Optimal. Leaf size=64 \[ -\frac {\sqrt {\frac {d x^6}{c}+1} F_1\left (-\frac {1}{3};2,\frac {1}{2};\frac {2}{3};-\frac {b x^6}{a},-\frac {d x^6}{c}\right )}{2 a^2 x^2 \sqrt {c+d x^6}} \]
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Rubi [A] time = 0.09, antiderivative size = 64, 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 = {465, 511, 510} \[ -\frac {\sqrt {\frac {d x^6}{c}+1} F_1\left (-\frac {1}{3};2,\frac {1}{2};\frac {2}{3};-\frac {b x^6}{a},-\frac {d x^6}{c}\right )}{2 a^2 x^2 \sqrt {c+d x^6}} \]
Antiderivative was successfully verified.
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Rule 465
Rule 510
Rule 511
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^2 \left (a+b x^3\right )^2 \sqrt {c+d x^3}} \, dx,x,x^2\right )\\ &=\frac {\sqrt {1+\frac {d x^6}{c}} \operatorname {Subst}\left (\int \frac {1}{x^2 \left (a+b x^3\right )^2 \sqrt {1+\frac {d x^3}{c}}} \, dx,x,x^2\right )}{2 \sqrt {c+d x^6}}\\ &=-\frac {\sqrt {1+\frac {d x^6}{c}} F_1\left (-\frac {1}{3};2,\frac {1}{2};\frac {2}{3};-\frac {b x^6}{a},-\frac {d x^6}{c}\right )}{2 a^2 x^2 \sqrt {c+d x^6}}\\ \end {align*}
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Mathematica [B] time = 0.37, size = 226, normalized size = 3.53 \[ \frac {-5 x^6 \left (a+b x^6\right ) \sqrt {\frac {d x^6}{c}+1} \left (3 a^2 d^2-15 a b c d+8 b^2 c^2\right ) F_1\left (\frac {2}{3};\frac {1}{2},1;\frac {5}{3};-\frac {d x^6}{c},-\frac {b x^6}{a}\right )+20 a \left (c+d x^6\right ) \left (3 a^2 d-3 a b \left (c-d x^6\right )-4 b^2 c x^6\right )+2 b d x^{12} \left (a+b x^6\right ) \sqrt {\frac {d x^6}{c}+1} (4 b c-3 a d) F_1\left (\frac {5}{3};\frac {1}{2},1;\frac {8}{3};-\frac {d x^6}{c},-\frac {b x^6}{a}\right )}{120 a^3 c x^2 \left (a+b x^6\right ) \sqrt {c+d x^6} (b c-a d)} \]
Warning: Unable to verify antiderivative.
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fricas [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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{6} + a\right )}^{2} \sqrt {d x^{6} + c} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.46, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{6}+a \right )^{2} \sqrt {d \,x^{6}+c}\, x^{3}}\, dx \]
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 \[ \int \frac {1}{{\left (b x^{6} + a\right )}^{2} \sqrt {d x^{6} + c} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{x^3\,{\left (b\,x^6+a\right )}^2\,\sqrt {d\,x^6+c}} \,d x \]
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 \[ \int \frac {1}{x^{3} \left (a + b x^{6}\right )^{2} \sqrt {c + d x^{6}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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