Optimal. Leaf size=86 \[ -\frac {\sqrt {\frac {b x^3}{a}+1} \sqrt {\frac {d x^3}{c}+1} F_1\left (-\frac {1}{3};\frac {1}{2},\frac {1}{2};\frac {2}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{x \sqrt {a+b x^3} \sqrt {c+d x^3}} \]
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Rubi [A] time = 0.10, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {511, 510} \[ -\frac {\sqrt {\frac {b x^3}{a}+1} \sqrt {\frac {d x^3}{c}+1} F_1\left (-\frac {1}{3};\frac {1}{2},\frac {1}{2};\frac {2}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{x \sqrt {a+b x^3} \sqrt {c+d x^3}} \]
Antiderivative was successfully verified.
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Rule 510
Rule 511
Rubi steps
\begin {align*} \int \frac {1}{x^2 \sqrt {a+b x^3} \sqrt {c+d x^3}} \, dx &=\frac {\sqrt {1+\frac {b x^3}{a}} \int \frac {1}{x^2 \sqrt {1+\frac {b x^3}{a}} \sqrt {c+d x^3}} \, dx}{\sqrt {a+b x^3}}\\ &=\frac {\left (\sqrt {1+\frac {b x^3}{a}} \sqrt {1+\frac {d x^3}{c}}\right ) \int \frac {1}{x^2 \sqrt {1+\frac {b x^3}{a}} \sqrt {1+\frac {d x^3}{c}}} \, dx}{\sqrt {a+b x^3} \sqrt {c+d x^3}}\\ &=-\frac {\sqrt {1+\frac {b x^3}{a}} \sqrt {1+\frac {d x^3}{c}} F_1\left (-\frac {1}{3};\frac {1}{2},\frac {1}{2};\frac {2}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{x \sqrt {a+b x^3} \sqrt {c+d x^3}}\\ \end {align*}
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Mathematica [B] time = 0.26, size = 189, normalized size = 2.20 \[ \frac {5 x^3 \sqrt {\frac {b x^3}{a}+1} \sqrt {\frac {d x^3}{c}+1} (a d+b c) F_1\left (\frac {2}{3};\frac {1}{2},\frac {1}{2};\frac {5}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+8 b d x^6 \sqrt {\frac {b x^3}{a}+1} \sqrt {\frac {d x^3}{c}+1} F_1\left (\frac {5}{3};\frac {1}{2},\frac {1}{2};\frac {8}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )-20 \left (a+b x^3\right ) \left (c+d x^3\right )}{20 a c x \sqrt {a+b x^3} \sqrt {c+d x^3}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.85, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x^{3} + a} \sqrt {d x^{3} + c}}{b d x^{8} + {\left (b c + a d\right )} x^{5} + a c x^{2}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {b x^{3} + a} \sqrt {d x^{3} + c} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.18, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b \,x^{3}+a}\, \sqrt {d \,x^{3}+c}\, x^{2}}\, dx \]
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 \[ \int \frac {1}{\sqrt {b x^{3} + a} \sqrt {d x^{3} + c} x^{2}}\,{d x} \]
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 \[ \int \frac {1}{x^2\,\sqrt {b\,x^3+a}\,\sqrt {d\,x^3+c}} \,d x \]
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 \[ \int \frac {1}{x^{2} \sqrt {a + b x^{3}} \sqrt {c + d x^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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