Optimal. Leaf size=115 \[ \frac {b^2 \tan ^{-1}\left (\frac {x^3 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^6}}\right )}{3 a^{5/2} \sqrt {b c-a d}}+\frac {\sqrt {c+d x^6} (2 a d+3 b c)}{9 a^2 c^2 x^3}-\frac {\sqrt {c+d x^6}}{9 a c x^9} \]
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Rubi [A] time = 0.16, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {465, 480, 583, 12, 377, 205} \[ \frac {b^2 \tan ^{-1}\left (\frac {x^3 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^6}}\right )}{3 a^{5/2} \sqrt {b c-a d}}+\frac {\sqrt {c+d x^6} (2 a d+3 b c)}{9 a^2 c^2 x^3}-\frac {\sqrt {c+d x^6}}{9 a c x^9} \]
Antiderivative was successfully verified.
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Rule 12
Rule 205
Rule 377
Rule 465
Rule 480
Rule 583
Rubi steps
\begin {align*} \int \frac {1}{x^{10} \left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x^4 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx,x,x^3\right )\\ &=-\frac {\sqrt {c+d x^6}}{9 a c x^9}+\frac {\operatorname {Subst}\left (\int \frac {-3 b c-2 a d-2 b d x^2}{x^2 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx,x,x^3\right )}{9 a c}\\ &=-\frac {\sqrt {c+d x^6}}{9 a c x^9}+\frac {(3 b c+2 a d) \sqrt {c+d x^6}}{9 a^2 c^2 x^3}-\frac {\operatorname {Subst}\left (\int -\frac {3 b^2 c^2}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx,x,x^3\right )}{9 a^2 c^2}\\ &=-\frac {\sqrt {c+d x^6}}{9 a c x^9}+\frac {(3 b c+2 a d) \sqrt {c+d x^6}}{9 a^2 c^2 x^3}+\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx,x,x^3\right )}{3 a^2}\\ &=-\frac {\sqrt {c+d x^6}}{9 a c x^9}+\frac {(3 b c+2 a d) \sqrt {c+d x^6}}{9 a^2 c^2 x^3}+\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x^3}{\sqrt {c+d x^6}}\right )}{3 a^2}\\ &=-\frac {\sqrt {c+d x^6}}{9 a c x^9}+\frac {(3 b c+2 a d) \sqrt {c+d x^6}}{9 a^2 c^2 x^3}+\frac {b^2 \tan ^{-1}\left (\frac {\sqrt {b c-a d} x^3}{\sqrt {a} \sqrt {c+d x^6}}\right )}{3 a^{5/2} \sqrt {b c-a d}}\\ \end {align*}
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Mathematica [C] time = 4.65, size = 253, normalized size = 2.20 \[ -\frac {\left (\frac {d x^6}{c}+1\right ) \left (-\frac {8 x^6 \left (c+d x^6\right )^2 (b c-a d) \, _3F_2\left (2,2,2;1,\frac {5}{2};\frac {(b c-a d) x^6}{c \left (b x^6+a\right )}\right )}{a+b x^6}+\frac {3 c \left (c^2-4 c d x^6-8 d^2 x^{12}\right ) \sin ^{-1}\left (\sqrt {\frac {x^6 (b c-a d)}{c \left (a+b x^6\right )}}\right )}{\sqrt {\frac {a x^6 \left (c+d x^6\right ) (b c-a d)}{c^2 \left (a+b x^6\right )^2}}}+\frac {24 d x^{12} \left (c+d x^6\right ) (a d-b c) \, _2F_1\left (2,2;\frac {5}{2};\frac {(b c-a d) x^6}{c \left (b x^6+a\right )}\right )}{a+b x^6}\right )}{27 c^3 x^9 \left (a+b x^6\right ) \sqrt {c+d x^6}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 1.06, size = 416, normalized size = 3.62 \[ \left [-\frac {3 \, \sqrt {-a b c + a^{2} d} b^{2} c^{2} x^{9} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{12} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{6} + a^{2} c^{2} - 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{9} - a c x^{3}\right )} \sqrt {d x^{6} + c} \sqrt {-a b c + a^{2} d}}{b^{2} x^{12} + 2 \, a b x^{6} + a^{2}}\right ) - 4 \, {\left ({\left (3 \, a b^{2} c^{2} - a^{2} b c d - 2 \, a^{3} d^{2}\right )} x^{6} - a^{2} b c^{2} + a^{3} c d\right )} \sqrt {d x^{6} + c}}{36 \, {\left (a^{3} b c^{3} - a^{4} c^{2} d\right )} x^{9}}, \frac {3 \, \sqrt {a b c - a^{2} d} b^{2} c^{2} x^{9} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{6} - a c\right )} \sqrt {d x^{6} + c} \sqrt {a b c - a^{2} d}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{9} + {\left (a b c^{2} - a^{2} c d\right )} x^{3}\right )}}\right ) + 2 \, {\left ({\left (3 \, a b^{2} c^{2} - a^{2} b c d - 2 \, a^{3} d^{2}\right )} x^{6} - a^{2} b c^{2} + a^{3} c d\right )} \sqrt {d x^{6} + c}}{18 \, {\left (a^{3} b c^{3} - a^{4} c^{2} d\right )} x^{9}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {1}{\left (b \,x^{6}+a \right ) \sqrt {d \,x^{6}+c}\, x^{10}}\, dx \]
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 \[ \int \frac {1}{{\left (b x^{6} + a\right )} \sqrt {d x^{6} + c} x^{10}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^{10}\,\left (b\,x^6+a\right )\,\sqrt {d\,x^6+c}} \,d x \]
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 \[ \int \frac {1}{x^{10} \left (a + b x^{6}\right ) \sqrt {c + d x^{6}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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