}\left (\sqrt {\frac {x^8 (b c-a d)}{c \left (a+b x^8\right )}}\right )}{c \sqrt {\frac {a x^8 \left (c+d x^8\right ) (b c-a d)}{c^2 \left (a+b x^8\right )^2}}}\right )}{4 x^4 \left (a+b x^8\right ) \sqrt {c+d x^8}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.80, size = 142, normalized size = 1.78 \begin {gather*} \frac {b \sqrt {b c-a d} \tan ^{-1}\left (\frac {b \sqrt {d} x^8}{\sqrt {a} \sqrt {b c-a d}}+\frac {b x^4 \sqrt {c+d x^8}}{\sqrt {a} \sqrt {b c-a d}}+\frac {\sqrt {a} \sqrt {d}}{\sqrt {b c-a d}}\right )}{4 a^{3/2} (a d-b c)}-\frac {\sqrt {c+d x^8}}{4 a c x^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 332, normalized size = 4.15 \begin {gather*} \left [-\frac {\sqrt {-a b c + a^{2} d} b c x^{4} \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 ) + 4 \, \sqrt {d x^{8} + c} {\left (a b c - a^{2} d\right )}}{16 \, {\left (a^{2} b c^{2} - a^{3} c d\right )} x^{4}}, -\frac {\sqrt {a b c - a^{2} d} b c x^{4} \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 ) + 2 \, \sqrt {d x^{8} + c} {\left (a b c - a^{2} d\right )}}{8 \, {\left (a^{2} b c^{2} - a^{3} c d\right )} x^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 116, normalized size = 1.45 \begin {gather*} \frac {1}{4} \, d^{\frac {3}{2}} {\left (\frac {b \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 )}{\sqrt {a b c d - a^{2} d^{2}} a d} + \frac {2}{{\left ({\left (\sqrt {d} x^{4} - \sqrt {d x^{8} + c}\right )}^{2} - c\right )} a d}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.66, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (b \,x^{8}+a \right ) \sqrt {d \,x^{8}+c}\, x^{5}}\, dx \end {gather*}
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 \begin {gather*} \int \frac {1}{{\left (b x^{8} + a\right )} \sqrt {d x^{8} + c} x^{5}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^5\,\left (b\,x^8+a\right )\,\sqrt {d\,x^8+c}} \,d x \end {gather*}
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 \begin {gather*} \int \frac {1}{x^{5} \left (a + b x^{8}\right ) \sqrt {c + d x^{8}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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